Epitext for Differential Logic and Dynamic Systems
| Stand and unfold yourself. | Hamlet: Francsico—1.1.2 |
Purpose
Review and Transition
A Functional Conception of Propositional Calculus
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Out of the dimness opposite equals advance . . . . | |
| — Walt Whitman, Leaves of Grass, [Whi, 28] |
Qualitative Logic and Quantitative Analogy
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Logical, however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions. |
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| — John Dewey, How We Think, [Dew, 56] | ||
Philosophy of Notation : Formal Terms and Flexible Types
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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. |
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| — W.V. Quine, Mathematical Logic, [Qui, 7] | ||
Special Classes of Propositions
Basis Relativity and Type Ambiguity
The Analogy Between Real and Boolean Types
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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. |
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| — W.V. Quine, Mathematical Logic, [Qui, 7] | ||
Theory of Control and Control of Theory
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You will hardly know who I am or what I mean, | |
| — Walt Whitman, Leaves of Grass, [Whi, 88] |
Propositions as Types and Higher Order Types
Reality at the Threshold of Logic
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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. |
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| — W.V. Quine, Mathematical Logic, [Qui, 7] | ||
Tables of Propositional Forms
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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. |
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| — W.V. Quine, Mathematical Logic, [Qui, 7–8] | ||
A Differential Extension of Propositional Calculus
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Fire over water: | |
| — I Ching, Hexagram 64, [Wil, 249] |
Differential Propositions : The Qualitative Analogues of Differential Equations
An Interlude on the Path
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There would have been no beginnings: instead, speech would proceed from me, while I stood in its path – a slender gap – the point of its possible disappearance. |
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| — Michel Foucault, The Discourse on Language, [Fou, 215] | ||
The Extended Universe of Discourse
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At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me. |
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| — Michel Foucault, The Discourse on Language, [Fou, 215] | ||
Intentional Propositions
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Do you guess I have some intricate purpose? | |
| — Walt Whitman, Leaves of Grass, [Whi, 45] |
Life on Easy Street
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Failing to fetch me at first keep encouraged, | |
| — Walt Whitman, Leaves of Grass, [Whi, 88] |
Back to the Beginning : Some Exemplary Universes
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I would have preferred to be enveloped in words, borne way beyond all possible beginnings. |
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| — Michel Foucault, The Discourse on Language, [Fou, 215] | ||
A One-Dimensional Universe
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There was never any more inception than there is now, | |
| — Walt Whitman, Leaves of Grass, [Whi, 28] |
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The clock indicates the moment . . . . but what does | |
| — Walt Whitman, 'Leaves of Grass', [Whi, 79] |
Example 1. A Square Rigging
Urge and urge and urge,
Always the procreant urge of the world.Walt Whitman, Leaves of Grass, [Whi, 28]
Back to the Feature
I guess it must be the flag of my disposition, out of hopeful
green stuff woven.Walt Whitman, Leaves of Grass, [Whi, 31]
Tacit Extensions
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.
Michel Foucault, The Discourse on Language, [Fou, 215]
Example 2. Drives and Their Vicissitudes
I open my scuttle at night and see the far-sprinkled systems,
And all I see, multiplied as high as I can cipher, edge but
the rim of the farther systems.Walt Whitman, Leaves of Grass, [Whi, 81]
Transformations of Discourse
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the fons et origo of an unfathomable transformation.
Robert Musil, The Man Without Qualities, [Mus, 39]
Foreshadowing Transformations : Extensions and Projections of Discourse
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.
Gaston Leroux, The Phantom of the Opera, [Ler, 126]
Extension from 1 to 2 Dimensions
Extension from 2 to 4 Dimensions
Thematization of Functions : And a Declaration of Independence for Variables
And as imagination bodies forth
The forms of things unknown, the poet's pen
Turns them to shapes, and gives to airy nothing
A local habitation and a name.A Midsummer Night's Dream, 5.1.18
Thematization : Venn Diagrams
The known universe has one complete lover and that is the greatest poet.
He consumes an eternal passion and is indifferent which chance happens
and which possible contingency of fortune or misfortune and persuades
daily and hourly his delicious pay.Walt Whitman, Leaves of Grass, [Whi, 11-12]
Thematization : Truth Tables
That which distorts honest shapes or which creates unearthly
beings or places or contingencies is a nuisance and a revolt.Walt Whitman, Leaves of Grass, [Whi, 19]
Propositional Transformations
If only the word 'artificial' were associated with the idea of art, or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that logical refers to artificial thought.
John Dewey, How We Think, [Dew, 56-57]
Alias and Alibi Transformations
Transformations of General Type
Es ist passiert, "it just sort of happened", people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.
Robert Musil, The Man Without Qualities, [Mus, 34]
Analytic Expansions : Operators and Functors
Consider what effects that might conceivably
have practical bearings you conceive the
objects of your conception to have. Then,
your conception of those effects is the
whole of your conception of the object.C.S. Peirce, "The Maxim of Pragmatism", CP 5.438
Operators on Propositions and Transformations
Differential Analysis of Propositions and Transformations
The resultant metaphysical problem now is this:
Does the man go round the squirrel or not?
William James, Pragmatism, [Jam, 43]
The Secant Operator : E
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.
William James, Pragmatism, [Jam, 46]
The Radius Operator : e
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.
William James, Pragmatism, [Jam, 46]
The Phantom of the Operators : η
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, which was playing the most perfect music!
Gaston Leroux, The Phantom of the Opera, [Ler, 81]
The Chord Operator : D
What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.
William James, Pragmatism, [Jam, 45]
The Tangent Operator : T
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.
William James, Pragmatism, [Jam, 300]
Transformations of Type B2 → B1
Analytic Expansion of Conjunction
In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a soul.
What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.
Robert Musil, The Man Without Qualities, [Mus, 118]
Tacit Extension of Conjunction
I teach straying from me, yet who can stray from me?
I follow you whoever you are from the present hour;
My words itch at your ears till you understand them.Walt Whitman, Leaves of Grass, [Whi, 83]
Enlargement Map of Conjunction
No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.
Robert Musil, The Man Without Qualities, [Mus, 62]
Digression : Reflection on Use and Mention
Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked — almost as one might turn a stone over to see what its hidden side is like or what is covered by it.
John Dewey, How We Think, [Dew, 57]
The well-known capacity that thoughts have — as doctors have discovered — for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.
Robert Musil, The Man Without Qualities, [Mus, 130]
Difference Map of Conjunction
"It doesn't matter what one does", the Man Without Qualities said to himself, shrugging his shoulders. "In a tangle of forces like this it doesn't make a scrap of difference." He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.
Robert Musil, The Man Without Qualities, [Mus, 8]
Differential of Conjunction
By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.
Michel Foucault, The Archaeology of Knowledge, [Fou, 143]
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.
Robert Musil, The Man Without Qualities, [Mus, 144]
Remainder of Conjunction
I bequeath myself to the dirt to grow from the grass I love,
If you want me again look for me under your bootsoles.You will hardly know who I am or what I mean,
But I shall be good health to you nevertheless,
And filter and fibre your blood.Failing to fetch me at first keep encouraged,
Missing me one place search another,
I stop some where waiting for youWalt Whitman, Leaves of Grass, [Whi, 88]
Summary of Conjunction
Analytic Series : Coordinate Method
And if he is told that something is the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could "just as easily" be, and to attach no more importance to what is than to what is not.
Robert Musil, The Man Without Qualities, [Mus, 12]
Analytic Series : Recap
Terminological Interlude
Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been "starred", in spite of their solidity …
Gaston Leroux, The Phantom of the Opera, [Ler, 230]
End of Perfunctory Chatter : Time to Roll the Clip!
Operator Maps : Areal Views
Operator Maps : Box Views
Operator Diagrams for the Conjunction J = uv
Taking Aim at Higher Dimensional Targets
The past and present wilt . . . . I have filled them and
emptied them,
And proceed to fill my next fold of the future.Walt Whitman, Leaves of Grass, [Whi, 87]
Transformations of Type B2 → B2
Epilogue, Enchoiry, Exodus
It is time to explain myself . . . . let us stand up.
Walt Whitman, Leaves of Grass, [Whi, 79]
