The Way The Cookie Uncrumbles Itself

Posted on April 16, 2013 by Jon Awbrey
I put down the cup and turn to my mind.  It is up to my mind to find the truth.  But how?  What grave uncertainty, whenever the mind feels overtaken by itself;  when it, the seeker, is also the obscure country where it must seek and where all its baggage will be nothing to it.  Seek?  Not only that:  create.  It is face to face with something that does not yet exist and that only it can accomplish, and bring into its light.

🙞 Proust • In Search of Lost Time • 1.48

Marcel Proust (1913–1927), In Search of Lost Time, Christopher Prendergast (ed.), Penguin Books, London, UK, 2002, 6 volumes:

  1. The Way by Swann’s (1913), Lydia Davis (trans.)
  2. In the Shadow of Young Girls in Flower (1919), James Grieve (trans.)
  3. The Guermantes Way (1920–1921), Mark Treharne (trans.)
  4. Sodom and Gomorrah (1921–1922), John Sturrock (trans.)
  5. The Prisoner (1923), Carol Clark (trans.)
    The Fugitive (1925), Peter Collier (trans.)
  6. Finding Time Again (1927), Ian Patterson (trans.)
Posted in Anamnesis, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Madeleine, Memory, Proust, Time | Tagged , , , , , , , | 1 Comment

What It Is

Posted on April 13, 2013 by Jon Awbrey

Re: Gil Kalai

If I remember my long ago readings well enough, Jimmy the Ancient Greek could lay odds as well as any modern bookmaker on the outcomes of Olympic contests, but that was not really the point of Zeno’s humble homilies. Read in philosophical context, they had to do with a contention between two schools of thought about the relation of eternal being to secular becoming. Followers of Parmenides like Zeno would say that whatever it is that really is, is one, eternal, and unchanging. They would not be impressed that it took us a couple of millennia to “save the appearances” of change, since the appearances are only illusions anyway.

Posted in Change, Heraclitus, Infinity, Logic, Mathematics, Motion, Paradox, Parmenides, Phenomenology, Zeno | Tagged , , , , , , , , , | Leave a comment

Slip Slidin’ Away

Posted on April 11, 2013 by Jon Awbrey

And you give me the choice between a description that is sure but that teaches me nothing and hypotheses that claim to teach me but that are not sure.

— Albert Camus • The Myth of Sisyphus

Re: R.J. Lipton and K.W. Regan • Zeno Proof Paradox

The classical paradoxes of change and motion really have to do with a disconnect that exists between two realms —

On the one hand we have the phenomenology.  There is no problem there since we obviously observe all sorts of Achillean runners passing all sorts of Tortoises all sorts of times, the respective handicaps of heels and hulls notwithstanding.

On the other hand we have the logical theories and mathematical models that we bring to bear on the phenomena by way of trying to describe and explain them.

There’s the rub.  Get a model or theory that “saves the appearances” (solves the phenomena) and the paradox disappears.

Transpose the phenomena from a classical mode to a quantum‑mechanical, information‑theoretic, or ordinary logical key — and the note that resolves the chord is a trifle harder to find.

In a related development, we could hardly complete this course without mentioning the logical version of Zeno’s Paradox given by Lewis Carroll —

Posted in Albert Camus, C.S. Peirce, Change, Differential Logic, Infinity, Lewis Carroll, Logic, Mathematics, Meno, Modus Ponens, Motion, Paradox, Phenomenology, Sisyphus, Syllogism, Time, Zeno | Tagged , , , , , , , , , , , , , , , , | 3 Comments

Meno Meno Tekel Upharsin

Posted on April 1, 2013 by Jon Awbrey

Re: Theodora Goss

Can we ever become what we weren’t in eternity?
Can we ever learn what we weren’t born knowing?
Can we ever share what we never had in common?

Lately I’ve begun to see that these ancient riddles of change, coming to know, and communication all spring from a common root.

Posted in Anamnesis, Change, Communication, Education, Eternity, Innate Ideas, Learning, Meno, Tabula Rasa, Teaching, Time, Universal Harmony | Tagged , , , , , , , , , , , | Leave a comment

The Present Is Big With The Future

Posted on April 1, 2013 by Jon Awbrey
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.

Posted in Causality, Certainty, Chance, Contingency, Determination, Determinism, Differential Calculus, Differential Logic, Evil, Free Will, Freedom, Hologrammautomaton, Infinitesimals, Leibniz, Preëstablished Harmony, Theodicy | Tagged , , , , , , , , , , , , , , , | 17 Comments

Pragmatism Meets Absurdity

Posted on March 30, 2013 by Jon Awbrey

At the streetcorner …

At any streetcorner the feeling of absurdity can strike any man in the face. As it is, in its distressing nudity, in its light without effulgence, it is elusive. But that very difficulty deserves reflection. It is probably true that a man remains forever unknown to us and that there is in him something irreducible that escapes us. But practically I know men and recognize them by their behavior, by the totality of their deeds, by the consequences caused in life by their presence. Likewise, all those irrational feelings which offer no purchase to analysis. I can define them practically, appreciate them practically, by gathering together the sum of their consequences in the domain of the intelligence, by seizing and noting all their aspects, by outlining their universe. (10–11)

Albert Camus, The Myth of Sisyphus and Other Essays, Justin O’Brien (trans.), Random House, New York, NY, 1991. Originally published in France as Le Mythe de Sisyphe by Librairie Gallimard, 1942. First published in the United States by Alfred A. Knopf, 1955.

Posted in Absurdity, Albert Camus, Existentialism, Peirce, Pragmatic Maxim, Pragmatism, Sisyphus | Tagged , , , , , , | Leave a comment

A Determined Soul

Posted on March 26, 2013 by Jon Awbrey

Selections from Albert Camus, “The Myth of Sisyphus”

Everything considered, a determined soul will always manage. (41)
⁂  
To a man devoid of blinders, there is no finer sight than that of the intelligence at grips with a reality that transcends it. (55)
⁂  
At any streetcorner the feeling of absurdity can strike any man in the face. As it is, in its distressing nudity, in its light without effulgence, it is elusive. But that very difficulty deserves reflection. It is probably true that a man remains forever unknown to us and that there is in him something irreducible that escapes us. But practically I know men and recognize them by their behavior, by the totality of their deeds, by the consequences caused in life by their presence. Likewise, all those irrational feelings which offer no purchase to analysis. I can define them practically, appreciate them practically, by gathering together the sum of their consequences in the domain of the intelligence, by seizing and noting all their aspects, by outlining their universe. (10–11)
⁂  
And you give me the choice between a description that is sure but that teaches me nothing and hypotheses that claim to teach me but that are not sure. (20)
⁂  
I said that the world was absurd, but I was too hasty. This world in itself is not reasonable, that is all that can be said. But what is absurd is the confrontation of this irrational and the wild longing for clarity whose call echoes in the human heart. The absurd depends as much on man as on the world. For the moment it is all that links them together. It binds them one to the other as only hatred can weld two creatures together. (21)
⁂  
The absurd is born of this confrontation between the human need and the unreasonable silence of the world. This must not be forgotten. This must be clung to because the whole consequence of a life can depend on it. The irrational, the human nostalgia, and the absurd that is born of their encounter — these are the three characters in the drama that must necessarily end with all the logic of which an existence is capable. (28)

Albert Camus, The Myth of Sisyphus and Other Essays, Justin O’Brien (trans.), Random House, New York, NY, 1991. Originally published in France as Le Mythe de Sisyphe by Librairie Gallimard, 1942. First published in the United States by Alfred A. Knopf, 1955.

Notes

I started out trying to read this as philosophy, but the strain of doing that was making me ill, so I had to switch over and read it as a novel, a stream of consciousness narrative of a man fighting his way through a storm of mental impressions.

Posted in Albert Camus, Sisyphus | Tagged , | Leave a comment

τὰ δὲ μοι παθήματα ἐόντα ἀχάριτα μαθήματα γέγονε

Posted on March 24, 2013 by Jon Awbrey

Croesus, addressing Cyrus, says —

τὰ δὲ μοι παθήματα ἐόντα ἀχάριτα μαθήματα γέγονε.

My sufferings, though painful, have been my lessons.

— Herodotus • 1.207

Notes • (1) • (2) • (3) • (4) • (5)

Posted in Anthem | Tagged | 2 Comments

A Meno Acid

Posted on March 21, 2013 by Jon Awbrey

What answers to the Meno Paradox
Comes in the moment of realizing —
Gathering together the building blocks
Is just the beginning of the building.

Posted in Artificial Intelligence, Cognitive Science, Education, Inquiry, Learning, Meno, Philosophy, Plato, Teaching, Verse | Tagged , , , , , , , , , | Leave a comment

Indicator Functions • 1

Posted on March 13, 2013 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • Who Invented Boolean Functions?

One of the things it helps to understand about 19th Century mathematicians, and those who built the bridge to the 20th, is that they were capable of high abstraction — in Peirce’s case a cut above what is common today — and yet they remained close enough to the point where abstract forms are teased away from the concrete materials of mathematical inquiry to maintain a sense of connection between the two.  There are few better places to see this connection than in the medium of venn diagrams.  But venn diagrams are such familiar pictures that it’s easy to overlook their subtleties, so it may be useful to spend some time developing the finer points of what they picture.

There are actually several types of boolean functions depicted in the typical venn diagram.  Each has the boolean domain \mathbb{B} = \{ 0, 1 \} or one of its powers \mathbb{B}^k as its functional codomain but its functional domain need not be limited to a finite cardinality.  To sort their variety, consider the array of functional arrows in the following figure.

Indicator Functions

Suppose X is a universe of discourse represented by the rectangular area of a Venn diagram.  Note that the set X itself may have any cardinality.  The most general type of Boolean function is a map f : X \to \mathbb{B}.  This is known as a Boolean-valued function since only its functional values need be in \mathbb{B}.

A function of the type f : X \to \mathbb{B} is called a characteristic function in set theory or an indicator function in probability and statistics since it can be taken to characterize or indicate a particular subset S of X, namely, the fiber or inverse image of the value 1, for which we have the notation and definition f^{-1}(1) = \{ x \in X : f(x) = 1 \}.

The notation f_S is often used for the characteristic function of a subset S of X.  Putting all the pieces together then, we have f_S^{-1}(1) = S \subseteq X.

To be continued …

Posted in Abstraction, Boole, Boolean Functions, C.S. Peirce, Category Theory, Characteristic Functions, Euler, Indicator Functions, John Venn, Logic, Mathematics, Peirce, Propositional Calculus, Set Theory, Venn Diagrams, Visualization | Tagged , , , , , , , , , , , , , , , | Leave a comment