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## Contents

- 1 Collated Material
- 2 Chronological Material
- 2.1 2005
- 2.1.1 IDS • Inquiry Into Information (2005 Jan 12)
- 2.1.1.1 Texts
- 2.1.1.1.1 DET. Determination
- 2.1.1.1.2 EXC. Excuses
- 2.1.1.1.3 EII. Extension, Intension, Information
- 2.1.1.1.4 FOR. Inquiry Into Formalization
- 2.1.1.1.5 INF. Inquiry Into Information
- 2.1.1.1.6 SYM. Inquiry Into Symbolization
- 2.1.1.1.7 LAS. Logic As Semiotic
- 2.1.1.1.8 MOSI. Manifolds Of Sensuous Impressions
- 2.1.1.1.9 MOD. Model Theory
- 2.1.1.1.10 SR. Sign Relations
- 2.1.1.1.11 Future Agenda

- 2.1.1.2 Links

- 2.1.1.1 Texts

- 2.1.1 IDS • Inquiry Into Information (2005 Jan 12)

- 2.1 2005

## Collated Material

## Chronological Material

### 2005

#### IDS • Inquiry Into Information (2005 Jan 12)

##### Texts

- Contents
- DET. Determination
- EXC. Excuses
- EII. Extension, Intension, Information
- FOR. Inquiry Into Formalization
- INF. Inquiry Into Information
- LAS. Logic As Semiotic
- MOD. Model Theory
- MOS. Manifolds Of Sense
- SYM. Inquiry Into Symbolization

###### DET. Determination

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Now that I have proved sufficiently that everything | comes to pass according to determinate reasons, there | cannot be any more difficulty over these principles | of God's foreknowledge. Although these determinations | do not compel, they cannot but be certain, and they | foreshadow what shall happen. | | It is true that God sees all at once the whole sequence | of this universe, when he chooses it, and that thus he | has no need of the connexion of effects and causes in | order to foresee these effects. But since his wisdom | causes him to choose a sequence in perfect connexion, | he cannot but see one part of the sequence in the other. | | It is one of the rules of my system of general harmony, | 'that the present is big with the future', and that he | who sees all sees in that which is that which shall be. | | What is more, I have proved conclusively that God sees in | each portion of the universe the whole universe, owing to | the perfect connexion of things. He is infinitely more | discerning than Pythagoras, who judged the height of | Hercules by the size of his footprint. There must | therefore be no doubt that effects follow their | causes determinately, in spite of contingency | and even of freedom, which nevertheless exist | together with certainty or determination. | | Gottfried Wilhelm (Freiherr von) Leibniz, |'Theodicy: Essays on the Goodness of God, | the Freedom of Man, and the Origin of Evil', | Edited with an Introduction by Austin Farrer, | Translated by E.M. Huggard from C.J. Gerhardt's | Edition of the 'Collected Philosophical Works', | 1875-1890. Routledge 1951. Open Court 1985. | Paragraph 360, page 341. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Earlier this century in 'The Open Universe: An Argument for Indeterminism', | Karl Popper wrote, "Common sense inclines, on the one hand, to assert that | every event is caused by some preceding events, so that every event can be | explained or predicted. ... On the other hand, ... common sense attributes | to mature and sane human persons ... the ability to choose freely between | alternative possibilities of acting." This "dilemma of determinism", as | William James called it, is closely related to the meaning of time. Is the | future given, or is it under perpetual construction? A profound dilemma for | all of mankind, as time is the fundamental dimension of our existence. | | Ilya Prigogine (In Collaboration with Isabelle Stengers), |'The End of Certainty: Time, Chaos, and the New Laws of Nature', | The Free Press, New York, NY, 1997, p. 1. Originally published as: |'La Fin des Certitudes', Éditions Odile Jacob, 1996. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 3 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Of triadic Being the multitude of forms | is so terrific that I have usually shrunk | from the task of enumerating them; and for | the present purpose such an enumeration would | be worse than superfluous: it would be a great | inconvenience. In another paper, I intend to | give the formal definition of a sign, which I | have worked out by arduous and long labour. | I will omit the explanation of it here. | Suffice it to say that a sign endeavors | to represent, in part at least, an Object, | which is therefore in a sense the cause, or | determinant, of the sign even if the sign | represents its object falsely. But to say | that it represents its Object implies that | it affects a mind, and so affects it as, | in some respect, to determine in that mind | something that is mediately due to the Object. | That determination of which the immediate cause, | or determinant, is the Sign, and of which the | mediate cause is the Object may be termed the | 'Interpretant' ... | | Charles Sanders Peirce, CP 6.347 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 4 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | That whatever action is brute, unintelligent, and unconcerned | with the result of it is purely dyadic is either demonstrable | or is too evident to be demonstrable. But in case that dyadic | action is merely a member of a triadic action, then so far from | its furnishing the least shade of presumption that all the action | in the physical universe is dyadic, on the contrary, the entire and | triadic action justifies a guess that there may be other and more marked | examples in the universe of the triadic pattern. No sooner is the guess | made than instances swarm upon us amply verifying it, and refuting the | agnostic position; while others present new problems for our study. | With the refutation of agnosticism, the agnostic is shown to be | a superficial neophyte in philosophy, entitled at most to | an occasional audience on special points, yet infinitely | more respectable than those who seek to bolster up what | is really true by sophistical arguments -- the traitors | to truth that they are ... | | Charles Sanders Peirce, CP 6.332 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 5 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Accurate writers have apparently made a distinction | between the 'definite' and the 'determinate'. A subject | is 'determinate' in respect to any character which inheres | in it or is (universally and affirmatively) predicated of | it, as well as in respect to the negative of such character, | these being the very same respect. In all other respects it | is 'indeterminate'. The 'definite' shall be defined presently. | | A sign (under which designation I place every kind of thought, | and not alone external signs), that is in any respect objectively | indeterminate (i.e., whose object is undetermined by the sign itself) | is objectively 'general' in so far as it extends to the interpreter | the privilege of carrying its determination further. 'Example': | "Man is mortal." To the question, What man? the reply is that the | proposition explicitly leaves it to you to apply its assertion to | what man or men you will. | | A sign that is objectively indeterminate in any respect | is objectively 'vague' in so far as it reserves further | determination to be made in some other conceivable sign, | or at least does not appoint the interpreter as its deputy | in this office. 'Example': "A man whom I could mention seems | to be a little conceited." The 'suggestion' here is that the | man in view is the person addressed; but the utterer does not | authorize such an interpretation or 'any' other application of | what she says. She can still say, if she likes, that she does | 'not' mean the person addressed. Every utterance naturally | leaves the right of further exposition in the utterer; and | therefore, in so far as a sign is indeterminate, it is vague, | unless it is expressly or by a well-understood convention | rendered general. | | Charles Sanders Peirce, 'Collected Papers', CP 5.447 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 6 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Perhaps a more scientific pair of definitions would be | that anything is 'general' in so far as the principle of | the excluded middle does not apply to it and is 'vague' | in so far as the principle of contradiction does not | apply to it. | | Thus, although it is true that "Any proposition | you please, 'once you have determined its identity', | is either true or false"; yet 'so long as it remains | indeterminate and so without identity', it need neither | be true that any proposition you please is true, nor that | any proposition you please is false. | | So likewise, while it is false that "A proposition 'whose | identity I have determined' is both true and false", yet | until it is determinate, it may be true that a proposition | is true and that a proposition is false. | | Charles Sanders Peirce, 'Collected Papers', CP 5.448 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 7 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | These remarks require supplementation. Determination, in general, is not | defined at all; and the attempt at defining the determination of a subject | with respect to a character only covers (or seems only to cover) explicit | propositional determination. The incidental remark [5.447] to the effect | that words whose meaning should be determinate would leave "no latitude of | interpretation" is more satisfactory, since the context makes it plain that | there must be no such latitude either for the interpreter or for the utterer. | The explicitness of the words would leave the utterer no room for explanation | of his meaning. This definition has the advantage of being applicable to a | command, to a purpose, to a medieval substantial form; in short to anything | capable of indeterminacy. (That everything indeterminate is of the nature | of a sign can be proved inductively by imagining and analyzing instances of | the surdest description. Thus, the indetermination of an event which should | happen by pure chance without cause, 'sua sponte', as the Romans mythologically | said, 'spontanément' in French (as if what was done of one's own motion were sure | to be irrational), does not belong to the event -- say, an explosion -- 'per se', | or as an explosion. Neither is it by virtue of any real relation: it is by | virtue of a relation of reason. Now what is true by virtue of a relation of | reason is representative, that is, is of the nature of a sign. A similar | consideration applies to the indiscriminate shots and blows of a Kentucky | free fight.) Even a future event can only be determinate in so far as it | is a consequent. Now the concept of a consequent is a logical concept. | It is derived from the concept of the conclusion of an argument. But an | argument is a sign of the truth of its conclusion; its conclusion is the | rational 'interpretation' of the sign. This is in the spirit of the Kantian | doctrine that metaphysical concepts are logical concepts applied somewhat | differently from their logical application. The difference, however, is | not really as great as Kant represents it to be, and as he was obliged to | represent it to be, owing to his mistaking the logical and metaphysical | correspondents in almost every case. | | Charles Sanders Peirce, 'Collected Papers', CP 5.448, note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 8 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Another advantage of this definition is that it saves us | from the blunder of thinking that a sign is indeterminate | simply because there is much to which it makes no reference; | that, for example, to say, "C.S. Peirce wrote this article", | is indeterminate because it does not say what the color of | the ink used was, who made the ink, how old the father of | the ink-maker when his son was born, nor what the aspect | of the planets was when that father was born. By making | the definition turn upon the interpretation, all that is | cut off. | | Charles Sanders Peirce, 'Collected Papers', CP 5.448, note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 9 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | At the same time, it is tolerably evident that the definition, | as it stands, is not sufficiently explicit, and further, that | at the present stage of our inquiry cannot be made altogether | satisfactory. For what is the interpretation alluded to? | To answer that convincingly would be either to establish | or to refute the doctrine of pragmaticism. | | Still some explanations may be made. Every sign has a single object, | though this single object may be a single set or a single continuum | of objects. No general description can identify an object. But the | common sense of the interpreter of the sign will assure him that the | object must be one of a limited collection of objects. [Long example]. | | [And so] the latitude of interpretation which constitutes the | indeterminacy of a sign must be understood as a latitude which | might affect the achievement of a purpose. For two signs whose | meanings are for all possible purposes equivalent are absolutely | equivalent. This, to be sure, is rank pragmaticism; for a purpose | is an affection of action. | | Charles Sanders Peirce, 'Collected Papers', CP 5.448, note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 10 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | The October remarks [i.e. those in the above paper] made the | proper distinction between the two kinds of indeterminacy, viz.: | indefiniteness and generality, of which the former consists in | the sign's not sufficiently expressing itself to allow of an | indubitable determinate interpretation, while the [latter] | turns over to the interpreter the right to complete the | determination as he please. | | It seems a strange thing, when one comes to ponder over it, that a sign | should leave its interpreter to supply a part of its meaning; but the | explanation of the phenomenon lies in the fact that the entire universe -- | not merely the universe of existents, but all that wider universe, | embracing the universe of existents as a part, the universe which | we are all accustomed to refer to as "the truth" -- that all this | universe is perfused with signs, if it is not composed exclusively | of signs. Let us note this in passing as having a bearing upon the | question of pragmaticism. | | The October remarks, with a view to brevity, omitted to mention that | both indefiniteness and generality might primarily affect either the | logical breadth or the logical depth of the sign to which it belongs. | It now becomes pertinent to notice this. When we speak of the depth, | or signification, of a sign we are resorting to hypostatic abstraction, | that process whereby we regard a thought as a thing, make an interpretant | sign the object of a sign. It has been a butt of ridicule since Molière's | dying week, and the depth of a writer on philosophy can conveniently be | sounded by his disposition to make fun of the basis of voluntary inhibition, | which is the chief characteristic of mankind. For cautious thinkers will | not be in haste to deride a kind of thinking that is evidently founded | upon observation -- namely, upon observation of a sign. At any rate, | whenever we speak of a predicate we are representing a thought as | a thing, as a 'substantia', since the concepts of 'substance' and | 'subject' are one, its concomitants only being different in the two | cases. It is needful to remark this in the present connexion, because, | were it not for hypostatic abstraction, there could be no generality of | a predicate, since a sign which should make its interpreter its deputy to | determine its signification at his pleasure would not signify anything, | unless 'nothing' be its significate. | | Charles Sanders Peirce, 'Collected Papers', CP 5.448, note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 11 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Concepts, or terms, are, in logic, conceived to have | 'subjective parts', being the narrower terms into which | they are divisible, and 'definitive parts', which are the | higher terms of which their definitions or descriptions are | composed: these relationships constitute "quantity". | | This double way of regarding a class-term as a whole of parts | is remarked by Aristotle in several places (e.g., 'Metaphysics', | D. xxv. 1023 b22). It was familiar to logicians of every age. | ... and it really seems to have been Kant who made these ideas | pervade logic and who first expressly called them quantities. | But the idea was old. Archbishop Thomson, W.D. Wilson, and | C.S. Peirce endeavor to make out a third quantity of terms. | The last calls his third quantity "information", and defines | it as the "sum of synthetical propositions in which the symbol | is subject or predicate", antecedent or consequent. The word | "symbol" is here employed because this logician regards the | quantities as belonging to propositions and to arguments, | as well as to terms. | | A distinction of 'extensive' and 'comprehensive distinctness' is | due to Scotus ('Opus Oxon.', I. ii. 3): namely, the usual effect | upon a term of an increase of information will be either to increase | its breadth without without diminishing its depth, or to increase its | depth without diminishing its breadth. But the effect may be to show | that the subjects to which the term was already known to be applicable | include the entire breadth of another another term which had not been | known to be so included. In that case, the first term has gained in | 'extensive distinctness'. Or the effect may be to teach that the | marks already known to be predicable of the term include the | entire depth of another term not previously known to be so | included, thus increasing the 'comprehensive distinctness' | of the former term. | | The passage of thought from a broader to a narrower concept | without change of information, and consequently with increase | of depth, is called 'descent'; the reverse passage, 'ascent'. | | For various purposes, we often imagine our information to be less than | it is. When this has the effect of diminishing the breadth of a term | without increasing its depth, the change is called 'restriction'; | just as when, by an increase of real information, a term gains | breadth without losing depth, it is said to gain extension. | This is, for example, a common effect of 'induction'. | In such case, the effect is called generalization. | | A decrease of supposed information may have the effect | of diminishing the depth of a term without increasing its | information. This is often called 'abstraction'; but it is | far better to call it 'prescission'; for the word 'abstraction' | is wanted as the designation of an even far more important procedure, | whereby a transitive element of thought is made substantive, as in the | grammatical change of an adjective into an abstract noun. This may be | called the principal engine of mathematical thought. | | When an increase of real information has the effect of increasing the | depth of a term without diminishing the breadth, the proper word for the | process is 'amplification'. In ordinary language, we are inaccurately said | to 'specify', instead of to 'amplify', when we add to information in this way. | The logical operation of forming a hypothesis often has this effect, which may, | in such case, be called 'supposition'. Almost any increase of depth may be called | 'determination'. | | Charles Sanders Peirce, 'Collected Papers', CP 2.364 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 12 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Determine. | | The 'termination' is an ending, and a 'term' is | a period (that comes to an end). 'Terminal' was | first (and still may be) an adjective; The Latin | noun 'terminus' has come directly into English: | Latin 'terminare, terminat-', to end; 'terminus', | boundary. From the limit itself, as in 'term' of | office or imprisonment, 'term' grew to mean the | limiting conditions (the 'terms' of an agreement); | hence, the 'defining' (Latin 'finis', end; compare | 'finance') of the idea, as in a 'term' of reproach; | 'terminology'. To 'determine' is to set down limits | or bounds to something, as when you 'determine' to | perform a task, or as 'determinism' pictures limits | set to man's freedom. 'Predetermined' follows this | sense; but 'extermination' comes later. Otherwise, | existence would be 'interminable'. | | Joseph T. Shipley, 'Dictionary of Word Origins', | Rowman & Allanheld, Totowa, NJ, 1967, 1985. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 13 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | To determine means to make a circumstance different from what | it might have been otherwise. For example, a drop of rain | falling on a stone determines it to be wet, provided the | stone may have been dry before. But if the fact of | a whole shower half an hour previous is given, | then one drop does not determine the stone to | be wet; for it would be wet, at any rate. | | CSP, CE 1, pages 245-246. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 14 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Taking it for granted, then, that the inner and outer worlds are | superposed throughout, without possibility of separation, let us | now proceed to another point. There is a third world, besides the | inner and the outer; and all three are coëxtensive and contain every | experience. Suppose that we have an experience. That experience has | three determinations -- three different references to a substratum or | substrata, lying behind it and determining it. In the first place, | it is a determination of an object external to ourselves -- we feel | that it is so because it is extended in space. Thereby it is in the | external world. In the second place, it is a determination of our own | soul, it is 'our' experience; we feel that it is so because it lasts in | time. Were it a flash of sensation, there for less than an instant, and | then utterly gone from memory, we should not have time to think it ours. | But while it lasts, and we reflect upon it, it enters into the internal | world. We have now considered that experience as a determination of the | modifying object and of the modified soul; now, I say, it may be and is | naturally regarded as also a determination of an idea of the Universal | mind; a preëxistent, archetypal Idea. Arithmetic, the law of number, | 'was' before anything to be numbered or any mind to number had been | created. It 'was' though it did not 'exist'. It was not 'a fact' | nor a thought, but it was an unuttered word. 'En arche en o logos'. | We feel an experience to be a determination of such an archetypal | Logos, by virtue of its // 'depth of tone' / logical intension //, | and thereby it is in the 'logical world'. | | Note the great difference between this view and Hegel's. | Hegel says, logic is the science of the pure idea. I should | describe it as the science of the laws of experience in virtue | of its being a determination of the idea, or in other words as | the formal science of the logical world. | | In this point of view, efforts to ascertain precisely how the | intellect works in thinking, -- that is to say investigation | of internal characterictics -- is no more to the purpose which | logical writers as such, however vaguely have in view, than | would be the investigation of external characteristics. | | CSP, CE 1, pages 168-169. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 15 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | But not to follow this subject too far, we have | now established three species of representations: | 'copies', 'signs', and 'symbols'; of the last of | which only logic treats. A second approximation to | a definition of it then will be, the science of symbols | in general and as such. But this definition is still | too broad; this might, indeed, form the definition of | a certain science which would be a branch of Semiotic | or the general science of representations which might | be called Symbolistic, and of this logic would be | a species. But logic only considers symbols | from a particular point of view. | | A symbol in general and as such has three relations. | The first is its relation to the pure Idea or Logos | and this (from the analogy of the grammatical terms | for the pronouns I, It, Thou) I call its relation | of the first person, since it is its relation to | its own essence. The second is its relation to | the Consciousness as being thinkable, or to any | language as being translatable, which I call its | relation to the second person, since it refers to | its power of appealing to a mind. The third is its | relation to its object, which I call its relation to | the third person or It. Every symbol is subject to | three distinct systems of formal law as conditions | of its taking up these three relations. If it | violates either one of these three codes, the | condition of its having either of the three | relations, it ceases to be a symbol and makes | 'nonsense'. Nonsense is that which has a certain | resemblance to a symbol without being a symbol. But | since it simulates the symbolic character it is usually | only one of the three codes which it violates; at any rate, | flagrantly. Hence there should be at least three different kinds | of nonsense. And accordingly we remark that that we call nonsense | meaningless, absurd, or quibbling, in different cases. If a symbol | violates the conditions of its being a determination of the pure | Idea or logos, it may be so nearly a determination thereof as | to be perfectly intelligible. If for instance instead | of 'I am' one should say 'I is'. | 'I is' is in itself meaningless, | it violates the conditions of its | relation to the form it is meant | to embody. Thus we see that the | conditions of the relation of the | first person are the laws of grammar. | | I will now take another example. I know my opinion is false, still I hold it. | This is grammatical, but the difficulty is that it violates the conditions | of its having an object. Observe that this is precisely the difficulty. | It not only cannot be a determination of this or that object, but it | cannot be a determination of any object, whatever. This is the | whole difficulty. I say that, I receive contradictories into | one opinion or symbolical representation; now this implies | that it is a symbol of nothing. Here is another example: | This very proposition is false. This is a proposition to | which the law of excluded middle namely that every symbol | must be false or true, does not apply. For if it is false it | is thereby true. And if not false it is thereby not true. Now | why does not this law apply to this proposition. Simply because it | does itself state that it has no object. It talks of itself and only | of itself and has no external relation whatever. These examples show | that logical laws only hold good, as conditions of a symbol's having | an object. The fact that it has often been called the science of | truth confirms this view. | | I define logic therefore as the science of the conditions | which enable symbols in general to refer to objects. | | At the same time 'symbolistic' in general gives a trivium consisting of | Universal Grammar, Logic, and Universal Rhetoric, using this last term to | signify the science of the formal conditions of intelligibility of symbols. | | CSP, CE 1, pages 174-175. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 16 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | The consideration of this imperfect datum leads us to make | a fundamental observation; namely, that the problem how we | can make an induction is one and the same with the problem how | we can make any general statement, with reason; for there is | no way left in which such a statement can originate except from | induction or pure fiction. Hereby, we strike down at once all | attempts at solving the problem as involve the supposition of | a major premiss as a datum. Such explanations merely show | that we can arrive at one general statement by deduction | from another, while they leave the real question, | untouched. The peculiar merit of Aristotle's | theory is that after the objectionable portion | of it is swept away and after it has thereby been | left utterly powerless to account for any certainty | or even probability in the inference from induction, | we still retain these 'forms' which show what the | 'actual process' is. | | And what is this process? We have in the apodictic conclusion, | some most extraordinary observation, as for example that a great | number of animals -- namely neat and deer, feed only upon vegetables. | This proposition, be it remarked, need not have had any generality; if | the animals observed instead of being all 'neat' had been so very various | that we knew not what to say of them except that they were 'herbivora' and | 'cloven-footed', the effect would have been to render the argument simply | irresistable. In addition to this datum, we have another; namely that | these same animals are all cloven-footed. Now it would not be so very | strange that all cloven-footed animals should be herbivora; animals | of a particular structure very likely may use a particular food. | But if this be indeed so, then all the marvel of the conclusion | is explained away. So in order to avoid a marvel which must in | some form be accepted, we are led to believe what is easy to | believe though it is entirely uncertain. | | CSP, CE 1, page 179. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 17 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | There is a large class of reasonings which are neither deductive nor inductive. | I mean the inference of a cause from its effect or reasoning to a physical hypothesis. | I call this reasoning 'à posteriori'. If I reason that certain conduct is wise because | it has a character which belongs 'only' to wise things, I reason 'à priori'. If I think | it is wise because it once turned out to be wise, that is if I infer that it is wise on | this occasion because it was wise on that occasion, I reason inductively. But if | I think it is wise because a wise man does it, I then make the pure hypothesis | that he does it because he is wise, and I reason 'à posteriori'. The form | this reasoning assumes, is that of an inference of a minor premiss in | any of the figures. The following is an example. | | Light gives certain fringes. | Ether waves give certain fringes. | Ether waves gives these fringes. | Light is ether waves. | .: Light is ether waves. | .: Light gives these fringes. | | CSP, CE 1, page 180. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 18 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | We come now to the question, what is the 'rationale' of these three kinds | of reasoning. And first let us understand precisely what we intend by this. | It is clear then that it is none of our business to inquire in what manner we | think when we reason, for we have already seen that logic is wholly separate | from psychology. What we seek is an explicit statement of the logical ground | of these different kinds of inference. This logical ground will have two parts, | 1st the ground of possibility and 2nd the ground of proceedure. The ground of | possibility is the special property of symbols upon which every inference of | a certain kind rests. The ground of proceedure is the property of symbols | which makes a certain inference possible from certain premisses. The | ground of possibility must be both discovered and demonstrated, fully. | The ground of proceedure must be exhibited in outline, but it is not | requisite to fill up all the details of this subject, especially | as that would lead us too far into the technicalities of logic. | | As the three kinds of reasoning are entirely distinct, each must have | a different ground of possibility; and the principle of each kind must | be proved by that same kind of inference for it would be absurd to attempt | to rest it on a weaker kind of inference and to rest it on one as strong as | itself would be simply to reduce it to that other kind of reasoning. Moreover, | these principles must be logical principles because we do not seek any other | ground now, than a logical ground. As logical principles, they will not | relate to the symbol in itself or in its relation to equivalent symbols | but wholly in its relation to what it symbolizes. In other words | it will relate to the symbolization of objects. | | CSP, CE 1, page 183. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 19 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Now all symbolization is of three objects, at once; the first is a possible thing, | the second is a possible form, the third is a possible symbol. It will be objected | that the two latter are not properly objects. We have hitherto regarded the symbol | as 'standing for' the thing, as a concrete determination of its form, and addressing | a symbol; and it is true that it is only by referring to a possible thing that a | symbol has an objective relation, it is only by bearing in it a form that it has | any subjective relation, and it is only by equaling another symbol that it has any | tuistical relation. But this objective relation once given to a symbol is at once | applicable to all to which it necessarily refers; and this is shown by the fact | of our regarding every symbol as 'connotative' as well as 'denotative', and by our | regarding one word as standing for another whenever we endeavor to clear up a little | obscurity of meaning. And the reason that this is so is that the possible symbol and | the possible form to which a symbol is related each relate also to that thing which | is its immediate object. Things, forms, and symbols, therefore, are symbolized in | every symbolization. And this being so, it is natural to suppose that our three | principles of inference which we know already refer to some three objects of | symbolization, refer to these. | | That such really is the case admits of proof. For the principle of inference 'à priori' | must be established 'à priori'; that is by reasoning analytically from determinant to | determinate, in other words from definition. But this can only be applied to an object | whose characteristics depend upon its definition. Now of most things the definition | depends upon the character, the definition of a symbol alone determines its character. | Hence the principle of inference 'à priori' must relate to symbols. The principle of | inference 'à posteriori' must be established 'à posteriori', that is by reasoning from | determinate to determinant. This is only applicable to that which is determined by what | it determines; in other words, to that which is only subject to the truth and falsehood | which affects its determinant and which in itself is mere 'zero'. But this is only true | of pure forms. Hence the principle of inference 'à posteriori' must relate to pure form. | The principle of inductive inference must be established inductively; that is by reasoning | from parts to whole. This is only applicable to that whose whole is given in the sum of the | parts; and this is only the case with things. Hence the principle of inductive inference | must relate to things. | | CSP, CE 1, pages 183-184. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 20 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Is there any knowledge 'à priori'? All our thought begins with | experience, the mind furnishes no material for thought whatever. | This is acknowledged by all the philosophers with whom we need concern | ourselves at all. The mind only works over the materials furnished by | sense; no dream is so strange but that all its elementary parts are | reminiscences of appearance, the collocation of these alone are we | capable of originating. In one sense, therefore, everything may | be said to be inferred from experience; everything that we know, | or think or guess or make up may be said to be inferred by some | process valid or fallacious from the impressions of sense. But | though everything in this loose sense is inferred from experience, | yet everything does not require experience to be as it is in order | to afford data for the inference. Give me the relations of 'any' | geometrical intuition you please and you give me the data for proving | all the propositions of geometry. In other words, everything is not | determined by experience. And this admits of proof. For suppose | there may be universal and necessary judgements; as for example | the moon must be made of green cheese. But there is no element of | necessity in an impression of sense for necessity implies that things | would be the same as they are were certain accidental circumstances | different from what they are. I may here note that it is very common | to misstate this point, as though the necessity here intended were a | necessity of thinking. But it is not meant to say that what we feel | compelled to think we are absolutely compelled to think, as this would | imply; but that if we think a fact 'must be' we cannot have observed | that it 'must be'. The principle is thus reduced to an analytical one. | In the same way universality implies that the event would be the same | were the things within certain limits different from what they are. | Hence universal and necessary elements of experience are not determined | from without. But are they, therefore, determined from within? Are they | determined at all? Does not this very conception of determination imply | causality and thus beg the whole question of causality at the very outset? | Not at all. The determination here meant is not real determination but | logical determination. A cognition 'à priori' is one which any experience | contains reason for and therefore which no experience determines but which | contains elements such as the mind introduces in working up the materials | of sense, or rather as they are not new materials, they are the working up. | | CSP, CE 1, pages 246-247. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 21 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | The terms 'à priori' and 'à posteriori' in their ancient sense | denote respectively reasoning from an antecedent to a consequent | and from a consequent to an antecedent. Thus suppose we know that | every incompetent general will meet with defeat. Then if we reason | that because a given general is incompetent that he must meet with | a defeat, we reason 'à priori'; but if we reason that because a | general is defeated he was a bad one, we reason 'à posteriori'. | | Kant however uses these terms in another and derived sense. He did not | entirely originate their modern use, for his contemporaries were already | beginning to apply them in the same way, but he fixed their 'meaning' in | the new application and made them household words in subsequent philosophy. | | If one judges that a house falls down on the testimony of his eyesight | then it is clear that he reasons 'à posteriori' because he infers the | fact from an effect of it on his eyes. If he judges that a house falls | because he knows that the props have been removed he reasons 'à priori'; | yet not purely 'à priori' for his premisses were obtained from experience. | But if he infers it from axioms innate in the constitution of the mind, | he may be said to reason purely 'à priori'. All this had been said | previously to Kant. I will now state how he modified the meaning | of the terms while preserving this application of them. What is | known from experience must be known 'à posteriori', because the | thought is determined from without. To determine means to make | a circumstance different from what it might have been otherwise. | For example, a drop of rain falling on a stone determines it to | be wet, provided the stone may have been dry before. But if the | fact of a whole shower half an hour previous is given, then one | drop does not determine the stone to be wet; for it would be wet, | at any rate. Now, it is said that the results of experience are | inferred 'à posteriori', for this reason that they are determined | from without the mind by something not previously present to it; | being so determined their determinants or //causes/reasons// are | not present to the mind and of course could not be reasoned from. | Hence, a thought determined from without by something not in | consciousness even implicitly is inferred 'à posteriori'. | | Kant, accordingly, uses the term 'à posteriori' as meaning what | is determined from without. The term 'à priori' he uses to mean | determined from within or involved implicitly in the whole of what | is present to consciousness (or in a conception which is the logical | condition of what is in consciousness). The twist given to the words | is so slight that their application remains almost exactly the same. | If there is any change it is this. A primary belief is 'à priori' | according to Kant; for it is determined from within. But it is not | 'inferred' at all and therefore neither of the terms is applicable in | their ancient sense. And yet as an explicit judgment it is inferred | and inferred 'à priori'. | | CSP, CE 1, pages 245-246. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o DET. Note 22 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Though I talk of forms as something independent of the mind, | I only mean that the mind so conceives them and that that | conception is valid. I thus say that all the qualities | we know are determinations of the pure idea. But that | we have any further knowledge of the idea or that | this is to know it in itself I entirely deny. | | CSP, CE 1, page 256. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

###### EXC. Excuses

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Note 1. Excuses o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | We must regard classical mathematics | as a combinatorial game played with the | primitive symbols, and we must determine | in a finitary combinatorial way to which | combinations of primitive symbols the | construction methods or "proofs" lead. | | JVN, TFFOM, page 62. | | Johann Von Neumann, |"The Formalist Foundations of Mathematics", | from a "Symposium on the Foundations of Mathematics", | originally published in 'Erkenntnis' (1931), pp 91-121; | translated from German by Erna Putnam & Gerald J. Massey, | reprinted in Paul Benacerraf & Hilary Putnam (editors), |'Philosophy of Mathematics, Selected Readings', 2nd ed., | Cambridge University Press, Cambridge, UK, 1983. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Note 2. Exergues o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | It is a great honor and at the same time a necessity for me to round out and | develop my thoughts on the foundations of mathematics, which I expounded here | one day five years ago and which since then have constantly kept me most actively | occupied. With this new way of providing a foundation for mathematics, which we may | appropriately call a proof theory, I pursue a significant goal, for I should like to | eliminate once and for all the questions regarding the foundations of mathematics, in | the form in which they are now posed, by turning every mathematical proposition into | a formula that can be concretely exhibited and strictly derived, thus recasting | mathematical definitions and inferences in such a way that they are unshakable | and yet provide an adequate picture of the whole science. I believe that I | can attain this goal completely with my proof theory, even if a great deal | of work must still be done before it is fully developed. | | DH, TFOM, page 464. | | David Hilbert, |"The Foundations Of Mathematics", | address delivered to the Hamburg Mathematical Seminar, July 1927, | reprinted in Jean van Heijenoort (ed.), 'From Frege To Gödel, | A Source Book in Mathematical Logic', 1879-1931', | Harvard University Press, Cambridge, MA, 1967. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Note 3. Exercises o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Take this remark out of context, please. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Note 4. Exorabilities o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | No more than any other science can mathematics be founded by logic alone; | rather, as a condition for the use of logical inferences and the performance | of logical operations, something must already be given to us in our faculty of | representation [in der Vorstellung], certain extralogical concrete objects that | are intuitively [anschaulich] present as immediate experience prior to all thought. | If logical inference is to be reliable, it must be possible to survey these objects | completely in all their parts, and the fact that they occur, that they differ from one | another, and that they follow each other, or are concatenated, is immediately given | intuitively, together with the objects, as something that neither can be reduced to | anything else nor requires reduction. This is the basic philosophical position | that I regard as requisite for mathematics and, in general, for all scientific | thinking, understanding, and communication. And in mathematics, in particular, | what we consider is the concrete signs themselves, whose shape, according to | the conception we have adopted, is immediately clear and recognizable. | This is the very least that must be presupposed; no scientific thinker | can dispense with it, and therefore everyone must maintain it, | consciously or not. | | DH, TFOM, pages 464-465. | | David Hilbert, |"The Foundations Of Mathematics", | address delivered to the Hamburg Mathematical Seminar, July 1927, | reprinted in Jean van Heijenoort (ed.), 'From Frege To Gödel, | A Source Book in Mathematical Logic', 1879-1931', | Harvard University Press, Cambridge, MA, 1967. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Note 5. Excisions o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | There is no break between my painting and my cut-outs. | Only with something more of the abstract and the | absolute, I have arrived at a distillation of | form ... of this or that object which I used | to present in all its complexity in space, | I now keep only the sign which suffices, | necessary for its existence in its | own form, for the composition | as I conceive it. | | Henri Matisse o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Note 6. Exordia o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Carmina qui quondam studio florente peregi, | Flebilis heu maestos cogor inire modos. | Ecce mihi lacerae dictant scribenda camenae | Et veris elegi fletibus ora rigant. | Has saltem nullus potuit pervincere terror, | Ne nostrum comites prosequerentur iter. | Gloria felicis olim viridisque iuventae | Solantur maesti nunc mea fata senis. | Venit enim properata malis inopina senectus | Et dolor aetatem iussit inesse suam. | Intempestivi funduntur vertice cani | Et tremit effeto corpore laxa cutis. | Mors hominum felix quae se nec dulcibus annis | Inserit et maestis saepe vocata venit. | Eheu quam surda miseros avertitur aure | Et flentes oculos claudere saeva negat. | Dum levibus male fida bonis fortuna faveret, | Paene caput tristis merserat hora meum. | Nunc quia fallacem mutavit nubila vultum, | Protrahit ingratas impia vita moras. | Quid me felicem totiens iactastis amici? | Qui cecidit, stabili non erat ille gradu. | | Verses I made once glowing with content; | Tearful, alas, sad songs must I begin. | See how the Muses grieftorn bid me write, | And with unfeigned tears these elegies drench my face. | But them at least my fear that my friends might tread my path | Companions still | Could not keep me silent: they were once | My green youth's glory; now in my sad old age | They comfort me. | For age has come unlooked for, hastened by ills, | And anguish sternly adds its years to mine; | My head is white before its time, my skin hangs loose | About my tremulous frame: I am worn out. | Death, if he come | Not in the years of sweetness | But often called to those who want to end their misery | Is welcome. My cries he does not hear; | Cruel he will not close my weeping eyes. | While fortune favoured me -- | How wrong to count on swiftly-fading joys -- | Such an hour of bitterness might have bowed my head. | Now that her clouded, cheating face is changed | My cursed life drags on its long, unwanted days. | Ah why, my friends, | Why did you boast so often of my happiness? | How faltering even then the step | Of one now fallen. | | Boethius (Anicius Manlius Severinus Boetius, c.480-524 A.D.), |'The Consolation of Philosophy', Translation by S.J. Tester, | New Edition, Loeb Classical Library, Harvard/Heinemann, 1973. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Note 7. Experiments o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Now, Gentlemen, it behooves me, at the outset of this course, | to confess to you that in this respect I stand before you an | Aristotelian and a scientific man, condemning with the whole | strength of conviction the Hellenic tendency to mingle | Philosophy and Practice. | | There are sciences, of course, many of whose results are almost immediately | applicable to human life, such as physiology and chemistry. But the true | scientific investigator completely loses sight of the utility of what he | is about. It never enters his mind. Do you think that the physiologist | who cuts up a dog reflects while doing so, that he may be saving a human | life? Nonsense. If he did, it would spoil him for a scientific man; | and 'then' the vivisection would become a crime. However, in physiology | and in chemistry, the man whose brain is occupied with utilities, though | he will not do much for science, may do a great deal for human life. | But in philosophy, touching as it does upon matters which are, and | ought to be, sacred to us, the investigator who does not stand | aloof from all intent to make practical applications, will not | only obstruct the advance of the pure science, but what is | infinitely worse, he will endanger his own moral integrity | and that of his readers. | | CSP, RATLOT, 107. | | Charles Sanders Peirce, |'Reasoning and the Logic of Things', |'The Cambridge Conferences Lectures of 1898', | Edited by Kenneth Laine Ketner, Introduction | by Kenneth Laine Ketner and Hilary Putnam, | Harvard University Press, Cambridge, MA, 1992. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Note 8. Exquisitions o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | While I was thinking these thoughts to myself in silence, | and set my pen to record this tearful complaint, there seemed | to stand above my head a woman. Her look filled me with awe; | her burning eyes penetrated more deeply than those of ordinary men; | her complexion was fresh with an ever-lively bloom, yet she seemed | so ancient that none would think her of our time. It was difficult | to say how tall she might be, for at one time she seemed to confine | herself to the ordinary measure of man, and at another the crown of | her head touched the heavens; and when she lifted her head higher | yet, she penetrated the heavens themselves, and was lost to the | sight of men. Her dress was made of very fine, imperishable thread, | of delicate workmanship: she herself wove it, as I learned later, | for she told me. Its form was shrouded by a kind of darkness of | forgotten years, like a smoke-blackened family statue in the atrium. | On its lower border was woven the Greek letter Pi, and on the upper, | Theta, and between the two letters steps were marked like a ladder, | by which one might climb from the lower letter to the higher. | But violent hands had ripped this dress and torn away what | bits they could. In her right hand she carried a book, | and in her left, a sceptre. | | Boethius (Anicius Manlius Severinus Boetius, c.480-524 A.D.), |'The Consolation of Philosophy', Translation by S.J. Tester, | New Edition, Loeb Classical Library, Harvard/Heinemann, 1973. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Note 9. Exorcisms o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Now when she saw the Muses of poetry standing by my bed, | helping me to find words for my grief, she was disturbed | for a moment, and then cried out with fiercely blazing eyes: | "Who let these theatrical tarts in with this sick man? Not only | have they no cures for his pain, but with their sweet poison they | make it worse. These are they who choke the rich harvest of the | fruits of reason with the barren thorns of passion. They accustom | a man's mind to his ills, not rid him of them. If your enticements | were distracting merely an unlettered man, as they usually do, I should | not take it so seriously -- after all, it would do no harm to us in our | task -- but to distract this man, reared on a diet of Eleatic and Academic | thought! Get out, you Sirens, beguiling men straight to their destruction! | Leave him to 'my' Muses to care for and restore to health." Thus upbraided, | that company of the Muses dejectedly hung their heads, confessing their shame | by their blushes, and dismally left my room. I myself, since my sight was | so dimmed with tears that I could not clearly see who this woman was of | such commanding authority, was struck dumb, my eyes cast down; and | I went on waiting in silence to see what she would do next. Then | she came closer and sat on the end of my bed, and seeing my face | worn with weeping and cast down with sorrow, she bewailed my | mind's confusion bitterly in these verses: ... | | Boethius (Anicius Manlius Severinus Boetius, c.480-524 A.D.), |'The Consolation of Philosophy', Translation by S.J. Tester, | New Edition, Loeb Classical Library, Harvard/Heinemann, 1973. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Note 10. Expositions o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Actors, taught not to let any embarrassment show | on their faces, put on a mask. I will do the same. | So far, I have been a spectator in this theatre which | is the world, but I am now about to mount the stage, | and I come forward masked. | | René Descartes, 'Praeambula', CSM 1, page 2. | | René Descartes, 'The Philosophical Writings of Descartes', Volume 1, | Translated by John Cottingham, Robert Stoothoff, Dugald Murdoch, | Cambridge University Press, Cambridge, UK, 1985. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Note 11. Exponents o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | The sciences are at present masked, but if the | masks were taken off, they would be revealed | in all their beauty. If we could see how the | sciences are linked together, we would find | them no harder to retain in our minds than | the series of numbers. | | René Descartes, 'Praeambula', CSM 1, page 3. | | René Descartes, 'The Philosophical Writings of Descartes', Volume 1, | Translated by John Cottingham, Robert Stoothoff, Dugald Murdoch, | Cambridge University Press, Cambridge, UK, 1985. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Note 12. Experimenta o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | I use the term "vices" to refer to the diseases of the mind, | which are not so easy to recognize as diseases of the body. | This is because we have frequently experienced sound bodily | health, but have never known true health of the mind. | | René Descartes, 'Experimenta', CSM 1, page 3. | | René Descartes, 'The Philosophical Writings of Descartes', Volume 1, | Translated by John Cottingham, Robert Stoothoff, Dugald Murdoch, | Cambridge University Press, Cambridge, UK, 1985. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Note 13. Exilaration o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Just as the imagination employs figures in order to conceive of bodies, | so, in order to frame ideas of spiritual things, the intellect makes | use of certain bodies which are perceived through the senses, such | as wind and light. By this means we may philosophize in a more | exalted way, and develop the knowledge to raise our minds to | lofty heights. | | It may seem surprising to find weighty judgements in the writings | of the poets rather than the philosophers. The reason is that the | poets were driven to write by enthusiasm and the force of imagination. | We have within us the sparks of knowledge, as in a flint: philosophers | extract them through reason, but poets force them out through the sharp | blows of the imagination, so that they shine more brightly. | | René Descartes, 'Olympica', CSM 1, page 4. | | René Descartes, 'The Philosophical Writings of Descartes', Volume 1, | Translated by John Cottingham, Robert Stoothoff, Dugald Murdoch, | Cambridge University Press, Cambridge, UK, 1985. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Note 14. Exsertion o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Where number is irrelevant, regimented mathematical technique | has hitherto tended to be lacking. Thus it is that the progress | of natural science has depended so largely upon the discernment of | measurable quantity of one sort or another. Measurement consists | in correlating our subject matter with the series of real numbers; | and such correlations are desirable because, once they are set up, | all the well-worked theory of numerical mathematics lies ready at | hand as a tool for our further reasoning. But no science can rest | entirely on measurement, and many scientific investigations are | quite out of reach of that device. To the scientist longing for | non-quantitative techniques, then, mathematical logic brings hope. | It provides explicit techniques for manipulating the most basic | ingredients of discourse. Its yield for science may be expected to | consist also in a contribution of rigor and clarity -- a sharpening of | the concepts of science. Such sharpening of concepts should serve both | to disclose hitherto hidden consequences of given scientific hypotheses, | and to obviate subtle errors which may stand in the way of scientific | progress. | | Quine, 'Mathematical Logic', pages 7-8. | | Quine, Willard Van Orman, |'Mathematical Logic', Revised Edition, | Harvard University Press, Cambridge, MA, | 1940, 1951, 1981. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

###### EII. Extension, Intension, Information

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EII. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Subj: Free Range Chickens! Date: Wed, 01 Nov 2000 16:00:49 -0500 From: Jon Awbrey <jawbrey@oakland.edu> To: Chris Partridge <mail@ChrisPartridge.com> CC: John Sowa <sowa@bestweb.net>, Matthew West <Matthew.R.West@is.shell.com>, Stand Up Ontology <standard-upper-ontology@ieee.org> CP = Chris Partridge CP: As you know your points assume that the extension only ranges over the actual world -- if it were allowed to range over possible worlds/entities then "featherless bipeds" including plucked chickens would never be co-extensive with humans (a strategy followed by David Lewis, for example). CP: This would not make the 'method of definition' and the 'extension' give the same result. For example, equilateral and equiangular triangles would have the same extension but different definitions. For mathematicians, at least, this extensional approach 'works' as equilateral triangles are provably equiangular triangles. CP: Also introducing the method of definition raises the problem of giving a criteria for 'identity' or 'equality' of definitions. Something not as simple as for extensions. It is easy to say when two extensions are the same - it is not so easy to say when two definitions are equivalent. In my experience, when people speak of "the actual universe" (TAU) -- pretending to the throne of knowledge that would be entitled to prefix that definite, all too definite article to it -- they are more likely to be talking about a universe much more personally familiar to them, the universe that they imagine to fit their own description of it. To actually talk about the actual universe is speak in the hopeful lights of a future perfect perspective, a future contingent retrospective, if you will: | The universe that will have been found to be actual, when and if | the end of inquiry into actualities will have become actualized. Thus questions of extension (the extensions that we know about), plus questions of intension (the intensions that we know about), can be regarded as independent of questions of epistemology and inquiry only in the ideal, the imaginary, the "entelectual" end of all perspectives. In the meantime, these issues are indexed to their issuers, intentional and otherwise, and their intended interpreters, like you and I. By the way, my memory is dim here once again, but it seems like there are probably geometries where the concepts of equiangular triangles and equilateral triangles diverge. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EII. Note 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Subj: Extension, Intension, Information Date: Wed, 01 Nov 2000 21:42:07 -0500 From: Jon Awbrey <jawbrey@oakland.edu> To: Stand Up Ontology <standard-upper-ontology@ieee.org> CC: Chris Partridge <mail@ChrisPartridge.com>, John F Sowa <sowa@bestweb.net>, Matthew West <Matthew.R.West@is.shell.com> CP = Chris Partridge JS = John Sowa CP: As you know your points assume that the extension only ranges over the actual world -- if it were allowed to range over possible worlds/entities then "featherless bipeds" including plucked chickens would never be co-extensive with humans (a strategy followed by David Lewis, for example). JS: That is true. But I suggest that for KIF (and SUO in general) we avoid metaphysically loaded terms such as "possible worlds" and use the more formal, less confusing term "models". CP: Also introducing the method of definition raises the problem of giving a criteria for "identity" or "equality" of definitions. Something not as simple as for extensions. It is easy to say when two extensions are the same -- it is not so easy to say when two definitions are equivalent. JS: Alonzo Church made the point that term "equality by extension" is univocal for functions, predicates, etc., but the term "equality by intension" has as many possible meanings as there are possible methods for defining intensions. To make that term precise, he proposed the lambda calculus as a method for defining what it means for two definitions to be "the same". Definition A is equivalent to definition B iff there is some sequence of lambda conversions that transform A to B (and vice-versa). JS: Church also made the point that there was nothing special about lambda conversions and said that other formally defined translation methods could also be used. That is one of the motivations for my proposed "meaning-preserving translations" that I discussed in a previous note. The following web page has a definition of MPT functions, excerpted from my KR book: JS: http://www.bestweb.net/~sowa/ontology/meaning.htm JS: The definition of MPT functions includes lambda conversions as a special case for translations from some language L into L, but they are general enough to include translations from one logic-based language, such as KIF, to or from many other formal languages, such as CGs, DLs, SUO-CE, etc. As you may well expect, Peirce had a different way of looking at this problem of the relationship between extension and intension, the latter of which he preferred to refer to as "comprehension", though my exaggerated need for syntactic symmetry will prevent me from following him on that score. In point of fact, one of the reasons why Peirce invented the subject that he dubbed the "Theory of Information", first presented in lectures in 1865, though, sad to say, in such an obscure out-of-the-way academic backwater as Harvard College that it could hardly be expected to make much of a splash in the intellectual currents of the times, was precisely to address this problem and to bring about an integrated understanding of the exchange relations between these two aspects of what he came to recognize was a much more fundamental notion, related to an interpretive agent's state of knowledge about an objective situation, to wit, this very "information" of which he spoke, of which I speak. And you can look it up! So what? So I think that maybe we ought to go look it up, in CE 1, for a start, because I think that we may just find, if try sometimes, with a little help from our fieri friend Peirce, that there may, after all -- well, actually, before all -- be a way out of these Fregean deadlocks and these Russellian mortemains. Just Maybe ... o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

###### FOR. Inquiry Into Formalization

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o FOR. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o While looking into my dissertation for concrete illustrations of sign relations, I found to my amazement that the set-up for the "Story of A and B" contains not a few reflections that may throw a different light on our recent animadversions over the topic of formalization. So here 'tis ... 1.3.4 Discussion of Formalization: Concrete Examples The previous section outlined a variety of general issues surrounding the concept of formalization. The following section will plot the specific objectives of this project in constructing formal models of intellectual processes. In this section I wish to take a breather between these abstract discussions in order to give their main ideas a few points of contact with terra firma. To do this, I examine a selection of concrete examples, artificially constructed to approach the minimum levels of non-trivial complexity, that are intended to illustrate the kinds of mathematical objects I have in mind using as formal models. 1.3.4.1 Formal Models: A Sketch To sketch the features of the modeling activity that are relevant to the immediate purpose: The modeler begins with a "phenomenon of interest" or a "process of interest" (POI) and relates it to a formal "model of interest" (MOI), the whole while working within a particular "interpretive framework" (IF) and relating the results from one "system of interpretation" (SOI) to another, or to a subsequent development of the same SOI. The POI's that define the intents and the purposes of this project are the closely related processes of inquiry and interpretation, so the MOI's that must be formulated are models of inquiry and interpretation, species of formal systems that are even more intimately bound up than usual with the IF's employed and the SOI's deployed in their ongoing development as models. Since all of the interpretive systems and all of the process models that are being mentioned here come from the same broad family of mathematical objects, the different roles that they play in this investigation are mainly distinguished by variations in their manner and degree of formalization: 1. The typical POI comes from natural sources and casual conduct. It is not formalized in itself but only in the form of its image or model, and just to the extent that aspects of its structure and function are captured by a formal MOI. But the richness of any natural phenomenon or realistic process seldom falls within the metes and bounds of any final or finite formula. 2. Beyond the initial stages of investigation, the MOI is postulated as a completely formalized object, or is quickly on its way to becoming one. As such, it serves as a pivotal fulcrum and a point of application poised between the undefined reaches of "phenomena" and "noumena", respectively, terms that serve more as directions of pointing than as denotations of entities. What enables the MOI to grasp these directions is the quite felicitous mathematical circumstance that there can be well-defined and finite relations between entities that are infinite and even indefinite in themselves. Indeed, exploiting this handle on infinity is the main trick of all computational models and effective procedures. It is how a "finitely informed creature" (FIC) can "make infinite use of finite means". Thus, my reason for calling the MOI cardinal or pivotal is that it forms a model in two senses, loosely analogical and more strictly logical, integrating twin roles of the model concept in a single focus. 3. Finally, the IF's and the SOI's always remain partly out of sight, caught up in various stages of explicit notice between casual informality and partial formalization, with no guarantee or even much likelihood of a completely articulate formulation being forthcoming or even possible. Still, it is usually worth the effort to try lifting one edge or another of these frameworks and backdrops into the light, at least for a time. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o FOR. Note 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Subj: Inquiry Into Bodaciousness -- Anti Matter & Pro Forma Went To Sea ... Date: Wed, 05 Sep 2001 09:52:15 -0400 From: Jon Awbrey <jawbrey@oakland.edu> To: Arisbe <arisbe@stderr.org>, Generic Ontology Group <ontology@ieee.org> CC: Organization Complexity Autonomy <oca@cc.newcastle.edu.au> Must Being Pro Forma Be Being Anti Matter? To notice a form at the stern of the ship making a plank of wood and a bell of clay is scarcely to atomize the gang aft agley, to demolish a form's material supposition. The moat rimes the eye of the interpreter, the stigmata of whose antismantism illude the eye's motives to persist in imagining that being pro forma is being anti matter, to demolish a form's material supposition. Is Being Pro Forma Being Ever Anti Matter? Is Being Pro Forma Ever Being Anti Matter? Would necessity be the mother of invention, Possibility must be the new bairn's father. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

###### INF. Inquiry Into Information

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o INF. Inquiry Into Information o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o INF. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Review Of Earlier Material On Determination DET. http://suo.ieee.org/ontology/msg03172.html DET. http://stderr.org/pipermail/inquiry/2004-December/thread.html#2197 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o INF. Note 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | In order to understand how these principles of 'à posteriori' | and inductive inference can be put into practice, we must | consider by itself the substitution of one symbol for | another. Symbols are alterable and comparable in | three ways. In the first place they may denote | more or fewer possible differing things; in this | regard they are said to have 'extension'. In the | second place, they may imply more or less as to | the quality of these things; in this respect | they are said to have 'intension'. In the | third place they may involve more or less | real knowledge; in this respect they | have 'information' and 'distinctness'. | Logical writers generally speak only | of extension and intension and Kant | has laid down the law that these | quantities are inverse in respect | of each other. | | CSP, CE 1, page 187. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o INF. Note 3 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o I am going to run through the series of concrete illustrations that Peirce lays out to explain his take on the conceptions of extension, intension, and information. It is a mite long, but helps better than anything else I know to bring what Peirce is talking about down to earth. For ease of comprehension I will divide this extended paragraph into more moderate-sized chunks. | For example, take 'cat'; now increase the extension of that greatly -- | 'cat' or 'rabbit' or 'dog'; now apply to this extended class the | additional intension 'feline'; -- 'feline cat' or 'feline rabbit' | or 'feline dog' is equal to 'cat' again. This law holds good as | long as the information remains constant, but when this is changed | the relation is changed. Thus 'cats' are before we know about them | separable into 'blue cats" and 'cats not blue' of which classes 'cats' | is the most extensive and least intensive. But afterwards we find out | that one of those classes cannot exist; so that 'cats' increases its | intension to equal 'cats not blue' while 'cats not blue' increases its | extension to equal 'cats'. | | Again, to give a better case, 'rational animal' is divisible into 'mortal rational animal' | and 'immortal rational animal'; but upon information we find that no 'rational animal' | is 'immortal' and this fact is symbolized in the word 'man'. 'Man', therefore, has at | once the extension of 'rational animal' with the intension of 'mortal rational animal', | and far more beside, because it involves more 'information' than either of the previous | symbols. 'Man' is more 'distinct' than 'rational animal', and more 'formal' than | 'mortal rational animal'. | | Now of two statements both of which are true, it is obvious that | that contains the most truth which contains the most information. | If two predicates of the same intension, therefore, are true of | the same subject, the most formal one contains the most truth. | | Thus, it is better to say Socrates is a man, than to say Socrates | is an animal who is rational mortal risible biped &c. because | the former contains all the last and in addition it forms | the synthesis of the whole under a definite 'form'. | | On the other hand if the same predicate is applicable | to two equivalent subjects, that one is to be preferred | which is the most 'distinct'; thus it conveys more truth | to say All men are born of women, than All rational animals | are born of women, because the former has at once as much | extension as the latter, and a much closer reference to | the things spoken of. | | CSP, CE 1, pages 187-188. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o INF. Note 4 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Let us now take the two statements, S is P, T is P; | let us suppose that T is much more distinct than S and | that it is also more extensive. But we 'know' that S is P. | Now if T were not more extensive than S, T is P would contain | more truth than S is P; being more extensive it 'may' contain | more truth and it may also introduce a falsehood. Which of these | probabilities is the greatest? T by being more extensive becomes | less intensive; it is the intension which introduces truth and the | extension which introduces falsehood. If therefore T increases the | intension of S more than its extension, T is to be preferred to S; | otherwise not. Now this is the case of induction. Which contains | most truth, 'neat' and 'deer' are herbivora, or cloven-footed | animals are herbivora? | | In the two statements, S is P, S is Q, let Q be at once more 'formal' and | more 'intensive' than P; and suppose we only 'know' that S is P. In this | case the increase of formality gives a chance of additional truth and the | increase of intension a chance of error. If the extension of Q is more | increased than than its intension, then S is Q is likely to contain more | truth than S is P and 'vice versa'. This is the case of 'à posteriori' | reasoning. We have for instance to choose between | | Light gives fringes of such and such a description | | and | | Light is ether-waves. | | CSP, CE 1, pages 188-189. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o INF. Note 5 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Thus the process of information disturbs the relations | of extension and comprehension for a moment and the | class which results from the equivalence of two | others has a greater intension than one and | a greater extension than the other. Hence, | we may conveniently alter the formula for the | relations of extension and comprehension; thus, | instead of saying that one is the reciprocal of | the other, or | | comprehension x extension = constant, | | we may say | | comprehension x extension = information. | | We see then that all symbols besides their denotative and connotative objects have another; | their informative object. The denotative object is the total of possible things denoted. | The connotative object is the total of symbols translated or implied. The informative | object is the total of forms manifested and is measured by the amount of intension the | term has, over and above what is necessary for limiting its extension. For example, | the denotative object of 'man' is such collections of matter the word knows while it | knows them, i.e., while they are organized. The connotative object of 'man' is the | total form which the word expresses. The informative object of 'man' is the total | fact which it embodies; or the value of the conception which is its equivalent | symbol. | | CSP, CE 1, page 276. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o INF. Note 6 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | The difference between connotation, denotation, and information | supplies the basis for another division of terms and propositions; | a division which is related to the one we have just considered in | precisely the same way as the division of syllogism into 3 figures | is related to the division into Deduction, Induction, and Hypothesis. | | Every symbol which has connotation and denotation has also information. | For by the denotative character of a symbol, I understand application | to objects implied in the symbol itself. The existence therefore of | objects of a certain kind is implied in every connotative denotative | symbol; and this is information. | | Now there are certain imperfect or false symbols produced by the combination | of true symbols which have lost either their denotation or their connotation. | When symbols are combined together in extension as for example in the compound | term "cats and dogs", their sum possesses denotation but no connotation or at least | no connotation which determines their denotation. Hence, such terms, which I prefer | to call 'enumerative' terms, have no information and it remains unknown whether there | be any real kind corresponding to cats and dogs taken together. On the other hand | when symbols are combined together in comprehension as for example in the compound | "tailed men" the product possesses connotation but no denotation, it not being | therein implied that there may be any 'tailed men'. Such conjunctive terms | have therefore no information. Thirdly there are names purporting to be of | real kinds as 'men'; and these are perfect symbols. | | Enumerative terms are not truly symbols but only signs; and | Conjunctive terms are copies; but these copies and signs must | be considered in symbolistic because they are composed of symbols. | | When an enumerative term forms the subject of a grammatical proposition, | as when we say "cats and dogs have tails", there is no logical unity in the | proposition at all. Logically, therefore, it is two propositions and not one. | The same is the case when a conjunctive proposition forms the predicate of a | sentence; for to say that "hens are feathered bipeds" is simply to predicate | two unconnected marks of them. | | When an enumerative term as such is the predicate of a proposition, that proposition | cannot be a denotative one, for a denotative proposition is one which merely analyzes | the denotation of its predicate, but the denotation of an enumerative term is analyzed | in the term itself; hence if an enumerative term as such were the predicate of a | proposition that proposition would be equivalent in meaning to its own predicate. | On the other hand, if a conjunctive term as such is the subject of a proposition, | that proposition cannot be connotative, for the connotation of a conjunctive term | is already analyzed in the term itself, and a connotative proposition does no more | than analyze the connotation of its subject. Thus we have | | Conjunctive Simple Enumerative | | propositions so related to | | Denotative Informative Connotative | | propositions that what is on the left hand | of one line cannot be on the right hand of | the other. | | CSP, CE 1, pages 278-279. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o INF. Note 7 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | We are now in a condition to discuss the question | of the grounds of scientific inference. This | problem naturally divides itself into parts: | | 1st To state and prove the principles | upon which the possibility in general | of each kind of inference depends, | | 2nd To state and prove the rules | for making inferences | in particular cases. | | The first point I shall discuss in the remainder of this lecture; | the second I shall scarcely be able to touch upon in these lectures. | | Inference in general obviously supposes symbolization; and | all symbolization is inference. For every symbol as we have seen | contains information. And in the last lecture we saw that all kinds | of information involve inference. Inference, then, is symbolization. | They are the same notions. Now we have already analyzed the notion | of a 'symbol', and we have found that it depends upon the possibility | of representations acquiring a nature, that is to say an immediate | representative power. This principle is therefore the ground | of inference in general. | | CSP, CE 1, pages 279-280. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o INF. Note 8 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | But there are three distinct kinds of inference; | inconvertible and different in their conception. | There must, therefore, be three different principles | to serve for their grounds. These three principles | must also be indemonstrable; that is to say, each | of them so far as it can be proved must be proved | by means of that kind of inference of which it | is the ground. For if the principle of either | kind of inference were proved by another kind | of inference, the former kind of inference | would be reduced to the latter; and since | the different kinds of inference are in | all respects different this cannot be. | You will say that it is no proof of | these principles at all to support | them by that which they themselves | support. But I take it for granted | at the outset, as I said at the beginning | of my first lecture, that induction and hypothesis | have their own validity. The question before us is 'why' | they are valid. The principles, therefore, of which we | are in search, are not to be used to prove that the | three kinds of inference are valid, but only to | show how they come to be valid, and the proof | of them consists in showing that they | determine the validity of the | three kinds of inference. | | CSP, CE 1, page 280. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o INF. Note 9 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | But these three principles must have this in common that they refer to 'symbolization' | for they are principles of inference which is symbolization. As grounds of the | possibility of inference they must refer to the possibility of symbolization or | symbolizability. And as logical principles they must relate to the reference | of symbols to objects; for logic has been defined as the science of the | general conditions of the relations of symbols to objects. But as three | different principles they must state three different relations of | symbols to objects. Now we already found that a symbol has three | different relations to objects; namely, connotation, denotation, | and information, which are its relations to the object considered | as a thing, a form, and an equivalent representation. Hence, | it is obvious that these three principles must relate to | the symbolizability of things, of forms, and of symbols. | | CSP, CE 1, pages 280-281. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o INF. Note 10 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Our next business is to find out which is which. | For this purpose we must consider that each principle | is to be proved by the kind of inference which it supports. | | The ground of deductive inference then must be established deductively; | that is by reasoning from determinant to determinate, or in other words | by reasoning from definition. But this kind of reasoning can only be | applied to an object whose character depends upon its definition. | Now of most objects it is the definition which depends upon the | character; and so the definition must therefore itself rest on | induction or hypothesis. But the principle of deduction must | rest on nothing but deduction, and therefore it must relate | to something whose character depends upon its definition. | Now the only objects of which this is true are symbols; | they indeed are created by their definition; while | neither forms nor things are. Hence, the principle | of deduction must relate to the symbolizability of | symbols. | | The principle of hypothetic inference must be established hypothetically, | that is by reasoning from determinate to determinant. Now it is clear that | this kind of reasoning is applicable only to that which is determined by what | it determines; or that which is only subject to truth and falsehood so far as | its determinate is, and is thus of itself pure 'zero'. Now this is the case with | nothing whatever except the pure forms; they indeed are what they are only in so | far as they determine some symbol or object. Hence the principle of hypothetic | inference must relate to the symbolizability of forms. | | The principle of inductive inference must be established inductively, | that is by reasoning from parts to whole. This kind of reasoning can | apply only to those objects whose parts collectively are their whole. | Now of symbols this is not true. If I write 'man' here and 'dog' here | that does not constitute the symbol of 'man and dog', for symbols have | to be reduced to the unity of symbolization which Kant calls the unity | of apperception and unless this be indicated by some special mark they | do not constitute a whole. In the same way forms have to determine the | same matter before they are added; if the curtains are green and the | wainscot yellow that does not make a 'yellow-green'. But with things | it is altogether different; wrench the blade and handle of a knife | apart and the form of the knife has dissappeared but they are the | same thing -- the same matter -- that they were before. Hence, | the principle of induction must relate to the symbolizability | of things. | | All these principles must as principles be universal. | Hence they are as follows: -- | | All things, forms, symbols are symbolizable. | | CSP, CE 1, pages 281-282. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

###### SYM. Inquiry Into Symbolization

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SYM. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o I think that it is time for me to take a little break -- to try and pose my current guess as to what Peirce is going on about with all that guff about symbolization. I find our pupallary Peirce trying to wriggle his way out of the Cartesian/Kantian cocoon in which he finds himself encased at his phase of metamorphosis in view. Remember that for Peirce "concepts are a species of symbols", and so to talk about "symbolization" and "symbolizability" is tantamount to invoking a generalization of "conceptualization" and "conceivability". So the whole scene in question is taking place on the stage set by Kant, whose depiction of the Creation, Development, and Elimination operators that work on concept-ions Peirce has already intoned in his prologue to the entire drama: | The essential of a thing -- the character of it -- | is the unity of the manifold therein contained. | 'Id est', the logical principle, from which as | major premiss the facts thereof can be deduced. | | What are called a man's principles however | are only certain beliefs of his that he may or | may not carry out. They therefore do not compose | his character, but the general expression of the facts -- | the ACTS OF HIS SOUL -- does. | | What he does is important. | How he feels is incidental. | | CSP, CE 1, page 6. | | Charles Sanders Peirce, |"Private Thoughts, Principally On The Conduct Of Life" (Number 37, August 1860), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. If that is a bit too oracular, then he will echo it again in the interlude to come. | This paper is based upon the theory already established, that the function of | conceptions is to reduce the manifold of sensuous impressions to unity, and that | the validity of a conception consists in the impossibility of reducing the content | of consciousness to unity without the introduction of it. (CSP, CP 1.545, CE 2.49). Before I can say any more about this business I will have to dig up some old essays of mine on the relationship between artificial sets and natural kinds. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SYM. Note 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o I am going to try and explain one of the conceptual schemes that I use to interpret what Peirce says about symbolization and symbolizability in his First "Logic Of Science" Lectures. To do this I need to discuss the relations between two lattices or partial orders, one being a lattice of "arbitrary sets" (SET), the other being a lattice of "natural kinds" (NAT). For the time being, I limit myself to concrete, discrete, even finite universes of discourse, where all of the sets in view are subsets of a set X. Here is a little essay in which I first broached this subject to the Peirce Forum last year, in what I once thought was an amusing manner. Subj: HOPE's & FEAR's Date: Tue, 23 May 2000 12:20:58 -0400 From: Jon Awbrey <jawbrey@oakland.edu> To: Peirce Discussion Forum <peirce-l@lyris.acs.ttu.edu> Day 1 Bright and early Monday morning I woke up with the idea of returning to a peculiar theme in Peirce's work, one that persists in nagging me, often while I am reading something else, or working on another topic to which it seems fairly incidental and all too tangential, but a theme, nevertheless, that I can barely catch a glimmer of here and there, not even get so much as a firm handle on it, though it continues to plague me with that not-to-be-denied sense of its vague imports and general importance. Moreover, it seems like every attempt that I have made to raise this sunken ship of a topic, whether in conversations here or elsewhere, has soon gone down in flames or subsided with nary a whimper -- I cannot decide which is worse, but either way it has been a fruitless, a frustrating, and a vastly unsatisfactory experience just to try to express it. Now, with that sort of build-up I think that you all must be terribly -- yes, "terribly" is precisely the word that fits -- excited about the prospects of my bringing it up -- whatever the heck it is -- again, but, in spite of this all too painful suspense, on both our parts, I woke up, as I said, on that day, rather enthusiastic about my prospects this time. Without further ado -- well, thank you! -- let me just say that this is the topic of Peirce's Notes on "A Limited Universe of Marks" (ALUOM), appearing for the first time in the volume by Peirce and his students entitled 'Studies in Logic', published by Little, Brown, and Company, of Boston, MA, in 1883, (CE 4, 405-466, specifically for this note, 450-453), and reworked again for the "Grand Logic", in 1893, (CP 2.517-531). The original book of studies was republished as a kind of Centenary Edition in 1983, but I have apparently misplaced my copy of this work in one of my last few of many geographic relocations. Since these two versions of the remarks on ALUOM, as I will call it, may enjoy widely varying levels of accessibility across the breadth of this Forum, I think that it might benefit discussion to copy as much of them as I think necessary into the Sources thread. The reason why I prefer to do things this way is to separate my own remarks and speculations as much as possible from the texts themselves, and further, to leave the texts available in a maximally uncluttered fashion for use in the light of what may well turn out to be a variety of different purposes. To this task I will now turn, but before I leave this bit of preamble behind for what for may well turn out, or maybe not so well turn out, to be a lengthy interval of time, I probably ought to say a few more words in explanation of my current tag-line for this thread. On what do I feel that the theme of ALUOM has a bearing? Well, on many things, easy to collect but difficult to classify, at least, in any thoroughly rationalized and schematized way. Natural kinds, the logic of inquiry, especially the teasing apart of abduction and induction, "giving a rule to abduction" (GARTA), constraints, and thus information, innate or acquired, on admssible hype and on permissible hypotheses, the nature of the human, the Pragmatic Cosmos that orders the normative sciences into a concentrically focusing and yet climbing spiral of ascent, aspiration, reflection, and regardedness, from aesthetics via ethics to logic in their turns -- you will no doubt begin to think that I am merely free-associating or spinning out topics at random if I even begin to unroll my rigamarole shopping list of things that I think are encompassed here. Nevertheless, what little form of organization that I can e-spy in this seeming chaos and this teeming cornucopia of pragmatic commonplaces is enough to get me cranking, at least, on some sort of exposition, no matter how chancy and risky it may start out being at first. In this vein, I find what appears to be two distinct ways, perhaps a couple of dual ways to approach the instigatory question of it all, the question of "GARTA", of what sort of limits may exist on our admission, creation, generation, and imagination of propositions to describe our worlds of experience, including as a special case the propositions that we may choose to employ as explanations of striking phenomena. One "way of thinking" (WOT) is the one that I will dub as the way of "Higher Order Propositional Expressions" (HOPE's). The other WOT that I can see, at least, in so far as I can see any other way at all, is the one that I will dub, fittingly enough, as the way of "Framed Extensions And Restrictions" (FEAR's). With apologies to Pandora, I will choose to introduce the HOPE's first, and put off the FEAR's until later. With gratitude to Shirazad, I will choose to make these HOPE's the story of yet another day, as who knows what any day brings? Day 2 If one treats hypotheses as any other propositions, as so many simple closed circles in a venn diagram, as so many logical variables in a truth table, then one way of talking about constraints on hypotheses is by making use of propositions about propositions, or "higher order propositions" (HO propositions), which are naturally expressed in the formulas of "higher order propositional expressions" (HOPE's), telling what propositions, in general, hypotheses or not, are admitted to the universe of discourse, curtailing discussion to "a limited number of marks", as Peirce had a habit of putting it, in his studied and exquisitely classical way. The dizzying hypes of these orders of abstraction makes it advisable to begin with a concrete and a memorable case, if a rather ridiculous example. Here, understand that we are only concerned with the purified form, and not the ignoble content, of this artificially simple example. Don't bother to try and stop me if you have already heard this one, as I think that it is likely to be one of Aristotle's most outlandish jokes, because I already have in mind another end altogether that I hope will eventually serve to redeem the evident absurdity of it. Consider the humorous definition of a Human Being as a Featherless Two-legged Critter, to schematize it, if not utterly to traumatize it, let me express the subject matter in the following way: | A = Apterous (featherless) | B = Bipedal (two-legged) | C = Critter (animal) | H = Human (human being) Now, I had been planning to introduce some venn diagrams at this point, but after wasting two days of trying, and trying the patience of all concerned, and not concerned, I am afraid that will have to forego, for now, that brand of diagram -- what conceivable significance could iconic diagrams have in philosophy, anyway? -- at least, until I can figure out a way to arrive at a non-distorted form of representation without the occasional experiment or two, indeed, short of a persistent, persevering, indefinite series of experiments. Of course, if I could figure out how to do that, in full generality and without loss of geniality, as they will say, what need would there be for any inquiry at all, much less any brand of theory concerned with the logic and the practice of actual inquiry? But never mind all that. Where there is a will, there will be a way, whether the pathways of the requistite varieties of reactions have all of their semiotic catalysts in all of the most optimal places or not. There is always one way or another to go forward, even if one's active duty status as an exponent of Peirce and one's interim role as an interpretant of Peirce must abdicate a few of the iconic attributes that remain most fitting to these tasks, and even if we must relegate ourselves to symbolically talking about the kinds of diagrams that Peirce regarded as important to actually, brutally, crudely, deliberately, existentially, faithfully take a modicum of trouble to draw, for all that one can learn from the concrete and practical process of going through the exercise to do it. In short, short of the facilities of the graphic medium, I will have to require you to exercise your imagination to a somewhat greater extent than you otherwise might have to, and I will count myself fortunate in the circumstance, that when it comes to imagination, you folks have no shortage of that! But I did have a bit more luck with a somewhat simpler class of diagrams -- What double-edged luck, indeed, that it should have encouraged me to go on ahead and rush in blindly where even angles and anglers fear to leave the marks of their treads! -- but never mind all that. These rather more tractable diagrams, although they lack that one critically important and crucially iconic property of continuously reminding the viewer, not only of the conceivably-continuously-supporting extensions of human concepts, but also of the arbitrariness of the heraldic distinctions that humans are wont to mark upon the underlying fields of existential experience, as if the divisions we impose in our own conceits and in our images were capable of placing any brand of demand on Nature at its joints that Nature could not cast off as quick as Nature can dispose of us. But never mind all that. The sort of diagram that I have been able to draw on these walls, at least, so far, are none other than the "logical lattice", the "propositional partial order", or, with a tip of the hat to Tom Gollier, the "implicative food chain" type of diagram. And so I will satisfice with these for now. So let me try to draw you a picture of the situation that I want to discuss, if it must be one that requires the viewer to "connect the dots" just a bit. Figure 1 outlines the subject matter, to wit, the category "human being" (H) here defined as falling under the head of an "apterous biped" (G = A |^| B), hence bound by the set-theoretic intersection G of the respective extensions of the two concepts, "apterous" (= featherless) and "bipedal" (= two-legged). Now the wise-cracking sort of person, that everyone among us has encountered before, will naturally be compelled to say, ignoring the natural and implicit constraints of the discussion to what are often described as "natural kinds", "But what of the plucked chicken? -- a two-legged critter without feathers? -- Is that your model of a genuine Mensch?" (To get the full effect, you have to imagine this being said in a Woody Allen voice.) And so you patiently go about explaining, as if your interrupreter did not already know this -- such is the role of a straightman in this genre of commodious eristic, as you know that you'll get your turn sooner or later, hopefully in the very next bit -- all about how a "plucked chicken", along with many other hype-o-thetical and hi-pathetical creatures that might be abduced, construed, confabulated, and otherwise plucked from thin air, is what one calls an "artificial kind", in "essence", or the lack thereof, not really a "kind" (Greek 'genus') at all. | A B | o o | |\ /| | |.\ /.| | | \ / | | | . \ / . | | | \ / | | | . \ / . | | | G | | | . / \ . | | | / \ | | | / . . \ | | | / . . \ | | | / . . \ | | |/. .\| | o o | H P | | Figure 1. On Being Human | | A = Apterous (featherless animal) | B = Bipedal (two-legged being) | C = Critter (creature, creation) | G = GLB = Intersection of A and B | H = HB = Human Being | P = PC = Plucked Chicken Okay, I think that will do to set up the joke. I will save the explanation and the resolution of it -- not nearly so fun a task -- till next we meet in this space. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SYM. Note 3 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Note A. On A Limited Universe Of Marks | | Boole, De Morgan, and their followers, frequently speak of | a "limited universe of discourse" in logic. An unlimited universe | would comprise the whole realm of the logically possible. In such | a universe, every universal proposition, not tautologous, is false; | every particular proposition, not absurd, is true. Our discourse | seldom relates to this universe: we are either thinking of the | physically possible, or of the historically existent, or of | the world of some romance, or of some other limited universe. | | But besides its universe of objects, our discourse also refers to | a universe of characters. Thus, we might naturally say that virtue | and an orange have nothing in common. It is true that the English | word for each is spelt with six letters, but this is not one of the | marks of the universe of our discourse. | | A universe of things is unlimited in which every combination of characters, | short of the whole universe of characters, occurs in some object. In like | manner, the universe of characters is unlimited in case every aggregate | of things short of the whole universe of things possesses in common one | of the characters of the universe of characters. The conception of | ordinary syllogistic is so unclear that it would hardly be accurate | to say that it supposes an unlimited universe of characters; but | it comes nearer to that than to any other consistent view. The | non-possession of any character is regarded as implying the | possession of another character the negative of the first. | | In our ordinary discourse, on the other hand, not only are both universes limited, but, | further than that, we have nothing to do with individual objects nor simple marks; | so that we have simply the two distinct universes of things and marks related to | one another, in general, in a perfectly indeterminate manner. The consequence | is, that a proposition concerning the relations of two groups of marks is not | necessarily equivalent to any proposition concerning classes of things; so | that the distinction between propositions in extension and propositions in | comprehension is a real one, separating two kinds of facts, whereas in the | view of ordinary syllogistic the distinction only relates to two modes of | considering any fact. To say that every object of the class S is included | among the class of P's, of course must imply that every common character of | the P's is a common character of the S's. But the converse implication is by | no means necessary, except with an unlimited universe of marks. The reasonings | in depth of which I have spoken, suppose, of course, the absence of any general | regularity about the relations of marks and things. (CSP, SIL, 182-183). | | CSP, SIL, pages 182-186. (Cf. CE 4, pages 450-453, CP 2.517-531). | | Charles Sanders Peirce, "Note A. On A Limited Universe Of Marks" (1883), | CSP (ed.), 'Studies in Logic, by Members of the Johns Hopkins University', | Reprinted with an Introduction by Max H. Fisch & a Preface by Achim Eschbach, | in 'Foundations of Semiotics, Volume 1', John Benjamins, Amsterdam, NL, 1983. | |'Writings of Charles S. Peirce: A Chronological Edition, Volume 4, 1879-1884', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1986. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SYM. Note 4 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o This is one of those puzzles that I have been puzzling away at for almost as long as I can remember. I have gotten fairly well acquainted with the various pieces of the puzzle, but haven't quite figured out yet how they ought to fit together. It all seems to have something to do with an intricate relationship among concepts, kinds (of the natural kind, naturally), and symbols. I know that I am always reminded of it when I read what Peirce says on the issues of "symbolization" and "symbolizability". And I have the impression that there is a vast order of generalization in the works here, taking the topics of "observation" and "observables", along with "computation" and "computables", and even "conception" and "conceivables" under its wing with plenty of room left over. Still, the best that I seem able to do at this juncture in time is just to keep assembling the pieces together and just to keep staring at them till the right sorts of connections occur to me. The pieces of the puzzle are these: 1. Remember that for Peirce "concepts are a species of symbols", and so to talk about "symbolization" and "symbolizability" is tantamount to invoking a generalization of "conceptualization" and "conceivability". So the whole scene in question is taking place on the stage set by Kant, whose depiction of the creation, development, and elimination operators that work on conceptions Peirce has already intoned in his prologue to the entire drama: | The essential of a thing -- the character of it -- | is the unity of the manifold therein contained. | 'Id est', the logical principle, from which as | major premiss the facts thereof can be deduced. | | CSP, CE 1, page 6. | | Charles Sanders Peirce, |"Private Thoughts, Principally On The Conduct Of Life" (Number 37, August 1860), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. 2. The idea about the function of conceptions that Peirce obtained from Kant: | This paper is based upon the theory already established, that the function of | conceptions is to reduce the manifold of sensuous impressions to unity, and that | the validity of a conception consists in the impossibility of reducing the content | of consciousness to unity without the introduction of it. (CSP, CP 1.545, CE 2.49). | | http://www.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm 3. Peirce's "Note On A Limited Universe Of Marks" (NOALUOM). | http://suo.ieee.org/ontology/msg03204.html | | CSP, SIL, pages 182-186. (Cf. CE 4, pages 450-453, CP 2.517-531). | | Charles Sanders Peirce, "Note A. On A Limited Universe Of Marks" (1883), | CSP (ed.), 'Studies in Logic, by Members of the Johns Hopkins University', | Reprinted with an Introduction by Max H. Fisch & a Preface by Achim Eschbach, | in 'Foundations of Semiotics, Volume 1', John Benjamins, Amsterdam, NL, 1983. | |'Writings of Charles S. Peirce: A Chronological Edition, Volume 4, 1879-1884', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1986. 4. An image that I have about the relationship between artificial kinds and natural kinds in terms of a mapping, a morphism, a restriction, or a quotient relation between two lattices. I worked this out once in application to the "apterous biped" definition of the human, but I Kant quite recall the punchline. Let me begin again with this last bit to see if I can get a little bit further this time through it. Consider the joke definition of a Human Being as a Featherless Two-legged Critter. By way of a schematic formalization, I set out the matter in the following manner: | A B | o o | |\. ./| | | \ . . / | | | \ . . / | | | \ . . / | | | \ / | | | . \ / . | | | G | | | . / \ . | | | / \ | | | . / \ . | | | / \ | | |./ \.| | |/ \| | o o | H P | | Figure 1. On Being Human | | A = Apterous = featherless animal | B = Bipedal = two-legged being | C = Critter = creature, creation | G = glb(A, B) = A |^| B | H = Human Being | P = Plucked Chicken Figure 1 outlines the subject matter, to wit, the category "human being" (H) here defined as falling under the head of an "apterous biped" (G = A |^| B), hence bound by the set-theoretic intersection G of the respective extensions of the two concepts, "apterous" (= featherless) and "bipedal" (= two-legged). Now the wise-cracking sort of person, one who ignores the naturally implicit constraints of the discussion to what are often described as "natural kinds", will naturally be compelled to pipe up, "But what of the plucked chicken? -- a two-legged critter without feathers? -- is that your idea of human being?" Now, we know that the response to this witlesscism must invoke the distinction between what one calls an "artificial kind" and a "natural kind", respectively, even though it is difficult to say just how this difference makes a difference. Here is one possible way to view the situation: | SET NAT NAT | | A B A B A B | o o o o o o | |\. ./| | / \ / | | \ . . / | | / \ / | | \ . . / | | / \ / | | \ . . / | | / \ / | | \ / | | / \ / | | . \ / . | | / \ / | | G | | / G | | . / \ . | | / = | | / \ | | / = | | . / \ . | | / = | | / \ | | / = | |./ \.| | / = | |/ \| |/ = | o o o o | H P H H | | Figure 2. On Being Human, All Too Human Think of the initial set-up as being cast in a lattice of arbitrary sets. Within that setting, the "greatest lower bound" (glb) of the extensions of A and B is their set-theoretic intersection, G = glb(A, B) = A |^| B. This G covers the desired class H but also admits the risible category P. Now, suppose that we are clued into the fact that not all sets in SET are admissible, allowable, natural, pertinent, relevant, or whatever, to the aims of the discussion in view, and that only some mysterious 'je ne sais quoi' subset of "natural kinds", NAT c SET, is at stake, a limitation that, whatever else it does, excludes the set P and all of that ilk from beneath glb(A, B). Though we cannot quite say how we apply this information, we know it by its effects to give us the lattice structure in the next frame, where H = glb(A, B), and thus in this more natural setting the proposed definition works okay. An alternative way to look at the transformation of our views from the arbitrary lattice SET to the natural lattice NAT, is illustrated in the last frame, where the equal signs indicate that the nodes for G and H are identified. In this picture, the measure of the interval that once existed between G and H, now shrunk to nil, gives a rough indication of the quantity of information that went into forming the natural end result. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

###### LAS. Logic As Semiotic

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o LAS. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Logic, in its general sense, is, as I believe I have shown, only another name for |'semiotic' ([Greek: semeiotike]), the quasi-necessary, or formal, doctrine of signs. | By describing the doctrine as "quasi-necessary", or formal, I mean that we observe the | characters of such signs as we know, and from such an observation, by a process which | I will not object to naming Abstraction, we are led to statements, eminently fallible, | and therefore in one sense by no means necessary, as to what 'must be' the characters | of all signs used by a "scientific" intelligence, that is to say, by an intelligence | capable of learning by experience. As to that process of abstraction, it is itself | a sort of observation. The faculty which I call abstractive observation is one which | ordinary people perfectly recognize, but for which the theories of philosophers sometimes | hardly leave room. It is a familiar experience to every human being to wish for something | quite beyond his present means, and to follow that wish by the question, "Should I wish for | that thing just the same, if I had ample means to gratify it?" To answer that question, he | searches his heart, and in doing so makes what I term an abstractive observation. He makes | in his imagination a sort of skeleton diagram, or outline sketch, of himself, considers what | modifications the hypothetical state of things would require to be made in that picture, and | then examines it, that is, 'observes' what he has imagined, to see whether the same ardent | desire is there to be discerned. By such a process, which is at bottom very much like | mathematical reasoning, we can reach conclusions as to what 'would be' true of signs | in all cases, so long as the intelligence using them was scientific. (CP 2.227). | | Charles Sanders Peirce, 'Collected Papers', CP 2.227, | Editors' Note: From An Unidentified Fragment, c. 1897. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o LAS. Note 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Logic is an analysis of forms not a study of the mind. | It tells 'why' an inference follows not 'how' it arises | in the mind. It is the business therefore of the logician | to break up complicated inferences from numerous premisses | into the simplest possible parts and not to leave them | as they are. | | CSP, CE 1, page 217. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o LAS. Note 3 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Some reasons having now been given for adopting the | unpsychological conception of the science, let us now | seek to make this conception sufficiently distinct to | serve for a definition of logic. For this purpose we | must bring our 'logos' from the abstract to the concrete, | from the absolute to the dependent. There is no science | of absolutes. The metaphysical logos is no more to us | than the metaphysical soul or the metaphysical matter. | To the absolute Idea or Logos, the dependent or relative | 'word' corresponds. The word 'horse', is thought of as | being a word though it be unwritten, unsaid, and unthought. | It is true, it must be considered as having been thought; | but it need not have been thought by the same mind which | regards it as being a word. I can think of a word in | Feejee, though I can attach no definite articulation to | it, and do not guess what it would be like. Such a word, | abstract but not absolute, is no more than the genus of | all symbols having the same meaning. We can also think | of the higher genus which contains words of all meanings. | A first approximation to a definition, then, will be that | logic is the science of representations in general, whether | mental or material. This definition coincides with Locke's. | It is however too wide for logic does not treat of all kinds | of representations. The resemblance of a portrait to its | object, for example, is not logical truth. It is necessary, | therefore, to divide the genus representation according to | the different ways in which it may accord with its object. | | The first and simplest kind of truth is the resemblance of a copy. | It may be roughly stated to consist in a sameness of predicates. | Leibniz would say that carried to its highest point, it would | destroy itself by becoming identity. Whether that is true or | not, all known resemblance has a limit. Hence, resemblance | is always partial truth. On the other hand, no two things | are so different as to resemble each other in no particular. | Such a case is supposed in the proverb that Dreams go by | contraries, -- an absurd notion, since concretes have no | contraries. A false copy is one which claims to resemble | an object which it does not resemble. But this never fully | occurs, for two reasons; in the first place, the falsehood | does not lie in the copy itself but in the 'claim' which is | made for it, in the 'superscription' for instance; in the | second place, as there must be 'some' resemblance between | the copy and its object, this falsehood cannot be entire. | Hence, there is no absolute truth or falsehood of copies. | Now logical representations have absolute truth and | falsehood as we know 'à posteriori' from the law | of excluded middle. Hence, logic does not treat | of copies. | | The second kind of truth, is the denotation of a sign, | according to a previous convention. A child's name, for | example, by a convention made at baptism, denotes that person. | Signs may be plural but they cannot have genuine generality because | each of the objects to which they refer must have been fixed upon | by convention. It is true that we may agree that a certain sign | shall denote a certain individual conception, an individual act | of an individual mind, and that conception may stand for all | conceptions resembling it; but in this case, the generality | belongs to the 'conception' and not to the sign. Signs, | therefore, in this narrow sense are not treated of in | logic, because logic deals only with general terms. | | The third kind of truth or accordance of a representation | with its object, is that which inheres in the very nature | of the representation whether that nature be original or | acquired. Such a representation I name a 'symbol'. | | CSP, CE 1, pages 169-170. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o LAS. Note 4 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | How often do we think of the thing in algebra? | When we use the symbol of multiplication we do not | even think out the conception of multiplication, we think | merely of the laws of that symbol, which coincide with the | laws of the conception, and what is more to the purpose, | coincide with the laws of multiplication in the object. | Now, I ask, how is it that anything can be done with | a symbol, without reflecting upon the conception, | much less imagining the object that belongs to it? | It is simply because the symbol has acquired a nature, | which may be described thus, that when it is brought before | the mind certain principles of its use -- whether reflected on | or not -- by association immediately regulate the action of the | mind; and these may be regarded as laws of the symbol itself | which it cannot 'as a symbol' transgress. | | CSP, CE 1, page 173. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o LAS. Note 5 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o This text picks up from the point where I went tangential, a while ago under the "Determination" heading, at Note 19: http://suo.ieee.org/ontology/msg02673.html http://suo.ieee.org/ontology/msg02706.html http://suo.ieee.org/ontology/msg03172.html | Finally, these principles as principles applying not to this or that | symbol, form, thing, but to all equally, must be universal. And as | grounds of possibility they must state what is possible. Now what | is the universal principle of the possible symbolization of symbols? | It is that all symbols are symbolizable. And the other principles | must predicate the same thing of forms and things. | | These, then, are the three principles of inference. Our next business is | to demonstrate their truth. But before doing so, let me repeat that these | principles do not serve to prove that the kinds of inference are valid, since | their own proof, on the contrary, must rest on the assumption of that validity. | Their use is only to show what the condition of that validity is. Hence, the | only proof of the truth of these principles is this; to show, that if these | principles be admitted as sufficient, and if the validity of the several kinds | of inference be also admitted, that then the truth of these principles follows | by the respective kinds of inference which each establishes. | | CSP, CE 1, pages 184-185. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o LAS. Note 6 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | To prove then, first, that all symbols are symbolizable. | Every syllogism consists of three propositions with two terms | each, a subject and a predicate, and three terms in all each term | being used twice. It is obvious that one term must occur both as | subject and predicate. Now a predicate is a symbol of its subject. | Hence in all reasoning 'à priori' a symbol must be symbolized. | But as reasoning 'à priori' is possible about a statement | without reference to its predicate, all symbols must be | symbolizable. | | 2nd To prove that all forms are symbolizable. | Since this proposition relates to pure form it is | sufficient to show that its consequences are true. | Now the consequence will be that if a symbol of any | object be given, but if this symbol does not adequately | represent any form then another symbol more formal may | always be substituted for it, or in other words as soon | as we know what form it ought to symbolize the symbol may | be so changed as to symbolize that form. But this process | is a description of inference 'à posteriori'. Thus in the | example relating to light; the symbol of "giving such and | such phenomena" which is altogether inadequate to express a | form is replaced by "ether-waves" which is much more formal. | The consequence then of the universal symbolization of forms | is the inference 'à posteriori', and there is no truth or | falsehood in the principle except what appears in the | consequence. Hence, the consequence being valid, | the principle may be accepted. | | 3rd To prove that all things may be symbolized. | If we have a proposition, the subject of which is not | properly a symbol of the thing it signifies; then in case | everything may be symbolized, it is possible to replace this | subject by another which is true of it and which does symbolize | the subject. But this process is inductive inference. Thus having | observed of a great variety of animals that they all eat herbs, if I | substitute for this subject which is not a true symbol, the symbol | "cloven-footed animals" which is true of these animals, I make an | induction. Accordingly I must acknowledge that this principle | leads to induction; and as it is a principle of objects, | what is true of its subalterns is true of it; and since | induction is always possible and valid, this principle | is true. | | CSP, CE 1, pages 185-186. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o LAS. Note 7 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Having discovered and demonstrated the grounds of the possibility of | the three inferences, let us take a preliminary glance at the manner in | which additions to these principles may make them grounds of proceedure. | | The principle of inference 'à priori' has been apodictically demonstrated; | the principle of inductive inference has been shown upon sufficient evidence | to be true; the principle of inference 'à posteriori' has been shown to be one | which nothing can contradict. These three degrees of modality in the principles of | the three inferences show the amount of certainty which each is capable of affording. | Inference 'à priori' is as we all know the only apodictic proceedure; yet no one | thinks of questioning a good induction; while inference 'à posteriori' is | proverbially uncertain. 'Hypotheses non fingo', said Newton; striving | to place his theory on a firm inductive basis. Yet provisionally we | must make hypotheses; we start with them; the baby when he lies | turning his fingers before his eyes is testing a hypothesis he has | already formed, as to the connection of touch and sight. Apodictic | reasoning can only be applied to the manipulation of our knowledge; | it never can extend it. So that it is an induction which eventually | settles every question of science; and nine-tenths of the inferences | we draw in any hour not of study are of this kind. | | CSP, CE 1, page 186. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o LAS. Note 8 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | The first distinction we found it necessary to draw -- | the first set of conceptions we have to signalize -- | forms a triad | | Thing Representation Form. | | Kant you remember distinguishes in all mental representations the | matter and the form. The distinction here is slightly different. | In the first place, I do not use the word 'Representation' as | a translation of the German 'Vorstellung' which is the general | term for any product of the cognitive power. Representation, | indeed, is not a perfect translation of that term, because it | seems necessarily to imply a mediate reference to its object, | which 'Vorstellung' does not. I however would limit the term | neither to that which is mediate nor to that which is mental, | but would use it in its broad, usual, and etymological sense | for anything which is supposed to stand for another and which | might express that other to a mind which truly could understand | it. Thus our whole world -- that which we can comprehend -- is | a world of representations. | | No one can deny that there are representations, for every thought is one. | But with 'things' and 'forms' scepticism, though still unfounded, is at first | possible. The 'thing' is that for which a representation might stand prescinded | from all that would constitute a relation with any representation. The 'form' is | the respect in which a representation might stand for a thing, prescinded from both | thing and representation. We thus see that 'things' and 'forms' stand very differently | with us from 'representations'. Not in being prescinded elements, for representations | also are prescinded from other representations. But because we know representations | absolutely, while we only know 'forms' and 'things' through representations. Thus | scepticism is possible concerning 'them'. But for the very reason that they are | known only relatively and therefore do not belong to our world, the hypothesis | of 'things' and 'forms' introduces nothing false. For truth and falsity only | apply to an object as far as it can be known. If indeed we could know things | and forms in themselves, then perhaps our representations of them might | contradict this knowledge. But since all that we know of them we know | through representations, if our representations be consistent they | have all the truth that the case admits of. | | CSP, CE 1, pages 256-257. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o LAS. Note 9 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | We found representations to be of three kinds | | Signs Copies Symbols. | | By a 'copy', I mean a representation whose agreement with | its object depends merely upon a sameness of predicates. | | By a 'sign', I mean a representation whose reference to | its object is fixed by convention. | | By a 'symbol', I mean one which upon being presented to the mind -- | without any resemblance to its object and without any reference to | a previous convention -- calls up a concept. I consider concepts, | themselves, as a species of symbols. | | A symbol is subject to three conditions. First it must represent an object, | or informed and representable thing. Second it must be a manifestation of | a 'logos', or represented and realizable form. Third it must be translatable | into another language or system of symbols. | | The science of the general laws of relations of symbols to logoi is general grammar. | The science of the general laws of their relations to objects is logic. And the | science of the general laws of their relations to other systems of symbols is | general rhetoric. | | CSP, CE 1, pages 257-258. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o LAS. Note 10 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | When have then three different kinds of inference. | | Deduction or inference 'à priori', | | Induction or inference 'à particularis', and | | Hypothesis or inference 'à posteriori'. | | It is necessary now to examine this classification critically. | | And first let me specify what I claim for my invention. I do not claim that it is | a natural classification, in the sense of being right while all others are wrong. | I do not know that such a thing as a natural classification is possible in the | nature of the case. The science which most resembles logic is mathematics. | Now among mathematical forms there does not seem to be any natural classification. | It is true that in the solutions of quadratic equations, there are generally two | solutions from the positive and negative values of the root with an impossible | gulf between them. But this classing is owing to the forms being restricted | by the conditions of the problem; and I believe that all natural classes arise | from some problem -- something which was to be accomplished and which could be | accomplished only in certain ways. Required to make a musical instrument; | you must set either a plate or a string in vibration. Required to make | an animal; it must be either a vertebrate, an articulate, a mollusk, or | a radiate. However this may be, in Geometry we find ourselves free to make | several different classifications of curves, either of which shall be equally | good. In fact, in order to make any classification of them whatever we must | introduce the purely arbitrary element of a system of coördinates or something | of the kind which constitutes the point of view from which we regard the curves | and which determines their classification completely. Now it may be said that | one system of coördinates is more 'natural' than another; and it is obvious | that the conditions of binocular vision limit us in our use of our eyes to | the use of particular coördinates. But this fact that one such system | is more natural to us has clearly nothing to do with pure mathematics | but is merely introducing a problem; given two eyes, required to form | geometrical judgements, how can we do it? In the same way, I conceive | that the syllogism is nothing but the system of coördinates or method of | analysis which we adopt in logic. There is no reason why arguments should | not be analyzed just as correctly in some other way. It is a great mistake to | suppose that arguments as they are thought are often syllogisms, but even if this | were the case it would have no bearing upon pure logic as a formal science. It is | the principal business of the logician to analyze arguments into their elements just | as it is part of the business of the geometer to analyze curves; but the one is no | more bound to follow the natural process of the intellect in his analysis, than the | other is bound to follow the natural process of perception. | | CSP, CE 1, pages 267-268. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

###### MOSI. Manifolds Of Sensuous Impressions

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | This paper is based upon the theory already established, that the function of | conceptions is to reduce the manifold of sensuous impressions to unity, and that | the validity of a conception consists in the impossibility of reducing the content | of consciousness to unity without the introduction of it. (CSP, CP 1.545, CE 2.49). Let me read you a story from one of my favorite books of manifolds: | Serge Lang, |'Differential & Riemannian Manifolds', | Springer-Verlag, New York, NY, 1995. But first, a message from our medium: o~~~~~~~~~o~~~NOTATIONATE~NOTEFACTION~~~o~~~~~~~~~o In presenting this text I am obligated to change many Greek characters into Latin letters, and so by way of a slightly skewed form of compensation, I will convert Roman numerals to Arabic decimals. Notes from the translator (me) will be placed in square brackets, to ease the transits to English. Let "|_|", interfixed with extra space around it, or else "|_|<i>", antefixed, signify the union of two sets, or of the many sets indexed by i, respectively. Let "|^|", interfixed with extra space on either side, or else "|^|<i>", antefixed, signify the intersection of two sets, or of a family of many sets indexed by i, respectively. Let "o", interfixed with extra space around it, signify functional composition, interpreted in the sense that (f o g)(x) = f(g(x)). Note to critics who may happen to follow the style sheet of the APA ("American Pedantical Association"). The "we" that you see prevailing in this mannerism of mathematical writing is not of necessity the "we" of plural authorship, and of necessity not the "we" of birth through royal blood, as it was discovered years ago that there is no royal robe to mathematics, but it is the very democratic "we" of the participatory demonstracy, and it begins to lose its title to that with every citizen of this res publica who demurs from their reponsibility and their right to follow along. o~~~~~~~~~o~~~NOITCAFETON~ETANOITATON~~~o~~~~~~~~~o Good. Once upon a time ... o~~~~~~~~~o~~~~~~~~~o~RECITATIVE~o~~~~~~~~~o~~~~~~~~~o Chapt 2. Manifolds Starting with open subsets of Banach spaces [think R^n for the moment], one can glue them together with 'C^p'-isomorphisms [bijective mappings that are continuously differentiable up to at least as far as order p]. The result is called a manifold. We begin by giving the formal definition. We then make manifolds into a category, and discuss special types of morphisms. We define the tangent space at each point, and apply the criteria following the inverse function theorem to get a local splitting of a manifold when the tangent space splits at a point. We shall wait until the next chapter to give a manifold structure to the union of all the tangent spaces. 2.1. Atlases, Charts, Morphisms Let X be a set. An "atlas" of class C^p (p >= 0) on X is a collection of pairs (U<i>, q<i>) (i ranging in some indexing set), satisfying the following conditions: AT 1. Each U<i> is a subset of X and the U<i> cover X. AT 2. Each q<i> is a bijection of U<i> onto an open subset q<i>U<i> of some Banach space E<i> and for any i, j, [it is true that] q<i>(U<i> |^| U<j>) is open in E<i>. AT 3. The map q<j> o q<i>^-1 : q<i>(U<i> |^| U<j>) -> q<j>(U<i> |^| U<j>) is a 'C^p'-isomorphism for each pair of indices i, j. It is a trivial exercise in point set topology to prove that one can give X a topology in a unique way such that each U<i> is open, and the q<i> are topological isomorphisms. (Lang, DARM, 20-21). | Serge Lang, |'Differential And Riemannian Manifolds' (DARM), | Springer-Verlag, New York, NY, 1995, pp. 20-21. o~~~~~~~~~o~~~~~~~~~o~EVITATICER~o~~~~~~~~~o~~~~~~~~~o To be continued, and I dare say it, to be differentiated, up to some order as yet to be predestinately determinate. Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Out Of The Mouth Of The Horse: | Now the discovery of ideas as general as these is chiefly | the willingness to make a brash or speculative abstraction, | in this case supported by the pleasure of purloining words | from the philosophers: "Category" from Aristotle and Kant, | "Functor" from Carnap ('Logische Syntax der Sprache'), and | "natural transformation" from then current informal parlance. | | 'Cat.Work.Math.', pages 29-30. | | Saunders Mac Lane, | 'Categories for the Working Mathematician', | Springer-Verlag, New York, NY, 1971. Really, Folks! I could not be making 'all' of this stuff up! o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o I have been charged, if not tried and convicted, of scaring people and horses, on account of the the noise of this new-fangled contraption makes and all of that outlandish, unheard-of rumbling that emanates from under my bonnet and manifold. And so I am sentenced to the punishment that my grandparents once told me befit their times, of sending a forerunner ahead of the car, you know, the one whose tired new wheels are still yet to get themselves invented, a harbinger as it were, to wave a flag or ring a bell or cry the alarum, but softly, very softly. All kidding aside, I was getting to point of drawing you a picture, anyway, since it is just the thing that called for in order to reduce the manifold of symbolic ingressions to a unity of iconic complexion and due proportion. Here is the typical picture of their subject to which manifold theorists have become accustomed, that, were it to be drawn in a more fluid medium, and not so badly quartered in this e-current style, would be e-mediately recognizable as the "Planarian", more popularly, the "Flatworm Diagram". Here, again, for ease of reference, is the definition of an atlas of class C^p: | Let X be a set. An "atlas" of class C^p (p >= 0) on X is a collection | of pairs (U<i>, q<i>) (i ranging in some indexing set), satisfying the | following conditions: | | AT 1. Each U<i> is a subset of X and the U<i> cover X. | | AT 2. Each q<i> is a bijection of U<i> onto an open subset q<i>U<i> | of some Banach space E<i> and for any i, j, [it is true that] | q<i>(U<i> |^| U<j>) is open in E<i>. | | AT 3. The map | | q<j> o q<i>^-1 : q<i>(U<i> |^| U<j>) -> q<j>(U<i> |^| U<j>) | | is a 'C^p'-isomorphism for each pair of indices i, j. | | (Lang, DARM, page 20). And here is (a squared-off version of) the paradigmatic picture, capturing what is most of the essence in our manifold situation: o---------------------------------------o o-------------------o | X | | E<i> | | | | | | | | o | | | | / \ | | o | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ q<i> | | / q<i>U<i>\ | | / o---------------------->| o o o | | / \ | | \ / \ / | | / \ | | \ / \ / | | / U<i> \ | | o o | | / \ | | \ / | | / \ | | \ / | | o o o | | o | | \ / \ / | | | | \ / \ / | | | | \ / U<i>\ / | o---------|---------o | \ / \ / | | | o |^| o | q<j> o q<i>^-1 | / \ / \ | | | / \ U<j>/ \ | o---------v---------o | / \ / \ | | E<j> | | / \ / \ | | | | o o o | | o | | \ / | | / \ | | \ / | | / \ | | \ U<j> / | | o o | | \ / | | / \ / \ | | \ / | | / \ / \ | | \ o---------------------->| o o o | | \ / q<j> | | \ q<j>U<j>/ | | \ / | | \ / | | \ / | | \ / | | o | | \ / | | | | \ / | | | | o | | | | | | | | | o---------------------------------------o o-------------------o Figure 1. Manifold Of Sensuous Impressions o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Next time, I'll be transmitting to you from the other hemisphere of the brain. It won't be long till you long for the days when all I did is read you poetry! Until Then, Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 3 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Manifold SIG, It will be useful to keep at the ready a compendium of the most essential elements of our subject, and so I will maintain what is needed in a cumulative appendix to these notes. I continue the story with further readings from Lang's DARM, knitting up a few more strands of terminology into our yarn. Recall the definition of an atlas: An "atlas of class C^p (p >= 0)" on a set X is a collection of pairs (U_i, q_i), satisfying the conditions AT 1, AT 2, AT 3, (vide syllabus at end of this note). Naturally enough, however much artifice may have gone into its natural naming, an atlas is conceived and executed all in order to collect a number of charts: | Each pair (U_i, q_i) will be called a "chart" of the atlas. | If a point x of X lies in U_i, then we say that (U_i, q_i) | is a "chart at" x. We find next the need for a notion of "compatibility" among and between different atlases and their charts: | Suppose that we are given an open subset U of X and a topological isomorphism | q : U -> U' onto an open subset of some Banach space E. We shall say that | (U, q) is "compatible" with the atlas {(U_i, q_i)} if each map q_i q^-1 | (defined on a suitable intersection as in AT 3) is a C^p-isomorphism. | | Two atlases are said to be "compatible" if each chart of one is compatible with | the other atlas. One verifies immediately that the relation of compatibility | between atlases is an equivalence relation. An equivalence class of atlases | of class C^p on X is said to define a structure of "C^p-manifold" on X. | | If all the vector spaces E_i in some atlas are toplinearly isomorphic, | then we can always find an equivalent atlas for which they are all equal, | say to the vector space E. We then say that X is an "E-manifold" or that | X is "modeled" on E. | | Lang, DARM, page 21 E-nough For E-nonce ... o~~~~~~~~~o~~~~~~~~~o~SYLLABUS~o~~~~~~~~~o~~~~~~~~~o | 2. Manifolds | | 2.1. Atlases, Charts, Morphisms | | Let X be a set. An "atlas of class C^p (p >= 0)" on X is a collection | of pairs (U_i, q_i) (i ranging in some indexing set), satisfying the | following conditions: | | AT 1. Each U_i is a subset of X and the U_i cover X. | | AT 2. Each q_i is a bijection of U_i onto an open subset q_i U_i | of some Banach space E_i and for any i, j, [it holds that] | q_i (U_i |^| U_j) is open in E_i. | | AT 3. The map | | q_j o q_i^-1 : q_i (U_i |^| U_j) --> q_j (U_i |^| U_j) | | is a C^p-isomorphism for each pair of indices i, j. | | Lang, DARM, page 20. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | o---------------------------------------o o-------------------o | | X | | E_i | | | | | | | | | | o | | | | | / \ | | | o | | / \ | | | / \ | | / \ | | | / \ | | / \ | | | / \ q_i | | / q_i U_i \ | | | / o---------------------->| o o o | | | / \ | | \ / \ / | | | / \ | | \ / \ / | | | / U_i \ | | o o | | | / \ | | \ / | | | / \ | | \ / | | | o o o | | o | | | \ / \ / | | | | | \ / \ / | | | | | \ / U_i \ / | o---------|---------o | | \ / \ / | | | | o |^| o | q_j o q_i^-1 | | / \ / \ | | | | / \ U_j / \ | o---------v---------o | | / \ / \ | | E_j | | | / \ / \ | | | | | o o o | | o | | | \ / | | / \ | | | \ / | | / \ | | | \ U_j / | | o o | | | \ / | | / \ / \ | | | \ / | | / \ / \ | | | \ o---------------------->| o o o | | | \ / q_j | | \ q_j U_j / | | | \ / | | \ / | | | \ / | | \ / | | | o | | \ / | | | | | \ / | | | | | o | | | | | | | | | | | | o---------------------------------------o o-------------------o | | Figure 1. Manifold Of Coordinated Impressions o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 4 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Once the preambulatory props and supports have been set out in such a way as to establish our subject on the appropriate numbers and aerity of feet, my aim for the creature's tentaclive course is to chart a beeline for the tripod or trivet where our subject might well have been able to read that destiny from the outset, if our subject had but taken the trouble to read. It is time to introduce the concept of "coordinates": | If E = R^n for some fixed n, then we say that the manifold is "n-dimensional". | In this case, a chart | | q : U -> R^n | | is given by n coordinate functions q_1, ..., q_n. If 'P' denotes a point of U, | these functions are often written | | x_1(P), ..., x_n(P), | | or simply x_1, ..., x_n. They are called "local coordinates" on the manifold. | | If the integer p (which may also be infinity) is fixed throughout a discussion, | we also say that X is a manifold. | | Lang, DARM, page 21. o~~~~~~~~~o~~~~~~~~~o~SYLLABUS~o~~~~~~~~~o~~~~~~~~~o | 2. Manifolds | | 2.1. Atlases, Charts, Morphisms | | Let X be a set. An "atlas of class C^p (p >= 0)" on X is a collection | of pairs (U_i, q_i) (i ranging in some indexing set), satisfying the | following conditions: | | AT 1. Each U_i is a subset of X and the U_i cover X. | | AT 2. Each q_i is a bijection of U_i onto an open subset q_i U_i | of some Banach space E_i and for any i, j, [it holds that] | q_i (U_i |^| U_j) is open in E_i. | | AT 3. The map | | q_j o q_i^-1 : q_i (U_i |^| U_j) --> q_j (U_i |^| U_j) | | is a C^p-isomorphism for each pair of indices i, j. | | Lang, DARM, page 20. | | Each pair (U_i, q_i) will be called a "chart" of the atlas. | If a point x of X lies in U_i, then we say that (U_i, q_i) | is a "chart at" x. | | Suppose that we are given an open subset U of X and a topological isomorphism | q : U -> U' onto an open subset of some Banach space E. We shall say that | (U, q) is "compatible" with the atlas {(U_i, q_i)} if each map q_i q^-1 | (defined on a suitable intersection as in AT 3) is a C^p-isomorphism. | | Two atlases are said to be "compatible" if each chart of one is compatible with | the other atlas. One verifies immediately that the relation of compatibility | between atlases is an equivalence relation. An equivalence class of atlases | of class C^p on X is said to define a structure of "C^p-manifold" on X. | | If all the vector spaces E_i in some atlas are toplinearly isomorphic, | then we can always find an equivalent atlas for which they are all equal, | say to the vector space E. We then say that X is an "E-manifold" or that | X is "modeled" on E. | | Lang, DARM, page 21. | | Serge Lang, |'Differential & Riemannian Manifolds', | Springer-Verlag, New York, NY, 1995. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | o---------------------------------------o o-------------------o | | X | | E_i | | | | | | | | | | o | | | | | / \ | | | o | | / \ | | | / \ | | / \ | | | / \ | | / \ | | | / \ q_i | | / q_i U_i \ | | | / o---------------------->| o o o | | | / \ | | \ / \ / | | | / \ | | \ / \ / | | | / U_i \ | | o o | | | / \ | | \ / | | | / \ | | \ / | | | o o o | | o | | | \ / \ / | | | | | \ / \ / | | | | | \ / U_i \ / | o---------|---------o | | \ / \ / | | | | o |^| o | q_j o q_i^-1 | | / \ / \ | | | | / \ U_j / \ | o---------v---------o | | / \ / \ | | E_j | | | / \ / \ | | | | | o o o | | o | | | \ / | | / \ | | | \ / | | / \ | | | \ U_j / | | o o | | | \ / | | / \ / \ | | | \ / | | / \ / \ | | | \ o---------------------->| o o o | | | \ / q_j | | \ q_j U_j / | | | \ / | | \ / | | | \ / | | \ / | | | o | | \ / | | | | | \ / | | | | | o | | | | | | | | | | | | o---------------------------------------o o-------------------o | | Figure 1. Manifold Of Concreated Impressions o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 5 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | And let us also, to escape entanglement with | difficulties about the physical or psychical | nature of its "object", not call it a feeling | of fragrance or of any other determinate sort, | but limit ourselves to assuming that it is a | feeling of 'q'. | | William James, 'The Meaning Of Truth', | Longmans, Green, & Co., London, 1909, | page 3. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | The collection of C^p-manifolds will be denoted by 'Man^p'. | If we look only at those modeled on spaces in a category $U$ | then we write 'Man^p($U$)'. Those modeled on a fixed E will | be denoted by 'Man^p(E)'. We shall make these into categories | by defining morphisms below. | | Let X be manifold, and U an open subset of X. Then it is possible, | in the obvious way, to induce a manifold structure on U, by taking | as charts the intersections | | (U_i |^| U, q_i | (U_i |^| U)). | | [Notation. "f | S" indicates the function f as restricted to the set S.] | | If X is a topological space, covered by open subsets V_j, and if we are | given on each V_j a manifold structure such that for each pair j, j' the | induced structure on V_j |^| V_j' coincides, then it is clear that we can | give to X a unique manifold structure inducing the given ones on each V_j. | | Lang, DARM, pages 21-22. o~~~~~~~~~o~~~~~~~~~o~SYLLABUS~o~~~~~~~~~o~~~~~~~~~o | 2. Manifolds | | 2.1. Atlases, Charts, Morphisms | | Let X be a set. An "atlas of class C^p (p >= 0)" on X is a collection | of pairs (U_i, q_i) (i ranging in some indexing set), satisfying the | following conditions: | | AT 1. Each U_i is a subset of X and the U_i cover X. | | AT 2. Each q_i is a bijection of U_i onto an open subset q_i U_i | of some Banach space E_i and for any i, j, [it holds that] | q_i (U_i |^| U_j) is open in E_i. | | AT 3. The map | | q_j o q_i^-1 : q_i (U_i |^| U_j) --> q_j (U_i |^| U_j) | | is a C^p-isomorphism for each pair of indices i, j. | | Lang, DARM, page 20. | | Each pair (U_i, q_i) will be called a "chart" of the atlas. | If a point x of X lies in U_i, then we say that (U_i, q_i) | is a "chart at" x. | | Suppose that we are given an open subset U of X and a topological isomorphism | q : U -> U' onto an open subset of some Banach space E. We shall say that | (U, q) is "compatible" with the atlas {(U_i, q_i)} if each map q_i q^-1 | (defined on a suitable intersection as in AT 3) is a C^p-isomorphism. | | Two atlases are said to be "compatible" if each chart of one is compatible with | the other atlas. One verifies immediately that the relation of compatibility | between atlases is an equivalence relation. An equivalence class of atlases | of class C^p on X is said to define a structure of "C^p-manifold" on X. | | If all the vector spaces E_i in some atlas are toplinearly isomorphic, | then we can always find an equivalent atlas for which they are all equal, | say to the vector space E. We then say that X is an "E-manifold" or that | X is "modeled" on E. | | If E = R^n for some fixed n, then we say that the manifold is "n-dimensional". | In this case, a chart | | q : U -> R^n | | is given by n coordinate functions q_1, ..., q_n. If 'P' denotes a point of U, | these functions are often written | | x_1(P), ..., x_n(P), | | or simply x_1, ..., x_n. They are called "local coordinates" on the manifold. | | If the integer p (which may also be infinity) is fixed throughout a discussion, | we also say that X is a manifold. | | Lang, DARM, page 21. | | Serge Lang, |'Differential & Riemannian Manifolds', | Springer-Verlag, New York, NY, 1995. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | o---------------------------------------o o-------------------o | | X | | E_i | | | | | | | | | | o | | | | | / \ | | | o | | / \ | | | / \ | | / \ | | | / \ | | / \ | | | / \ q_i | | / q_i U_i \ | | | / o---------------------->| o o o | | | / \ | | \ / \ / | | | / \ | | \ / \ / | | | / U_i \ | | o o | | | / \ | | \ / | | | / \ | | \ / | | | o o o | | o | | | \ / \ / | | | | | \ / \ / | | | | | \ / U_i \ / | o---------|---------o | | \ / \ / | | | | o |^| o | q_j o q_i^-1 | | / \ / \ | | | | / \ U_j / \ | o---------v---------o | | / \ / \ | | E_j | | | / \ / \ | | | | | o o o | | o | | | \ / | | / \ | | | \ / | | / \ | | | \ U_j / | | o o | | | \ / | | / \ / \ | | | \ / | | / \ / \ | | | \ o---------------------->| o o o | | | \ / q_j | | \ q_j U_j / | | | \ / | | \ / | | | \ / | | \ / | | | o | | \ / | | | | | \ / | | | | | o | | | | | | | | | | | | o---------------------------------------o o-------------------o | | Figure 1. Manifold Of Concreated Impressions o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 6 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Now, if this feeling of 'q' be the only creation of | the god, it will of course form the entire universe. | | William James, 'The Meaning of Truth', | Longmans, Green, & Co., London, 1909, | pages 3-4. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | If X, Y are two manifolds, then one can give the | product X x Y a manifold structure in the obvious way. | If {(U_i, q_i)} and {(V_j, r_j)} are atlases for X, Y | respectively, then | | {(U_i x V_j, q_i x r_j)} | | is an atlas for the product, and the product of compatible | atlases gives rise to compatible atlases, so that we do get | a well-defined product structure. | | Let X, Y be two manifolds. Let f : X -> Y be a map. | We shall say that f is a "C^p-morphism" if, given x in X, | there exists a chart (U, q) at x and a chart (V, r) at f(x) | such that f(U) c V, and the map | | r o f o q^-1 : qU -> rV | | is a C^p-morphism in the sense of Chapter 1, Section 3. | One sees then immediately that this same condition holds | for any choice of charts (U, q) at x and (V, r) at f(x) | such that F(U) c V. | | It is clear that the composite of two C^p-morphisms is itself | a C^p-morphism (because it is true for open subsets of vector | spaces). The C^p-manifolds and C^p-morphisms form a category. | The notion of isomorphism is therefore defined ... | | If f : X -> Y is a morphism, and (U, q) is a chart | at a point x in X, while (V, r) is a chart at f(x), | then we shall also denote by | | f_V,U : qU -> rV | | the map rfq^-1 [that is, r o f o q^-1]. | | Lang, DARM, page 22. o~~~~~~~~~o~~~~~~~~~o~SYLLABUS~o~~~~~~~~~o~~~~~~~~~o | 2. Manifolds | | 2.1. Atlases, Charts, Morphisms | | Let X be a set. An "atlas of class C^p (p >= 0)" on X is a collection | of pairs (U_i, q_i) (i ranging in some indexing set), satisfying the | following conditions: | | AT 1. Each U_i is a subset of X and the U_i cover X. | | AT 2. Each q_i is a bijection of U_i onto an open subset q_i U_i | of some Banach space E_i and for any i, j, [it holds that] | q_i (U_i |^| U_j) is open in E_i. | | AT 3. The map | | q_j o q_i^-1 : q_i (U_i |^| U_j) --> q_j (U_i |^| U_j) | | is a C^p-isomorphism for each pair of indices i, j. | | Lang, DARM, page 20. | | Each pair (U_i, q_i) will be called a "chart" of the atlas. | If a point x of X lies in U_i, then we say that (U_i, q_i) | is a "chart at" x. | | Suppose that we are given an open subset U of X and a topological isomorphism | q : U -> U' onto an open subset of some Banach space E. We shall say that | (U, q) is "compatible" with the atlas {(U_i, q_i)} if each map q_i q^-1 | (defined on a suitable intersection as in AT 3) is a C^p-isomorphism. | | Two atlases are said to be "compatible" if each chart of one is compatible with | the other atlas. One verifies immediately that the relation of compatibility | between atlases is an equivalence relation. An equivalence class of atlases | of class C^p on X is said to define a structure of "C^p-manifold" on X. | | If all the vector spaces E_i in some atlas are toplinearly isomorphic, | then we can always find an equivalent atlas for which they are all equal, | say to the vector space E. We then say that X is an "E-manifold" or that | X is "modeled" on E. | | If E = R^n for some fixed n, then we say that the manifold is "n-dimensional". | In this case, a chart | | q : U -> R^n | | is given by n coordinate functions q_1, ..., q_n. If 'P' denotes a point of U, | these functions are often written | | x_1(P), ..., x_n(P), | | or simply x_1, ..., x_n. They are called "local coordinates" on the manifold. | | If the integer p (which may also be infinity) is fixed throughout a discussion, | we also say that X is a manifold. | | Lang, DARM, page 21. | | Let X be manifold, and U an open subset of X. Then it is possible, | in the obvious way, to induce a manifold structure on U, by taking | as charts the intersections | | (U_i |^| U, q_i | (U_i |^| U)). | | [Notation. "f | S" indicates the function f as restricted to the set S.] | | If X is a topological space, covered by open subsets V_j, and if we are | given on each V_j a manifold structure such that for each pair j, j' the | induced structure on V_j |^| V_j' coincides, then it is clear that we can | give to X a unique manifold structure inducing the given ones on each V_j. | | Lang, DARM, page 22. | | Serge Lang, |'Differential & Riemannian Manifolds', | Springer-Verlag, New York, NY, 1995. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | o---------------------------------------o o-------------------o | | X | | E_i | | | | | | | | | | o | | | | | / \ | | | o | | / \ | | | / \ | | / \ | | | / \ | | / \ | | | / \ q_i | | / q_i U_i \ | | | / o---------------------->| o o o | | | / \ | | \ / \ / | | | / \ | | \ / \ / | | | / U_i \ | | o o | | | / \ | | \ / | | | / \ | | \ / | | | o o o | | o | | | \ / \ / | | | | | \ / \ / | | | | | \ / U_i \ / | o---------|---------o | | \ / \ / | | | | o |^| o | q_j o q_i^-1 | | / \ / \ | | | | / \ U_j / \ | o---------v---------o | | / \ / \ | | E_j | | | / \ / \ | | | | | o o o | | o | | | \ / | | / \ | | | \ / | | / \ | | | \ U_j / | | o o | | | \ / | | / \ / \ | | | \ / | | / \ / \ | | | \ o---------------------->| o o o | | | \ / q_j | | \ q_j U_j / | | | \ / | | \ / | | | \ / | | \ / | | | o | | \ / | | | | | \ / | | | | | o | | | | | | | | | | | | o---------------------------------------o o-------------------o | | Figure 1. Manifold Of Concrescent Impressions o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 7 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Well now, can our little feeling, thus left alone in the universe, -- | for the god and we psychological critics may be supposed left out | of the account, -- can the feeling, I say, be said to have any sort | of a cognitive function? For it to 'know', there must be something | to be known. What is there, on the present supposition? | One may reply, "the feeling's content 'q'." | | William James, 'The Meaning of Truth', | Longmans, Green, & Co., London, 1909, | page 5. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | It is also convenient to have a local terminology. | Let U be an open set (of a manifold or a Banach space) | containing a point x_0. By a "local isomorphism" at x_0 | we mean an isomorphism | | f : U_1 -> V | | from some open set U_1 containing x_0 (and contained in U) | to an open set V (in some manifold or some Banach space). | Thus a local isomorphism is essentially a change of chart, | locally near a given point. | | Lang, DARM, page 23. | | Serge Lang, |'Differential & Riemannian Manifolds', | Springer-Verlag, New York, NY, 1995. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o References And Incidental Nuances (RAIN) http://www.philosophy.ru/library/kant/01/cr_pure_reason.html http://ez2www.com/go.php3?site=book&go=0387943382 http://hallmathematics.com/mathematics/1433.shtml http://hallmathematics.com/mathematics/630.shtml o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 8 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | But does it not seem more proper to call this the feeling's 'quality' than its content? | Does not the word "content" suggest that the feeling has already dirempted itself | as an act from its content as an object? And would it be quite safe to assume | so promptly that the quality 'q' of a feeling is one and the same thing | with a feeling of the quality 'q'? | | William James, 'The Meaning of Truth', | Longmans, Green, & Co., London, 1909, | page 5. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 2.2. Submanifolds, Immersions, Submersions | | Let X be a topological space, and Y a subset of X. | We say that Y is "locally closed" in X if every point | y in Y has an open neighborhood U in X such that Y |^| U | is closed in U. One verifies easily that a locally closed | subset is the intersection of an open set and a closed set. | For instance, any open subset of X is locally closed, and | any open interval is locally closed in the plane. | | Let X be a manifold (of class C^p with p >= 0). Let Y be a subset of X | and assume that for each point y in Y there exists a chart (V, r) at y | such that r gives an isomorphism of V with a product V_1 x V_2 where | V_1 is open in some space E_1 and V_2 is open in some space E_2, | and such that | | r(Y |^| V) = V_1 x a_2 | | for some point a_2 in V_2 (which we could take to be 0). Then it is clear | that Y is locally closed in X. Furthermore, the map r induces a bijection | | r_1 : Y |^| V -> V_1. | | The collection of pairs (Y |^| V, r_1) obtained in the above manner constitues | an atlas for Y, of class C^p. The verification of this assertion, whose formal | details we leave to the reader, depends on the following obvious fact. | | Lemma 2.1. Let U_1, U_2, V_1, V_2 be open subsets of Banach spaces, | | and g : U_1 x U_2 -> V_1 x V_2 a C^p-morphism. | | Let a_2 be in U_2 and b_2 be in V_2 | | and assume that g maps U_1 x a_2 into V_1 x b_2. | | Then the induced map | | g_1 : U_1 -> V_1 | | is also a morphism. | | Indeed, it is obtained as a composite map | | U -> U_1 x U_2 -> V_1 x V_2 -> V_1, | | the first map being an inclusion and the third a projection. | | We have therefore defined a C^p-structure on Y which will be called | a "submanifold" of X. This structure satisfies a universal mapping | property, which characterizes it, namely: | | | Given any map f : Z -> X from a manifold Z into X such that f(Z) is contained in Y. | | Let f_Y : Z -> Y be the induced map. Then f is a morphism if and only if f_Y is | | a morphism. | | The proof of this assertion depends on Lemma 2.1, and is trivial. | | Finally, we note that the inclusion of Y into X is a morphism. | | If Y is also a closed subspace of X, then | we say that it is a "closed submanifold". | | Lang, DARM, pages 23-24. o~~~~~~~~~o~~~~~~~~~o~SYLLABUS~o~~~~~~~~~o~~~~~~~~~o | 2. Manifolds | | 2.1. Atlases, Charts, Morphisms | | Let X be a set. An "atlas of class C^p (p >= 0)" on X is a collection | of pairs (U_i, q_i) (i ranging in some indexing set), satisfying the | following conditions: | | AT 1. Each U_i is a subset of X and the U_i cover X. | | AT 2. Each q_i is a bijection of U_i onto an open subset q_i U_i | of some Banach space E_i and for any i, j, [it holds that] | q_i (U_i |^| U_j) is open in E_i. | | AT 3. The map | | q_j o q_i^-1 : q_i (U_i |^| U_j) --> q_j (U_i |^| U_j) | | is a C^p-isomorphism for each pair of indices i, j. | | Lang, DARM, page 20. | | Each pair (U_i, q_i) will be called a "chart" of the atlas. | If a point x of X lies in U_i, then we say that (U_i, q_i) | is a "chart at" x. | | Suppose that we are given an open subset U of X and a topological isomorphism | q : U -> U' onto an open subset of some Banach space E. We shall say that | (U, q) is "compatible" with the atlas {(U_i, q_i)} if each map q_i q^-1 | (defined on a suitable intersection as in AT 3) is a C^p-isomorphism. | | Two atlases are said to be "compatible" if each chart of one is compatible with | the other atlas. One verifies immediately that the relation of compatibility | between atlases is an equivalence relation. An equivalence class of atlases | of class C^p on X is said to define a structure of "C^p-manifold" on X. | | If all the vector spaces E_i in some atlas are toplinearly isomorphic, | then we can always find an equivalent atlas for which they are all equal, | say to the vector space E. We then say that X is an "E-manifold" or that | X is "modeled" on E. | | If E = R^n for some fixed n, then we say that the manifold is "n-dimensional". | In this case, a chart | | q : U -> R^n | | is given by n coordinate functions q_1, ..., q_n. If 'P' denotes a point of U, | these functions are often written | | x_1(P), ..., x_n(P), | | or simply x_1, ..., x_n. They are called "local coordinates" on the manifold. | | If the integer p (which may also be infinity) is fixed throughout a discussion, | we also say that X is a manifold. | | Lang, DARM, page 21. | | Let X be manifold, and U an open subset of X. Then it is possible, | in the obvious way, to induce a manifold structure on U, by taking | as charts the intersections | | (U_i |^| U, q_i | (U_i |^| U)). | | [Notation. "f | S" indicates the function f as restricted to the set S.] | | If X is a topological space, covered by open subsets V_j, and if we are | given on each V_j a manifold structure such that for each pair j, j' the | induced structure on V_j |^| V_j' coincides, then it is clear that we can | give to X a unique manifold structure inducing the given ones on each V_j. | | If X, Y are two manifolds, then one can give the | product X x Y a manifold structure in the obvious way. | If {(U_i, q_i)} and {(V_j, r_j)} are atlases for X, Y | respectively, then | | {(U_i x V_j, q_i x r_j)} | | is an atlas for the product, and the product of compatible | atlases gives rise to compatible atlases, so that we do get | a well-defined product structure. | | Let X, Y be two manifolds. Let f : X -> Y be a map. | We shall say that f is a "C^p-morphism" if, given x in X, | there exists a chart (U, q) at x and a chart (V, r) at f(x) | such that f(U) c V, and the map | | r o f o q^-1 : qU -> rV | | is a C^p-morphism in the sense of Chapter 1, Section 3. | One sees then immediately that this same condition holds | for any choice of charts (U, q) at x and (V, r) at f(x) | such that F(U) c V. | | It is clear that the composite of two C^p-morphisms is itself | a C^p-morphism (because it is true for open subsets of vector | spaces). The C^p-manifolds and C^p-morphisms form a category. | The notion of isomorphism is therefore defined ... | | If f : X -> Y is a morphism, and (U, q) is a chart | at a point x in X, while (V, r) is a chart at f(x), | then we shall also denote by | | f_V,U : qU -> rV | | the map rfq^-1 [that is, r o f o q^-1]. | | Lang, DARM, page 22. | | It is also convenient to have a local terminology. | Let U be an open set (of a manifold or a Banach space) | containing a point x_0. By a "local isomorphism" at x_0 | we mean an isomorphism | | f : U_1 -> V | | from some open set U_1 containing x_0 (and contained in U) | to an open set V (in some manifold or some Banach space). | Thus a local isomorphism is essentially a change of chart, | locally near a given point. | | Lang, DARM, page 23. | | Serge Lang, |'Differential & Riemannian Manifolds', | Springer-Verlag, New York, NY, 1995. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | o---------------------------------------o o-------------------o | | X | | E_i | | | | | | | | | | o | | | | | / \ | | | o | | / \ | | | / \ | | / \ | | | / \ | | / \ | | | / \ q_i | | / q_i U_i \ | | | / o---------------------->| o o o | | | / \ | | \ / \ / | | | / \ | | \ / \ / | | | / U_i \ | | o o | | | / \ | | \ / | | | / \ | | \ / | | | o o o | | o | | | \ / \ / | | | | | \ / \ / | | | | | \ / U_i \ / | o---------|---------o | | \ / \ / | | | | o |^| o | q_j o q_i^-1 | | / \ / \ | | | | / \ U_j / \ | o---------v---------o | | / \ / \ | | E_j | | | / \ / \ | | | | | o o o | | o | | | \ / | | / \ | | | \ / | | / \ | | | \ U_j / | | o o | | | \ / | | / \ / \ | | | \ / | | / \ / \ | | | \ o---------------------->| o o o | | | \ / q_j | | \ q_j U_j / | | | \ / | | \ / | | | \ / | | \ / | | | o | | \ / | | | | | \ / | | | | | o | | | | | | | | | | | | o---------------------------------------o o-------------------o | | Figure 1. Manifold Of Coagitated Impressions o~~~~~~~~~o~~~~~~~~~o~SUBALLYS~o~~~~~~~~~o~~~~~~~~~o References And Incidental Nuances (RAIN) http://www.philosophy.ru/library/kant/01/cr_pure_reason.html http://ez2www.com/go.php3?site=book&go=0387943382 http://hallmathematics.com/mathematics/1433.shtml http://hallmathematics.com/mathematics/630.shtml o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 9 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | The quality 'q', so far, is an entirely subjective fact | which the feeling carries so to speak endogenously, or | in its pocket. If any one pleases to dignify so simple | a fact as this by the name of knowledge, of course | nothing can prevent him. But let us keep closer | to the path of common usage, and reserve the name | knowledge for the cognition of "realities", meaning | by realities things that exist independently of the | feeling through which their cognition occurs. If the | content of the feeling occur nowhere in the universe | outside of the feeling itself, and perish with the | feeling, common usage refuses to call it a reality, | and brands it as a subjective feature of the feeling's | constitution, or at the most as the feeling's 'dream'. | | William James, 'The Meaning of Truth', | Longmans, Green, & Co., London, 1909, | page 5-6. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Suppose that X is finite dimensional of dimension n, and that Y is a submanifold of dimension m. | Then from the definition we see that the local product structure in the neighborhood of a point | of Y can be expressed in terms of local coordinates as follows. Each point P of Y has an open | neighborhood U in X with local coordinates (x_1, ..., x_n) such that the points of Y in U are | precisely those whose last n - m coordinates are 0, that is, those points having coordinates | of type | | (x_1, ..., x_m, 0, ..., 0). | | Let f : Z -> X be a morphism, and let z be in Z. We shall say that f is | an "immersion" at z if there exists an open neighborhood Z_1 of z in Z | such that the restriction of f to Z_1 induces an isomorphism of Z_1 | onto a submanifold of X. We say that f is an "immersion" if it is | an immersion at every point. | | Note that there exist injective immersions | which are not isomorphisms onto submanifolds, | as given by the following example: | ________ | / \ | / \ | | | | | | | \ V | \___________________________________________ | | (The arrow means that the line approaches itself without touching.) | An immersion which does give an isomorphism onto a submanifold is | called an "embedding", and it is called a "closed embedding" if | this submanifold is closed. | | Lang, DARM, pages 24-25. o~~~~~~~~~o~~~~~~~~~o~SYLLABUS~o~~~~~~~~~o~~~~~~~~~o | 2. Manifolds | | 2.1. Atlases, Charts, Morphisms | | Let X be a set. An "atlas of class C^p (p >= 0)" on X is a collection | of pairs (U_i, q_i) (i ranging in some indexing set), satisfying the | following conditions: | | AT 1. Each U_i is a subset of X and the U_i cover X. | | AT 2. Each q_i is a bijection of U_i onto an open subset q_i U_i | of some Banach space E_i and for any i, j, [it holds that] | q_i (U_i |^| U_j) is open in E_i. | | AT 3. The map | | q_j o q_i^-1 : q_i (U_i |^| U_j) --> q_j (U_i |^| U_j) | | is a C^p-isomorphism for each pair of indices i, j. | | Lang, DARM, page 20. | | Each pair (U_i, q_i) will be called a "chart" of the atlas. | If a point x of X lies in U_i, then we say that (U_i, q_i) | is a "chart at" x. | | Suppose that we are given an open subset U of X and a topological isomorphism | q : U -> U' onto an open subset of some Banach space E. We shall say that | (U, q) is "compatible" with the atlas {(U_i, q_i)} if each map q_i q^-1 | (defined on a suitable intersection as in AT 3) is a C^p-isomorphism. | | Two atlases are said to be "compatible" if each chart of one is compatible with | the other atlas. One verifies immediately that the relation of compatibility | between atlases is an equivalence relation. An equivalence class of atlases | of class C^p on X is said to define a structure of "C^p-manifold" on X. | | If all the vector spaces E_i in some atlas are toplinearly isomorphic, | then we can always find an equivalent atlas for which they are all equal, | say to the vector space E. We then say that X is an "E-manifold" or that | X is "modeled" on E. | | If E = R^n for some fixed n, then we say that the manifold is "n-dimensional". | In this case, a chart | | q : U -> R^n | | is given by n coordinate functions q_1, ..., q_n. If 'P' denotes a point of U, | these functions are often written | | x_1(P), ..., x_n(P), | | or simply x_1, ..., x_n. They are called "local coordinates" on the manifold. | | If the integer p (which may also be infinity) is fixed throughout a discussion, | we also say that X is a manifold. | | Lang, DARM, page 21. | | Let X be manifold, and U an open subset of X. Then it is possible, | in the obvious way, to induce a manifold structure on U, by taking | as charts the intersections | | (U_i |^| U, q_i | (U_i |^| U)). | | [Notation. "f | S" indicates the function f as restricted to the set S.] | | If X is a topological space, covered by open subsets V_j, and if we are | given on each V_j a manifold structure such that for each pair j, j' the | induced structure on V_j |^| V_j' coincides, then it is clear that we can | give to X a unique manifold structure inducing the given ones on each V_j. | | If X, Y are two manifolds, then one can give the | product X x Y a manifold structure in the obvious way. | If {(U_i, q_i)} and {(V_j, r_j)} are atlases for X, Y | respectively, then | | {(U_i x V_j, q_i x r_j)} | | is an atlas for the product, and the product of compatible | atlases gives rise to compatible atlases, so that we do get | a well-defined product structure. | | Let X, Y be two manifolds. Let f : X -> Y be a map. | We shall say that f is a "C^p-morphism" if, given x in X, | there exists a chart (U, q) at x and a chart (V, r) at f(x) | such that f(U) c V, and the map | | r o f o q^-1 : qU -> rV | | is a C^p-morphism in the sense of Chapter 1, Section 3. | One sees then immediately that this same condition holds | for any choice of charts (U, q) at x and (V, r) at f(x) | such that F(U) c V. | | It is clear that the composite of two C^p-morphisms is itself | a C^p-morphism (because it is true for open subsets of vector | spaces). The C^p-manifolds and C^p-morphisms form a category. | The notion of isomorphism is therefore defined ... | | If f : X -> Y is a morphism, and (U, q) is a chart | at a point x in X, while (V, r) is a chart at f(x), | then we shall also denote by | | f_V,U : qU -> rV | | the map rfq^-1 [that is, r o f o q^-1]. | | Lang, DARM, page 22. | | It is also convenient to have a local terminology. | Let U be an open set (of a manifold or a Banach space) | containing a point x_0. By a "local isomorphism" at x_0 | we mean an isomorphism | | f : U_1 -> V | | from some open set U_1 containing x_0 (and contained in U) | to an open set V (in some manifold or some Banach space). | Thus a local isomorphism is essentially a change of chart, | locally near a given point. | | 2.2. Submanifolds, Immersions, Submersions | | Let X be a topological space, and Y a subset of X. | We say that Y is "locally closed" in X if every point | y in Y has an open neighborhood U in X such that Y |^| U | is closed in U. One verifies easily that a locally closed | subset is the intersection of an open set and a closed set. | For instance, any open subset of X is locally closed, and | any open interval is locally closed in the plane. | | Let X be a manifold (of class C^p with p >= 0). Let Y be a subset of X | and assume that for each point y in Y there exists a chart (V, r) at y | such that r gives an isomorphism of V with a product V_1 x V_2 where | V_1 is open in some space E_1 and V_2 is open in some space E_2, | and such that | | r(Y |^| V) = V_1 x a_2 | | for some point a_2 in V_2 (which we could take to be 0). Then it is clear | that Y is locally closed in X. Furthermore, the map r induces a bijection | | r_1 : Y |^| V -> V_1. | | The collection of pairs (Y |^| V, r_1) obtained in the above manner constitues | an atlas for Y, of class C^p. The verification of this assertion, whose formal | details we leave to the reader, depends on the following obvious fact. | | Lang, DARM, page 23. | | Lemma 2.1. Let U_1, U_2, V_1, V_2 be open subsets of Banach spaces, | | and g : U_1 x U_2 -> V_1 x V_2 a C^p-morphism. | | Let a_2 be in U_2 and b_2 be in V_2 | | and assume that g maps U_1 x a_2 into V_1 x b_2. | | Then the induced map | | g_1 : U_1 -> V_1 | | is also a morphism. | | Indeed, it is obtained as a composite map | | U_1 -> U_1 x U_2 -> V_1 x V_2 -> V_1, | | the first map being an inclusion and the third a projection. | | We have therefore defined a C^p-structure on Y which will be called | a "submanifold" of X. This structure satisfies a universal mapping | property, which characterizes it, namely: | | | Given any map f : Z -> X from a manifold Z into X such that | | f(Z) is contained in Y. Let f_Y : Z -> Y be the induced map. | | Then f is a morphism if and only if f_Y is a morphism. | | The proof of this assertion depends on Lemma 2.1, and is trivial. | | Finally, we note that the inclusion of Y into X is a morphism. | | If Y is also a closed subspace of X, then | we say that it is a "closed submanifold". | | Lang, DARM, page 24. | | Serge Lang, |'Differential & Riemannian Manifolds', | Springer-Verlag, New York, NY, 1995. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | o---------------------------------------o o-------------------o | | X | | E_i | | | | | | | | | | o | | | | | / \ | | | o | | / \ | | | / \ | | / \ | | | / \ | | / \ | | | / \ q_i | | / q_i U_i \ | | | / o---------------------->| o o o | | | / \ | | \ / \ / | | | / \ | | \ / \ / | | | / U_i \ | | o o | | | / \ | | \ / | | | / \ | | \ / | | | o o o | | o | | | \ / \ / | | | | | \ / \ / | | | | | \ / U_i \ / | o---------|---------o | | \ / \ / | | | | o |^| o | q_j o q_i^-1 | | / \ / \ | | | | / \ U_j / \ | o---------v---------o | | / \ / \ | | E_j | | | / \ / \ | | | | | o o o | | o | | | \ / | | / \ | | | \ / | | / \ | | | \ U_j / | | o o | | | \ / | | / \ / \ | | | \ / | | / \ / \ | | | \ o---------------------->| o o o | | | \ / q_j | | \ q_j U_j / | | | \ / | | \ / | | | \ / | | \ / | | | o | | \ / | | | | | \ / | | | | | o | | | | | | | | | | | | o---------------------------------------o o-------------------o | | Figure 1. Manifold Of Coagitant Impressions o~~~~~~~~~o~~~~~~~~~o~SUBALLYS~o~~~~~~~~~o~~~~~~~~~o References And Incidental Nuances (RAIN) http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Riemann.html http://www.philosophy.ru/library/kant/01/cr_pure_reason.html http://ez2www.com/go.php3?site=book&go=0387943382 http://hallmathematics.com/mathematics/1433.shtml http://hallmathematics.com/mathematics/630.shtml o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 10 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | For the feeling to be cognitive in the specific sense, then, | it must be self-transcendent; and we must prevail upon the | god to 'create a reality outside of it' to correspond to its | intrinsic quality 'q'. Thus only can it be redeemed from the | condition of being a solipsism. If now the new-created reality | 'resemble' the feeling's quality 'q', I say that the feeling may | be held by us 'to be cognizant of that reality'. | | William James, 'The Meaning of Truth', | Longmans, Green, & Co., London, 1909, | page 6. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | A morphism f : X -> Y will be called a "submersion" at a point x in X | if there exists a chart (U, q) at x and a chart (V, r) at f(x) such that | q gives an isomorphism of U on a product U_1 x U_2 (U_1 and U_2 open in | some Banach spaces), and such that the map | | rfq^-1 = f_V,U : U_1 x U_2 -> V | | is a projection. One sees then that the image of a submersion is | an open subset (a submersion is in fact an open mapping). We say | that f is a "submersion" if it is a submersion at every point. | | For manifolds modeled on Banach spaces, we have the usual criterion | for immersions and submersions in terms of the derivative. | | Proposition 2.2. Let X, Y be manifolds of class C^p (p >= 1) modeled on Banach spaces. | | Let f : X -> Y be a C^p-morphism. Let x be in X. Then: | | 1. f is an immersion at x if and only if | | there exists a chart (U, q) at x and (V, r) at f(x) | | such that f'_V,U(qx) is injective and splits. | | 2. f is a submersion at x if and only if | | there exists a chart (U, q) at x and (V, r) at f(x) | | such that f'_V,U(qx) is surjective and its kernel splits. | | Proof. This is an immediate consequence | of Corollaries 5.4 and 5.6 of | the inverse mapping theorem. | | The conditions expressed in [Propositions 2.2.1 and 2.2.2] depend only on the | derivative [f'], and if they hold for one choice of charts (U, q) and (V, r), | respectively, then they hold for every choice of such charts. It is therefore | convenient to introduce a terminology in order to deal with such properties. | | Lang, DARM, page 25. o~~~~~~~~~o~~~~~~~~~o~SYLLABUS~o~~~~~~~~~o~~~~~~~~~o | 2. Manifolds | | 2.1. Atlases, Charts, Morphisms | | Let X be a set. An "atlas of class C^p (p >= 0)" on X is a collection | of pairs (U_i, q_i) (i ranging in some indexing set), satisfying the | following conditions: | | AT 1. Each U_i is a subset of X and the U_i cover X. | | AT 2. Each q_i is a bijection of U_i onto an open subset q_i U_i | of some Banach space E_i and for any i, j, [it holds that] | q_i (U_i |^| U_j) is open in E_i. | | AT 3. The map | | q_j o q_i^-1 : q_i (U_i |^| U_j) --> q_j (U_i |^| U_j) | | is a C^p-isomorphism for each pair of indices i, j. | | Lang, DARM, page 20. | | Each pair (U_i, q_i) will be called a "chart" of the atlas. | If a point x of X lies in U_i, then we say that (U_i, q_i) | is a "chart at" x. | | Suppose that we are given an open subset U of X and a topological isomorphism | q : U -> U' onto an open subset of some Banach space E. We shall say that | (U, q) is "compatible" with the atlas {(U_i, q_i)} if each map q_i q^-1 | (defined on a suitable intersection as in AT 3) is a C^p-isomorphism. | | Two atlases are said to be "compatible" if each chart of one is compatible with | the other atlas. One verifies immediately that the relation of compatibility | between atlases is an equivalence relation. An equivalence class of atlases | of class C^p on X is said to define a structure of "C^p-manifold" on X. | | If all the vector spaces E_i in some atlas are toplinearly isomorphic, | then we can always find an equivalent atlas for which they are all equal, | say to the vector space E. We then say that X is an "E-manifold" or that | X is "modeled" on E. | | If E = R^n for some fixed n, then we say that the manifold is "n-dimensional". | In this case, a chart | | q : U -> R^n | | is given by n coordinate functions q_1, ..., q_n. If 'P' denotes a point of U, | these functions are often written | | x_1(P), ..., x_n(P), | | or simply x_1, ..., x_n. They are called "local coordinates" on the manifold. | | If the integer p (which may also be infinity) is fixed throughout a discussion, | we also say that X is a manifold. | | Lang, DARM, page 21. | | Let X be manifold, and U an open subset of X. Then it is possible, | in the obvious way, to induce a manifold structure on U, by taking | as charts the intersections | | (U_i |^| U, q_i | (U_i |^| U)). | | [Notation. "f | S" indicates the function f as restricted to the set S.] | | If X is a topological space, covered by open subsets V_j, and if we are | given on each V_j a manifold structure such that for each pair j, j' the | induced structure on V_j |^| V_j' coincides, then it is clear that we can | give to X a unique manifold structure inducing the given ones on each V_j. | | If X, Y are two manifolds, then one can give the | product X x Y a manifold structure in the obvious way. | If {(U_i, q_i)} and {(V_j, r_j)} are atlases for X, Y | respectively, then | | {(U_i x V_j, q_i x r_j)} | | is an atlas for the product, and the product of compatible | atlases gives rise to compatible atlases, so that we do get | a well-defined product structure. | | Let X, Y be two manifolds. Let f : X -> Y be a map. | We shall say that f is a "C^p-morphism" if, given x in X, | there exists a chart (U, q) at x and a chart (V, r) at f(x) | such that f(U) c V, and the map | | r o f o q^-1 : qU -> rV | | is a C^p-morphism in the sense of Chapter 1, Section 3. | One sees then immediately that this same condition holds | for any choice of charts (U, q) at x and (V, r) at f(x) | such that F(U) c V. | | It is clear that the composite of two C^p-morphisms is itself | a C^p-morphism (because it is true for open subsets of vector | spaces). The C^p-manifolds and C^p-morphisms form a category. | The notion of isomorphism is therefore defined ... | | If f : X -> Y is a morphism, and (U, q) is a chart | at a point x in X, while (V, r) is a chart at f(x), | then we shall also denote by | | f_V,U : qU -> rV | | the map rfq^-1 [that is, r o f o q^-1]. | | Lang, DARM, page 22. | | It is also convenient to have a local terminology. | Let U be an open set (of a manifold or a Banach space) | containing a point x_0. By a "local isomorphism" at x_0 | we mean an isomorphism | | f : U_1 -> V | | from some open set U_1 containing x_0 (and contained in U) | to an open set V (in some manifold or some Banach space). | Thus a local isomorphism is essentially a change of chart, | locally near a given point. | | 2.2. Submanifolds, Immersions, Submersions | | Let X be a topological space, and Y a subset of X. | We say that Y is "locally closed" in X if every point | y in Y has an open neighborhood U in X such that Y |^| U | is closed in U. One verifies easily that a locally closed | subset is the intersection of an open set and a closed set. | For instance, any open subset of X is locally closed, and | any open interval is locally closed in the plane. | | Let X be a manifold (of class C^p with p >= 0). Let Y be a subset of X | and assume that for each point y in Y there exists a chart (V, r) at y | such that r gives an isomorphism of V with a product V_1 x V_2 where | V_1 is open in some space E_1 and V_2 is open in some space E_2, | and such that | | r(Y |^| V) = V_1 x a_2 | | for some point a_2 in V_2 (which we could take to be 0). Then it is clear | that Y is locally closed in X. Furthermore, the map r induces a bijection | | r_1 : Y |^| V -> V_1. | | The collection of pairs (Y |^| V, r_1) obtained in the above manner constitues | an atlas for Y, of class C^p. The verification of this assertion, whose formal | details we leave to the reader, depends on the following obvious fact. | | Lang, DARM, page 23. | | Lemma 2.1. Let U_1, U_2, V_1, V_2 be open subsets of Banach spaces, | | and g : U_1 x U_2 -> V_1 x V_2 a C^p-morphism. | | Let a_2 be in U_2 and b_2 be in V_2 | | and assume that g maps U_1 x a_2 into V_1 x b_2. | | Then the induced map | | g_1 : U_1 -> V_1 | | is also a morphism. | | Indeed, it is obtained as a composite map | | U_1 -> U_1 x U_2 -> V_1 x V_2 -> V_1, | | the first map being an inclusion and the third a projection. | | We have therefore defined a C^p-structure on Y which will be called | a "submanifold" of X. This structure satisfies a universal mapping | property, which characterizes it, namely: | | | Given any map f : Z -> X from a manifold Z into X such that | | f(Z) is contained in Y. Let f_Y : Z -> Y be the induced map. | | Then f is a morphism if and only if f_Y is a morphism. | | The proof of this assertion depends on Lemma 2.1, and is trivial. | | Finally, we note that the inclusion of Y into X is a morphism. | | If Y is also a closed subspace of X, then | we say that it is a "closed submanifold". | | Suppose that X is finite dimensional of dimension n, and that Y is a submanifold of dimension m. | Then from the definition we see that the local product structure in the neighborhood of a point | of Y can be expressed in terms of local coordinates as follows. Each point P of Y has an open | neighborhood U in X with local coordinates (x_1, ..., x_n) such that the points of Y in U are | precisely those whose last n - m coordinates are 0, that is, those points having coordinates | of type | | (x_1, ..., x_m, 0, ..., 0). | | Let f : Z -> X be a morphism, and let z be in Z. We shall say that f is | an "immersion" at z if there exists an open neighborhood Z_1 of z in Z | such that the restriction of f to Z_1 induces an isomorphism of Z_1 | onto a submanifold of X. We say that f is an "immersion" if it is | an immersion at every point. | | Note that there exist injective immersions | which are not isomorphisms onto submanifolds, | as given by the following example: | ________ | / \ | / \ | | | | | | | \ V | \___________________________________________ | | (The arrow means that the line approaches itself without touching.) | An immersion which does give an isomorphism onto a submanifold is | called an "embedding", and it is called a "closed embedding" if | this submanifold is closed. | | Lang, DARM, pages 24-25. | | Serge Lang, |'Differential & Riemannian Manifolds', | Springer-Verlag, New York, NY, 1995. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | o---------------------------------------o o-------------------o | | X | | E_i | | | | | | | | | | o | | | | | / \ | | | o | | / \ | | | / \ | | / \ | | | / \ | | / \ | | | / \ q_i | | / q_i U_i \ | | | / o---------------------->| o o o | | | / \ | | \ / \ / | | | / \ | | \ / \ / | | | / U_i \ | | o o | | | / \ | | \ / | | | / \ | | \ / | | | o o o | | o | | | \ / \ / | | | | | \ / \ / | | | | | \ / U_i \ / | o---------|---------o | | \ / \ / | | | | o |^| o | q_j o q_i^-1 | | / \ / \ | | | | / \ U_j / \ | o---------v---------o | | / \ / \ | | E_j | | | / \ / \ | | | | | o o o | | o | | | \ / | | / \ | | | \ / | | / \ | | | \ U_j / | | o o | | | \ / | | / \ / \ | | | \ / | | / \ / \ | | | \ o---------------------->| o o o | | | \ / q_j | | \ q_j U_j / | | | \ / | | \ / | | | \ / | | \ / | | | o | | \ / | | | | | \ / | | | | | o | | | | | | | | | | | | o---------------------------------------o o-------------------o | | Figure 1. Manifold Of Cognitive Impressions o~~~~~~~~~o~~~~~~~~~o~SUBALLYS~o~~~~~~~~~o~~~~~~~~~o References And Incidental Nuances (RAIN) http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Riemann.html http://www.philosophy.ru/library/kant/01/cr_pure_reason.html http://ez2www.com/go.php3?site=book&go=0387943382 http://hallmathematics.com/mathematics/1433.shtml http://hallmathematics.com/mathematics/630.shtml o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 11 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Some persons will immediately cry out, "How 'can' | a reality resemble a feeling?" Here we find how | wise we were to name the quality of the feeling | by an algebraic letter 'q'. We flank the whole | difficulty of resemblance between an inner state | and an outward reality, by leaving it free to any | one to postulate as the reality whatever sort of | thing he thinks 'can' resemble a feeling, -- if | not an outward thing, then another feeling like | the first one, -- the mere feeling 'q' in the | critic's mind for example. | | William James, 'The Meaning of Truth', | Longmans, Green, & Co., London, 1909, | page 8. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o And now the real fun begins ... | Let X be a manifold of class C^p (p >= 1). | Let x be a point of X. We consider triples (U, q, v) | where (U, q) is a chart at x and v is an element of the | vector space in which qU lies. We say that two such triples | (U, q, v) and (V, r, w) are "equivalent" if the derivative of | rq^-1 at qx maps v on w. The formula reads: | | (rq^-1)'(qx)v = w | | (obviously an equivalence relation by the chain rule). | | An equivalence class of such triples is called a "tangent vector" of X at x. | The set of such tangent vectors is called the "tangent space" of X at x and | is denoted by 'T_x(X)'. Each chart (U, q) determines a bijection of T_x(X) | on a Banach space, namely the equivalence class of (U, q, v) corresponds to | the vector v. By means of such a bijection it is possible to transport to | T_x(X) the structure of topological vector space given by the chart, and | it is immediate that this structure is independent of the chart selected. | | Lang, DARM, pages 25-26. o~~~~~~~~~o~~~~~~~~~o~SYLLABUS~o~~~~~~~~~o~~~~~~~~~o | 2. Manifolds | | 2.1. Atlases, Charts, Morphisms | | Let X be a set. An "atlas of class C^p (p >= 0)" on X is a collection | of pairs (U_i, q_i) (i ranging in some indexing set), satisfying the | following conditions: | | AT 1. Each U_i is a subset of X and the U_i cover X. | | AT 2. Each q_i is a bijection of U_i onto an open subset q_i U_i | of some Banach space E_i and for any i, j, [it holds that] | q_i (U_i |^| U_j) is open in E_i. | | AT 3. The map | | q_j o q_i^-1 : q_i (U_i |^| U_j) --> q_j (U_i |^| U_j) | | is a C^p-isomorphism for each pair of indices i, j. | | Lang, DARM, page 20. | | Each pair (U_i, q_i) will be called a "chart" of the atlas. | If a point x of X lies in U_i, then we say that (U_i, q_i) | is a "chart at" x. | | Suppose that we are given an open subset U of X and a topological isomorphism | q : U -> U' onto an open subset of some Banach space E. We shall say that | (U, q) is "compatible" with the atlas {(U_i, q_i)} if each map q_i q^-1 | (defined on a suitable intersection as in AT 3) is a C^p-isomorphism. | | Two atlases are said to be "compatible" if each chart of one is compatible with | the other atlas. One verifies immediately that the relation of compatibility | between atlases is an equivalence relation. An equivalence class of atlases | of class C^p on X is said to define a structure of "C^p-manifold" on X. | | If all the vector spaces E_i in some atlas are toplinearly isomorphic, | then we can always find an equivalent atlas for which they are all equal, | say to the vector space E. We then say that X is an "E-manifold" or that | X is "modeled" on E. | | If E = R^n for some fixed n, then we say that the manifold is "n-dimensional". | In this case, a chart | | q : U -> R^n | | is given by n coordinate functions q_1, ..., q_n. If 'P' denotes a point of U, | these functions are often written | | x_1(P), ..., x_n(P), | | or simply x_1, ..., x_n. They are called "local coordinates" on the manifold. | | If the integer p (which may also be infinity) is fixed throughout a discussion, | we also say that X is a manifold. | | Lang, DARM, page 21. | | Let X be manifold, and U an open subset of X. Then it is possible, | in the obvious way, to induce a manifold structure on U, by taking | as charts the intersections | | (U_i |^| U, q_i | (U_i |^| U)). | | [Notation. "f | S" indicates the function f as restricted to the set S.] | | If X is a topological space, covered by open subsets V_j, and if we are | given on each V_j a manifold structure such that for each pair j, j' the | induced structure on V_j |^| V_j' coincides, then it is clear that we can | give to X a unique manifold structure inducing the given ones on each V_j. | | If X, Y are two manifolds, then one can give the | product X x Y a manifold structure in the obvious way. | If {(U_i, q_i)} and {(V_j, r_j)} are atlases for X, Y | respectively, then | | {(U_i x V_j, q_i x r_j)} | | is an atlas for the product, and the product of compatible | atlases gives rise to compatible atlases, so that we do get | a well-defined product structure. | | Let X, Y be two manifolds. Let f : X -> Y be a map. | We shall say that f is a "C^p-morphism" if, given x in X, | there exists a chart (U, q) at x and a chart (V, r) at f(x) | such that f(U) c V, and the map | | r o f o q^-1 : qU -> rV | | is a C^p-morphism in the sense of Chapter 1, Section 3. | One sees then immediately that this same condition holds | for any choice of charts (U, q) at x and (V, r) at f(x) | such that F(U) c V. | | It is clear that the composite of two C^p-morphisms is itself | a C^p-morphism (because it is true for open subsets of vector | spaces). The C^p-manifolds and C^p-morphisms form a category. | The notion of isomorphism is therefore defined ... | | If f : X -> Y is a morphism, and (U, q) is a chart | at a point x in X, while (V, r) is a chart at f(x), | then we shall also denote by | | f_V,U : qU -> rV | | the map rfq^-1 [that is, r o f o q^-1]. | | Lang, DARM, page 22. | | It is also convenient to have a local terminology. | Let U be an open set (of a manifold or a Banach space) | containing a point x_0. By a "local isomorphism" at x_0 | we mean an isomorphism | | f : U_1 -> V | | from some open set U_1 containing x_0 (and contained in U) | to an open set V (in some manifold or some Banach space). | Thus a local isomorphism is essentially a change of chart, | locally near a given point. | | 2.2. Submanifolds, Immersions, Submersions | | Let X be a topological space, and Y a subset of X. | We say that Y is "locally closed" in X if every point | y in Y has an open neighborhood U in X such that Y |^| U | is closed in U. One verifies easily that a locally closed | subset is the intersection of an open set and a closed set. | For instance, any open subset of X is locally closed, and | any open interval is locally closed in the plane. | | Let X be a manifold (of class C^p with p >= 0). Let Y be a subset of X | and assume that for each point y in Y there exists a chart (V, r) at y | such that r gives an isomorphism of V with a product V_1 x V_2 where | V_1 is open in some space E_1 and V_2 is open in some space E_2, | and such that | | r(Y |^| V) = V_1 x a_2 | | for some point a_2 in V_2 (which we could take to be 0). Then it is clear | that Y is locally closed in X. Furthermore, the map r induces a bijection | | r_1 : Y |^| V -> V_1. | | The collection of pairs (Y |^| V, r_1) obtained in the above manner constitues | an atlas for Y, of class C^p. The verification of this assertion, whose formal | details we leave to the reader, depends on the following obvious fact. | | Lang, DARM, page 23. | | Lemma 2.1. Let U_1, U_2, V_1, V_2 be open subsets of Banach spaces, | | and g : U_1 x U_2 -> V_1 x V_2 a C^p-morphism. | | Let a_2 be in U_2 and b_2 be in V_2 | | and assume that g maps U_1 x a_2 into V_1 x b_2. | | Then the induced map | | g_1 : U_1 -> V_1 | | is also a morphism. | | Indeed, it is obtained as a composite map | | U_1 -> U_1 x U_2 -> V_1 x V_2 -> V_1, | | the first map being an inclusion and the third a projection. | | We have therefore defined a C^p-structure on Y which will be called | a "submanifold" of X. This structure satisfies a universal mapping | property, which characterizes it, namely: | | | Given any map f : Z -> X from a manifold Z into X such that | | f(Z) is contained in Y. Let f_Y : Z -> Y be the induced map. | | Then f is a morphism if and only if f_Y is a morphism. | | The proof of this assertion depends on Lemma 2.1, and is trivial. | | Finally, we note that the inclusion of Y into X is a morphism. | | If Y is also a closed subspace of X, then | we say that it is a "closed submanifold". | | Suppose that X is finite dimensional of dimension n, and that Y is a submanifold of dimension m. | Then from the definition we see that the local product structure in the neighborhood of a point | of Y can be expressed in terms of local coordinates as follows. Each point P of Y has an open | neighborhood U in X with local coordinates (x_1, ..., x_n) such that the points of Y in U are | precisely those whose last n - m coordinates are 0, that is, those points having coordinates | of type | | (x_1, ..., x_m, 0, ..., 0). | | Let f : Z -> X be a morphism, and let z be in Z. We shall say that f is | an "immersion" at z if there exists an open neighborhood Z_1 of z in Z | such that the restriction of f to Z_1 induces an isomorphism of Z_1 | onto a submanifold of X. We say that f is an "immersion" if it is | an immersion at every point. | | Lang, DARM, page 24. | | Note that there exist injective immersions | which are not isomorphisms onto submanifolds, | as given by the following example: | ________ | / \ | / \ | | | | | | | \ V | \___________________________________________ | | (The arrow means that the line approaches itself without touching.) | An immersion which does give an isomorphism onto a submanifold is | called an "embedding", and it is called a "closed embedding" if | this submanifold is closed. | | A morphism f : X -> Y will be called a "submersion" at a point x in X | if there exists a chart (U, q) at x and a chart (V, r) at f(x) such that | q gives an isomorphism of U on a product U_1 x U_2 (U_1 and U_2 open in | some Banach spaces), and such that the map | | rfq^-1 = f_V,U : U_1 x U_2 -> V | | is a projection. One sees then that the image of a submersion is | an open subset (a submersion is in fact an open mapping). We say | that f is a "submersion" if it is a submersion at every point. | | For manifolds modeled on Banach spaces, we have the usual criterion | for immersions and submersions in terms of the derivative. | | Proposition 2.2. Let X, Y be manifolds of class C^p (p >= 1) modeled on Banach spaces. | | Let f : X -> Y be a C^p-morphism. Let x be in X. Then: | | 1. f is an immersion at x if and only if | | there exists a chart (U, q) at x and (V, r) at f(x) | | such that f'_V,U(qx) is injective and splits. | | 2. f is a submersion at x if and only if | | there exists a chart (U, q) at x and (V, r) at f(x) | | such that f'_V,U(qx) is surjective and its kernel splits. | | Proof. This is an immediate consequence | of Corollaries 5.4 and 5.6 of | the inverse mapping theorem. | | The conditions expressed in [Propositions 2.2.1 and 2.2.2] depend only on the | derivative [f'], and if they hold for one choice of charts (U, q) and (V, r), | respectively, then they hold for every choice of such charts. It is therefore | convenient to introduce a terminology in order to deal with such properties. | | Lang, DARM, page 25. | | Serge Lang, |'Differential & Riemannian Manifolds', | Springer-Verlag, New York, NY, 1995. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | o---------------------------------------o o-------------------o | | X | | E_i | | | | | | | | | | o | | | | | / \ | | | o | | / \ | | | / \ | | / \ | | | / \ | | / \ | | | / \ q_i | | / q_i U_i \ | | | / o---------------------->| o o o | | | / \ | | \ / \ / | | | / \ | | \ / \ / | | | / U_i \ | | o o | | | / \ | | \ / | | | / \ | | \ / | | | o o o | | o | | | \ / \ / | | | | | \ / \ / | | | | | \ / U_i \ / | o---------|---------o | | \ / \ / | | | | o |^| o | q_j o q_i^-1 | | / \ / \ | | | | / \ U_j / \ | o---------v---------o | | / \ / \ | | E_j | | | / \ / \ | | | | | o o o | | o | | | \ / | | / \ | | | \ / | | / \ | | | \ U_j / | | o o | | | \ / | | / \ / \ | | | \ / | | / \ / \ | | | \ o---------------------->| o o o | | | \ / q_j | | \ q_j U_j / | | | \ / | | \ / | | | \ / | | \ / | | | o | | \ / | | | | | \ / | | | | | o | | | | | | | | | | | | o---------------------------------------o o-------------------o | | Figure 1. Manifold Of Dissensual Impressions o~~~~~~~~~o~~~~~~~~~o~SUBALLYS~o~~~~~~~~~o~~~~~~~~~o References And Incidental Nuances (RAIN) http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Riemann.html http://www.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm http://www.philosophy.ru/library/kant/01/cr_pure_reason.html http://ez2www.com/go.php3?site=book&go=0387943382 http://hallmathematics.com/mathematics/1433.shtml http://hallmathematics.com/mathematics/630.shtml o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 12 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Our little supposed feeling, whatever it may be, | from the cognitive point of view, whether a bit of | knowledge or a dream, is certainly no psychical zero. | It is a most positively and definitely qualified inner | fact, with a complexion all its own. Of course there | are many mental facts which it is 'not'. It knows 'q', | if 'q' be a reality, with a very minimum of knowledge. | It neither dates nor locates it. It neither classes nor | names it. And it neither knows itself as a feeling, nor | contrasts itself with other feelings, nor estimates its | own duration or intensity. It is, in short, if there | is no more of it than this, a most dumb and helpless | and useless kind of thing. | | William James, 'The Meaning of Truth', | Longmans, Green, & Co., London, 1909, | page 10. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | If U, V are open in Banach spaces, then to every morphism of class C^p (p >= 1) | we can associate its derivative Df(x). If now f : X -> Y is a morphism of | one manifold into another, and x a point of X, then by means of charts | we can interpret the derivative of f on each chart at x as a mapping | | df(x) = T_x f : T_x(X) -> T_f(x)(Y). | | Indeed, this map T_x f is the unique linear map having the following property. | If (U, q) is a chart at x and (V, r) is a chart at f(x) such that f(U) c V | and ^v^ is a tangent vector at x represented by v in the chart (U, q), then | | T_x f(^v^) | | is the tangent vector at f(x) represented by Df_V,U(x)v. | The representation of T_x f on the spaces of charts can | be given in the form of a diagram | | T_x(X) o-------->o E | | | | T_x f | | f'_V,U(x) | v v | T_f(x)(Y) o-------->o F | | The map T_x f is obviously continuous and linear for the structure of | topological vector space which we have placed on T_x(X) and T_f(x)(Y). | | As a matter of notation, we shall sometimes write f_*,x instead of T_x f. | | The operation T satisfies an obvious functorial property, | namely, if f : X -> Y and g : Y -> Z are morphisms, then | | T_x(g o f) = T_f(x)(g) o T_x(f). | | T_x(id) = id. | | We may reformulate Proposition 2.2: | | Proposition 2.3. Let X, Y be manifolds of class C^p (p >= 1) modeled on Banach spaces. | | Let f : X -> Y be a C^p-morphism. Let x be in X. Then: | | 1. f is an immersion at x if and only if | | the map T_x f is injective and splits. | | 2. f is a submersion at x if and only if | | the map T_x f is surjective and its kernel splits. | | Note. If X, Y are finite dimensional, then the condition that T_x f splits | is superfluous. Every subspace of a finite dimensional vector space splits. | | Lang, DARM, pages 26-27. o~~~~~~~~~o~~~~~~~~~o~SYLLABUS~o~~~~~~~~~o~~~~~~~~~o | 2. Manifolds | | 2.1. Atlases, Charts, Morphisms | | Let X be a set. An "atlas of class C^p (p >= 0)" on X is a collection | of pairs (U_i, q_i) (i ranging in some indexing set), satisfying the | following conditions: | | AT 1. Each U_i is a subset of X and the U_i cover X. | | AT 2. Each q_i is a bijection of U_i onto an open subset q_i U_i | of some Banach space E_i and for any i, j, [it holds that] | q_i (U_i |^| U_j) is open in E_i. | | AT 3. The map | | q_j o q_i^-1 : q_i (U_i |^| U_j) --> q_j (U_i |^| U_j) | | is a C^p-isomorphism for each pair of indices i, j. | | Lang, DARM, page 20. | | Each pair (U_i, q_i) will be called a "chart" of the atlas. | If a point x of X lies in U_i, then we say that (U_i, q_i) | is a "chart at" x. | | Suppose that we are given an open subset U of X and a topological isomorphism | q : U -> U' onto an open subset of some Banach space E. We shall say that | (U, q) is "compatible" with the atlas {(U_i, q_i)} if each map q_i q^-1 | (defined on a suitable intersection as in AT 3) is a C^p-isomorphism. | | Two atlases are said to be "compatible" if each chart of one is compatible with | the other atlas. One verifies immediately that the relation of compatibility | between atlases is an equivalence relation. An equivalence class of atlases | of class C^p on X is said to define a structure of "C^p-manifold" on X. | | If all the vector spaces E_i in some atlas are toplinearly isomorphic, | then we can always find an equivalent atlas for which they are all equal, | say to the vector space E. We then say that X is an "E-manifold" or that | X is "modeled" on E. | | If E = R^n for some fixed n, then we say that the manifold is "n-dimensional". | In this case, a chart | | q : U -> R^n | | is given by n coordinate functions q_1, ..., q_n. If 'P' denotes a point of U, | these functions are often written | | x_1(P), ..., x_n(P), | | or simply x_1, ..., x_n. They are called "local coordinates" on the manifold. | | If the integer p (which may also be infinity) is fixed throughout a discussion, | we also say that X is a manifold. | | Lang, DARM, page 21. | | Let X be manifold, and U an open subset of X. Then it is possible, | in the obvious way, to induce a manifold structure on U, by taking | as charts the intersections | | (U_i |^| U, q_i | (U_i |^| U)). | | [Notation. "f | S" indicates the function f as restricted to the set S.] | | If X is a topological space, covered by open subsets V_j, and if we are | given on each V_j a manifold structure such that for each pair j, j' the | induced structure on V_j |^| V_j' coincides, then it is clear that we can | give to X a unique manifold structure inducing the given ones on each V_j. | | If X, Y are two manifolds, then one can give the | product X x Y a manifold structure in the obvious way. | If {(U_i, q_i)} and {(V_j, r_j)} are atlases for X, Y | respectively, then | | {(U_i x V_j, q_i x r_j)} | | is an atlas for the product, and the product of compatible | atlases gives rise to compatible atlases, so that we do get | a well-defined product structure. | | Let X, Y be two manifolds. Let f : X -> Y be a map. | We shall say that f is a "C^p-morphism" if, given x in X, | there exists a chart (U, q) at x and a chart (V, r) at f(x) | such that f(U) c V, and the map | | r o f o q^-1 : qU -> rV | | is a C^p-morphism in the sense of Chapter 1, Section 3. | One sees then immediately that this same condition holds | for any choice of charts (U, q) at x and (V, r) at f(x) | such that F(U) c V. | | It is clear that the composite of two C^p-morphisms is itself | a C^p-morphism (because it is true for open subsets of vector | spaces). The C^p-manifolds and C^p-morphisms form a category. | The notion of isomorphism is therefore defined ... | | If f : X -> Y is a morphism, and (U, q) is a chart | at a point x in X, while (V, r) is a chart at f(x), | then we shall also denote by | | f_V,U : qU -> rV | | the map rfq^-1 [that is, r o f o q^-1]. | | Lang, DARM, page 22. | | It is also convenient to have a local terminology. | Let U be an open set (of a manifold or a Banach space) | containing a point x_0. By a "local isomorphism" at x_0 | we mean an isomorphism | | f : U_1 -> V | | from some open set U_1 containing x_0 (and contained in U) | to an open set V (in some manifold or some Banach space). | Thus a local isomorphism is essentially a change of chart, | locally near a given point. | | 2.2. Submanifolds, Immersions, Submersions | | Let X be a topological space, and Y a subset of X. | We say that Y is "locally closed" in X if every point | y in Y has an open neighborhood U in X such that Y |^| U | is closed in U. One verifies easily that a locally closed | subset is the intersection of an open set and a closed set. | For instance, any open subset of X is locally closed, and | any open interval is locally closed in the plane. | | Let X be a manifold (of class C^p with p >= 0). Let Y be a subset of X | and assume that for each point y in Y there exists a chart (V, r) at y | such that r gives an isomorphism of V with a product V_1 x V_2 where | V_1 is open in some space E_1 and V_2 is open in some space E_2, | and such that | | r(Y |^| V) = V_1 x a_2 | | for some point a_2 in V_2 (which we could take to be 0). Then it is clear | that Y is locally closed in X. Furthermore, the map r induces a bijection | | r_1 : Y |^| V -> V_1. | | The collection of pairs (Y |^| V, r_1) obtained in the above manner constitues | an atlas for Y, of class C^p. The verification of this assertion, whose formal | details we leave to the reader, depends on the following obvious fact. | | Lang, DARM, page 23. | | Lemma 2.1. Let U_1, U_2, V_1, V_2 be open subsets of Banach spaces, | | and g : U_1 x U_2 -> V_1 x V_2 a C^p-morphism. | | Let a_2 be in U_2 and b_2 be in V_2 | | and assume that g maps U_1 x a_2 into V_1 x b_2. | | Then the induced map | | g_1 : U_1 -> V_1 | | is also a morphism. | | Indeed, it is obtained as a composite map | | U_1 -> U_1 x U_2 -> V_1 x V_2 -> V_1, | | the first map being an inclusion and the third a projection. | | We have therefore defined a C^p-structure on Y which will be called | a "submanifold" of X. This structure satisfies a universal mapping | property, which characterizes it, namely: | | | Given any map f : Z -> X from a manifold Z into X such that | | f(Z) is contained in Y. Let f_Y : Z -> Y be the induced map. | | Then f is a morphism if and only if f_Y is a morphism. | | The proof of this assertion depends on Lemma 2.1, and is trivial. | | Finally, we note that the inclusion of Y into X is a morphism. | | If Y is also a closed subspace of X, then | we say that it is a "closed submanifold". | | Suppose that X is finite dimensional of dimension n, and that Y is a submanifold of dimension m. | Then from the definition we see that the local product structure in the neighborhood of a point | of Y can be expressed in terms of local coordinates as follows. Each point P of Y has an open | neighborhood U in X with local coordinates (x_1, ..., x_n) such that the points of Y in U are | precisely those whose last n - m coordinates are 0, that is, those points having coordinates | of type | | (x_1, ..., x_m, 0, ..., 0). | | Let f : Z -> X be a morphism, and let z be in Z. We shall say that f is | an "immersion" at z if there exists an open neighborhood Z_1 of z in Z | such that the restriction of f to Z_1 induces an isomorphism of Z_1 | onto a submanifold of X. We say that f is an "immersion" if it is | an immersion at every point. | | Lang, DARM, page 24. | | Note that there exist injective immersions | which are not isomorphisms onto submanifolds, | as given by the following example: | ________ | / \ | / \ | | | | | | | \ V | \___________________________________________ | | (The arrow means that the line approaches itself without touching.) | An immersion which does give an isomorphism onto a submanifold is | called an "embedding", and it is called a "closed embedding" if | this submanifold is closed. | | A morphism f : X -> Y will be called a "submersion" at a point x in X | if there exists a chart (U, q) at x and a chart (V, r) at f(x) such that | q gives an isomorphism of U on a product U_1 x U_2 (U_1 and U_2 open in | some Banach spaces), and such that the map | | rfq^-1 = f_V,U : U_1 x U_2 -> V | | is a projection. One sees then that the image of a submersion is | an open subset (a submersion is in fact an open mapping). We say | that f is a "submersion" if it is a submersion at every point. | | For manifolds modeled on Banach spaces, we have the usual criterion | for immersions and submersions in terms of the derivative. | | Proposition 2.2. Let X, Y be manifolds of class C^p (p >= 1) modeled on Banach spaces. | | Let f : X -> Y be a C^p-morphism. Let x be in X. Then: | | 1. f is an immersion at x if and only if | | there exists a chart (U, q) at x and (V, r) at f(x) | | such that f'_V,U(qx) is injective and splits. | | 2. f is a submersion at x if and only if | | there exists a chart (U, q) at x and (V, r) at f(x) | | such that f'_V,U(qx) is surjective and its kernel splits. | | Proof. This is an immediate consequence | of Corollaries 5.4 and 5.6 of | the inverse mapping theorem. | | The conditions expressed in [Propositions 2.2.1 and 2.2.2] depend only on the | derivative [f'], and if they hold for one choice of charts (U, q) and (V, r), | respectively, then they hold for every choice of such charts. It is therefore | convenient to introduce a terminology in order to deal with such properties. | | Let X be a manifold of class C^p (p >= 1). | Let x be a point of X. We consider triples (U, q, v) | where (U, q) is a chart at x and v is an element of the | vector space in which qU lies. We say that two such triples | (U, q, v) and (V, r, w) are "equivalent" if the derivative of | rq^-1 at qx maps v on w. The formula reads: | | (rq^-1)'(qx)v = w | | (obviously an equivalence relation by the chain rule). | | An equivalence class of such triples is called a "tangent vector" of X at x. | The set of such tangent vectors is called the "tangent space" of X at x and | is denoted by 'T_x(X)'. Each chart (U, q) determines a bijection of T_x(X) | on a Banach space, namely the equivalence class of (U, q, v) corresponds to | the vector v. By means of such a bijection it is possible to transport to | T_x(X) the structure of topological vector space given by the chart, and | it is immediate that this structure is independent of the chart selected. | | Lang, DARM, pages 25-26. | | Serge Lang, |'Differential & Riemannian Manifolds', | Springer-Verlag, New York, NY, 1995. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | o---------------------------------------o o-------------------o | | X | | E_i | | | | | | | | | | o | | | | | / \ | | | o | | / \ | | | / \ | | / \ | | | / \ | | / \ | | | / \ q_i | | / q_i U_i \ | | | / o---------------------->| o o o | | | / \ | | \ / \ / | | | / \ | | \ / \ / | | | / U_i \ | | o o | | | / \ | | \ / | | | / \ | | \ / | | | o o o | | o | | | \ / \ / | | | | | \ / \ / | | | | | \ / U_i \ / | o---------|---------o | | \ / \ / | | | | o |^| o | q_j o q_i^-1 | | / \ / \ | | | | / \ U_j / \ | o---------v---------o | | / \ / \ | | E_j | | | / \ / \ | | | | | o o o | | o | | | \ / | | / \ | | | \ / | | / \ | | | \ U_j / | | o o | | | \ / | | / \ / \ | | | \ / | | / \ / \ | | | \ o---------------------->| o o o | | | \ / q_j | | \ q_j U_j / | | | \ / | | \ / | | | \ / | | \ / | | | o | | \ / | | | | | \ / | | | | | o | | | | | | | | | | | | o---------------------------------------o o-------------------o | | Figure 1. Manifold Of Consensual Impressions o~~~~~~~~~o~~~~~~~~~o~SUBALLYS~o~~~~~~~~~o~~~~~~~~~o References And Incidental Nuances (RAIN) http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Riemann.html http://www.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm http://www.philosophy.ru/library/kant/01/cr_pure_reason.html http://ez2www.com/go.php3?site=book&go=0387943382 http://hallmathematics.com/mathematics/1433.shtml http://hallmathematics.com/mathematics/630.shtml o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 13 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Now obviously if our supposed feeling of 'q' | is (if knowledge at all) only knowledge of the | mere acquaintance-type, it is milking a he-goat, | as the ancients would have said, to try to extract | from it any deliverance 'about' anything under the sun, | even about itself. And it is as unjust, after our failure, | to turn upon it and call it a psychical nothing, as it would be, | after our fruitless attack upon the billy-goat, to proclaim the | non-lactiferous character of the whole goat-tribe. | | William James, 'The Meaning of Truth', | Longmans, Green, & Co., London, 1909, | page 12. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | If W is a submanifold of a manifold Y of class C^p (p >= 1), then the inclusion | | i : W -> Y | | induces a map | | T_w i : T_w(W) -> T_w(Y) | | which is in fact an injection. From the definition of a submanifold, one sees | immediately that the image of T_w i splits. It will be convenient to identify | T_w(W) in T_w(Y) if no confusion can result. | | A morphism f : X -> Y will be said to be "transversal" over the submanifold W of Y | if the following condition is satisfied. | | Let x in X be such that f(x) is in W. Let (V, r) be a chart at f(x) such that | r : V -> V_1 x V_2 is an isomorphism on a product, with | | r(f(x)) = (0, 0) and r(W |^| V) = V_1 x 0. | | Then there exists an open neighborhood U of x such that the composite map | | f r pr | U -----> V -----> V_1 x V_2 ------> V_2 | | is a submersion. | | In particular, if f is transversal over W, then f^-1(W) is a submanifold of X, | because the inverse image of 0 by our local composite map | | pr o r o f | | is equal to the inverse image of W |^| V by r. | | Lang, DARM, page 27. o~~~~~~~~~o~~~~~~~~~o~SYLLABUS~o~~~~~~~~~o~~~~~~~~~o | 2. Manifolds | | 2.1. Atlases, Charts, Morphisms | | Let X be a set. An "atlas of class C^p (p >= 0)" on X is a collection | of pairs (U_i, q_i) (i ranging in some indexing set), satisfying the | following conditions: | | AT 1. Each U_i is a subset of X and the U_i cover X. | | AT 2. Each q_i is a bijection of U_i onto an open subset q_i U_i | of some Banach space E_i and for any i, j, [it holds that] | q_i (U_i |^| U_j) is open in E_i. | | AT 3. The map | | q_j o q_i^-1 : q_i (U_i |^| U_j) --> q_j (U_i |^| U_j) | | is a C^p-isomorphism for each pair of indices i, j. | | Lang, DARM, page 20. | | Each pair (U_i, q_i) will be called a "chart" of the atlas. | If a point x of X lies in U_i, then we say that (U_i, q_i) | is a "chart at" x. | | Suppose that we are given an open subset U of X and a topological isomorphism | q : U -> U' onto an open subset of some Banach space E. We shall say that | (U, q) is "compatible" with the atlas {(U_i, q_i)} if each map q_i q^-1 | (defined on a suitable intersection as in AT 3) is a C^p-isomorphism. | | Two atlases are said to be "compatible" if each chart of one is compatible with | the other atlas. One verifies immediately that the relation of compatibility | between atlases is an equivalence relation. An equivalence class of atlases | of class C^p on X is said to define a structure of "C^p-manifold" on X. | | If all the vector spaces E_i in some atlas are toplinearly isomorphic, | then we can always find an equivalent atlas for which they are all equal, | say to the vector space E. We then say that X is an "E-manifold" or that | X is "modeled" on E. | | If E = R^n for some fixed n, then we say that the manifold is "n-dimensional". | In this case, a chart | | q : U -> R^n | | is given by n coordinate functions q_1, ..., q_n. If 'P' denotes a point of U, | these functions are often written | | x_1(P), ..., x_n(P), | | or simply x_1, ..., x_n. They are called "local coordinates" on the manifold. | | If the integer p (which may also be infinity) is fixed throughout a discussion, | we also say that X is a manifold. | | Lang, DARM, page 21. | | Let X be manifold, and U an open subset of X. Then it is possible, | in the obvious way, to induce a manifold structure on U, by taking | as charts the intersections | | (U_i |^| U, q_i | (U_i |^| U)). | | [Notation. "f | S" indicates the function f as restricted to the set S.] | | If X is a topological space, covered by open subsets V_j, and if we are | given on each V_j a manifold structure such that for each pair j, j' the | induced structure on V_j |^| V_j' coincides, then it is clear that we can | give to X a unique manifold structure inducing the given ones on each V_j. | | If X, Y are two manifolds, then one can give the | product X x Y a manifold structure in the obvious way. | If {(U_i, q_i)} and {(V_j, r_j)} are atlases for X, Y | respectively, then | | {(U_i x V_j, q_i x r_j)} | | is an atlas for the product, and the product of compatible | atlases gives rise to compatible atlases, so that we do get | a well-defined product structure. | | Let X, Y be two manifolds. Let f : X -> Y be a map. | We shall say that f is a "C^p-morphism" if, given x in X, | there exists a chart (U, q) at x and a chart (V, r) at f(x) | such that f(U) c V, and the map | | r o f o q^-1 : qU -> rV | | is a C^p-morphism in the sense of Chapter 1, Section 3. | One sees then immediately that this same condition holds | for any choice of charts (U, q) at x and (V, r) at f(x) | such that F(U) c V. | | It is clear that the composite of two C^p-morphisms is itself | a C^p-morphism (because it is true for open subsets of vector | spaces). The C^p-manifolds and C^p-morphisms form a category. | The notion of isomorphism is therefore defined ... | | If f : X -> Y is a morphism, and (U, q) is a chart | at a point x in X, while (V, r) is a chart at f(x), | then we shall also denote by | | f_V,U : qU -> rV | | the map rfq^-1 [that is, r o f o q^-1]. | | Lang, DARM, page 22. | | It is also convenient to have a local terminology. | Let U be an open set (of a manifold or a Banach space) | containing a point x_0. By a "local isomorphism" at x_0 | we mean an isomorphism | | f : U_1 -> V | | from some open set U_1 containing x_0 (and contained in U) | to an open set V (in some manifold or some Banach space). | Thus a local isomorphism is essentially a change of chart, | locally near a given point. | | 2.2. Submanifolds, Immersions, Submersions | | Let X be a topological space, and Y a subset of X. | We say that Y is "locally closed" in X if every point | y in Y has an open neighborhood U in X such that Y |^| U | is closed in U. One verifies easily that a locally closed | subset is the intersection of an open set and a closed set. | For instance, any open subset of X is locally closed, and | any open interval is locally closed in the plane. | | Let X be a manifold (of class C^p with p >= 0). Let Y be a subset of X | and assume that for each point y in Y there exists a chart (V, r) at y | such that r gives an isomorphism of V with a product V_1 x V_2 where | V_1 is open in some space E_1 and V_2 is open in some space E_2, | and such that | | r(Y |^| V) = V_1 x a_2 | | for some point a_2 in V_2 (which we could take to be 0). Then it is clear | that Y is locally closed in X. Furthermore, the map r induces a bijection | | r_1 : Y |^| V -> V_1. | | The collection of pairs (Y |^| V, r_1) obtained in the above manner constitues | an atlas for Y, of class C^p. The verification of this assertion, whose formal | details we leave to the reader, depends on the following obvious fact. | | Lang, DARM, page 23. | | Lemma 2.1. Let U_1, U_2, V_1, V_2 be open subsets of Banach spaces, | | and g : U_1 x U_2 -> V_1 x V_2 a C^p-morphism. | | Let a_2 be in U_2 and b_2 be in V_2 | | and assume that g maps U_1 x a_2 into V_1 x b_2. | | Then the induced map | | g_1 : U_1 -> V_1 | | is also a morphism. | | Indeed, it is obtained as a composite map | | U_1 -> U_1 x U_2 -> V_1 x V_2 -> V_1, | | the first map being an inclusion and the third a projection. | | We have therefore defined a C^p-structure on Y which will be called | a "submanifold" of X. This structure satisfies a universal mapping | property, which characterizes it, namely: | | | Given any map f : Z -> X from a manifold Z into X such that | | f(Z) is contained in Y. Let f_Y : Z -> Y be the induced map. | | Then f is a morphism if and only if f_Y is a morphism. | | The proof of this assertion depends on Lemma 2.1, and is trivial. | | Finally, we note that the inclusion of Y into X is a morphism. | | If Y is also a closed subspace of X, then | we say that it is a "closed submanifold". | | Suppose that X is finite dimensional of dimension n, and that Y | is a submanifold of dimension m. Then from the definition we see that | the local product structure in the neighborhood of a point of Y can be | expressed in terms of local coordinates as follows. Each point P of Y | has an open neighborhood U in X with local coordinates (x_1, ..., x_n) | such that the points of Y in U are precisely those whose last n - m | coordinates are 0, that is, those points having coordinates of type | | (x_1, ..., x_m, 0, ..., 0). | | Let f : Z -> X be a morphism, and let z be in Z. We shall say that f is | an "immersion" at z if there exists an open neighborhood Z_1 of z in Z | such that the restriction of f to Z_1 induces an isomorphism of Z_1 | onto a submanifold of X. We say that f is an "immersion" if it is | an immersion at every point. | | Lang, DARM, page 24. | | Note that there exist injective immersions | which are not isomorphisms onto submanifolds, | as given by the following example: | ________ | / \ | / \ | | | | | | | \ V | \___________________________________________ | | (The arrow means that the line approaches itself without touching.) | An immersion which does give an isomorphism onto a submanifold is | called an "embedding", and it is called a "closed embedding" if | this submanifold is closed. | | A morphism f : X -> Y will be called a "submersion" at a point x in X | if there exists a chart (U, q) at x and a chart (V, r) at f(x) such that | q gives an isomorphism of U on a product U_1 x U_2 (U_1 and U_2 open in | some Banach spaces), and such that the map | | rfq^-1 = f_V,U : U_1 x U_2 -> V | | is a projection. One sees then that the image of a submersion is | an open subset (a submersion is in fact an open mapping). We say | that f is a "submersion" if it is a submersion at every point. | | For manifolds modeled on Banach spaces, we have the usual criterion | for immersions and submersions in terms of the derivative. | | Proposition 2.2. Let X, Y be manifolds of class C^p (p >= 1) modeled on Banach spaces. | | Let f : X -> Y be a C^p-morphism. Let x be in X. Then: | | 1. f is an immersion at x if and only if | | there exists a chart (U, q) at x and (V, r) at f(x) | | such that f'_V,U(qx) is injective and splits. | | 2. f is a submersion at x if and only if | | there exists a chart (U, q) at x and (V, r) at f(x) | | such that f'_V,U(qx) is surjective and its kernel splits. | | Proof. This is an immediate consequence | of Corollaries 5.4 and 5.6 of | the inverse mapping theorem. | | The conditions expressed in [Propositions 2.2.1 and 2.2.2] depend only on the | derivative [f'], and if they hold for one choice of charts (U, q) and (V, r), | respectively, then they hold for every choice of such charts. It is therefore | convenient to introduce a terminology in order to deal with such properties. | | Let X be a manifold of class C^p (p >= 1). | Let x be a point of X. We consider triples (U, q, v) | where (U, q) is a chart at x and v is an element of the | vector space in which qU lies. We say that two such triples | (U, q, v) and (V, r, w) are "equivalent" if the derivative of | rq^-1 at qx maps v on w. The formula reads: | | (rq^-1)'(qx)v = w | | (obviously an equivalence relation by the chain rule). | | Lang, DARM, page 25. | | An equivalence class of such triples is called a "tangent vector" of X at x. | The set of such tangent vectors is called the "tangent space" of X at x and | is denoted by 'T_x(X)'. Each chart (U, q) determines a bijection of T_x(X) | on a Banach space, namely the equivalence class of (U, q, v) corresponds to | the vector v. By means of such a bijection it is possible to transport to | T_x(X) the structure of topological vector space given by the chart, and | it is immediate that this structure is independent of the chart selected. | | If U, V are open in Banach spaces, then to every morphism of class C^p (p >= 1) | we can associate its derivative Df(x). If now f : X -> Y is a morphism of | one manifold into another, and x a point of X, then by means of charts | we can interpret the derivative of f on each chart at x as a mapping | | df(x) = T_x f : T_x(X) -> T_f(x)(Y). | | Indeed, this map T_x f is the unique linear map having the following property. | If (U, q) is a chart at x and (V, r) is a chart at f(x) such that f(U) c V | and ^v^ is a tangent vector at x represented by v in the chart (U, q), then | | T_x f(^v^) | | is the tangent vector at f(x) represented by Df_V,U(x)v. | The representation of T_x f on the spaces of charts can | be given in the form of a diagram | | T_x(X) o-------->o E | | | | T_x f | | f'_V,U(x) | v v | T_f(x)(Y) o-------->o F | | The map T_x f is obviously continuous and linear for the structure of | topological vector space which we have placed on T_x(X) and T_f(x)(Y). | | As a matter of notation, we shall sometimes write f_*,x instead of T_x f. | | The operation T satisfies an obvious functorial property, | namely, if f : X -> Y and g : Y -> Z are morphisms, then | | T_x(g o f) = T_f(x)(g) o T_x(f). | | T_x(id) = id. | | We may reformulate Proposition 2.2: | | Proposition 2.3. Let X, Y be manifolds of class C^p (p >= 1) modeled on Banach spaces. | | Let f : X -> Y be a C^p-morphism. Let x be in X. Then: | | 1. f is an immersion at x if and only if | | the map T_x f is injective and splits. | | 2. f is a submersion at x if and only if | | the map T_x f is surjective and its kernel splits. | | Note. If X, Y are finite dimensional, then the condition that T_x f splits | is superfluous. Every subspace of a finite dimensional vector space splits. | | Lang, DARM, page 26. | | Serge Lang, |'Differential & Riemannian Manifolds', | Springer-Verlag, New York, NY, 1995. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | o---------------------------------------o o-------------------o | | X | | E_i | | | | | | | | | | o | | | | | / \ | | | o | | / \ | | | / \ | | / \ | | | / \ | | / \ | | | / \ q_i | | / q_i U_i \ | | | / o---------------------->| o o o | | | / \ | | \ / \ / | | | / \ | | \ / \ / | | | / U_i \ | | o o | | | / \ | | \ / | | | / \ | | \ / | | | o o o | | o | | | \ / \ / | | | | | \ / \ / | | | | | \ / U_i \ / | o---------|---------o | | \ / \ / | | | | o |^| o | q_j o q_i^-1 | | / \ / \ | | | | / \ U_j / \ | o---------v---------o | | / \ / \ | | E_j | | | / \ / \ | | | | | o o o | | o | | | \ / | | / \ | | | \ / | | / \ | | | \ U_j / | | o o | | | \ / | | / \ / \ | | | \ / | | / \ / \ | | | \ o---------------------->| o o o | | | \ / q_j | | \ q_j U_j / | | | \ / | | \ / | | | \ / | | \ / | | | o | | \ / | | | | | \ / | | | | | o | | | | | | | | | | | | o---------------------------------------o o-------------------o | | Figure 1. Manifold Of Chimerical Impressions o~~~~~~~~~o~~~~~~~~~o~SUBALLYS~o~~~~~~~~~o~~~~~~~~~o References And Incidental Nuances (RAIN) http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Riemann.html http://www.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm http://www.philosophy.ru/library/kant/01/cr_pure_reason.html http://ez2www.com/go.php3?site=book&go=0387943382 http://hallmathematics.com/mathematics/1433.shtml http://hallmathematics.com/mathematics/630.shtml o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 14 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Manifold SIG, I am planning to parade before you only a few more squibs of Lang's snappy presentation of DARM's before I lay into it with my own hamlet-fausted soliloquy on what it all means to me, but I think that I can now safely vouchsafe to your long-suffering souls one key of importance to its imports. I hope that this will serve to suggest at least a hint of a connection to the business of inquiry, modeling, semiotics, and sign relations, especially with regard to many pressing questions about change and diversity in our conceptual and symbolic systems, including the problems of designing interoperable perspectives and mutually intelligible codes for the worlds we vorpally construe. Let's view our archetype of a manifold, the Figure of a space X and a couple of charts (U_i, q_i) and (U_j, q_j) from its atlas: | o---------------------------------------o o-------------------o | | X | | E_i | | | | | | | | | | o | | | | | / \ | | | o | | / \ | | | / \ | | / \ | | | / \ | | / \ | | | / \ q_i | | / q_i U_i \ | | | / o---------------------->| o o o | | | / \ | | \ / \ / | | | / \ | | \ / \ / | | | / U_i \ | | o Eij o | | | / \ | | \ / | | | / \ | | \ / | | | o o o | | o | | | \ / \ / | | | | | \ / \ / | | | | | \ / \ / | o---------|---------o | | \ / \ / | | | | o Uij o | q_j o q_i^-1 | | / \ / \ | | | | / \ / \ | o---------v---------o | | / \ / \ | | E_j | | | / \ / \ | | | | | o o o | | o | | | \ / | | / \ | | | \ / | | / \ | | | \ U_j / | | o Eji o | | | \ / | | / \ / \ | | | \ / | | / \ / \ | | | \ o---------------------->| o o o | | | \ / q_j | | \ q_j U_j / | | | \ / | | \ / | | | \ / | | \ / | | | o | | \ / | | | | | \ / | | | | | o | | | | | | | | | | | | o---------------------------------------o o-------------------o | | Figure 1. Manifold Of Sensuous Impressions For ease of reverence, I resuscitate the revelant liturgy: o~~~~~~~~~o~~~~~~~~~o~DEFINITION~o~~~~~~~~~o~~~~~~~~~o | 2. Manifolds | | 2.1. Atlases, Charts, Morphisms | | Let X be a set. An "atlas" of class C^p (p >= 0) on X is a collection | of pairs (U_i, q_i) (i ranging in some indexing set), satisfying the | following conditions: | | AT 1. Each U_i is a subset of X and the U_i cover X. | | AT 2. Each q_i is a bijection of U_i onto an open subset q_i(U_i) | of some Banach space E_i and for any i, j, [it holds that] | q_i (U_i |^| U_j) is open in E_i. | | AT 3. The map | | q_j o q_i^-1 : q_i (U_i |^| U_j) --> q_j (U_i |^| U_j) | | is a C^p-isomorphism for each pair of indices i, j. | | Each pair (U_i, q_i) will be called a "chart" of the atlas. | If a point x of X lies in U_i, then we say that (U_i, q_i) | is a "chart at" x. | | Lang, DARM, pages 20-21. o~~~~~~~~~o~~~~~~~~~o~NOITINIFED~o~~~~~~~~~o~~~~~~~~~o Backing away from all of the pointy little pointillistic details a bit, let us now take in a grandly more impressionistic view of this picture. Regard all of that busy-ness about Banach this and C^p that as nothing more than somebody or another's personal aesthetic with regard to what they think it might be that makes a space "pretty" or a mapping "nice". Now think of X as being the "object space", the "real" space in which all of us are really the most interested, at least, if we know what's good for us, and consider E_i and E_j to be the spaces of, let us say, my impressions, measurements, nomenclature, senses, signs, symbology, terminology, utterances, vocabulary, whatever, and yours, respectively. Let us now focus on the subsets of X, E_i, E_j that are indicated as follows: | 0. U_ij = U_i |^| U_j c X | | 1. E_ij = q_i (U_ij) c E_i | | 2. E_ji = q_j (U_ij) c E_j The mapping of the form q_j o q_i^-1 is what does the work of partially translating my code into yours, to the extent that it is possible to do so by flipping charts. This is easier to see if one lays out the maps in a straight line presentation: | q_i^-1 q_j | E_ij ------------> U_ij ------------> E_ji | Hence, maps of the form q_j o q_i^-1 are called "transition" or "translation" maps. A helpful hint in this regard is to read "(q_j o q_i^-1)(s)" in either one of the following ways, according to which reading best suits the occasion: | (q_j o q_i^-1)(s) = thy own name for what I usually call s. | | (q_j o q_i^-1)(s) = the new name for what I used to call s. In other words, as one says, we are talking about an objective interpretive situation, with the sign s and the interpretant sign t = (q_j o q_i^-1)(s) for the shared object x = (q_i^-1)(s). Next question: Does this manifold picture capture the most generic brand of objective interpretive situation? Exercise for the interpreter ... Jon Awbrey o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 15 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Apology to 'q' | It is always the "speechlessness" of sensation, its inability | to make any "statement", that is held to make the very notion | of it meaningless, and to justify the student of knowledge in | scouting it out of existence. "Significance", in the sense | of standing as the sign of other mental states, is taken | to be the sole function of what mental states we have; | and from the perception that our little primitive | sensation has as yet no significance in this | literal sense, it is an easy step to call it | first meaningless, next senseless, then | vacuous, and finally to brand it as | absurd and inadmissible. But in | this universal liquidation, this | everlasting slip, slip, slip, | of direct acquaintance into | knowledge-'about', until at | last nothing is left about | which the knowledge can be | supposed to obtain, does | not all "significance" | depart from the | situation? | And when our knowledge about things has reached its never so complicated perfection, | must there not needs abide alongside of it and inextricably mixed in with it | some acquaintance with 'what' things all this knowledge is about? | | James, "Func of Cog", pages 13-14. | | William James, "The Function Of Cognition", | Read before the Aristotelian Society, 1 Dec 1884. | First published in 'Mind', 10 (1885). Reprinted in |'The Meaning Of Truth: A Sequel To "Pragmatism"', | Longmans, Green, & Company, London, UK, 1909. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 16 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Now, our supposed little feeling gives a 'what'; | and if other feelings should succeed which remember the first, | its 'what' may stand as subject or predicate of some piece of knowledge-about, | of some judgment, perceiving relations between it and other 'whats' which the other | feelings may know. The hitherto dumb 'q' will then receive a name and be no longer speechless. | | James, "Func of Cog", page 14. | | William James, "The Function Of Cognition", | Read before the Aristotelian Society, 1 Dec 1884. | First published in 'Mind', 10 (1885). Reprinted in |'The Meaning Of Truth, A Sequel To "Pragmatism"', | Longmans, Green, & Company, London, UK, 1909. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | As with immersions and submersions, we have a characterization | of transversal maps in terms of tangent spaces. | | Proposition 2.4. Let X, Y be manifolds of class C^p (p >= 1) modeled on Banach spaces. | | Let f : X -> Y be a C^p-morphism, and W a submanifold of Y. | | The map f is transversal over W | | if and only if | | for each x in X such that f(x) lies in W, | | the composite map | | T_x(f) | T_x(X) --------> T_w(Y) ------> T_w(Y)/T_w(W) | | with w = f(x) is surjective and its kernel splits. | | Proof. If f is transversal over W, then for each point x in X such that f(x) lies in W, | we choose charts as in the definition, and reduce the question to one of maps | of open subsets of Banach spaces. In that case, the conclusion concerning the | tangent spaces follows at once from the assumed direct product decompositions. | Conversely, assume our condition on the tangent map. The question being local, | we can assume that Y = V_1 x V_2 is a product of open sets in Banach spaces | such that W = V_1 x 0, and we can also assume that X = U is open in some | Banach space, x = 0. Then we let g : U -> V_2 be the map pi o f, where | pi is the projection, and note that our assumption means that g'(0) is | surjective and its kernel splits. Furthermore, g^-1(0) = f^-1(W). | We can then use Corollary 5.7 of the inverse mapping theorem | to conclude the proof. | | Lang, DARM, pages 27-28. o~~~~~~~~~o~~~~~~~~~o~SYLLABUS~o~~~~~~~~~o~~~~~~~~~o | 2. Manifolds | | 2.1. Atlases, Charts, Morphisms | | Let X be a set. An "atlas of class C^p (p >= 0)" on X is a collection | of pairs (U_i, q_i) (i ranging in some indexing set), satisfying the | following conditions: | | AT 1. Each U_i is a subset of X and the U_i cover X. | | AT 2. Each q_i is a bijection of U_i onto an open subset q_i U_i | of some Banach space E_i and for any i, j, [it holds that] | q_i (U_i |^| U_j) is open in E_i. | | AT 3. The map | | q_j o q_i^-1 : q_i (U_i |^| U_j) --> q_j (U_i |^| U_j) | | is a C^p-isomorphism for each pair of indices i, j. | | Lang, DARM, page 20. | | Each pair (U_i, q_i) will be called a "chart" of the atlas. | If a point x of X lies in U_i, then we say that (U_i, q_i) | is a "chart at" x. | | Suppose that we are given an open subset U of X and a topological isomorphism | q : U -> U' onto an open subset of some Banach space E. We shall say that | (U, q) is "compatible" with the atlas {(U_i, q_i)} if each map q_i q^-1 | (defined on a suitable intersection as in AT 3) is a C^p-isomorphism. | | Two atlases are said to be "compatible" if each chart of one is compatible with | the other atlas. One verifies immediately that the relation of compatibility | between atlases is an equivalence relation. An equivalence class of atlases | of class C^p on X is said to define a structure of "C^p-manifold" on X. | | If all the vector spaces E_i in some atlas are toplinearly isomorphic, | then we can always find an equivalent atlas for which they are all equal, | say to the vector space E. We then say that X is an "E-manifold" or that | X is "modeled" on E. | | If E = R^n for some fixed n, then we say that the manifold is "n-dimensional". | In this case, a chart | | q : U -> R^n | | is given by n coordinate functions q_1, ..., q_n. If 'P' denotes a point of U, | these functions are often written | | x_1(P), ..., x_n(P), | | or simply x_1, ..., x_n. They are called "local coordinates" on the manifold. | | If the integer p (which may also be infinity) is fixed throughout a discussion, | we also say that X is a manifold. | | Lang, DARM, page 21. | | Let X be manifold, and U an open subset of X. Then it is possible, | in the obvious way, to induce a manifold structure on U, by taking | as charts the intersections | | (U_i |^| U, q_i | (U_i |^| U)). | | [Notation. "f | S" indicates the function f as restricted to the set S.] | | If X is a topological space, covered by open subsets V_j, and if we are | given on each V_j a manifold structure such that for each pair j, j' the | induced structure on V_j |^| V_j' coincides, then it is clear that we can | give to X a unique manifold structure inducing the given ones on each V_j. | | If X, Y are two manifolds, then one can give the | product X x Y a manifold structure in the obvious way. | If {(U_i, q_i)} and {(V_j, r_j)} are atlases for X, Y | respectively, then | | {(U_i x V_j, q_i x r_j)} | | is an atlas for the product, and the product of compatible | atlases gives rise to compatible atlases, so that we do get | a well-defined product structure. | | Let X, Y be two manifolds. Let f : X -> Y be a map. | We shall say that f is a "C^p-morphism" if, given x in X, | there exists a chart (U, q) at x and a chart (V, r) at f(x) | such that f(U) c V, and the map | | r o f o q^-1 : qU -> rV | | is a C^p-morphism in the sense of Chapter 1, Section 3. | One sees then immediately that this same condition holds | for any choice of charts (U, q) at x and (V, r) at f(x) | such that F(U) c V. | | It is clear that the composite of two C^p-morphisms is itself | a C^p-morphism (because it is true for open subsets of vector | spaces). The C^p-manifolds and C^p-morphisms form a category. | The notion of isomorphism is therefore defined ... | | If f : X -> Y is a morphism, and (U, q) is a chart | at a point x in X, while (V, r) is a chart at f(x), | then we shall also denote by | | f_V,U : qU -> rV | | the map rfq^-1 [that is, r o f o q^-1]. | | Lang, DARM, page 22. | | It is also convenient to have a local terminology. | Let U be an open set (of a manifold or a Banach space) | containing a point x_0. By a "local isomorphism" at x_0 | we mean an isomorphism | | f : U_1 -> V | | from some open set U_1 containing x_0 (and contained in U) | to an open set V (in some manifold or some Banach space). | Thus a local isomorphism is essentially a change of chart, | locally near a given point. | | 2.2. Submanifolds, Immersions, Submersions | | Let X be a topological space, and Y a subset of X. | We say that Y is "locally closed" in X if every point | y in Y has an open neighborhood U in X such that Y |^| U | is closed in U. One verifies easily that a locally closed | subset is the intersection of an open set and a closed set. | For instance, any open subset of X is locally closed, and | any open interval is locally closed in the plane. | | Let X be a manifold (of class C^p with p >= 0). Let Y be a subset of X | and assume that for each point y in Y there exists a chart (V, r) at y | such that r gives an isomorphism of V with a product V_1 x V_2 where | V_1 is open in some space E_1 and V_2 is open in some space E_2, | and such that | | r(Y |^| V) = V_1 x a_2 | | for some point a_2 in V_2 (which we could take to be 0). Then it is clear | that Y is locally closed in X. Furthermore, the map r induces a bijection | | r_1 : Y |^| V -> V_1. | | The collection of pairs (Y |^| V, r_1) obtained in the above manner constitues | an atlas for Y, of class C^p. The verification of this assertion, whose formal | details we leave to the reader, depends on the following obvious fact. | | Lang, DARM, page 23. | | Lemma 2.1. Let U_1, U_2, V_1, V_2 be open subsets of Banach spaces, | | and g : U_1 x U_2 -> V_1 x V_2 a C^p-morphism. | | Let a_2 be in U_2 and b_2 be in V_2 | | and assume that g maps U_1 x a_2 into V_1 x b_2. | | Then the induced map | | g_1 : U_1 -> V_1 | | is also a morphism. | | Indeed, it is obtained as a composite map | | U_1 -> U_1 x U_2 -> V_1 x V_2 -> V_1, | | the first map being an inclusion and the third a projection. | | We have therefore defined a C^p-structure on Y which will be called | a "submanifold" of X. This structure satisfies a universal mapping | property, which characterizes it, namely: | | | Given any map f : Z -> X from a manifold Z into X such that | | f(Z) is contained in Y. Let f_Y : Z -> Y be the induced map. | | Then f is a morphism if and only if f_Y is a morphism. | | The proof of this assertion depends on Lemma 2.1, and is trivial. | | Finally, we note that the inclusion of Y into X is a morphism. | | If Y is also a closed subspace of X, then | we say that it is a "closed submanifold". | | Suppose that X is finite dimensional of dimension n, and that Y | is a submanifold of dimension m. Then from the definition we see that | the local product structure in the neighborhood of a point of Y can be | expressed in terms of local coordinates as follows. Each point P of Y | has an open neighborhood U in X with local coordinates (x_1, ..., x_n) | such that the points of Y in U are precisely those whose last n - m | coordinates are 0, that is, those points having coordinates of type | | (x_1, ..., x_m, 0, ..., 0). | | Let f : Z -> X be a morphism, and let z be in Z. We shall say that f is | an "immersion" at z if there exists an open neighborhood Z_1 of z in Z | such that the restriction of f to Z_1 induces an isomorphism of Z_1 | onto a submanifold of X. We say that f is an "immersion" if it is | an immersion at every point. | | Lang, DARM, page 24. | | Note that there exist injective immersions | which are not isomorphisms onto submanifolds, | as given by the following example: | ________ | / \ | / \ | | | | | | | \ V | \___________________________________________ | | (The arrow means that the line approaches itself without touching.) | An immersion which does give an isomorphism onto a submanifold is | called an "embedding", and it is called a "closed embedding" if | this submanifold is closed. | | A morphism f : X -> Y will be called a "submersion" at a point x in X | if there exists a chart (U, q) at x and a chart (V, r) at f(x) such that | q gives an isomorphism of U on a product U_1 x U_2 (U_1 and U_2 open in | some Banach spaces), and such that the map | | rfq^-1 = f_V,U : U_1 x U_2 -> V | | is a projection. One sees then that the image of a submersion is | an open subset (a submersion is in fact an open mapping). We say | that f is a "submersion" if it is a submersion at every point. | | For manifolds modeled on Banach spaces, we have the usual criterion | for immersions and submersions in terms of the derivative. | | Proposition 2.2. Let X, Y be manifolds of class C^p (p >= 1) modeled on Banach spaces. | | Let f : X -> Y be a C^p-morphism. Let x be in X. Then: | | 1. f is an immersion at x if and only if | | there exists a chart (U, q) at x and (V, r) at f(x) | | such that f'_V,U(qx) is injective and splits. | | 2. f is a submersion at x if and only if | | there exists a chart (U, q) at x and (V, r) at f(x) | | such that f'_V,U(qx) is surjective and its kernel splits. | | Proof. This is an immediate consequence | of Corollaries 5.4 and 5.6 of | the inverse mapping theorem. | | The conditions expressed in [Propositions 2.2.1 and 2.2.2] depend only on the | derivative [f'], and if they hold for one choice of charts (U, q) and (V, r), | respectively, then they hold for every choice of such charts. It is therefore | convenient to introduce a terminology in order to deal with such properties. | | Let X be a manifold of class C^p (p >= 1). | Let x be a point of X. We consider triples (U, q, v) | where (U, q) is a chart at x and v is an element of the | vector space in which qU lies. We say that two such triples | (U, q, v) and (V, r, w) are "equivalent" if the derivative of | rq^-1 at qx maps v on w. The formula reads: | | (rq^-1)'(qx)v = w | | (obviously an equivalence relation by the chain rule). | | Lang, DARM, page 25. | | An equivalence class of such triples is called a "tangent vector" of X at x. | The set of such tangent vectors is called the "tangent space" of X at x and | is denoted by 'T_x(X)'. Each chart (U, q) determines a bijection of T_x(X) | on a Banach space, namely the equivalence class of (U, q, v) corresponds to | the vector v. By means of such a bijection it is possible to transport to | T_x(X) the structure of topological vector space given by the chart, and | it is immediate that this structure is independent of the chart selected. | | If U, V are open in Banach spaces, then to every morphism of class C^p (p >= 1) | we can associate its derivative Df(x). If now f : X -> Y is a morphism of | one manifold into another, and x a point of X, then by means of charts | we can interpret the derivative of f on each chart at x as a mapping | | df(x) = T_x f : T_x(X) -> T_f(x)(Y). | | Indeed, this map T_x f is the unique linear map having the following property. | If (U, q) is a chart at x and (V, r) is a chart at f(x) such that f(U) c V | and ^v^ is a tangent vector at x represented by v in the chart (U, q), then | | T_x f(^v^) | | is the tangent vector at f(x) represented by Df_V,U(x)v. | The representation of T_x f on the spaces of charts can | be given in the form of a diagram | | T_x(X) o-------->o E | | | | T_x f | | f'_V,U(x) | v v | T_f(x)(Y) o-------->o F | | The map T_x f is obviously continuous and linear for the structure of | topological vector space which we have placed on T_x(X) and T_f(x)(Y). | | As a matter of notation, we shall sometimes write f_*,x instead of T_x f. | | The operation T satisfies an obvious functorial property, | namely, if f : X -> Y and g : Y -> Z are morphisms, then | | T_x(g o f) = T_f(x)(g) o T_x(f). | | T_x(id) = id. | | We may reformulate Proposition 2.2: | | Proposition 2.3. Let X, Y be manifolds of class C^p (p >= 1) modeled on Banach spaces. | | Let f : X -> Y be a C^p-morphism. Let x be in X. Then: | | 1. f is an immersion at x if and only if | | the map T_x f is injective and splits. | | 2. f is a submersion at x if and only if | | the map T_x f is surjective and its kernel splits. | | Note. If X, Y are finite dimensional, then the condition that T_x f splits | is superfluous. Every subspace of a finite dimensional vector space splits. | | Lang, DARM, page 26. | | If W is a submanifold of a manifold Y of class C^p (p >= 1), then the inclusion | | i : W -> Y | | induces a map | | T_w i : T_w(W) -> T_w(Y) | | which is in fact an injection. From the definition of a submanifold, one sees | immediately that the image of T_w i splits. It will be convenient to identify | T_w(W) in T_w(Y) if no confusion can result. | | A morphism f : X -> Y will be said to be "transversal" over the submanifold W of Y | if the following condition is satisfied. | | Let x in X be such that f(x) is in W. Let (V, r) be a chart at f(x) such that | r : V -> V_1 x V_2 is an isomorphism on a product, with | | r(f(x)) = (0, 0) and r(W |^| V) = V_1 x 0. | | Then there exists an open neighborhood U of x such that the composite map | | f r pr | U -----> V -----> V_1 x V_2 ------> V_2 | | is a submersion. | | In particular, if f is transversal over W, then f^-1(W) is a submanifold of X, | because the inverse image of 0 by our local composite map | | pr o r o f | | is equal to the inverse image of W |^| V by r. | | Lang, DARM, page 27. | | Serge Lang, |'Differential & Riemannian Manifolds', | Springer-Verlag, New York, NY, 1995. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | o---------------------------------------o o-------------------o | | X | | E_i | | | | | | | | | | o | | | | | / \ | | | o | | / \ | | | / \ | | / \ | | | / \ | | / \ | | | / \ q_i | | / q_i U_i \ | | | / o---------------------->| o o o | | | / \ | | \ / \ / | | | / \ | | \ / \ / | | | / U_i \ | | o Eij o | | | / \ | | \ / | | | / \ | | \ / | | | o o o | | o | | | \ / \ / | | | | | \ / \ / | | | | | \ / \ / | o---------|---------o | | \ / \ / | | | | o Uij o | q_j o q_i^-1 | | / \ / \ | | | | / \ / \ | o---------v---------o | | / \ / \ | | E_j | | | / \ / \ | | | | | o o o | | o | | | \ / | | / \ | | | \ / | | / \ | | | \ U_j / | | o Eji o | | | \ / | | / \ / \ | | | \ / | | / \ / \ | | | \ o---------------------->| o o o | | | \ / q_j | | \ q_j U_j / | | | \ / | | \ / | | | \ / | | \ / | | | o | | \ / | | | | | \ / | | | | | o | | | | | | | | | | | | o---------------------------------------o o-------------------o | | Figure 1. Manifold Of Connotative Impressions o~~~~~~~~~o~~~~~~~~~o~SUBALLYS~o~~~~~~~~~o~~~~~~~~~o References And Incidental Nuances (RAIN) http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Riemann.html http://www.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm http://www.philosophy.ru/library/kant/01/cr_pure_reason.html http://hallmathematics.com/mathematics/1433.shtml http://hallmathematics.com/mathematics/630.shtml o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 17 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | But every name, as students of logic know, has its "denotation"; and the | denotation always means some reality or content, relationless 'ab extra' | or with its internal relations unanalyzed, like the 'q' which our | primitive sensation is supposed to know. No relation-expressing | proposition is possible except on the basis of a preliminary | acquaintance with such "facts", with such contents, as this. | Let the 'q' be fragrance, let it be toothache, or let it be | a more complex kind of feeling, like that of the full-moon | swimming in her blue abyss, it must first come in that | simple shape, and be held fast in that first intention, | before any knowledge 'about' it can be attained. | The knowledge 'about' it is 'it' with a context | added. Undo 'it', and what is added cannot | be 'con'-text. | | James, "Func of Cog", pages 14-15. | | William James, "The Function Of Cognition", | Read before the Aristotelian Society, 1 Dec 1884. | First published in 'Mind', 10 (1885). Reprinted in |'The Meaning Of Truth, A Sequel To "Pragmatism"', | Longmans, Green, & Company, London, UK, 1909. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | If E is a Banach space, then the diagonal @D@ in ExE is | a closed subspace and splits: Either factor Ex0 or 0xE | is a closed complement. Consequently, the diagonal is | a closed submanifold of ExE. If X is any manifold of | class C^p, p >= 1, then the diagonal is therefore also | a submanifold. (It is closed of course if and only if | X is Hausdorff.) | | Let f : X -> Z and g : Y -> Z be two C^p-morphisms, p >= 1. | We say that they are 'transversal' if the morphism | | f x g : X x Y -> Z x Z | | is transversal over the diagonal. We remark right away | that the surjectivity of the map in Proposition 2.4 can be | expressed in two ways. Given two points x in X and y in Y | such that f(x) = g(y) = z, the condition | | Im(T_x f) + Im(T_y g) = T_z(Z) | | is equivalent to the condition | | Im(T_(x,y)(f x g)) + T_(z,z)(@D@) = T_(z,z)(Z x Z). | | Thus in the finite dimensional case, we could | take it as the definition of transversality. | | Lang, DARM, pages 28-29. | | Serge Lang, |'Differential & Riemannian Manifolds', | Springer-Verlag, New York, NY, 1995. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | o---------------------------------------o o-------------------o | | X | | E_i | | | | | | | | | | o | | | | | / \ | | | o | | / \ | | | / \ | | / \ | | | / \ | | / \ | | | / \ q_i | | / q_i U_i \ | | | / o---------------------->| o o o | | | / \ | | \ / \ / | | | / \ | | \ / \ / | | | / U_i \ | | o Eij o | | | / \ | | \ / | | | / \ | | \ / | | | o o o | | o | | | \ / \ / | | | | | \ / \ / | | | | | \ / \ / | o---------|---------o | | \ / \ / | | | | o Uij o | q_j o q_i^-1 | | / \ / \ | | | | / \ / \ | o---------v---------o | | / \ / \ | | E_j | | | / \ / \ | | | | | o o o | | o | | | \ / | | / \ | | | \ / | | / \ | | | \ U_j / | | o Eji o | | | \ / | | / \ / \ | | | \ / | | / \ / \ | | | \ o---------------------->| o o o | | | \ / q_j | | \ q_j U_j / | | | \ / | | \ / | | | \ / | | \ / | | | o | | \ / | | | | | \ / | | | | | o | | | | | | | | | | | | o---------------------------------------o o-------------------o | | Figure 1. Manifold Of Contextural Impressions o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o References And Incidental Nuances (RAIN) http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Riemann.html http://www.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm http://www.philosophy.ru/library/kant/01/cr_pure_reason.html http://ez2www.com/go.php3?site=book&go=0387943382 http://hallmathematics.com/mathematics/1433.shtml http://hallmathematics.com/mathematics/630.shtml o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 18 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | Let us say no more then about this objection, but enlarge our thesis, thus: | If there be in the universe a 'q' other than the 'q' in the feeling, | the latter may have acquaintance with an entity ejective to itself; | an acquaintance moreover, which, as mere acquaintance, it would be | hard to imagine susceptible either of improvement or increase, | being in its way complete; and which would oblige us (so long | as we refuse not to call acquaintance knowledge) to say not | only that the feeling is cognitive, but that all qualities | of feeling, 'so long as there is anything outside of them | which they resemble', are feelings 'of' qualities of | existence, and perceptions of outward fact. | | James, "Func of Cog", pages 15-16. | | William James, "The Function Of Cognition", | Read before the Aristotelian Society, 1 Dec 1884. | First published in 'Mind', 10 (1885). Reprinted in |'The Meaning Of Truth, A Sequel To "Pragmatism"', | Longmans, Green, & Company, London, UK, 1909. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | We use transversality as a sufficient condition under which the fiber product | of two morphisms exists. We recall that in any category, the 'fiber product' | of two morphisms f : X -> Z and g : Y -> Z over Z consists of an object P | and two morphisms | | g_1 : P -> X and g_2 : P -> Y | | such that f o g_1 = g o g_2, and satisfying the universal mapping property: | | Given an object S and two morphisms | | u_1 : S -> X and u_2 : S -> Y | | such that f o u_1 = g o u_2, there exists a unique morphism u : S -> P | making the following diagram commutative: | | S | o | /|\ | / | \ | / | \ | u_1 / u \ u_2 | / | \ | / | \ | v v v | X o<------P------>o Y | \ g_1 g_2 / | \ / | \ / | f \ / g | \ / | \ / | v v | o | Z | | The triple (P, g_1, g_2) is uniquely determined, | up to a unique isomorphism (in the obvious sense), | and P is also denoted by X x_Z Y. | | Lang, DARM, page 29. | | Serge Lang, |'Differential & Riemannian Manifolds', | Springer-Verlag, New York, NY, 1995. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | o---------------------------------------o o-------------------o | | X | | E_i | | | | | | | | | | o | | | | | / \ | | | o | | / \ | | | / \ | | / \ | | | / \ | | / \ | | | / \ q_i | | / q_i U_i \ | | | / o---------------------->| o o o | | | / \ | | \ / \ / | | | / \ | | \ / \ / | | | / U_i \ | | o Eij o | | | / \ | | \ / | | | / \ | | \ / | | | o o o | | o | | | \ / \ / | | | | | \ / \ / | | | | | \ / \ / | o---------|---------o | | \ / \ / | | | | o Uij o | q_j o q_i^-1 | | / \ / \ | | | | / \ / \ | o---------v---------o | | / \ / \ | | E_j | | | / \ / \ | | | | | o o o | | o | | | \ / | | / \ | | | \ / | | / \ | | | \ U_j / | | o Eji o | | | \ / | | / \ / \ | | | \ / | | / \ / \ | | | \ o---------------------->| o o o | | | \ / q_j | | \ q_j U_j / | | | \ / | | \ / | | | \ / | | \ / | | | o | | \ / | | | | | \ / | | | | | o | | | | | | | | | | | | o---------------------------------------o o-------------------o | | Figure 1. Manifold Of Ejective Impressions o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o References And Incidental Nuances (RAIN) http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Riemann.html http://www.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm http://www.philosophy.ru/library/kant/01/cr_pure_reason.html http://ez2www.com/go.php3?site=book&go=0387943382 http://hallmathematics.com/mathematics/1433.shtml http://hallmathematics.com/mathematics/630.shtml o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 19 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Apostrophe to 'q' | The point of this vindication of the cognitive function | of the first feeling lies, it will be noticed, in the | discovery that 'q' does exist elsewhere than in it. | In case this discovery were not made, we could not | be sure the feeling was cognitive; and in case | there were nothing outside to be discovered, | we should have to call the feeling a dream. | But the feeling itself cannot make the | discovery. Its own 'q' is the only | 'q' it grasps; and its own nature | is not a particle altered by | having the self-transcendent | function of cognition either | added to it or taken away. | The function is accidental; | synthetic, not analytic; | and falls outside and | not inside its being. | | James, "Func of Cog", page 16. | | William James, "The Function Of Cognition", | Read before the Aristotelian Society, 1 Dec 1884. | First published in 'Mind', 10 (1885). Reprinted in |'The Meaning Of Truth, A Sequel To "Pragmatism"', | Longmans, Green, & Company, London, UK, 1909. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 20 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o JA = Jon Awbrey SR = Seth Russell JA: In assuming a distinction between objects that are inherently subjects and subject to concepts that are intrinsically predicates, you have not only made a significantly charged ontological assumption, that is to say, taken up a non-trivially non-neutral position on the categories of being, ... SR: Agree. JA: you have also committed the transparent projection error that is commonly known as "confusing the map with the territory", to wit, "reifying syntax". SR: I don't understand this sin. *If* I have an ontology of representations, and that ontology admits objects *and* relations, and I have contrived some mechanism to manipulate these representations to model something; how is it a sin for that mechanism to manipulate the representations of the relations just as easily as it manipulates representations of non-relational objects? Where is the confusion in such a mechanism? It is not a sin, it's a guess. The mistake lies in mistaking the guess for a fact, and not even trying to imagine how it might all turn out otherwise. An ontology is not "what it is" -- it's a theory -- that is to say, it's "how you guess it might be". Human beings are pretty good guessers, or none of us would be here to debate, so we tend to take the establishment of current best guesses for granted, but every millennium or so it gets to be time to make a correction. Physics and parts of math have just been through the ringer. Metaphysics, in particular ontology, would save a bit of grief if they paid due attention to what others have already learned. There are one or two obstacles to that. I am trying to address them in my work. SR: Or have I totally misunderstood your allegation? Not sure yet. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Note 21 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o JA = Jon Awbrey SR = Seth Russell JA: In assuming a distinction between objects that are inherently subjects and subject to concepts that are intrinsically predicates, you have not only made a significantly charged ontological assumption, that is to say, taken up a non-trivially non-neutral position on the categories of being, you have also committed the transparent projection error that is commonly known as "confusing the map with the territory", to wit, "reifying syntax". SR: I don't understand this sin. *If* I have an ontology of representations, and that ontology admits objects *and* relations, and I have contrived some mechanism to manipulate these representations to model something; how is it a sin for that mechanism to manipulate the representations of the relations just as easily as it manipulates representations of non-relational objects? Where is the confusion in such a mechanism? JA: It is not a sin, it's a guess. The mistake lies in mistaking the guess for a fact, and not even trying to imagine how it might all turn out otherwise. SR: Oh !! ... do you know a method to tell whether a representation is a guess or whether it is a fact? Nota Bene. Just by way of explaining the way that I currently understand the words, "Facthood" and "Guessness" name no essences, and they are certainly not mutually exclusive, since every Fact first understudies as a Guess. So they are pragmatic roles or maybe even just theatrical parts in which we choose to cast the various premisses that we contemplate. The show must go on, at least for a while, but in the end the whole production lives or closes depending on whether we cast or mis-cast them. If you can't take a risk, don't try to be on the side of the angels. My Bet: They are all guesses. My Hedge: There is a way that seems to have shown some utility so far. Call it "scientific method", or "semio-automated ontology development", or just plain "inquiry" -- it is the way that I and many other lemmas have chosen to proceed toward our for-hoped, not for-lorn theorematas. SR: An ontology is not "what it is" -- it's a theory -- that is to say, it's "how you guess it might be". SR: Agree ... but that does not answer my particular query which asked what the big difference was between gussing about relations as opposed to guessing about not relational objects. Sorry, I cannot get what you are asking here. You had this big *If* in the starring role, and so I could not tell if you wanted me to contemplate some hypothetical case, or what. From previous experience, my chief difficulty interpreting what you say is that you use all sorts of crucial words in ways that are different from me, and when I ask you explain your meanings you typically decline, dissemble, or disappear. The last time that we chatted at length, I asked you the following questions: | 1. What do you mean by words like "arity" ("adicity", "valence")? | | 2. What do you mean by words like "composed" and "composition"? | | 3. What do you mean by words like "relation"? | | 4. What do you mean by words like "arrow"? | | 5. What do you mean by words like "node"? | | 6. What do you mean by words like "represent"? | | 7. What do you mean by words like "structure"? | | http://suo.ieee.org/ontology/msg03740.html I notice that at least two of these, "relation" and "represent", have recurred in your present query, and I fear that I can see no way to answer your question till I have a clue what it means. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

###### MOD. Model Theory

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOD. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o I see that we are developing a bit of a communication breakdown along both the "Inquiry Into Inquiry" and "Laws Of Logic" lines, so I will attempt to bridge this burgeoning language barrier by compiling a stock of standard options for nomenclature. Please bear with me for a while, as this will take some time to anchor. | 1.2 Model Theory for Sentential Logic | | Classical sentential logic is designed to study a set $S$ of simple statements, | and the compound statements built up from them. At the most intuitive level, | an intended interpretation of these statements is a "possible world", in | which each statement is either true or false. We wish to replace these | intuitive interpretations by a collection of precise mathematical objects | which we may use as our models. The first thing which comes to mind | is a function F which associates with each simple statement S one of | the truth values "true" or "false". Stripping away the inessentials, | we shall instead take a model to be a subset A of $S$; the idea is | that "S in A" indicates that the simple statement S is true, and | that "S not in A" indicates that the simple statement S is false. | | 1.2.1 By a 'model' A for $S$ we simply mean a subset A of $S$. | | Thus the set of all models has the power 2^|$S$|. Several relations and | operations between models come to mind; for example, A c B, $S$ - A, and | the intersection |^|_<i in I> A_i of a set {A_i : i in I} of models. | Two distinguished models are the empty set (/) and the set $S$ itself. | | We now set up the sentential logic as a formal language. | The symbols of our language are as follows: | | 1. connectives "&" (and), "~" (not). | 2. parentheses "(", ")". | 3. a nonempty set $S$ of sentence symbols. | | Intuitively, the sentence symbols stand for simple statements, and | the connectives "&", "~" stand for the words used to combine simple | statements into compound statements. Formally, the 'sentences' of | $S$ are defined as follows: | | 1.2.2 [Definition of 'sentence'] | | 1. Every sentence symbol S is a sentence. | 2. If q is a sentence then (~q) is a sentence. | 3. If q, r are sentences, then (q & r) is a sentence. | 4. A finite sequence of symbols is a sentence only if | it can be shown to be a sentence by a finite number | of applications of 1.2.2.1-3. | | C.C. Chang & H.J. Keisler, 'Model Theory', | North-Holland, Amsterdam, Netherlands, 1973, | pages 4-5. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOD. Note 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2 Model Theory for Sentential Logic (cont.) | | Hereafter we shall use the single symbol $S$ to denote both the | set of sentence symbols and the language built on these symbols. | There is no fear of confusion in this double usage since the | language is determined uniquely, modulo the connectives, | by the sentence symbols. | | We are now ready to build a bridge between the language $S$ and its models, | with the definition of the truth of a sentence in a model. We shall express | the fact that a sentence q is true in a model A succinctly by the special | notation | | A |= q. | | The relation A |= q is defined as follows: | | 1.2.3 [Definition of A |= q, that is, A is a 'model' of q, or q 'holds' in A] | | 1. If q is a sentence symbol S, then A |= q holds if and only if S is in A. | 2. If q is (r & s), then A |= q if and only if both A |= r and A |= s. | 3. If q is (~r), then A |= q if and only if it is not the case that A |= r. | | When A |= q, we say that q is 'true' in A, or that q 'holds' in A, or | that A is a 'model' of q. When it is not the case that A |= q, we say | that q is 'false' in A, or that q 'fails' in A. The above definition of | the relation A |= q is an example of a recursive definition based on 1.2.2. | The proof that the definition is unambiguous for each sentence q is, of course, | a proof by induction based on 1.2.2. | | C.C. Chang & H.J. Keisler, 'Model Theory', | North-Holland, Amsterdam, Netherlands, 1973, | page 7. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOD. Note 3 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2 Model Theory for Sentential Logic (cont.) | | An especially important kind of sentence is a 'valid sentence'. | A sentence q is called 'valid', in symbols, |= q, iff q holds | in all models for $S$, that is, iff A |= q for all A. ... | | At first glance, it seems that we have to examine uncountably many | different infinite models A in order to find out whether a sentence q | is valid. This is because validity is a semantical notion, defined in | terms of models. However, as the reader surely knows, there is a simple | and uniform test by which we can find out in only finitely many steps | whether or not a given sentence q is valid. | | This decision procedure for validity is based on a syntactical notion, | the notion of a tautology. Let q be a sentence such that all the sentence | symbols which occur in q are among the n+1 symbols S_0, S_1, ..., S_n. | Let a_0, a_1, ..., a_n be a sequence made up of the two letters t, f. | We shall call such a sequence an 'assignment'. | | 1.2.4 The 'value' of a sentence q for the assignment a_0, ..., a_n | is defined recursively as follows: | | 1. If q is the sentence symbol S_m, m =< n, then the value of q is a_m. | 2. If q is (~r), then the value of q is the opposite of the value of r. | 3. If q is (r & s), then the value of q is t if the values of r and s | are both t, and otherwise the value of q is f. | | Note how similar Definitions 1.2.3 and 1.2.4 are. The only | essential difference is that 1.2.3 involves an infinite model A, | while 1.2.4 involves only a finite assignment a_0, ..., a_n. | | 1.2.5 Let q be a sentence and let S_0, ..., S_n be all the sentence symbols | occurring in q. The sentence q is said to be a 'tautology', in symbols, | |- q, iff q has the value t for every assignment a_0, ..., a_n. | | We shall use both of the symbols |=, |- in many ways throughout this book. | To keep things straight, remember this: |= is used for semantical ideas, | and |- is used for syntactical ideas. | | The value of a sentence q for an assignment a_0, ..., a_n | may be very easily computed. We first find the values of the | sentence symbols occurring in q and then work our way through the | smaller sentences used in building up the sentence q. A table showing | the value of q for each possible assignment a_0, ..., a_n is called a | 'truth table' of q. We shall assume that truth tables are already | quite familiar to the reader, and that he [or she] knows how to | construct a truth table of a sentence. Truth tables provide a | simple and purely mechanical procedure to determine whether | a sentence q is a tautology -- simply write down the truth | table for q and check to see whether q has the value t for | every assignment. | | Proposition 1.2.6. Suppose that all the sentence symbols occurring in q | are among S_0, S_1, ..., S_n. Then the value of q | for an assignment a_0, a_1, ..., a_n, ..., a_(n+m) | is the same as the value of q for an assignment | a_0, a_1, ..., a_n. | | C.C. Chang & H.J. Keisler, 'Model Theory', | North-Holland, Amsterdam, Netherlands, 1973, | pages 7-8. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOD. Note 4 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2 Model Theory for Sentential Logic (cont.) | | We now prove the first of a series of theorems | which state that a certain syntactical condition | is equivalent to a semantical condition. | | Theorem 1.2.7 Completeness Theorem. |- q if and only if |= q, in words, | a sentence is a tautology if and only if it is valid. | | Proof. Let q be a sentence and let all the sentence symbols in q | be among S_0, ..., S_n. Consider an arbitrary model A. | For m = 0, 1, ..., n, put a_m = t if S_m is in A, | and a_m = f if S_m is not in A. This gives us | an assignment a_0, a_1, ..., a_n. We claim: | | 1. A |= q if and only if the value of q for | the assignment a_0, a_1, ..., a_n is t. | | This can be readily proved by induction. It is immediate | if q is a sentence symbol S_m. Assuming that (1) holds | for q = r and for q = s, we see at once that (1) holds | for q = (~r) and q = (r & s). | | Now let S_0, ..., S_n be all the sentence symbols occurring in q. | If q is a tautology, then by (1), q is valid. Since every assignment | a_0, a_1, ..., a_n can be obtained from some model A, it follows from (1) | that, if q is valid, then q is a tautology. -| | | Our decision procedure for |- q now can be used to decide whether q is valid. | Several times we shall have an occasion to use the fact that a particular | sentence is a tautology, or is valid. We shall never take the trouble | actually to give the proof that a sentence of $S$ is valid, because | the proof is always the same -- we simply look at the truth table. | | C.C. Chang & H.J. Keisler, 'Model Theory', | North-Holland, Amsterdam, Netherlands, 1973, | pages 8-9. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOD. Note 5 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2 Model Theory for Sentential Logic (cont.) | | We shall introduce abbreviations to our language in the usual way, | in order to make sentences more readable. The symbols "v" (or), | "=>" (implies), and "<=>" (if and only if) are abbreviations | defined as follows: | | (q v r) for (~((~q) & (~r))), | | (q => r) for ((~q) v r), | | (q <=> r) for ((q => r) & (r => q)). | | Of course, "v", "=>", and "<=>" could just as well have been included | in our list of symbols as three more connectives. However, there are | certain advantages to keeping our list of symbols short. For instance, | 1.2.2 and proofs by induction based on it are shorter this way. At the | other extreme, we could have managed with only a single connective, whose | English translation is "neither ... nor ...". We did not do this because | "neither ... nor ..." is a rather unnatural connective. | | C.C. Chang & H.J. Keisler, 'Model Theory', | North-Holland, Amsterdam, Netherlands, 1973, | page 6. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOD. Note 6 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2 Model Theory for Sentential Logic (cont.) | | Let us now introduce the notion of a formal deduction in our logic $S$. | | The 'Rule of Detachment' (or 'Modus Ponens') states: | | From r and (r => q) infer q. | | We say that q is 'inferred' from r, s by detachment | iff s is the sentence (r => q). | | Now consider a finite or infinite set @S@ of $S$. | | A sentence q is 'deducible' from @S@, in symbols, @S@ |- q, | iff there is a finite sequence r_0, r_1, ..., r_n of sentences | such that q = r_n, and each sentence r_m is either a tautology, | belongs to @S@, or is inferred from two earlier sentences of the | sequence by detachment. The sequence r_0, r_1, ..., r_n is called | a 'deduction' of q from @S@. Note that q is deducible from the | empty set of sentences if and only if q is a tautology. | | We shall say that @S@ is 'inconsistent' | iff we have @S@ |- q for all sentences q. | Otherwise, we say that @S@ is 'consistent'. | | Finally, we say that @S@ is 'maximal consistent' | iff @S@ is consistent, but the only consistent | set of sentences which includes @S@ is @S@ | itself. The proposition below contains | facts which can be found in most | elementary logic texts. | | Proposition 1.2.8 [Deductive Closure Properties of Consistent Sets] | | 1. If @S@ is consistent and @G@ is the set | of all sentences deducible from @S@, | then @G@ is consistent. | | 2. If @S@ is maximal consistent and @S@ |- q, | then q is an element of the set @S@. | | 3. @S@ is inconsistent if and only if | @S@ |- (S & (~S)) for any S in $S$. | | 4. Deduction Theorem. | If @S@ |_| {r} |- q, then @S@ |- (r => q). | | C.C. Chang & H.J. Keisler, 'Model Theory', | North-Holland, Amsterdam, Netherlands, 1973, | pages 9-10. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOD. Note 7 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2 Model Theory for Sentential Logic (cont.) | | Lemma 1.2.9 Lindenbaum's Theorem. Any consistent set @S@ of sentences | can be enlarged to a maximal consistent set @G@ of sentences. | | Proof. [C&K, page 10]. | | Lemma 1.2.10 Properties of Maximal Consistent Sets of Sentences. | Suppose @G@ is a maximal consistent set of sentences in $S$. Then: | | 1. For each sentence q, exactly one of the sentences q, (~q) belongs to @G@. | | 2. For each pair of sentences q, r, we have that (q & r) belongs to @G@ | if and only if both q and r belong to @G@. | | We leave the proof as an exercise. | | Now consider a set @S@ of sentences of $S$. | | We shall say that A is a 'model' of @S@, in symbols, A |= @S@, | iff every sentence q in @S@ is true in A. | | The set @S@ of sentences is said to be 'satisfiable' | iff it has at least one model. | | We now prove the most important theorem of sentential logic, | which is a criterion for a set @S@ to be satisfiable. | | Theorem 1.2.11 Extended Completeness Theorem. | A set @S@ of sentences of $S$ is consistent | if and only if @S@ is satisfiable. | | Proof. [C&K, page 11]. | | We can obtain a purely semantical corollary. | @S@ is said to be 'finitely satisfiable' iff | every finite subset of @S@ is satisfiable. | | Corollary 1.2.12 Compactness Theorem. | If @S@ is finitely satisfiable, | then @S@ is satisfiable. | | Proof. [C&K, page 11]. | | Note that the converse of the compactness theorem is trivially true, | i.e., every satisfiable set of sentences is finitely satisfiable. | | C.C. Chang & H.J. Keisler, 'Model Theory', | North-Holland, Amsterdam, Netherlands, 1973, | pages 10-11. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOD. Note 8 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2 Model Theory for Sentential Logic (cont.) | | We say that q is a 'consequence' of @S@, in symbols, @S@ |= q, | iff every model of @S@ is a model of q. | | Corollary 1.2.13 [Truth & Consequences] | | 1. @S@ |- q if and only if @S@ |= q. | | 2. If @S@ |= q, then there is a finite subset @S@_0 of @S@ such that @S@_0 |= q. | | We shall conclude our model theory for sentential logic with a few applications of | the compactness theorem. In these applications, the true spirit of model theory | will appear, but at a very rudimentary level. Since we shall often wish to | combine a finite set of sentences into a single sentence, we shall use | expressions like | | q_1 & q_2 & ... & q_n | | and | | q_1 v q_2 v ... v q_n. | | In these expressions the parentheses are assumed, | for the sake of definiteness, to be associated | to the right; for instance, | | q_1 & q_2 & q_3 = q_1 & (q_2 & q_3). | | First we introduce a bit more terminology. | | A set @G@ of sentences is called a 'theory'. | | A theory @G@ is said to be 'closed' | iff every consequence of @G@ belongs to @G@. | | A set @D@ of sentences is said to | be a 'set of axioms' for a theory @G@ | iff @G@ and @D@ have the same consequences. | | A theory is called 'finitely axiomatizable' | iff it has a finite set of axioms. | | Since we may form the conjunction of a finite | set of axioms, a finitely axiomatizable theory | actually always has a single axiom. | | The set @G@^c of all consequences of @G@ is the | unique closed theory which has @G@ as a set of axioms. | | Proposition 1.2.14 Relation Between Sets of Axioms and Sets of Models. | @D@ is a set of axioms for a theory @G@ iff | @D@ has exactly the same models as @G@. | | Corollary 1.2.15 Relation Between Finite Axiom Sets and Sets of Models. | Let @G@_1 and @G@_2 be two theories such that the | set of all models of @G@_2 is the complement of the | set of all models of @G@_1. Then @G@_1 and @G@_2 | are both finitely axiomatizable. | | Proof. [C&K, page 12]. | | C.C. Chang & H.J. Keisler, 'Model Theory', | North-Holland, Amsterdam, Netherlands, 1973, | pages 11-12. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOD. Note 9 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | 1.2 Model Theory for Sentential Logic (cont.) | | C.C. Chang & H.J. Keisler, 'Model Theory', | North-Holland, Amsterdam, Netherlands, 1973, | pages 12-13. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

###### SR. Sign Relations

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SR. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Systems Engineering: Interest Statement Jon Awbrey, September 1, 1992 1.2.2.3 Pragmatic Theory of Signs The theory of signs that I find most useful was developed over several decades in the last century by C.S. Peirce, the founder of modern American pragmatism. Signs are defined pragmatically, not by any essential substance, but by the role that they play within a three-part relationship of signs, interpreting signs, and referent objects. It is a tenet of pragmatism that all thought takes place in signs. Thought is not placed under any preconceived limitation or prior restriction to symbolic domains. It is merely noted that a certain analysis of the processes of perception and reasoning finds them to resolve into formal elements which possess the characters and participate in the relations that a definition will identify as distinctive of signs. One version of Peirce's sign definition is especially useful for the present purpose. It establishes for signs a fundamental role in logic and is stated in terms of abstract relational properties that are flexible enough to be interpreted in the materials of dynamic systems. Peirce gave this definition of signs in his 1902 "Application to the Carnegie Institution": | Logic is 'formal semiotic'. A sign is something, 'A', which brings | something, 'B', its 'interpretant' sign, determined or created by it, | into the same sort of correspondence (or a lower implied sort) with | something, 'C', its 'object', as that in which itself stands to 'C'. | This definition no more involves any reference to human thought than | does the definition of a line as the place within which a particle lies | during a lapse of time. (Peirce, NEM 4, 54). | | It is from this definition, together with a definition of "formal", | that I deduce mathematically the principles of logic. I also make | a historical review of all the definitions and conceptions of logic, | and show, not merely that my definition is no novelty, but that my | non-psychological conception of logic has 'virtually' been quite | generally held, though not generally recognized. (Peirce, NEM 4, 21). A placement and appreciation of this theory in a philosophical context that extends from Aristotle's early treatise 'On Interpretation' through John Dewey's later elaborations and applications (Dewey, 1910, 1929, 1938) is the topic of (Awbrey & Awbrey, 1992). Here, only a few features of this definition will be noted that are especially relevant to the goal of implementing intelligent interpreters. One characteristic of Peirce's definition is crucial in supplying a flexible infrastructure that makes the formal and mathematical treatment of sign relations possible. Namely, this definition allows objects to be characterized in two alternative ways that are substantially different in the domains they involve but roughly equivalent in their information content. Namely, objects of signs, that may exist in a reality exterior to the sign domain, insofar as they fall under this definition, allow themselves to be reconstituted nominally or reconstructed rationally as equivalence classes of signs. This transforms the actual relation of signs to objects, the relation or correspondence that is preserved in passing from initial signs to interpreting signs, into the membership relation that signs bear to their semantic equivalence classes. This transformation of a relation between signs and the world into a relation interior to the world of signs may be regarded as a kind of representational reduction in dimensions, like the foreshortening and planar projections that are used in perspective drawing. This definition takes as its subject a certain three-place relation, the sign relation proper, envisioned to consist of a certain set of three-tuples. The pattern of the data in this set of three-tuples, the extension of the sign relation, is expressed here in the form: <Object, Sign, Interpretant>. As a schematic notation for various sign relations, the letters "s", "o", "i" serve as typical variables ranging over the relational domains of signs, objects, interpretants, respectively. There are two customary ways of understanding this abstract sign relation as its structure affects concrete systems. In the first version the agency of a particular interpreter is taken into account as an implicit parameter of the relation. As used here, the concept of interpreter includes everything about the context of a sign's interpretation that affects its determination. In this view a specification of the two elements of sign and interpreter is considered to be equivalent information to knowing the interpreting or the interpretant sign, that is, the affect that is produced 'in' or the effect that is produced 'on' the interpreting system. Reference to an object or to an objective, whether it is successful or not, involves an orientation of the interpreting system and is therefore mediated by affects 'in' and effects 'on' the interpreter. Schematically, a lower case "j" can be used to represent the role of a particular interpreter. Thus, in this first view of the sign relation the fundamental pattern of data that determines the relation can be represented in the form <o, s, j> or <s, o, j>, as one chooses. In the second version of the sign relation the interpreter is considered to be a hypostatic abstraction from the actual process of sign transformation. In other words, the interpreter is regarded as a convenient construct that helps to personify the action but adds nothing informative to what is more simply observed as a process involving successive signs. An interpretant sign is merely the sign that succeeds another in a continuing sequence. What keeps this view from falling into sheer nominalism is the relation with objects that is preserved throughout the process of transformation. Thus, in this view of the sign relation the fundamental pattern of data that constitutes the relationship can be indicated by the optional forms <o, s, i> or <s, i, o>. Viewed as a totality, a complete sign relation would have to consist of all of those conceivable moments -- past, present, prospective, or in whatever variety of parallel universes that one may care to admit -- when something means something to somebody, in the pattern <s, o, j>, or when something means something about something, in the pattern <s, i, o>. But this ultimate sign relation is not often explicitly needed, and it could easily turn out to be logically and set-theoretically ill-defined. In physics, it is important for theoretical completeness to regard the whole universe as a single physical system, but more common to work with "isolated" subsystems. Likewise in the theory of signs, only particular and well-bounded subsystems of the ultimate sign relation are likely to be the subjects of sensible discussion. It is helpful to view the definition of individual sign relations on analogy with another important class of three-place relations of broad significance in mathematics and far-reaching application in physics: namely, the binary operations or ternary relations that fall under the definition of abstract groups. Viewed as a definition of individual groups, the axioms defining a group are what logicians would call highly non-categorical, that is, not every two models are isomorphic (Wilder, p. 36). But viewing the category of groups as a whole, if indeed it can be said to form a whole (MacLane, 1971), the definition allows a vast number of non-isomorphic objects, namely, the individual groups. In mathematical inquiry the closure property of abstract groups mitigates most of the difficulties that might otherwise attach to the precision of their individual definition. But in physics the application of mathematical structures to the unknown nature of the enveloping world is always tentative. Starting from the most elemental levels of instrumental trial and error, this kind of application is fraught with intellectual difficulty and even the risk of physical pain. The act of abstracting a particular structure from a concrete situation is no longer merely abstract. It becomes, in effect, a hypothesis, a guess, a bet on what is thought to be the most relevant aspect of a current, potentially dangerous, and always ever-insistently pressing reality. And this hypothesis is not a paper belief but determines action in accord with its character. Consequently, due to the abyss of ignorance that always remains to our kind and the chaos that can result from acting on what little is actually known, risk and pain accompany the extraction of particular structures, attempts to isolate particular forms, or guesses at viable factorizations of phenomena. Likewise in semiotics, it is hard to find any examples of autonomous sign relations and to isolate them from their ulterior entanglements. This kind of extraction is often more painful because the full analysis of each element in a particular sign relation may involve references to other object-, sign-, or interpretant-systems outside of its ostensible, initially secure bounds. As a result, it is even more difficult with sign systems than with the simpler physical systems to find coherent subassemblies that can be studied in isolation from the rest of the enveloping universe. These remarks should be enough to convey the plan of this work. Progress can be made toward new resettlements of ancient regions where only turmoil has reigned to date. Existing structures can be rehabilitated by continuing to unify the terms licensing AI representations with the terms leasing free space over dynamic manifolds. A large section of habitable space for dynamically intelligent systems could be extended in the following fashion: The images of state and the agents of change that are customary in symbolic AI could be related to the elements and the operators which form familiar planks in the tangent spaces of dynamic systems. The higher order concepts that fill out AI could be connected with the more complex constructions that are accessible from the moving platforms of these tangent spaces. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SR. Note 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Here are a couple of concrete examples of sign relations that I use to illustrate the basic properties of the genus in my dissertation. These examples were deliberately chosen to be as simple as they possibly could and still be interesting, or at least to fill out the most elementary features that the ilk of sign relations in general can have. So please do not pick on them for being too simple-minded, as that is the responsibility of their designer. If you want complexer examples, well, I liberally have a gadshillion of them. Indeed, up until the moment when one of my dissertation advisors asked me to construct something approaching a minimal example, I had never even deigned to consider any finite models that might happen to fall under the definition of a sign relation. When this gets down to the business of language learning and logical modeling, all of the serious examples of sign relations have sign domains with infinite cardinalities. Peirce thought that the power of the continuum was probably the minimum meaningful count, but I, in my computable wisdom, will be content with countable infinities for a while. We can start out by imagining that we take a sample of a fragment of a dialogue between two people, A and B, in which their language is restricted to just their own proper names, "A" and "B", plus the first and second person pronouns, "I" and "you", which will here be schematized as "i" and "u", respectively. To specify a sign relation one has to give three domains, the Object, Sign, Interpretant domains, schematized here as O, S, I, respectively. For this example, let us take the two sign relations, L(A) and L(B), corresponding to the usages of the two "interpreters", A and B, respectively. | L(A) and L(B) are subsets of OxSxI, | written here as L(A), L(B) c OxSxI, | where O, S, I are given as follows: | | O = {A, B}, | | S = {"A", "B", "i", "u"}, | | I = {"A", "B", "i", "u"}. | | L(A) has the following eight triples | of the form <o, s, i> in OxSxI: | | <A, "A", "A"> | <A, "A", "i"> | <A, "i", "A"> | <A, "i", "i"> | <B, "B", "B"> | <B, "B", "u"> | <B, "u", "B"> | <B, "u", "u"> | | L(B) has the following eight triples | of the form <o, s, i> in OxSxI: | | <A, "A", "A"> | <A, "A", "u"> | <A, "u", "A"> | <A, "u", "u"> | <B, "B", "B"> | <B, "B", "i"> | <B, "i", "B"> | <B, "i", "i"> That's the basic set-up. Next time I will discuss the relevant properties of these two sign relations. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SR. Note 3 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Let me try to keep this radically impoverished couple of exemplary sign relations, L(A) and L(B), before our attentions for a while. I want to use them only for the sake of illustrating the basic features that all sign relations have in common, a focus that will demand of us that we not become too distracted by the peculiar properties that they bear as individual cases. | Imagine that we have selected a sample of a fragment of a dialogue | between two people, A and B, in which their language is restricted to | their own proper names, "A" and "B", plus the first and second person | pronouns, "I" and "you", which will here be schematized as "i" and "u", | respectively. | | To specify a sign relation one has to give three domains, | the Object, Sign, Interpretant domains, schematized here | as O, S, I, respectively. | | For this example, let us take the two sign relations, L(A) and L(B), | corresponding to the usages of the two "interpreters", A and B, | respectively. | | L(A) and L(B) are subsets of OxSxI, | written here as L(A), L(B) c OxSxI, | where O, S, I are given as follows: | | O = {A, B}, | | S = {"A", "B", "i", "u"}, | | I = {"A", "B", "i", "u"}. | | L(A) has the following eight triples | of the form <o, s, i> in OxSxI: | | <A, "A", "A"> | <A, "A", "i"> | <A, "i", "A"> | <A, "i", "i"> | <B, "B", "B"> | <B, "B", "u"> | <B, "u", "B"> | <B, "u", "u"> | | L(B) has the following eight triples | of the form <o, s, i> in OxSxI: | | <A, "A", "A"> | <A, "A", "u"> | <A, "u", "A"> | <A, "u", "u"> | <B, "B", "B"> | <B, "B", "i"> | <B, "i", "B"> | <B, "i", "i"> FAQ's that come to mind are these: 1. What happened to the interpretive agent? 2. What happened to ideas, signs in a mind? 3. It's all so static, where's the process? Answers that come to mind are these: 1. The interpreters are just the two people, A and B. As it happens, they are also the objects, but that is an independent consideration, due merely to the narrowness of this particular discursive universe. The best thing that we can say about the location of the interpreters A and B under these circumstances is that they are represented by the whole sign relations, L(A) and L(B), that we have taken as very partial models of the agents' overall interpretive activity and conduct. 2. On our own time, we usually prefer to contemplate the types of sign relations in which we imagine the "interpretant sign" slot to be filled in by affective impressions and mental ideas, but when we wish to communicate the form of what we have been thinking about to others, then we are forced to employ slightly shifted types of sign relations, ones that have the same form, but where the interpretant slot is now filled by patently effable and publicly observable signs. The nice thing, the practically essential thing, about the "formal" approach is that it allows us to do this on a routine basis, all the while preserving the eidos, form, pattern, shape, structure, and so on, of what we are trying to get across to others. 3. The process of semiosis would normally be regarded as transpiring in the SI plane, as it were, with a triple <o, s, i> in the sign relation codifying the fact that the sign s can transition to the interpretant sign i in regard to the object o. Our static presentation of the sign relation thus constrains or "informs" the conduct of the associated sign process, but there is no requirement in the sign definition that its "determinations", in Peirce's sense, should make the whole process "deterministic", in the ordinary causal sense. That should be enough to get the ball rolling, in a thoroughly non-classical way, off course. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SR. Note 4 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o While looking into my dissertation for concrete illustrations of sign relations, I found to my amazement that the set-up for the "Story of A and B" contains not a few reflections that may throw a different light on our recent animadversions over the topic of formalization. So here 'tis ... 1.3.4 Discussion of Formalization: Concrete Examples The previous section outlined a variety of general issues surrounding the concept of formalization. The following section will plot the specific objectives of this project in constructing formal models of intellectual processes. In this section I wish to take a breather between these abstract discussions in order to give their main ideas a few points of contact with terra firma. To do this, I examine a selection of concrete examples, artificially constructed to approach the minimum levels of non-trivial complexity, that are intended to illustrate the kinds of mathematical objects I have in mind using as formal models. 1.3.4.1 Formal Models: A Sketch To sketch the features of the modeling activity that are relevant to the immediate purpose: The modeler begins with a "phenomenon of interest" or a "process of interest" (POI) and relates it to a formal "model of interest" (MOI), the whole while working within a particular "interpretive framework" (IF) and relating the results from one "system of interpretation" (SOI) to another, or to a subsequent development of the same SOI. The POI's that define the intents and the purposes of this project are the closely related processes of inquiry and interpretation, so the MOI's that must be formulated are models of inquiry and interpretation, species of formal systems that are even more intimately bound up than usual with the IF's employed and the SOI's deployed in their ongoing development as models. Since all of the interpretive systems and all of the process models that are being mentioned here come from the same broad family of mathematical objects, the different roles that they play in this investigation are mainly distinguished by variations in their manner and degree of formalization: 1. The typical POI comes from natural sources and casual conduct. It is not formalized in itself but only in the form of its image or model, and just to the extent that aspects of its structure and function are captured by a formal MOI. But the richness of any natural phenomenon or realistic process seldom falls within the metes and bounds of any final or finite formula. 2. Beyond the initial stages of investigation, the MOI is postulated as a completely formalized object, or is quickly on its way to becoming one. As such, it serves as a pivotal fulcrum and a point of application poised between the undefined reaches of "phenomena" and "noumena", respectively, terms that serve more as directions of pointing than as denotations of entities. What enables the MOI to grasp these directions is the quite felicitous mathematical circumstance that there can be well-defined and finite relations between entities that are infinite and even indefinite in themselves. Indeed, exploiting this handle on infinity is the main trick of all computational models and effective procedures. It is how a "finitely informed creature" (FIC) can "make infinite use of finite means". Thus, my reason for calling the MOI cardinal or pivotal is that it forms a model in two senses, loosely analogical and more strictly logical, integrating twin roles of the model concept in a single focus. 3. Finally, the IF's and the SOI's always remain partly out of sight, caught up in various stages of explicit notice between casual informality and partial formalization, with no guarantee or even much likelihood of a completely articulate formulation being forthcoming or even possible. Still, it is usually worth the effort to try lifting one edge or another of these frameworks and backdrops into the light, at least for a time. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SR. Note 5 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Let us return to the "Story of A and B", to see what fresh insights we might see. Here is the subsection of my dissertation where I first introduce the A & B example. It should serve to outline many of the basic concepts that arise in this approach to pragmatic semiotics. 1.3.4.2 Sign Relations: A Primer To the extent that their structures and functions can be discussed at all, it is likely that all of the formal entities that are destined to develop in this approach to inquiry will be instances of a class of three-place relations called "sign relations". At any rate, all of the formal structures that I have examined so far in this area have turned out to be easily converted to or ultimately grounded in sign relations. This class of triadic relations constitutes the main study of the "pragmatic theory of signs", a branch of logical philosophy devoted to understanding all types of symbolic representation and communication. There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry. In fact, the correspondence between the two studies exhibits so many parallels and coincidences that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, "thinking" in its best sense, "is a term denoting the various ways in which things acquire significance" (Dewey). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view. Because the examples in this section have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Still, these examples have subtleties of their own, and their careful treatment will serve to illustrate important issues in the general theory of signs. Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: "Ann", "Bob", "I", "you". The "object domain" of this discussion fragment is the set of two people {Ann, Bob}. The "syntactic domain" or the "sign system" of their discussion is limited to the set of four signs {"Ann", "Bob", "I", "You"}. In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The "system of interpretation" (SOI) associated with each language user can be represented in the form of an individual three-place relation called the "sign relation" of that interpreter. Understood in terms of its set-theoretic extension, a sign relation L is a subset of a cartesian product OxSxI. Here, O, S, and I are three sets that are known as the "object domain", the "sign domain", and the "interpretant domain", respectively, of the sign relation L c OxSxI. In general, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations that are contemplated in a computational framework are usually constrained to having I c S. In this case, interpretants are just a special variety of signs, and this makes it convenient to lump signs and interpretants together into a "syntactic domain". In the forthcoming examples, S and I are identical as sets, so the very same elements manifest themselves in two distinct roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains O, S, I for a given sign relation L, one may call this the "world of L" and write W = W(L) = O U S U I. To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible when the examples become more complicated, I introduce the following abbreviations: | O = Object Domain. | S = Sign Domain. | I = Interpretant Domain. | | O = {Ann, Bob} = {A, B}. | | S = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"}. | | I = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"}. | | In the present examples, S = I = Syntactic Domain. Tables 1 and 2 give the sign relations associated with the interpreters A and B, respectively, putting them in the form of relational databases. Thus, the rows of each Table list the ordered triples of the form <o, s, i> that make up the corresponding sign relations: A, B c OxSxI. The issue of using the same names for objects and for relations involving these objects will be taken up later, after the less problematic features of these relations have been treated. These Tables codify a rudimentary level of interpretive practice for the agents A and B, and provide a basis for formalizing the initial semantics that is appropriate to their common syntactic domain. Each row of a Table names an object and two co-referent signs, making up an ordered triple of the form <o, s, i> that is called an "elementary relation", that is, one element of the relation's set-theoretic extension. Already in this elementary context, there are several different meanings that might attach to the project of a "formal semantics". In the process of discussing these alternatives, I will introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions. Table 1. Sign Relation of Interpreter A o---------------o---------------o---------------o | Object | Sign | Interpretant | o---------------o---------------o---------------o | A | "A" | "A" | | A | "A" | "i" | | A | "i" | "A" | | A | "i" | "i" | o---------------o---------------o---------------o | B | "B" | "B" | | B | "B" | "u" | | B | "u" | "B" | | B | "u" | "u" | o---------------o---------------o---------------o Table 2. Sign Relation of Interpreter B o---------------o---------------o---------------o | Object | Sign | Interpretant | o---------------o---------------o---------------o | A | "A" | "A" | | A | "A" | "u" | | A | "u" | "A" | | A | "u" | "u" | o---------------o---------------o---------------o | B | "B" | "B" | | B | "B" | "i" | | B | "i" | "B" | | B | "i" | "i" | o---------------o---------------o---------------o One aspect of semantics is concerned with the reference that a sign has to its object, which is called its "denotation". For signs in general, neither the existence nor the uniqueness of a denotation is guaranteed. Thus, the denotation of a sign can refer to a plural, a singular, or a vacuous number of objects. In the pragmatic theory of signs, these references are formalized as certain types of dyadic relations that are obtained by projection from the triadic sign relations. The dyadic relation that constitutes the "denotative component" of a sign relation L is denoted by "Den(L)". Information about the denotative component of semantics can be derived from L by taking its "dyadic projection" on the plane that is generated by the object and the sign domains, indicated by any one of the equivalent forms, "Proj_OS(L)", "L_OS", or "L_12", and defined as: Den(L) = Proj_OS(L) = L_OS = {<o, s> in OxS : <o, s, i> in L for some i in I}. Looking to the denotative aspects of the present example, various rows of the Tables specify that A uses "i" to denote A and "u" to denote B, whereas B uses "i" to denote B and "u" to denote A. It is utterly amazing that even these impoverished remnants of natural language use have properties that quickly bring the usual prospects of formal semantics to a screeching halt. The other dyadic aspects of semantics that might be considered concern the reference that a sign has to its interpretant and the reference that an interpretant has to its object. As before, either type of reference can be multiple, unique, or empty in its collection of terminal points, and both can be formalized as different types of dyadic relations that are obtained as planar projections of the triadic sign relations. The connection that a sign makes to an interpretant is called its "connotation". In the general theory of sign relations, this aspect of semantics includes the references that a sign has to affects, concepts, impressions, intentions, mental ideas, and to the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. This complex ecosystem of references is unlikely ever to be mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the "connotative" import of language. Given a particular sign relation L, the dyadic relation that constitutes the "connotative component" of L is denoted by "Con(L)". The bearing that an interpretant has toward a common object of its sign and itself has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this omits the mediational character of the whole transaction. Given the service that interpretants supply in furnishing a locus for critical, reflective, and explanatory glosses on objective scenes and their descriptive texts, it is easy to regard them as "annotations" of both objects and signs, but this function points in the opposite direction to what is needed in this connection. What does one call the inverse of the annotation function? More generally asked, what is the converse of the annotation relation? In light of these considerations, I find myself still experimenting with terms to suit this last-mentioned dimension of semantics. On a trial basis, I will refer to it as the "ideational", "intentional", or "canonical" component of the sign relation, and I will try calling the reference of an interpretant sign to an object its "ideation", "intention", or "conation". Given a particular sign relation L, the dyadic relation that constitutes the "intentional component" of L is denoted by "Int(L)". A full consideration of the connotative and intentional aspects of semantics would force a return to difficult questions about the true nature of the interpretant sign in the general theory of sign relations. It is best to defer these issues to a later discussion. Fortunately, omission of this material does not interfere with understanding the purely formal aspects of the present example. Formally, these new aspects of semantics present no additional problem: The connotative component of a sign relation L can be formalized as its dyadic projection on the sign and interpretant domains, defined as follows: Con(L) = Proj_SI(L) = L_SI = {<s, i> in SxI : <o, s, i> in L for some o in O}. The intentional component of semantics for a sign relation L, or its "second moment of denotation", is adequately captured by its dyadic projection on the object and interpretant domains, defined as follows: Int(L) = Proj_OI(L) = L_OI = {<o, i> in OxI : <o, s, i> in L for some s in S}. As it happens, the sign relations A and B in the present example are fully symmetric with respect to exchanging signs and interpretants, so all of the structure of A_OS and B_OS is merely echoed in A_OI and B_OI, respectively. The principal concern of this project is not with every conceivable sign relation but chiefly with those that are capable of supporting inquiry processes. In these, the relationship between the connotational and the denotational aspects of meaning is not wholly arbitrary. Instead, this relationship must be naturally constrained or deliberately designed in such a way that it: 1. Represents the embodiment of significant properties that have objective reality in the agent's domain. 2. Supports the achievement of particular purposes that have intentional value for the agent. Therefore, my attention is directed mainly toward understanding the forms of correlation, coordination, and cooperation among the various components of sign relations that form the necessary conditions for carrying out coherent inquiries. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SR. Note 6 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o I continue with the discussion of sign relations in the medium of concrete examples, as illustrated by the "Story of A and B". 1.3.4.3 Semiotic Equivalence Relations If one examines the sign relations L(A) and L(B) that are associated with the interpreters A and B, respectively, one observes that they have many contingent properties that are not possessed by sign relations in general. One nice property possessed by the sign relations L(A) and L(B) is that their connotative components A_SI and B_SI constitute a pair of equivalence relations on their common syntactic domain S = I. It is convenient to refer to such structures as "semiotic equivalence relations" (SER's) since they equate signs that mean the same thing to somebody. Each of the SER's, A_SI, B_SI c SxI = SxS, partitions the whole collection of signs into "semiotic equivalence classes" (SEC's). This makes for an especially strong form of representation in that the structure of the participants' common object domain is reflected or reconstructed, part for part, in the structure of each of their "semiotic partitions" (SEP's) of the syntactic domain. The main trouble with this notion of semantics in the present situation is that the two semiotic partitions for A and B are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view (POV). Information about the different forms of semiotic equivalence that are induced by the interpreters A and B is summarized in Tables 3 and 4. The form of these Tables should suffice to explain what is meant by saying that the SEP's for A and B are orthogonal to each other. Table 3. Semiotic Partition of Interpreter A o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o | "A" "i" | o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o | "u" "B" | o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o Table 4. Semiotic Partition of Interpreter B o~~~~~~~~~~~~~~~o~~~~~~~~~~~~~~~o | "A" | "i" | | | | | "u" | "B" | o~~~~~~~~~~~~~~~o~~~~~~~~~~~~~~~o To discuss this situation further, I introduce the square bracket notation "[x]_E" to denote "the equivalence class of the element x under the equivalence relation E". A statement that the elements x and y are equivalent under E is called an "equation", and can be written in either one of two ways, as "[x]_E = [y]_E" or as "x =_E y". In the application to sign relations I extend this notation in the following ways. When L is a sign relation whose "syntactic projection" or connotative component L_SI is an equivalence relation on S, then I write "[s]_L" for "the equivalence class of s under L_SI". A statement that the signs x and y are synonymous under a SER L_SI is called a "semiotic equation" (SEQ), and can be written in either of the forms: "[x]_L = [y]_L" or "x =_L y". In many situations there is a further adaptation of the square bracket notation that can be useful. Namely, when there is known to exist a particular triple <o, s, i> in L, it is permissible to use "[o]_L" to mean the same thing as "[s]_L". These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions. In these terms, the SER for interpreter A yields the semiotic equations: | ["A"]_A = ["i"]_A | | ["B"]_A = ["u"]_A or | "A" =_A "i" | | "B" =_A "u" and the semiotic partition: {{"A", "i"}, {"B", "u"}}. In contrast, the SER for interpreter B yields the semiotic equations: | ["A"]_B = ["u"]_B | | ["B"]_B = ["i"]_B or | "A" =_B "u" | | "B" =_B "i" and the semiotic partition: {{"A", "u"}, {"B", "i"}}. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SR. Note 7 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o I continue with the discussion of sign relations in the medium of concrete examples, as illustrated by the "Story of A and B". This episode sketches a variety of graph-theoretical pictures that can aid the imagination in thinking about sign relations. 1.3.4.4 Graphical Representations The dyadic components of sign relations can be given graph-theoretic representations, as "digraphs" (or "directed graphs"), that provide concise pictures of their structural and potential dynamic properties. By way of terminology, a directed edge <x, y> is called an "arc" from point x to point y, and a self-loop <x, x> is called a "sling" at x. The denotative components Den(A) and Den(B) can be represented as digraphs on the six points of their common world set W = O |_| S |_| I = {A, B, "A", "B", "i", "u"}. The arcs are given as follows: 1. Den(A) has an arc from each point of {"A", "i"} to A and from each point of {"B", "u"} to B. 2. Den(B) has an arc from each point of {"A", "u"} to A and from each point of {"B", "i"} to B. Den(A) and Den(B) can be interpreted as "transition digraphs" that chart the succession of steps or the connection of states in a computational process. If the graph is read this way, the denotational arcs summarize the "upshots" of the computations that are involved when the interpreters A and B evaluate the signs in S according to their own frames of reference. The connotative components Con(A) and Con(B) can be represented as digraphs on the four points of their common syntactic domain S = I = {"A", "B", "i", "u"}. Since Con(A) and Con(B) are SER's, their digraphs conform to the pattern that is manifested by all digraphs of equivalence relations. In general, a digraph of an equivalence relation falls into connected components that correspond to the parts of the associated partition, with a complete digraph on the points of each part, and no other arcs. In the present case, the arcs are given as follows: 1. Con(A) has the structure of a SER on S, with a sling at each of the points in S, two-way arcs between the points of {"A", "i"}, and two-way arcs between the points of {"B", "u"}. 2. Con(B) has the structure of a SER on S, with a sling at each of the points in S, two-way arcs between the points of {"A", "u"}, and two-way arcs between the points of {"B", "i"}. Taken as transition digraphs, Con(A) and Con(B) highlight the associations that are permitted between equivalent signs, as this equivalence is judged by the interpreters A and B, respectively. The theme running through the last three subsections, that associates different interpreters and different aspects of interpretation with different sorts of relational structures on the same set of points, heralds a topic that will be developed extensively in the sequel. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

###### Future Agenda

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | I may mention in passing ... | walking on stilts ... | And even for 'communicating' the use of words, | what can be more perfect than the method of examples? | CSP, CE 1, pages 173-174. | | Charles Sanders Peirce, "Harvard Lectures 'On the Logic of Science'", (1865), |'Writings of Charles S. Peirce: A Chronological Edition, Volume 1, 1857-1866', | Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

##### Links

DET. Determination 01. http://suo.ieee.org/ontology/msg02377.html 02. http://suo.ieee.org/ontology/msg02378.html 03. http://suo.ieee.org/ontology/msg02379.html 04. http://suo.ieee.org/ontology/msg02380.html 05. http://suo.ieee.org/ontology/msg02384.html 06. http://suo.ieee.org/ontology/msg02387.html 07. http://suo.ieee.org/ontology/msg02388.html 08. http://suo.ieee.org/ontology/msg02389.html 09. http://suo.ieee.org/ontology/msg02390.html 10. http://suo.ieee.org/ontology/msg02391.html 11. http://suo.ieee.org/ontology/msg02395.html 12. http://suo.ieee.org/ontology/msg02407.html 13. http://suo.ieee.org/ontology/msg02550.html 14. http://suo.ieee.org/ontology/msg02552.html 15. http://suo.ieee.org/ontology/msg02556.html 16. http://suo.ieee.org/ontology/msg02594.html 17. http://suo.ieee.org/ontology/msg02651.html 18. http://suo.ieee.org/ontology/msg02673.html 19. http://suo.ieee.org/ontology/msg02706.html 20. http://suo.ieee.org/ontology/msg03177.html 21. http://suo.ieee.org/ontology/msg03185.html 22. http://suo.ieee.org/ontology/msg03188.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EXC. Excuses 01. http://suo.ieee.org/ontology/msg03240.html 02. http://suo.ieee.org/ontology/msg03241.html 03. http://suo.ieee.org/ontology/msg03242.html 04. http://suo.ieee.org/ontology/msg03243.html 05. http://suo.ieee.org/ontology/msg03258.html 06. http://suo.ieee.org/ontology/msg03262.html 07. http://suo.ieee.org/ontology/msg03264.html 08. http://suo.ieee.org/ontology/msg03265.html 09. http://suo.ieee.org/ontology/msg03266.html 10. http://suo.ieee.org/ontology/msg03268.html 11. http://suo.ieee.org/ontology/msg03269.html 12. http://suo.ieee.org/ontology/msg03270.html 13. http://suo.ieee.org/ontology/msg03274.html 14. http://suo.ieee.org/ontology/msg03276.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o EII. Extension, Intension, Information 01. http://suo.ieee.org/email/msg01883.html 02. http://suo.ieee.org/email/msg01887.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o FOR. Inquiry Into Formalization 01. http://suo.ieee.org/ontology/msg03228.html 02. http://suo.ieee.org/ontology/msg03231.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o INF. Inquiry Into Information 01. http://suo.ieee.org/ontology/msg03172.html 02. http://suo.ieee.org/ontology/msg03174.html 03. http://suo.ieee.org/ontology/msg03175.html 04. http://suo.ieee.org/ontology/msg03176.html 05. http://suo.ieee.org/ontology/msg03186.html 06. http://suo.ieee.org/ontology/msg03194.html 07. http://suo.ieee.org/ontology/msg03198.html 08. http://suo.ieee.org/ontology/msg03199.html 09. http://suo.ieee.org/ontology/msg03200.html 10. http://suo.ieee.org/ontology/msg03203.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SYM. Inquiry Into Symbolization 01. http://suo.ieee.org/ontology/msg03201.html 02. http://suo.ieee.org/ontology/msg03202.html 03. http://suo.ieee.org/ontology/msg03204.html 04. http://suo.ieee.org/ontology/msg03234.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o LAS. Logic As Semiotic 01. http://suo.ieee.org/ontology/msg03070.html 02. http://suo.ieee.org/ontology/msg03171.html 03. http://suo.ieee.org/ontology/msg03178.html 04. http://suo.ieee.org/ontology/msg03179.html 05. http://suo.ieee.org/ontology/msg03184.html 06. http://suo.ieee.org/ontology/msg03187.html 07. http://suo.ieee.org/ontology/msg03189.html 08. http://suo.ieee.org/ontology/msg03190.html 09. http://suo.ieee.org/ontology/msg03192.html 10. http://suo.ieee.org/ontology/msg03193.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOSI. Manifolds Of Sensuous Impressions 01. http://suo.ieee.org/ontology/msg03045.html 02. http://suo.ieee.org/ontology/msg03046.html 03. http://suo.ieee.org/ontology/msg03049.html 04. http://suo.ieee.org/ontology/msg03065.html 05. http://suo.ieee.org/ontology/msg03066.html 06. http://suo.ieee.org/ontology/msg03074.html 07. http://suo.ieee.org/ontology/msg03075.html 08. http://suo.ieee.org/ontology/msg03079.html 09. http://suo.ieee.org/ontology/msg03083.html 10. http://suo.ieee.org/ontology/msg03131.html 11. http://suo.ieee.org/ontology/msg03144.html 12. http://suo.ieee.org/ontology/msg03147.html 13. http://suo.ieee.org/ontology/msg03169.html 14. http://suo.ieee.org/ontology/msg03205.html 15. http://suo.ieee.org/ontology/msg03208.html 16. http://suo.ieee.org/ontology/msg03233.html 17. http://suo.ieee.org/ontology/msg03260.html 18. http://suo.ieee.org/ontology/msg03331.html 19. http://suo.ieee.org/ontology/msg03333.html 20. http://suo.ieee.org/ontology/msg03839.html 21. http://suo.ieee.org/ontology/msg03841.html 22. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o MOD. Model Theory 01. http://suo.ieee.org/ontology/msg03246.html 02. http://suo.ieee.org/ontology/msg03247.html 03. http://suo.ieee.org/ontology/msg03248.html 04. http://suo.ieee.org/ontology/msg03249.html 05. http://suo.ieee.org/ontology/msg03250.html 06. http://suo.ieee.org/ontology/msg03251.html 07. http://suo.ieee.org/ontology/msg03252.html 08. http://suo.ieee.org/ontology/msg03254.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SEM. Semiotics Formalization 01. http://suo.ieee.org/email/msg00815.html 02. http://suo.ieee.org/email/msg00829.html 03. http://suo.ieee.org/email/msg00892.html 04. http://suo.ieee.org/email/msg00893.html 05. http://suo.ieee.org/email/msg00894.html 06. http://suo.ieee.org/email/msg01111.html 07. http://suo.ieee.org/email/msg01112.html 08. http://suo.ieee.org/email/msg01113.html 09. http://suo.ieee.org/email/msg01130.html 10. http://suo.ieee.org/email/msg02611.html 11. http://suo.ieee.org/email/msg02617.html 12. http://suo.ieee.org/email/msg02621.html 13. http://suo.ieee.org/email/msg03189.html 14. http://suo.ieee.org/email/msg04401.html 15. http://suo.ieee.org/email/msg05433.html 16. http://suo.ieee.org/email/msg05451.html 17. http://suo.ieee.org/email/msg05487.html 18. http://suo.ieee.org/email/msg05517.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o SR. Sign Relations SUO List 01. http://suo.ieee.org/email/msg00729.html 02. http://suo.ieee.org/email/msg01224.html 03. http://suo.ieee.org/email/msg03111.html 04. http://suo.ieee.org/email/msg04807.html 05. http://suo.ieee.org/email/msg04810.html Ontology List 01. http://suo.ieee.org/ontology/msg03214.html 02. http://suo.ieee.org/ontology/msg03215.html 03. http://suo.ieee.org/ontology/msg03218.html 04. http://suo.ieee.org/ontology/msg03228.html 05. http://suo.ieee.org/ontology/msg03229.html 06. http://suo.ieee.org/ontology/msg03230.html 07. http://suo.ieee.org/ontology/msg03232.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o