C.S. Peirce : Information = Comprehension × Extension

Posted on May 8, 2013 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • A Most Perplexing Mystery

The inverse relationship between symmetry and diversity — that we see for example in the lattice-inverting map of a Galois correspondence — is a variation on an old theme in logic called the “inverse proportionality of comprehension and extension”.

C.S. Peirce, in his probings of the “laws of information”, found this principle to be a special case of a more general formula, saying that the reciprocal relation holds only when the quantity of information is constant.

Readings

Posted in C.S. Peirce, Comprehension, Diversity, Extension, Information, Information = Comprehension × Extension, Inquiry, Intension, Logic, Logic of Science, Peirce, Reciprocity, Semiotics, Sign Relations, Symmetry, Variety | Tagged , , , , , , , , , , , , , , , | 13 Comments

Rock On

Posted on May 6, 2013 by Jon Awbrey
Elsewhere I have brought out the fact that human will had no other purpose than to maintain awareness.  But that could not do without discipline.  Of all the schools of patience and lucidity, creation is the most effective.  It is also the staggering evidence of man’s sole dignity:  the dogged revolt against his condition, perseverance in an effort considered sterile.  It calls for a daily effort, self-mastery, a precise estimate of the limits of truth, measure, and strength.  It constitutes an ascesis.  All that “for nothing”, in order to repeat and mark time.  But perhaps the great work of art has less importance in itself than in the ordeal it demands of a man and the opportunity it provides him of overcoming his phantoms and approaching a little closer to his naked reality. (115)
⁂  
All that remains is a fate whose outcome alone is fatal.  Outside of that single fatality of death, everything, joy or happiness, is liberty.  A world remains of which man is the sole master.  What bound him was the illusion of another world.  The outcome of his thought, ceasing to be renunciatory, flowers in images.  It frolics — in myths, to be sure, but myths with no other depth than that of human suffering and, like it, inexhaustible.  Not the divine fable that amuses and blinds, but the terrestrial face, gesture, and drama in which are summed up a difficult wisdom and an ephemeral passion. (117–118)
⁂  
I leave Sisyphus at the foot of the mountain!  One always finds one’s burden again.  But Sisyphus teaches the higher fidelity that negates the gods and raises rocks.  He too concludes that all is well.  This universe henceforth without a master seems to him neither sterile nor futile.  Each atom of that stone, each mineral flake of that night-filled mountain, in itself forms a world.  The struggle itself toward the heights is enough to fill a man’s heart.  One must imagine Sisyphus happy. (123)

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, Diversity, Existentialism, Freedom, Myth, Oedipus, Passion, Revolt, Sisyphus | Tagged , , , , , , , , , | 14 Comments

Finding a Needle in a Cactus Patch

Posted on April 30, 2013 by Jon Awbrey

Re: R.J. Lipton • Sex, Lies, And Quantum Computers

Don’t know much about quantum computation, but my ventures in graphical syntaxes for propositional calculus did turn up a logical operator whose evaluation process reminded me a little of the themes involved in the collapse of the wave function.

Here is the essential information —

Boolean formulas constructed from minimal negation operators can be given graph-theoretic representation as “decorated” or “painted” versions of rooted cactus graphs.

Here are a couple of places where you can see some pictures and a description of the Fundamental Evaluation Rule for cactus expressions of propositional formulas.

Posted in Boolean Functions, C.S. Peirce, Cactus Graphs, Computational Complexity, Graph Theory, Logic, Logical Graphs, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Quantum Computing, Semiotics | Tagged , , , , , , , , , , , , | 4 Comments

Revolt, Freedom, Passion

Posted on April 23, 2013 by Jon Awbrey
Thus I draw from the absurd three consequences, which are my revolt, my freedom, and my passion. By the mere activity of consciousness I transform into a rule of life what was an invitation to death — and I refuse suicide. I know, to be sure, the dull resonance that vibrates throughout these days. Yet I have but a word to say: that it is necessary. When Nietzsche writes: “It clearly seems that the chief thing in heaven and on earth is to obey at length and in a single direction: in the long run there results something for which it is worth the trouble of living on this earth as, for example, virtue, art, music, the dance, reason, the mind — something that transfigures, something delicate, mad, or divine,” he elucidates the rule of a really distinguished code of ethics. But he also points the way of the absurd man. Obeying the flame is both the easiest and the hardest thing to do. However, it is good for man to judge himself occasionally. He is alone in being able to do so. (64–65)

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, Freedom, Inquiry, Method, Nietzsche, Passion, Revolt, Sisyphus | Tagged , , , , , , , , , | 3 Comments

Absurdum Quid

Posted on April 22, 2013 by Jon Awbrey
I am thus justified in saying that the feeling of absurdity does not spring from the mere scrutiny of a fact or an impression, but that it bursts from the comparison between a bare fact and a certain reality, between an action and the world that transcends it. The absurd is essentially a divorce. It lies in neither of the elements compared; it is born of their confrontation.

In this particular case and on the plane of intelligence, I can therefore say that the Absurd is not in man (if such a metaphor could have a meaning) nor in the world, but in their presence together. For the moment it is the only bond uniting them. If I wish to limit myself to facts, I know what man wants, I know what the world offers him, and now I can say that I also know what links them. I have no need to dig deeper. A single certainty is enough for the seeker. He simply has to derive all the consequences from it.

The immediate consequence is also a rule of method. The odd trinity brought to light in this way is certainly not a startling discovery. But it resembles the data of experience in that it is both infinitely simple and infinitely complicated. Its first distinguishing feature in this regard is that it cannot be divided. To destroy one of its terms is to destroy the whole. There can be no absurd outside the human mind. Thus, like everything else, the absurd ends with death. But there can be no absurd outside this world either. And it is by this elementary criterion that I judge the notion of the absurd to be essential and consider that it can stand as the first of my truths.

 (30–31)

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, Inquiry, Method, Peirce, Pragmatic Maxim, Pragmatism, Sisyphus, Tertium Quid, Thirdness, Triadicity | Tagged , , , , , , , , , , , | 1 Comment

Infinite Uses → Finite Means

Posted on April 17, 2013 by Jon Awbrey
The idea that a language is based on a system of rules determining the interpretation of its infinitely many sentences is by no means novel.  Well over a century ago, it was expressed with reasonable clarity by Wilhelm von Humboldt in his famous but rarely studied introduction to general linguistics (Humboldt, 1836).  His view that a language “makes infinite use of finite means” and that its grammar must describe the processes that make this possible is, furthermore, an outgrowth of a persistent concern, within rationalistic philosophy of language and mind, with this “creative” aspect of language use. (v)
⁂  
Although it was well understood that linguistic processes are in some sense “creative”, the technical devices for expressing a system of recursive processes were simply not available until much more recently.  In fact, a real understanding of how a language can (in Humboldt’s words) “make infinite use of finite means” has developed only within the last thirty years, in the course of studies in the foundations of mathematics. (8)

Reference

  • Noam Chomsky (1965), Aspects of the Theory of Syntax, MIT Press, Cambridge, MA.

Resource

Posted in Automata, Chomsky, Descartes, Finite Means, Formal Grammars, Formal Languages, Foundations of Mathematics, Infinite Use, Innate Ideas, Linguistics, Pigeonhole Principle, Recursion, Syntax, Wilhelm von Humboldt | Tagged , , , , , , , , , , , , , | Leave a comment

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 , , , , , , , | 2 Comments

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