Mathematics

"Anaximander... taught that the primary substance was infinite, eternal and ageless and... encompassed the world. This... is transformed into the various substances... Theophrastus quotes from Anaximander: "Into that from which things take their rise they pass away once more... for they make reparation and satisfaction to one another for their injustice according to the ordering of time." In this... the antithesis of Being and Becoming plays the fundamental role. The primary substance, infinite... ageless... undifferentiated Being, degenerates into... forms which lead to endless struggles. ...Becoming is ...a ...debasement of the infinite Being—a disintegration into the struggle ultimately expiated by a return into that ...without shape or character. The struggle ...is the opposition between hot and cold, fire and water, wet and dry, etc. ...[T]emporary victory ...is the injustice for which they ...make reparation in the ordering of time. ...[T]here is "eternal motion," the creation and passing away of worlds from infinity to infinity."

"Game theory is logically demanding, but on a practical level, it requires surprisingly few mathematical techniques. Algebra, calculus, and basic probability theory suffice. ...the stress placed on game-theoretic rigor in recent years is misplaced. Theorists could worry more about the empirical relevance of their models and take less solace in mathematical elegance. ...[I]f a proposition is proved for a model with a finite number of agents, it is... irrelevant whether it is true for an ifinite number... There are... only a finite number of people, or even bacteria. Similarly, if something is true in games in which payoffs are finitely divisible... it does not matter whether it is true when payoffs are infinitely divisible. There are no payoffs in the universe... infinitely divisible. Even time... continuous in principle, can be measured only by devices with a finite number of s. Of course, models based on the real and complex numbers can be hugely useful, but they are just approximations... There is... no intrinsic value of a theorem that is true for a continuum of agents on a , if it is also true for a finite number of agents of a finite choice space."

"Although I might very rationally put it in dispute, whe­ther there be any such centre in nature, or no; being that neither you nor any one else hath ever proved, whether the World be fi­nite and figurate, or else infinite and interminate; yet nevertheless granting you, for the present, that it is finite, and of a terminate Spherical Figure, and that thereupon it hath its centre; it will be requisite to see how credible it is that the Earth, and not rather some other body, doth possesse the said centre."

"I must have recourse to a Philosophical distinction, and say that the understanding is to be taken two ways, that is intensivè, or extensivè; and that extensive, that is, as to the multitude of intel­ligibles, which are infinite, the understanding of man is as nothing, though he should understand a thousand propositions; for that a thousand, in respect of infinity is but as a cypher: but taking the understanding intensive, (in as much as that term imports) intensively, that is, perfectly some propositions, I say, that humane wis­dom understandeth some propositions so perfectly, and is as abso­lutely certain thereof, as Nature herself; and such are the pure Mathematical sciences, to wit, Geometry and Arithmetick: in which Divine Wisdom knows infinite more propositions, because it knows them all; but I believe that the knowledge of those few compre­hended by humane understanding, equalleth the divine, as to the certainty objectivè, for that it arriveth to comprehend the neces­sity thereof, than which there can be no greater certainty."

"Nature doth, and in it alone is discovered an infinite wisdom, so that Divine Wisdom may be concluded to be infinitely infinite."

"Neither one nor the other doth follow, for that both the assertions may be true. The Oracle adjudged Socrates the wi­sest of all men, whose knowledg is limited; Socrates acknowledgeth that he knew nothing in relation to absolute wisdome, which is infinite; and because of infinite, much is the same part as is little, and as is nothing (for to arrive... to the infinite number, it is all one to accumulate thousands, tens, or ciphers,) therefore Socrates well perceived his wisdom to be nothing, in comparison of the infinite knowledg which he wanted. But yet, because there is some knowledg found amongst men, and this not equally shared to all, Socrates might have a greater share thereof than others, and therefore verified the answer of the Oracle."

"I will now say something which may perhaps astonish you; it refers to the possibility of dividing a line into its infinitely small elements by following the same order which one employs in dividing the same line into forty, sixty, or a hundred parts, that is, by dividing it into two, four, etc. He who thinks that, by following this method, he can reach an infinite number of points is greatly mistaken; for if this process were followed to eternity there would still remain finite parts which were undivided. ... Indeed by such a method one is very far from reaching the goal of indivisibility; on the contrary he recedes from it and while he thinks that, by continuing this division and by multiplying the multitude of parts, he will approach infinity, he is... getting farther and farther away from it. My reason is this. In the preceding discussion we concluded that, in an infinite number, it is necessary that the squares and cubes should be as numerous as the totality of the natural numbers [tutti i numeri], because both of these are as numerous as their roots which constitute the totality of the natural numbers. Next we saw that the larger the numbers taken the more sparsely distributed were the squares, and still more sparsely the cubes; therefore it is clear that the larger the numbers to which we pass the farther we recede from the infinite number; hence it follows that since this process carries us farther and farther from the end sought, if on turning back we shall find that any number can be said to be infinite, it must be unity. Here indeed are satisfied all those conditions which are requisite for an infinite number; I mean that unity contains in itself as many squares as there are cubes and natural numbers [tutti i numeri]. ... There is no difficulty in the matter because unity is at once a square, a cube, a square of a square, and all the other powers [dignitā]; nor is there any essential peculiarity in squares or cubes which does not belong to unity; as, for example, the property of two square numbers that they have between them a mean proportional; take any square number you please as the first term and unity for the other, then you will always find a number which is a mean proportional. Consider the two square numbers, 9 and 4; then 3 is the mean proportional between 9 and 1 [\frac{1}{3} = \frac{3}{9}]; while 2 is a mean proportional between 4 and 1 [\frac{1}{2} = \frac{2}{4}]; between 9 and 4 we have 6 as a mean proportional [\frac{4}{6} = \frac{6}{9}]. A property of cubes is that they must have between them two mean proportional numbers; take 8 and 27; between them lie 12 and 18 [\frac{8}{12} = \frac{18}{27}]; while between 1 and 8 we have 2 and 4 intervening [\frac{1}{2} = \frac{4}{8}]; and between 1 and 27 there lie 3 and 9 [\frac{1}{3} = \frac{9}{27}]. Therefore we conclude that unity is the only infinite number. These are some of the marvels which our imagination cannot grasp and which should warn us against the serious error of those who attempt to discuss the infinite by assigning to it the same properties which we employ for the finite, the natures of the two having nothing in common."

"As to the query whether the finite parts of a limited continuum [continuo terminato] are finite infinite in number I will... answer that they are neither finite or infinite."

"If I should ask... how many squares there are one might reply truly that there are as many as the corresponding number of roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square. ... But if I inquire how many roots there are, it cannot be denied that there are as many as there are numbers because every number is a root of some square. This being granted we must say that there are as many squares as there are numbers because they are just as numerous as their roots, and all the numbers are roots. Yet at the outset we said there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers. ... So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former, and finally the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite quantities. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number. Or if I had replied to him that the points in one line were equal in number to the squares; in another, greater than the totality of numbers; and in the little one, as many as the number of cubes, might I not, indeed, have satisfied him by thus placing more points in one line than in another and yet maintaining an infinite number in each. So much for the first difficulty."

"The world sings of an infinite Love: how can we fail to care for it?"

"Anaximander gave up the idea that water or any other known substance might be the first principle, and held that this is of the nature of the infinite, that is, matter without any determinate property except that of being infinite. All things are developed out of this and return to it again, so that an infinite series of worlds have been generated and have in turn become again resolved into the abstract mass."

"Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so ad infinitum, And the great fleas themselves, in turn, have greater fleas to go on, While these again have greater still, and greater still, and so on."

"Although velocity was a relative in Newtonian science, yet there did not exist one definite velocity which was assumed to be absolute. This was the infinite velocity. It was assumed that a velocity that was infinite or instantaneous for one observer would remain infinite or instantaneous for all other observers. So far... as velocity was concerned, the sole difference between Einstein and Newton is that with Einstein the absolute or invariant velocity is no longer infinite. ...It is this difference between the invariant velocities of Newton and Einstein which is responsible for all the major differences between classical and Einsteinian science... it is this finiteness of the invariant velocity which precludes us from attaching any absolute value to shape and distance. Einstein's theory proves that molar matter can never move with the absolute speed of light. We are therefore perfectly justified in saying that the velocity of matter remains essentially relative, since it can never attain that critical velocity (i.e., that of light) which is absolute."

"Algorithms are the computational content of proofs."

"The issue with AI, to me, is a very simple one. It's like the term algorithm. We watch companies use algorithms, and now AI, as a means of evading responsibility for their actions… If we endorse the view that AI is all-powerful, we are endorsing the view that it can alleviate people of responsibility for their actions—militarily, socio­economically, whatever. The biggest danger of AI is that we attribute these godlike characteristics to it and therefore let ourselves off the hook. I don't know what the mythological underpinnings of this are, but throughout history there's this tendency of human beings to create false idols, to mold something in our own image and then say we've got godlike powers because we did that."

"Mathematics is what we want to keep for ourselves. When playing games, we stick to the rules (or we are changing the game...), but when doing serious mathematics (not executing algorithms) we make up the rules—definitions, axioms... even logics. ...[I]n arithmetic we find s, which are a whole new 'game'... [T]o identify mathematics with games would be one of those part-for-whole mistakes (like 'all geometry is projective geometry' or 'arithmetic is just logic' from the nineteenth century)... [M]y separation of game analysis from playing games tells... against the analogy of mathematics to the expert play of the game itself."

"Numerical analysis is very much an experimental science."

"Algorithms + Data Structures = Programs"

"People seem to flinch at the word probability—it has too many syllables. Besides, it sounds mathematical, and it’s become politically correct in our country to be proud of not knowing any mathematics. (We’re already paying the price for that.)"

"As is known, the question of the objectivity or the subjectivity of probability has divided the world of science into two camps. Some maintain that there exist two types of probability, as above, others, that only the subjective exists, because regardless of what is supposed to take place, we cannot have full knowledge of it. Therefore, some lay the uncertainty of future events at the door of our knowledge of them, whereas others place it within the realm of the events themselves."

"The epistemological value of probability theory is based on the fact that chance phenomena, considered collectively and on a grand scale, create non-random regularity."

"No human being can give an eternal resolution to another or take it from him; If someone objects that then one might just as well be silent if there is no probability of winning others, he thereby has merely shown that although his life very likely thrived and prospered in probability and everyone of his undertakings in the service of probability went forward, he has never really ventured and consequently has never had or given himself the opportunity to consider that probability is an illusion, but to venture the truth is what gives human life and the human situation pith and meaning, to venture is the fountainhead of inspiration, whereas probability is the sworn enemy of enthusiasm, the mirage whereby the sensate person drags out time and keeps the eternal away, whereby he cheats God, himself, and his generation: cheats God of the honor, himself of liberating annihilation, and his generation of the equality of conditions."

"They should have known better. The probability of a train derailment was infinitesimal. That meant it was only a matter of time."

"R. A. Fisher, J. Neyman, R. von Mises, W. Feller, and L. J. Savage denied vehemently that probability theory is an extension of logic, and accused Laplace and Jeffreys of committing metaphysical nonsense for thinking that it is."

"There is no Algebraist nor Mathematician so expert in his science, as to place entire confidence in any truth immediately upon his discovery of it, or regard it as any thing, but a mere probability. Every time he runs over his proofs, his confidence increases; but still more by the approbation of his friends; and is raised to its utmost perfection by the universal assent and applauses of the learned world."

"Probability is too important to be left to the experts. [...] The experts, by their very expert training and practice, often miss the obvious and distort reality seriously. [...] The desire of the experts to publish and gain credit in the eyes of their peers has distorted the development of probability theory from the needs of the average user. The comparatively late rise of the theory of probability shows how hard it is to grasp, and the many paradoxes show clearly that we, as humans, lack a well grounded intuition in the matter. Neither the intuition of the man in the street, nor the sophisticated results of the experts provides a safe basis for important actions in the world we live in."

"My thesis, paradoxically, and a little provocatively, but nonetheless genuinely, is simply this :PROBABILITY DOES NOT EXIST.The abandonment of superstitious beliefs about the existence of Phlogiston, the Cosmic Ether, Absolute Space and Time, ... , or Fairies and Witches, was an essential step along the road to scientific thinking. Probability, too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception, an illusory attempt to exteriorize or materialize our true probabilistic beliefs."

"Al right. I already see you turning off. I can see you say you don't understand me. You can't understand that it could be chance. "I don't like it!" Tough! I don't like it either, but that's the way it is! Ok? I don't understand it either. ..."It must be that Nature knows that it's going to go up or down." No, it must not be that nature knows! We are not to tell Nature what she's gotta be! That's what we found out. Every time we take a guess as how she's got to be, and go and measure... She's clever. She's always got better imagination than we have, and she finds a cleverer way to do it than we have thought of. And in this particular case, the clever way to do it is by probability, by odds. ...[L]ight works by probability."

"It is a truth very certain that, when it is not in our power to determine what is true, we ought to follow what is most probable"

"Probability is the very guide of life."

"This statistical regularity in moral affairs fully establishes their being under the presidency of law. Man is now seen to be an enigma only as an individual; in the mass he is a mathematical problem."

"How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?"

"Einstein's supreme greatness was in transforming physical thinking from that of the culmination of classical physics about 1900 to that of quantum mechanics starting about 1925. ...far more than anyone else, he caused physicists to think in terms of probabilities. He began to do this in his early work in thermodynamics, and he brought such thinking to its first great fruition in 1905 in his work on Brownian movement and in his first work on radiation, in which he introduced the concept of light quanta or photons. Its second, even greater fruition was in his famous paper on the quantum theory of radiation in 1917. That paper illustrated methods that have been in use almost without change ever since, even though the majority of the users have no knowledge that it was Einstein who propounded them. It was in this paper that Einstein postulated the various transition possibilities between two states of a quantized system. ...quantum theory has existed ever since precisely for the purpose of evaluating these probabilities. In this paper of 1917 Einstein postulated in particular the process known as stimulated emission, and inferred the properties of this process. This is the process employed in the... light maser or laser."

"In applying dynamical principles to the motion of immense numbers of atoms, the limitation of our faculties forces us to abandon the attempt to express the exact history of each atom, and to be content with estimating the average condition of a group of atoms large enough to be visible. This method... which I may call the statistical method, and which in the present state of our knowledge is the only available method of studying the properties of real bodies, involves an abandonment of strict dynamical principles, and an adoption of the mathematical methods belonging to the theory of probability. … If the actual history of Science had been different, and if the scientific doctrines most familiar to us had been those which must be expressed in this way, it is possible that we might have considered the existence of a certain kind of contingency a self evident truth, and treated the doctrine of philosophical necessity as a mere sophism."

"Feeling threatened is not the same as being threatened, but the difference gets lost. The danger from low levels of radiation is quite low, as expressed as morbidity statistics or probabilities, but there is an unfortunate lack of connection to probability in the average person. Low probabilities are a particular problem of perception. If they were not, then nobody would play the lottery and the gambling industry would collapse."

"The laws of probability are mighty powerful, and they never sleep. If this were more widely understood there’d be a lot less crowing about good luck, and a lot less guilt about bad luck. And we’d have a more civilized world. Some things really do happen by chance, and there is little we can do to change that."

"Since the end of World War II, I have been working on the many ramifications of the theory of messages. Besides the electrical engineering theory of the transmission of messages, there is a larger field which includes not only the study of language but the study of messages as a means of controlling machinery and society, the development of computing machines and other such automata, certain reflections upon psychology and the nervous system, and a tentative new theory of scientific method. This larger theory of messages is a probabilistic theory, an intrinsic part of the movement that owes its origin to Willard Gibbs and which I have described in the introduction."

"At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate."

"Why do [people] confuse probability and expectation, that is, probability and probability times payoff? Mainly because much... schooling comes from examples in symmetric environments... the... bell curve... is entirely symmetric."

"The theory of probability combines commonsense reasoning with calculation. It domesticates luck, making it subservient to reason."

"The point of academic attacks is not exhibiting practical breaks; the point is that only a trained cryptographer can tell whether a given algorithm is secure or not. The author of an algorithm says: "My cipher is secure, and trust me, I am an expert at this. And to prove that I am a real good expert, I challenge other experts to find even the most impractical, academic flaw in my cipher"."

"Just like glue. Commercial ads state that the foobar glue can stick an elephant to the ceiling. Who needs to stick an elephant to the ceiling? But if it can do that, people will trust its sticking strength."

"We didn't do this with just a pencil and some paper. Lots of our notes are in pen. We didn't need to erase much."

"If you think cryptography is the answer to your problem, then you don't know what your problem is."

"Few false ideas have more firmly gripped the minds of so many intelligent men than the one that, if they just tried, they could invent a cipher that no one could break."

"The obvious mathematical breakthrough would be development of an easy way to factor large prime numbers."

"The real work in an attack, at least an attack against a well-designed cipher, is modifying the attack technique so that it works. Knudsen's papers are an excellent example of this; he is a master at making an attack work where others have failed. Differentials work where characteristics don't. Truncated differentials work where normal differentials don't. Even this year's exciting find, impossible differentials, are simply another way at looking at a differential attack. A cryptanalyst with a "menu" would have never found any of those attacks, and would have broken far fewer ciphers."

"The NSA response was, "Well, that was interesting, but there aren't any ciphers like that.""

"This method, seemingly very clever, actually played into our hands! And so it often happens that an apparently ingenious idea is in fact a weakness which the scientific cryptographer seizes on for his solution."

"There is also a side benefit: the difference in strength made by even really subtle changes warns us just how tricky crypto can be..."