Mathematical Quotes

500+ Sourced quotes

Arthur Cayley

Eric Temple Bell

Roger Ascham

Hilary Putnam

George Oppen

Charles Babbage

Richard Courant

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

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.

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

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.

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

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

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

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

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!

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.

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

And thus many are ignorant of mathematical truths, not out of any imperfection of their faculties, or uncertainty in the things themselves, but for want of application in acquiring, examining, and by due ways comparing those ideas.

Ex-Professor Moriarty of mathematical celebrity…is the Napoleon of crime, Watson.

Arthur Conan Doyle

One should not forget that the functions, like all mathematical constructions, are only our own creation, and that when the definition with which one begins ceases to make sense, one should not ask, what is, but what is convenient to assume in order that it [remain] significant.

Carl Friedrich Gauss

I know that most men — not only those considered clever, but even those who are very clever and capable of understanding most difficult scientific, mathematical, or philosophic, problems — can seldom discern even the simplest and most obvious truth if it be such as obliges them to admit the falsity of conclusions they have formed, perhaps with much difficulty — conclusions of which they are proud, which they have taught to others, and on which they have built their lives.

As we speak of poetical beauty, so ought we to speak of mathematical beauty and medical beauty. But we do not do so; and that reason is that we know well what is the object of mathematics, and that it consists in proofs, and what is the object of medicine, and that it consists in healing. But we do not know in what grace consists, which is the object of poetry.

Evolution has ensured that our brains just aren't equipped to visualise 11 dimensions directly. However, from a purely mathematical point of view it's just as easy to think in 11 dimensions, as it is to think in three or four.

Stephen Hawking

The real trouble with this world of ours is not that it is an unreasonable world, nor even that it is a reasonable one. The commonest kind of trouble is that it is nearly reasonable, but not quite.... It looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wilderness lies in wait.

G. K. Chesterton

Our telescopes and our mathematical investigations assure us on every hand - notwithstanding the cant of the more ignorant of the priesthood - that space, and therefore that bulk, is an important consideration in the eyes of the Almighty.

Edgar Allan Poe

Life is not an illogicality; yet it is a trap for logicians. It looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wildness lies in wait.

G. K. Chesterton

We have heard much about the poetry of mathematics, but very little of it has as yet been sung. The ancients had a juster notion of their poetic value than we. The most distinct and beautiful statements of any truth must take at last the mathematical form. We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both.

Henry David Thoreau

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

Carl Friedrich Gauss

Our design, not respecting arts, but philosophy, and our subject, not manual, but natural powers, we consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces, whether attractive or impulsive; and therefore we offer this work as mathematical principles of philosophy; for all the difficulty of philosophy seems to consist in this — from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena...

If I hadn't an absolute faith in the harmony of creation, I wouldn't have tried for thirty years to express it in a mathematical formula. It is only man's consciousness of what he does with his mind that elevates him above the animals, and enables him to become aware of himself and his relationship to the universe.

Albert Einstein