Finding a Needle in a Cactus Patch

Posted on April 30, 2013 by

Re: 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 is a place where you can see some pictures and a description of the Fundamental Evaluation Rule for cactus expressions of propositional formulas.

Posted in Boolean Functions, Cactus Graphs, Graph Theory, Logic, Logical Graphs, Minimal Negation Operators, Peirce, Propositional Calculus | Tagged , , , , , , , | 4 Comments

Revolt, Freedom, Passion

Posted on April 23, 2013 by
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
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
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)

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

☞ Pigeonhole Principle

Posted in Chomsky, Finite Means, Formal Language Theory, Formal Languages, Foundations of Mathematics, Generative Grammar, Infinite Use, Linguistics, Pigeonhole Principle, Recursion, Syntax, Wilhelm von Humboldt | Tagged , , , , , , , , , , , | Leave a comment

The Way The Cookie Uncrumbles Itself

Posted on April 16, 2013 by

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 , , , , , , , | Leave a comment

What It Is

Posted on April 13, 2013 by

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

Re: Zeno Proof Paradox

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

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, Change, Infinity, Lewis Carroll, Logic, Mathematics, Meno, Modus Ponens, Motion, Paradox, Peirce, Phenomenology, Sisyphus, Syllogism, Zeno | Tagged , , , , , , , , , , , , , , | 2 Comments