math is beauty
For examples of beautiful math see this MathOverflow discussion
Select quotes made aware to me by J.S. Milne’s site:
Certainly the best times were when I was alone with mathematics, free of ambition and pretense, and indifferent to the world. (Langlands, Mathematicians: An Outer View of the Inner World, p 142)
Every mathematician worthy of the name has experienced, if only rarely, the state of lucid exaltation in which one thought succeeds another as if miraculously, and in which the unconscious (however one interprets that word) seems to play a role. (André Weil)
A mathematician is so rare an animal that he deserves to be preserved, be it only on the score of curiosity. (Simone Weil as quoted on p.396 of the Pétrement biography)
I knew then, sometimes, the fullness of contemplation, when all heard sounds contribute to a single vast harmony. But more often, what had been brought into the light immediately became the motivation and means for a new plunge into the mists, in pursuit of a new incarnation of That which remained forever mysterious, unknown, calling me constantly. (Grothendieck)
The ability to feel the beauty of mathematics is probably the most important ability to be a good mathematician. Sense of beauty is of course defined very differently between mathematicians. It is important that each mathematician have their own way to feel the beauty of mathematics. Otherwise we cannot be original. (Kenji Fukaya)
Mathematics has been for me, not only a profession, but also my preferred hobby. … Again and again I have been guided by a sense of the architecture of this edifice, to which we continue to add new wings and new floors while renovating the parts already constructed, into feeling that certain problems had priority as opening new perspectives or establishing a new foundation for future constructions. This is the professional point of view, but happily these problems were those that attracted me the most. In other instances I was not guided by such motives, being attracted only by curiosity, by the need to know the answer to an enigma, without reference to its importance in a general context. (Borel, Œuvres IV, p 376)
It is while doing mathematical research that one truly comes to see the beauty of mathematics. It faces you in those moments when the underlying simplicity of a question appears and its meaningless complications can be forgotten. In those moments a piece of a colossal logical structure is illuminated, and some of the meaning hidden in the nature of things is finally revealed. (Ruelle, The Mathematician’s Brain, p 130)
Other quotes I’ve encountered:
Mathematics…possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure… (Bertrand Russell,Mysticism and Logic)
The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head. (G.K. Chesterton, orthodoxy)
Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature. (Albert Einstein, Emmy Noether’s Obituary)
When I heard the learn’d astronomer, When the proofs, the figures, were ranged in columns before me, When I was shown the charts and diagrams, to add, divide, and measure them, When I sitting heard the astronomer where he lectured with much applause in the lecture-room, How soon unaccountable I became tired and sick, Till rising and gliding out I wander’d off by myself, In the mystical moist night-air, and from time to time,Look’d up in perfect silence at the stars. (Walt Whitman)
I asked: ‘What does mathematics mean to you?’, and some people answered: “The manipulation of numbers, the manipulation of structures.’ And if I had asked what music means to you, would you have answered: ’The manipulation of notes? (Serge Lang, The Beauty of Doing Mathematics)
Computer science is no more about studying computers than astronomy is about studying telescopes. (Edsger Dijkstra)
Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? (Paul Halmos, “I want to be a mathematician”)
We often hear that mathematics consists mainly of “proving theorems.” Is a writer’s job mainly that of “writing sentences?” (Gian-Carlo Rota)a
“The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps…” (Alexander Grothendieck, writing to Ronald Brown)
“It seems to me that the poet has only to perceive that which others do not perceive, to look deeper than others look. And the mathematician must do the same thing.” - Sofia Kovalevskay
“Mathematics requires a small dose, not of genius, but of an imaginative freedom which, in a larger dose, would be insanity” - Angus K Rodgers
Poincaré famously said that ‘mathematics is the art of giving the same name to different things.’ (To which a poet quipped: ‘Poetry is the art of giving different names to the same thing.’) — Francis Su
Mathematics is the science of patterns and the art of engaging the meaning of those patterns. — Francis Su
“[Algebraic geometry] seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate.” David Mumford
“The books or the music in which we thought the beauty was located will betray us if we trust to them; it was not in them, it only came through them, and what came through them was longing. These things — the beauty, the memory of our own past — are good images of what we really desire; but if they are mistaken for the thing itself they turn into dumb idols, breaking the hearts of their worshipers. For they are not the thing itself; they are only the scent of a flower we have not found, the echo of a tune we have not heard, news from a country we have never visited.” C.S. Lewis, The Weight of Glory
“What matters to any teacher of mathematics is the teaching of what mathematicians in their shoptalk informally refer to as the “truth” of a theory, a truth that has to do with the concordance of a statement with facts of the world, like the truth of any physical law. In the teaching of mathematics, the truth that is demanded by the students and provided by the teacher is such a factual/worldly truth, not the formal truth that one associates with the game of theorem-proving. A good teacher of mathematics is one who knows how to disclose the full light of such factual/worldly truth before students, while at the same time training them in the skills of carefully recording such truth.” Gian-Carlo Rota, “The Concept of Mathematical Truth,” Review of Metaphysics 44, no. 3 (1991)
“Every time that a human being succeeds in making an effort of attention with the sole idea of increasing his grasp of truth, he acquires a greater aptitude for grasping it, even if his effort produces no visible fruit.” Simone Weil “Reflections on the Right Use of School Studies…” (1942)
Post-script on The Little Prince
“A rock pile ceases to be a rock pile the moment a single man contemplates it, bearing within him the image of a cathedral” Antoine de Saint Exupèry, The Little Prince
“‘What makes the desert beautiful,’ said the Little Prince, ‘is that somewhere it hides a well’” Antoine de Saint Exupèry, The Little Prince
“‘It is the time you have wasted for your rose that makes your rose so important.’ ‘It is the time I have wasted for my rose—’ said the little prince, so that he would be sure to remember. ‘Men have forgotten this truth,’ said the fox. ‘But you must not forget it.’ Antoine de Saint Exupèry, The Little Prince
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