Mathematics
198 quotes
Biography
"Rather like the way the Hubble Space Telescope has made a significant contribution to astronomy in enabling astronomers to discover hidden structures and properties of our distant universe, dynamic geometry software has allowed new worlds to become viewable and tangible in mathematics and particularly in geometry."
"Any author who uses mathematics should always express in ordinary language the meaning of the assumptions he admits, as well as the significance of the results obtained. The more abstract his theory, the more imperative this obligation. In fact, mathematics are and can only be a tool to explore reality. In this exploration, mathematics do not constitute an end in itself, they are and can only be a means."
"In doing mathematics, I express something personal. It is a source of joy to know that, despite this personal aspect, the fruit of my work can be of interest to other mathematicians."
"It was mathematics, the non-empirical science par excellence, wherein the mind appears to play only with itself, that turned out to be the science of sciences, delivering the key to those laws of nature and the universe that are concealed by appearances."
"Now comes the Einstein–Podolsky–Rosen entangled state. Now I see faces, people saying, "Oh..?" Don't worry! When you go to the concert, you don't need to be able to read the music, to enjoy the music. ...So here... [are] equations. It's a pleasure for my colleague physicists. If you can't read the equation, listen to me. I'm not going to sing, but... listen to the words... the words are... a way of describing the equations, and you don't need to know the mathematics..."
"Confused is of course the best state a mathematician can be in; the struggle out of that state is the primary drive for progress."
"If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.<!-- Ch. 5, p. 211 -->"
"Where there is life there is a pattern, and where there is a pattern there is mathematics. Once that germ of rationality and order exists to turn a chaos into a cosmos, then so does mathematics. There could not be a non-mathematical Universe containing living observers.<!-- Ch. 5, p. 230 -->"
"We say that the string is 'random' if there is no other representation of the string which is shorter than itself. But we will say that it is 'non-random' if there does exist such an abbreviated representation. ...In general, the shorter the possible representation... the less random... On this view we recognize science to be the search for algorithmic compressions. ...It is simplest to think of mathematics as the catalogue of all possible patterns. ...When viewed in this way, it is inevitable that the world is described by mathematics. ...In many ways the search for a Theory of Everything is a manifestation of a faith that this compression goes all the way down to the bedrock of reality..."
"Mathematics became an experimental subject. Individuals could follow previously intractable problems by simply watching what happened when they were programmed into a personal computer. ...The PC revolution has made science more visual and more immediate ...by creating films of imaginary experiences of mathematical worlds. ...Words are no longer enough."
"Something more than impeccable logic is required in mathematics. An expert logician will not necessarily be a passable mathematician for all his skill in logic, any more than a scholarly prosodist will be a respectable poet for all his mastery of meter."
"A narrative of the decisive epochs in the development of mathematics was wanted. ...Numerous professionals... know from hard experience what mathematical invention means. ...Whoever has himself attempted to advance mathematics is inclined to be more skeptical than the average spectator toward any alleged anticipation of notable progress. ...often what looks like an anticipation ...was not even aimed in the right direction. ...when at length progress started; it proceeded along lines totally different from those which, in retrospect, it 'should' have followed.<!--pp.vi-viii-->"
"Nearly always it is the recondite and complicated which is elaborated first; and it is only when some relatively unsophisticated mind attacks a problem that its deep simplicity is revealed."
"Ruth felt that math was like sex—get all you can, but best not done in public."
"The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason."
"Over time you will get the wrong answer more times than you get the right answer. That’s not a problem! You’ve learnt what doesn’t work, just try again."
"I would myself say that the purely imaginary objects are the only realities, the ὂντως ὂντα [truest things], in regard to which the corresponding physical objects are as the shadows in the cave; and it is only by means of them that we are able to deny the existence of a corresponding physical object; and if there is no conception of straightness, then it is meaningless to deny the conception of a perfectly straight line."
"You don’t need anybody’s permission to be a great mathematician!"
"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."
"Geometry is that of mathematical science which is devoted to consideration of form and size, and may be said to be the best and surest guide to study of all sciences in which ideas of dimension or space are involved. Almost all the knowledge required by navigators, architects, surveyors, engineers, and opticians, in their respective occupations, is deduced from geometry and branches of mathematics. All works of art are constructed according to the rules which geometry involves; and we find the same laws observed in the works of nature. The study of mathematics, generally, is also of great importance in cultivating habits of exact reasoning; and in this respect it forms a useful auxiliary to logic."
"There is probably no other science which presents such different appearances to one who cultivates and one who does not, as mathematics. To [the non-mathematician] it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is yet in the purple of bloom of vigorous youth, everywhere stretching out after the "attainable but unattained," and full of the excitement of nascent thoughts; its logic is beset with ambiguities, and its analytic processes, like Bunyan's road, have a quagmire on one side and a deep ditch on the other, and branch off into innumerable by-paths that end in a wilderness."
"Mathematics is the study of anything that obeys the rules of logic, using the rules of logic."
"The theory of the nature of mathematics is extremely reactionary. We do not subscribe to the fairly recent notion that mathematics is an abstract language based, say, on set theory. In many ways, it is unfortunate that philosophers and mathematicians like Russell and Hilbert were able to tell such a convincing story about the meaning-free formalism of mathematics. In Greek, mathematics simply meant learning, and we have adapted this... to define the term as "learning to decide." Mathematics is a way of preparing for decisions through thinking. Sets and classes provide one way to subdivide a problem for decision preparation; a set derives its meaning from decision making, and not vice versa."
"[F]inding direct measurement so often impossible, we are compelled to devise means of doing it indirectly. Hence arose Mathematics."
"To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples..."