A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details.

. . . we have overcome the notion that mathematical truths have an existence independent and apart from our own minds. It is even strange to us that such a notion could ever have existed.

Fourier is a mathematical poem.

The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.

The control of large numbers is possible, and like unto that of small numbers, if we subdivide them.

Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformed. They are no exceptions to the rule that God always geometrizes. Their problems of form are in the first instance mathematical problems, their problems of growth are essentially physical problems, and the morphologist is, ipso facto, a student of physical science.

This paper gives wrong solutions to trivial problems. The basic error,however, is not new.

No Roman ever died in contemplation over a geometrical diagram.

A book on the new physics, if not purely descriptive of experimental work, must essentially be mathematical.

In presenting a mathematical argument the great thing is to give the educated reader the chance to catch on at once to the momentary point and take details for granted: two trivialities omitted can add up to an impasse). The unpractised writer, even after the dawn of a conscience, gives him no such chance; before he can spot the point he has to tease his way through a maze of symbols of which not the tiniest suffix can be skipped.

The constructs of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth.

In the beginning (if there was such a thing), God created Newton's laws of motion together with the necessary masses and forces. This is all; everything beyond this follows from the development of appropriate mathematical methods by means of deduction.

The understanding of mathematics is necessary for a sound grasp of ethics.

There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.

I must study politics and war that my sons may have liberty to study mathematics and philosophy.

I know that two and two make four - and should be glad to prove it too if I could - though I must say if by any sort of process I could convert 2 and 2 into five it would give me much greater pleasure.

The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.

It is not enough to have a good mind; the main thing is to use it well.

Each problem that I solved became a rule, which served afterwards to solve other problems.

Science without religion is lame, religion without science is blind.

I have had my results for a long time: but I do not yet know how I am to arrive at them.

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

The simplest schoolboy is now familiar with truths for which Archimedes would have sacrificed his life.

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.

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