Proposition Infinity Quotes

My anxieties as to behavior are futile, ever more so, to infinity. If the other, incidentally or negligently, gives the telephone number of a place where he or she can be reached at certain times, I immediately grow baffled: should I telephone or shouldn't I? (It would do no good to tell me that I can telephone - that is the objective, reasonable meaning of the message - for it is precisely this permission I don't know how to handle.) What is futile is what apparently has and will have no consequence. But for me, an amorous subject, everything which is new, everything which disturbs, is received not as a fact but in the aspect of a sign which must be interpreted. From the lover's point of view, the fact becomes consequential because it is immediately transformed into a sign: it is the sign, not the fact, which is consequential (by its aura). If the other has given me this new telephone number, what was that the sign of? Was it an invitation to telephone right away, for the pleasure of the call, or only should the occasion arise, out of necessity? My answer itself will be a sign, which the other will inevitably interpret, thereby releasing, between us, a tumultuous maneuvering of images. Everything signifies: by this proposition, I entrap myself, I bind myself in calculations, I keep myself from enjoyment. Sometimes, by dint of deliberating about "nothing" (as the world sees it), I exhaust myself; then I try, in reaction, to return -- like a drowning man who stamps on the floor of the sea -- to a spontaneous decision (spontaneity: the great dream: paradise, power, delight): go on, telephone, since you want to! But such recourse is futile: amorous time does not permit the subject to align impulse and action, to make them coincide: I am not the man of mere "acting out" -- my madness is tempered, it is not seen; it is right away that I fear consequences, any consequence: it is my fear -- my deliberation -- which is "spontaneous.

That is to say that the truth values of every proposition within the reach of the mathematical system are already determined, and mathematicians essentially explore the system to find those truth values. It has a real feeling of discovery to it, but the underlying axioms are where we made it up, to put it loosely. Since many of the simpler axioms are based on our “self-evident” experience of reality, the map closely matches the terrain, and it is easy to fall into the trap of thinking the description is reality.

Let us weigh the gain and the loss in wagering that God is [exists]. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is. . . There is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite. And so our proposition is of infinite force, when there is the finite to stake in a game where there are equal risks of gain and of loss, and the infinite to gain.

Many of the really great, famous proofs in the history of math have been reduction proofs. Here's an example. It is Euclid's proof of Proposition 20 in Book IX of the Elements. Prop. 20 concerns the primes, which-as you probably remember from school-are those integers that can't be divided into smaller integers w/o remainder. Prop. 20 basically states that there is no largest prime number. (What this means of course is that the number of prime numbers is really infinite, but Euclid dances all around this; he sure never says 'infinite'.) Here is the proof. Assume that there is in fact a largest prime number. Call this number Pn. This means that the sequence of primes (2,3,5,7,11,...,Pn) is exhaustive and finite: (2,3,5,7,11,...,Pn) is all the primes there are. Now think of the number R, which we're defining as the number you get when you multiply all the primes up to Pn together and then add 1. R is obviously bigger than Pn. But is R prime? If it is, we have an immediate contradiction, because we already assumed that Pn was the largest possible prime. But if R isn't prime, what can it be divided by? It obviously can't be divided by any of the primes in the sequence (2,3,5,...,Pn), because dividing R by any of these will leave the remainder 1. But this sequence is all the primes there are, and the primes are ultimately the only numbers that a non-prime can be divided by. So if R isn't prime, and if none of the primes (2,3,5,...,Pn) can divide it, there must be some other prime that divides R. But this contradicts the assumption that (2,3,5,...,Pn) is exhaustive of all the prime numbers. Either way, we have a clear contradiction. And since the assumption that there's a largest prime entails a contradiction, modus tollens dictates that the assumption is necessarily false, which by LEM means that the denial of the assumption is necessarily true, meaning there is no largest prime. Q.E.D.

It would take someone exempt from all these qualities so that, without preoccupation in his judgment, he would judge of these propositions as indifferent to him; and for this reason we would need a judge who never was. To judge appearances that we receive from subjects, we would need a judicatory instrument; to verify that instrument, we would need demonstration; to verify the demonstration, an instrument; here we are going round a circle. Since the senses cannot stop our dispute, being themselves full of uncertainty, it must be up to reason; no reason can be established without another reason: here we are regressing to infinity. Our imagination does not apply itself to foreign objects, but is formed through the mediation of the senses; and the senses do not understand a foreign object, but only their own passions; and thus what we imagine and what appears to us are not from the object, but only

Every conclusion supposes premises; these premises themselves either are self-evident and need no demonstration, or can be established only by relying upon other propositions, and since we can not go back thus to infinity, every deductive science, and in particular geometry, must rest on a certain number of undemonstrable axioms.

And I turned to the examination of that same theology which I had once rejected with such contempt as unnecessary. Formerly it seemed to me a series of unnecessary absurdities, when on all sides I was surrounded by manifestations of life which seemed to me clear and full of sense; now I should have been glad to throw away what would not enter a health head, but I had nowhere to turn to. On this teaching religious doctrine rests, or at least with it the only knowledge of the meaning of life that I have found is inseparably connected. However wild it may seem too my firm old mind, it was the only hope of salvation. It had to be carefully, attentively examined in order to understand it, and not even to understand it as I understand the propositions of science: I do not seek that, nor can I seek it, knowing the special character of religious knowledge. I shall not seek the explanation of everything. I know that the explanation of everything, like the commencement of everything, must be concealed in infinity. But I wish to understand in a way which will bring me to what is inevitably inexplicable. I wish to recognize anything that is inexplicable as being so not because the demands of my reason are wrong (they are right, and apart from them I can understand nothing), but because I recognize the limits of my intellect. I wish to understand in such a way that everything that is inexplicable shall present itself to me as being necessarily inexplicable, and not as being something I am under an arbitrary obligation to believe.