Ronald Graham Quotes

One cannot really argue with a mathematical theorem.

The computer is not, in our opinion, a good model of the mind, but it is as the trumpet is to the orchestra - you really need it. And so, we have very massive simulations in computers because the problem is, of course, very complex.

The mathematics is not there till we put it there.

And for mathematical science, he that doubts their certainty hath need of a dose of hellebore.

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?

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

… 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.

To me, Turing is as much of a philosopher as he is a mathematician because his ideas deal with what it means to think.

In math, you're either right or you're wrong.

I remember having this vague idea that what mathematicians did was that some authority, someone, gave them problems to solve, and they just sort of solved them.

Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight.

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

The man who has learned that three plus one are four doesn't have to go through a proof of that assertion with coins, or dice, or chess pieces, or pencils. He knows it, and that's that. He cannot conceive a different sum. There are mathematicians who say that three plus one is a tautology for four, a different way of saying "four" ... If three plus one can be two, or fourteen, then reason is madness.

You can't argue with math, Carter. You'll always lose.

I had made an empirical discovery and it carried all the weight of a mathematical proof.

Mathematical demonstrations being built upon the impregnable Foundations of Geometry and Arithmetick are the only truths that can sink into the Mind of Man, void of all Uncertainty; and all other Discourses participate more or less of Truth according as their Subjects are more or less capable of Mathematical Demonstration.

We cannot ... prove geometrical truths by arithmetic.

Proof, one might say, does not merely shew that it is like this, but: how it is like this. It shows how 13+14 yield 27.

I write rhymes with addition and algebra, mental geometry.

The human understanding is a revelation from its maker, which can never be disputed or doubted. There can be no scepticism, Pyrrhonism, or incredulity or infidelity here. No prophecies, no miracles are necessary to prove this celestical communication. This revelation has made it certain that two and one make three, and that one is not three nor can three be one. We can never be so certain of any prophecy, or the fulfilment of any prophecy, or of any miracle, or the design of any miracle, as we are from the revelation of nature, that is, nature's God, that two and two are equal to four.

The programmers have another saying: 'The question of whether a machine can think is no more interesting than the question of whether a submarine can swim.'

It's hard to guess how smart the machines are, but a good rule of thumb is that they're always smarter than you think.

For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock. And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems.

It's interesting that the greatest minds of computer science, the founding fathers, like Alan Turing and Claude Shannon and Norbert Wiener, they all looked at chess as the ultimate test. So they thought, "Oh, if a machine can play chess, and beat strong players, set aside a world champion, that would be the sign of a dawn of the AI era." With all due respect, they were wrong.

The question you raise, 'How can such a formulation lead to computations?' doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand - and it always turned out that understanding was all that mattered.