Mathematical Quotes

500+ Sourced quotes

Arthur Cayley

Eric Temple Bell

Roger Ascham

Hilary Putnam

George Oppen

Charles Babbage

Richard Courant

Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning... The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings.

It is therefore not unreasonable to suppose that some portion of the neglect of science in England, may be attributed to the system of education we pursue. A young man passes from our public schools to the universities, ignorant of almost every branch of useful knowledge; and at these latter establishments … classical and mathematical pursuits are nearly the sole objects proposed to the student's ambition.

Charles Babbage

I think that it is a relatively good approximation to truth — which is much too complicated to allow anything but approximations — that mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is … governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas.

John von Neumann

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. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.

Richard Courant

As for everything else, so for a mathematical theory: beauty can be perceived but not explained.

A pure mathematical series would be one in which each term is derived from the preceding term by a rule.

Out of fifty mathematical papers presented in brief at such a meeting, it is a rare mathematician indeed who really understands what more than half a dozen are about.

Eric Temple Bell

All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers.

James Clerk Maxwell

Mark all mathematical heads which be only and wholly bent on these sciences, how solitary they be themselves, how unfit to live with others, and how unapt to serve the world.

No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it gives the impression of also being beautiful.

Oddly enough, Putnam believes part of the attraction of some of these formalisms is their obscurity:

I think part of the appeal of mathematical logic is that the formulas look mysterious - you write backward Es!

Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself the more clearly.

A good civilisation spreads over us freely like a tree, varying and yielding because it is alive. A bad civilisation stands up and sticks out above us like an umbrella-artificial, mathematical in shape; not merely universal, but uniform.

G. K. Chesterton

Philosophical knowledge is the knowledge gained by reason from concepts; mathematical knowledge is the knowledge gained by reason from the construction of concepts.

Perspective is a most subtle discovery in mathematical studies, for by means of lines it causes to appear distant that which is near, and large that which is small.

Leonardo da Vinci

I am giving this winter two courses of lectures to three students, of which one is only moderately prepared, the other less than moderately, and the third lacks both preparation and ability. Such are the onera of a mathematical profession.

Carl Friedrich Gauss

One can prove or refute anything at all with words. Soon people will perfect language technology to such an extent that they'll be proving with mathematical precision that twice two is seven.

Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?

Stephen Hawking

The Christian's God does not consist merely of a God who is the author of mathematical truths and the order of elements... But a God of love and consolation.

Mathematical knowledge... is, in fact, merely verbal knowledge. "3" means "2+1", and "4" means "3+1". Hence it follows (though the proof is long) that "4" means the same as "2+2". Thus mathematical knowledge ceases to be mysterious.

Bertrand Russell

Mathematical discoveries, like springtime violets in the woods, have their season which no human can hasten or retard.

Carl Friedrich Gauss

Absolute, true, and mathematical time, in and of itself and of its own nature, without reference to anything external, flows uniformly and by another name is called duration. Relative, apparent, and common time is any sensible and external measure (precise or imprecise) of duration by means of motion; such as a measure-for example, an hour, a day, a month, a year-is commonly used instead of true time.

Mathematical judgments are always synthetical.

Physics too deals with mathematical concepts; however, these concepts attain physical content only by the clear determination of their relation to the objects of experience.

Albert Einstein

Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.