Quotes on Mathematics
Mathematics is the extension of common sense by other means.
— Jordan Ellenberg (2014), contemporary American mathematician, educator, author, popularizer of mathematics
Society is ever increasingly mathematised, but this operates at a level invisible to most of its members.
— Paul Ernest (2000), contemporary British mathematician, educator, author
The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it.
— Jacques Hadamard slight paraphrase (ca. 1914), influential French mathematician, educator, author
I think that mathematics will have to become more and more algorithmic if it is going to be active and vital in the creative life. This means it is necessary to rethink what we teach, in school, in college, and in graduate school.
In our emphasis on deductive reasoning and rigor we have been following the Greek tradition, but there are other traditions—Babylonian, Hindu, Chinese, Mayan—and these have all followed a more algorithmic, more numerical procedure.
After all, the word algorithm, like the word algebra, comes from Arabic. And the numerals we use come from Hindu mathematics via the Arabs.
We can’t regard Greek mathematics as the only source of great mathematics, and yet somehow in the last half century there has been such emphasis on the greatness of “pure” mathematics that the other possible forms of mathematics have been put down.
I don’t mean that it is necessary to put down the rigorous Greek style mathematics, but it is necessary to raise up the status of the numerical, the algorithmic, the discrete mathematics.
— Albert W. Tucker (1983), influential Canadian mathematician, educator, historian, author
You might think that the theorem you just proved is the greatest thing ever—a guaranteed “game changer” and an instant classic. But in the grand scheme of things, it’s merely one discrete achievement—a drop in the vast bucket we call mathematics.
When you combine that “drop” with all of your other achievements, you might have produced a small volume of water—perhaps a cupful (or pitcherful)—altogether.
That cup of water, as I see it, doesn’t just sit in an engraved mug, alongside the degrees and awards lining our office walls. Instead, it’s part of a great river that’s been flowing for a long time and, I hope, will continue to flow into the indefinite future.
When I do mathematics, I like to know, whenever possible, where that river has come from and where it is headed.
— Shing-Tung Yau (2013), contemporary Chinese-American mathematician, Fields medalist, educator, author, popularizer of math
The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics.
— Gian-Carlo Rota (1992), influential Italian/American mathematician, educator, philosopher, author
If there is one thing in mathematics that fascinates me more than anything else (and doubtless always has), it is neither "number" nor "size", but always form.
And among the thousand-and-one faces whereby form chooses to reveal itself to us, the one that fascinates me more than any other and continues to fascinate me, is the structure hidden in mathematical things.
— Alexander Grothendieck (1992), influential French-German mathematician, Fields Medalist, author
Mathematics can be like archaeology. You might find a corner of something and decide it’s of interest. Then you dig somewhere else and you find another corner that looks very similar and think there may be some deep connection. You keep digging and finally discover the structure underneath. You have a thrill of discovery when something finally makes sense.
— Terence Tao (2013), contemporary Australian-American mathematician, Fields Medalist, educator, author
"Last time, 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 (1982), influential French-American mathematician, educator, author, activist
…The entire mathematical community would become much more productive if we open our eyes to the real values in what we are doing. [Some] propose a system of recognized roles divided into “speculation” and “proving”.
Such a division only perpetuates the myth that our progress is measured in units of standard theorems deduced.
…What we are producing is human understanding. We have many different ways to understand and many different processes that contribute to our understanding.
We will be more satisfied, more productive and happier if we recognize and focus on this.
— William P. Thurston slight paraphrase (1994), influential American mathematician, Fields medalist, educator, author
One must guard…against confusing the presentation of mathematics with the content of mathematics. An axiomatic presentation of a mathematical fact differs from the fact that is being presented just as medicine differs from food.
It is true that this particular medicine is necessary to keep the mathematician at a safe distance from the self-delusions of the mind. Nonetheless, understanding mathematics means being able to forget the medicine, and to enjoy the food.
— Gian-Carlo Rota (1996), influential Italian/American mathematician, educator, philosopher, author