Only two things are infinite, the universe and human stupidity, and I'm not sure about the former.
The real voyage of discovery consists not in seeking new landscapes, but in having new eyes.
In mathematics the art of proposing a question must be held of higher value than solving it.
Mathematics is the science which draws necessary conclusions.
Imagination is more important than knowledge.
Teaching is the royal road to learning.
Do I contradict myself? Very well, then I contradict myself, I am large, I contain multitudes.
Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.
There are many questions which fools can ask that wise men cannot answer.
I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind.
The study of mathematics is apt to commence in disappointment... We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it.
The method of "postulating" what we want has many advantages; they are the same as the advantages of theft over honest toil.
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.
Histories make men wise; poets, witty; the mathematics, subtle; natural philosophy, deep; moral, grave; logic and rhetoric, able to contend.
I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply the notes of our observations.
Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.
Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.
We could use up two Eternities in learning all that is to be learned about our own world and the thousands of nations that have arisen and flourished and vanished from it. Mathematics alone would occupy me eight million years.
Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems.
Mathematics began to seem too much like puzzle solving. Physics is puzzle solving, too, but of puzzles created by nature, not by the mind of man.
Some mathematician, I believe, has said that true pleasure lies not in the discovery of truth, but in the search for it.
Somebody came up to me after a talk I had given, and say, "You make mathematics seem like fun." I was inspired to reply, "If it isn't fun, why do it?"
We are justified in calling numbers a free creation of the human mind.
Before beginning [to try to prove Fermat's Last Theorem] I should have to put in three years of intensive study, and I haven't that much time to squander on a probable failure.
The Good Lord made all the integers; the rest is man's doing.
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