Recycled from a question I asked on MathOverFlow.
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.
| 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) |
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| 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) |
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| 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.
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.
| 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.
