Abraham Robinson Quotes

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.

There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation.

As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.

At the beginning of the new millennium, we still do not know why mathematics is true and whether it is certain. But we know what we do not know in an immeasurably richer way than we did. And learning this has been a remarkable achievement-among the greatest and least-known of the modern era.

Mathematics is a body of knowledge, but it contains no truths.

I am not a mathematician, but I was aware that for centuries, mathematics was considered the queen of the sciences because it claimed certainty. It was grounded on some fundamental certainties - axioms - that led to others.

You can't criticize geometry. It's never wrong.

If indeed, as Hilbert asserted, mathematics is a meaningless game played with meaningless marks on paper, the only mathematical experience to which we can refer is the making of marks on paper.

Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then suchand such another proposition is true of that thing.... Thus 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 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.

But twice-two-makes-four is for all that a most insupportable thing. Twice-two-makes-four is, in my humble opinion, nothing but a piece of impudence. Twice-two-makes-four is a farcical, dressed-up fellow who stands across your path with arms akimbo and spits at you.

Geometry is knowledge that appears to be produced by human beings, yet whose meaning is totally independent of them.

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

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.

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?

Do mathematics have a relation to reality or are they only a mathematical symbol?

Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional. In E. T. Bell Men of Mathematics, New York: Simona and Schuster, 1937.

It was easier to know it than to explain why I know it. If you were asked to prove that two and two made four, you might find some difficulty, and yet you are quite sure of the fact. ~ Sherlock Holmes

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.

It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention.

The three main medieval points of view regarding universals are designated by historians as realism, conceptualism, and nominalism. Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics under the new names logicism, intuitionism, and formalism.

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

I believe that two and two are four and that four and four are eight.

Some things you just can't question. Like you can't question why two plus two is four. So don't question it, don't try to look it up. I don't know who made it, all I know is it was put in my head that two plus two is four. So certain things happen. Why does it rain? Why am I so sexy? I don't know.

It is difficult even to attach a precise meaning to the term "scientific truth." So different is the meaning of the word "truth" according to whether we are dealing with a fact of experience, a mathematical proposition or a scientific theory. "Religious truth" conveys nothing clear to me at all.