G. H. Hardy Quotes

A mathematical proof must be perspicuous.

Proof is an idol before whom the pure mathematician tortures himself.

There is no royal road to geometry.

A mathematical theorem can be true whether or not it is realized in the world.

Irrefragability, thy name is mathematics.

A elegantly executed proof is a poem in all but the form in which it is written.

Mathematics, in its widest significance, is the development of all types of formal, necessary, deductive reasoning.

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

Arithmetic is where the content lies, and not logic; but logic prompts certainty, and not arithmetic.

The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience

Now all orators effect their demonstrative proofs by allegation either of enthymems or examples, and, besides these, in no other way whatever.

Euclid taught me that without assumptions there is no proof. Therefore, in any argument, examine the assumptions.

When you come up with a theory, you fall in love with the beauty the simplicity and elegance of it. But then you have to get a sheet of paper and pencil and crack out all the details. Hundreds and hundreds of pages. Because you have to prove it.

Even the greatest mathematicians, the ones that we would put into our mythology of great mathematicians, had to do a great deal of leg work in order to get to the solution in the end.

I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where half proof = 0, and it is demanded for proof that every doubt becomes impossible.

The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science.

Mathematicians aren't satisfied because they know there are no solutions up to four million or four billion, they really want to know that there are no solutions up to infinity.

if you do not appreciate the mathematics, you cannot see, among the great variety of facts, that logic permits you to go from one to the other

If you have ever studied geometry, you remember that by a course of reasoning Euclid proves that all the angles in a triangle are equal to two right angles. Euclid has shown you how to work it out. Now, if you undertake to disprove that proposition, and to show that it is erroneous, would you prove it to be false by calling Euclid a liar?

Mathematics seems to endow one with something like a new sense.

… 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.

I am never satisfied until I can make a mathematical model of what I am studying. If I succeed in making one, I understand. If I cannot, I do not understand.

In practical life we are compelled to follow what is most probable ; in speculative thought we are compelled to follow truth. A man would perish of hunger and thirst, if he refused to eat or drink, till he had obtained positive proof that food and drink would be good for him. But in philosophic reflection this is not so. On the contrary, we must take care not to admit as true anything, which is only probable. For when one falsity has been let in, infinite others follow.

I believe only in the mathematics.

I think I still like science and art better, but geometry is a big improvement over algebra.