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G. H. Hardy and Other Mathematician Quotations

All quotes are Hardy unless otherwise noted.

“A mathematician, like a painter or poet, is a maker of patterns. If [the mathematician’s] patterns are more permanent than theirs, it is because they are made with ideas.”

“Real mathematics must be justified as art if it can be justified at all.”

“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.”

“Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity”

“The seriousness of a theorem, of course, does not lie in its consequences, which are merely the evidence for its seriousness.”

“Pure mathematics is on the whole distinctly more useful than applied For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.”

“In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.”

“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 the notes of our observations.”

“Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.”

“It is not worth an intelligent [person’s] time to be in the majority. By definition, there are already enough people to do that.”

“It’s like asking why is Ludwig van Beethoven’s Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is.”

-Erdős

“In a way, mathematics is the only infinite human activity. It is conceivable that humanity could eventually learn everything in physics or biology. But humanity certainly won't ever be able to find out everything in mathematics, because the subject is infinite. Numbers themselves are infinite. That's why mathematics is really my only interest.”

-Erdős

"You don't have to believe in [the SF], but you should believe in The Book."

-Erdős

“Another roof, another proof”

-Erdős

“Some French socialist said that private property was theft . . . I say that private property is a nuisance.”

-Erdős

“The art of doing mathematics consists in finding that special case which contains all the germs of generality.”

-Hilbert

“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.”

-John von Neumann

“If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.”

-Paul Lockhart

“In math some authors call this ‘mathematical tourism’ with undertones of disdain. But I think tourism is fine—it would be a shame if the only options for traveling were to move somewhere to live there or else stay at home.”

-Eugenia Cheng

“...will probably always*
*I am tempted to say ‘always’ but my precise mathematical brain prevents me from making absolute statements without some sort of qualification such as ‘probably’ or ‘I believe’ or ‘it is almost certainly true that.’”

-Eugenia Cheng

“It is standard in math books to exhort the reader to work through exercises, but I believe this is offputting to many non-mathematicians, as well as some mathematicians (including me).”

-Eugenia Cheng

“I’ve always found the metaphor of a double-edged sword a bit strange, as it doesn’t seem to me that the two edges of a double-edged sword work in opposition to one another.”

-Eugenia Cheng

“[I like] the idea of starting sentences with ‘there is a sense in which…’ because math isn’t about right and wrong, it’s not about absolute truth; it’s about different contexts in which different things can be true, and about different senses in which different things can be valid. Abstraction in mathematics is about making precise which sense we mean, so that instead of having divisive arguments, whether it’s about abstract theories or about homophobic bakers, we can investigate more effectively what is causing certain outcomes to arise.”

-Eugenia Cheng

“A typical math education is a series of increasingly tall hurdles. If these were really hurdles it would make sense not to try higher ones if you’re unable to clear the lower ones. However, math is really more like an interconnected web of ideas;… everything is connected to everything else, and thus there are many possible routes around this web depending on what sort of brain you have. Some people do need to build up gradually through concrete examples towards abstract ideas. But not everyone is like that. For some people, the concrete examples don’t make sense until they’ve grasped the abstract ideas or, worse, the concrete examples are so offputting that they will give up if presented with those first.”

-Eugenia Cheng

“I have confirmed from my many years of teaching abstract mathematics to art students that I am not the only one who prefers to use abstract ideas to illuminate concrete examples rather than the other way round. Many of these art students consider that they’re bad at math because they were bad at memorizing times tables, because they’re bad at mental arithmetic, and they can’t solve equations. But this doesn’t mean they’re bad at math—it just means they’re not very good at times tables, mental arithmetic and equations, an absolutely tiny part of mathematics that hardly counts as abstract at all. It turns out that they do not struggle nearly as much when we get to abstract things such as higher dimensional spaces, subtle notions of equivalence, and category theory structures. Their blockage on mental arithmetic becomes irrelevant.”

-Eugenia Cheng

“Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality.”

-Richard Courant, German-American mathematician

“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.”

-Einstein ᔛ

“There should be no such thing as boring mathematics.”

-Edsger W. Dijkstra, Dutch systems scientist

“Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty’s sake and pulls it down to earth.”

-Marston Morse, American mathematician

“In mathematics the art of proposing a question must be held of higher value than solving it.”

-Cantor

“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.”

-John von Neumann

“In mathematics, you don’t understand things. You just get used to them.”

-John von Neumann

“The essence of math is not to make simple things complicated, but to make complicated things simple.”

-Stan Gudder, American mathematician

“Other sciences seek to discover the laws that God has chosen; mathematicians seek to discover the laws that God has to obey.”

Raoul Bott, Hungarian-American mathematician (possibly paraphrased, said to Bott by Jean-Pierre Serre)

“It is through logic that we prove, but through intuition that we discover.”

-Poincaré

“A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.”

-Stefan Banach, Polish mathematician

“Mathematicians are like Frenchmen: when you talk to them, they translate it into their own language, and then it is something entirely different.”

-J. W. von Goethe, German polymath

“The main application of Pure Mathematics is to make you happy.”

-Hendrik Lenstra, Dutch mathematician

Mathematics is “the science of analogy and the widespread applicability of mathematics in the natural sciences ... arises from the fundamental role which comparisons play in the mental process we refer to as 'understanding.’”

-Michael Atiyah, British-Lebanese mathematician

“The essence of mathematics lies precisely in its freedom.”

-Cantor

“Iterative methods give the exact answer to almost the right question.”

-Unknown