Aforismos e Citações

"A society which is uninterested in metaphysics will have no theoretical science."
(Mary B. Hesse, in Forces and Fields)

"The laws of nature are but the mathematical thoughts of God."

"The highest form of pure thought is in mathematics."

"The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful."

"A mathematician is a machine for turning coffee into theorems."
(Alfréd Rényi)

"Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever."
(Niels Hendrik Abel)

"Anyone attempting to generate random numbers by deterministic means is, of course, living in a state of sin."
(John von Neumann)

"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
(John von Neumann)

"But every error is due to extraneous factors (such as emotion and education); reason itself does not err."
(Kurt Gödel)

"One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them."
(Heinrich Hertz)

"On this view, everything in the physical universe is indeed governed in completely precise detail by mathematical principles - perhaps by equations, (...) or perhaps by some future mathematical notions fundamentally different from those which we would label by the term 'equations'. If this is right, then even our own physical actions would be entirely subject to such ultimate mathematical control, where 'control' might still allow for some random behaviour governed by strict probabilistic principles. Many people fell uncomfortable with contentions of this kind, and I must confess to having some unease with it myself. Nonetheless, my personal prejudices are indeed to favour a viewpoint of this general nature, since it is hard to see how any line can be drawn to separate physical actions under mathematical control from those which might lie beyond it. In my own view, the unease that many readers may share with me on this issue partly arises from a very limited notion of what 'mathematical control' might entail."
(Roger Penrose, in Road to Reality)

"In mathematics, it is indeed imperative to be absolutely clear that one's equations make strick sense. However, it is equaly important not to be insensitive to 'things going on behind the scenes' which may ultimately lead to deeper insights. It is easy to lose sight of such things by adhering too rigidly to what appears to be strictly logical..."
(Roger Penrose, in Road to Reality)

"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
(Albert Einstein)

"I wished to show that space-time is not necessarily something to which one can ascribe a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept 'empty space' loses its meaning."
(Albert Einstein, in 'Relativity: The Special and the General Theory')

"The scientist does not study nature because it is useful, he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would be not worth knowing, and if nature were not worth knowing, life would not be worth living."
(Henry Poincare)

"Physics can't exist without mathematics which provides it with the only language in which it can speak. Thus, services are continuously exchanged between pure mathematical analysis and physics. It is really remarkable that among works of analysis most useful for physics were those cultivated for their own beauty. In exchange, physics, exposing new problems, is as useful for mathematics as it is a model for an artist".
(Henry Poincare)

"The profound study of nature is the most fertile source of mathematical discoveries." (Joseph Fourier) "It seems to be one of the fundamental features of nature that fundamental physics laws are described in terms of great beauty and power. As time goes on, it becomes increasingly evident that the rules that the mathematician finds interesting are the same as those that nature has chosen."
(Paul Dirac)

"Astronomy itself is a venerable science, and might become a stepping stone to something more august, a science which I think is a convenient passage to mystical theology, for the happy body of heaven has matter underneath it, and its motion has seemed to the leaders in philosophy to be an imitation of mind. It proceeds to its demonstrations in no uncertain way, for it uses as its servants geometry and arithmetic, which it would not be improper to call a fixed standard of truth."
(Synesius of Cyrene, who was a pupil of Hypatia of Alexandria) "

'Immortality' may be a silly word, but probably a mathematician has the best chance of whatever it may mean."
(G. H. Hardy, in A Mathematician's Apology)

"A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. (...) The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics."
(G. H. Hardy, in A Mathematician's Apology)

"I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply our notes of our observations"
(G. H. Hardy, in A Mathematician's Apology)

"Hilbert's formalism was just one example (...) of the spirit that lies at the very heart of the twentieth century. At its core it involves the dominance of form over content, syntax over semantics, proof over truth. It is no surprise that the principal embodiment of a formal system, the computer, a pure syntax machine..., would become the century's dominant mechanical device."
(Palle Yourgrau)

"But also we know from Faust that 'All theory, dear friend, is gray, and the golden tree of life is green.' Gray is the ascetic aspect of color. Such is its symbolic value in ordinary language, and it is to this symbol that Goethe alludes. To be gray is the most that color can do when it desires to renounce its role as color; on the other hand, life is a green tree - which is an extravagant statement - and moreover that green tree of life turns out to be gilded, which is a still greater extravagance. This elegant wish to take refuge in gray when confronted with the marvelous and contradictory chromatic extravagance of life leads us to theorize. In theory we exchange reality for its ghost; this is what concepts are. Rather than living life we think about it. Yet who knows whether under this apparent asceticism and withdrawal from life, which is what hard thinking is, there may not be hidden a maximum form of vitality, its supreme luxury! Who knows if thinking about life may not be a way of adding to the ingenuous process of living a magnificent eagerness for super-living"
(José Ortega Y Gasset, in What is Philosophy?, pp 120-121)

" fact mathematicians are like theologians: we regard existence as the prime attribute of what we study. But unlike theologians, we need not always rely upon faith alone."
(Lawrence C. Evans, in Partial Differential Equations, pp 9)

"If Euler were alive today, he wouldn't be proving existence theorems."
(Lloyd N. Trefethen, in Numerical Linear Algebra, pp 324)

"One good theory is worth a thousand computer runs."
(Luenberger & Ye, in Linear and Nonlinear Programming, pp 6)

" ...although nature begins with the cause and ends with the experience we must follow the opposite course, namely ...begin with the experience and by means of it end with the cause."
(Leonardo da Vinci)

"Our science is from the watching of shadows."
(Ezra Pound)

" ...we have to remember that what we observe is not nature in itself but nature exposed to our method of questioning."
(Werner Heisenberg)

"Philosophy is written in that great book which ever lies before our gaze -- I mean the universe ...The book is written in the mathematical language ...without which one wanders in vain through a dart labyrinth."
(Galileo Galilei)

"All of exact science is dominated by the idea of approximation."
(Bertrand Russell)

"...numerical precision is the very soul of science..."
(D'Arcy Wentworth Thompson)

"Eventually, we reach the dim boundary ...There, we measure shadows..."
(Edwin Hubble)

"Sans les mathématiques on ne pénètre point au fond de la philosophie. Sans la philosophie on ne pénètre point au fond des mathématiques. Sans les deux on ne pénètre au fond de rien."