There can never be surprises in logic.

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

The only way to learn mathematics is to do mathematics.

If I am given a formula, and I am ignorant of its meaning, it cannot teach me anything, but if I already know it what does the formula teach me?

All science requires mathematics. The knowledge of mathematical things is almost innate in us. This is the easiest of sciences, a fact which is obvious in that no one's brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon.

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.

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

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

Mathematics is written for mathematicians.

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.

The mathematician has reached the highest rung on the ladder of human thought.

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.

Measure what is measurable, and make measurable what is not so.

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

We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori.

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.

Nature is an infinite sphere of which the center is everywhere and the circumference nowhere.

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.

Mathematics takes us into the region of absolute necessity, to which not only the actual word, but every possible word, must conform.

Mathematics is the only good metaphysics.

Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat-flour from peascods, so pages of formulae will not get a definite result out of loose data.

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.

One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.

A mathematician is a blind man in a dark room looking for a black cat which isn't there.

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