Ludwig Wittgenstein Quotes

One cannot really argue with a mathematical theorem.

It would be very discouraging if somewhere down the line you could ask a computer if the Riemann hypothesis is correct and it said, 'Yes, it is true, but you won't be able to understand the proof.'

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

But it is impossible to divide a cube into two cubes, or a fourth power into fourth powers, or generally any power beyond the square into like powers; of this I have found a remarkable demonstration. This margin is too narrow to contain it.

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 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?

I'm a max player. Don't get mad at me because I'm telling you what's real. One plus one is two, all day long, and it's never gonna change. And that's factorial.

317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.

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.

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.

Irrefragability, thy name is mathematics.

God made the integers, man made the rest.

Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians.

My mathematics is simple: one plus one = one.

The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such a discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful.

We are on the verge: Today our program proved Fermat's next-to-last theorem.

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.

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.

... mathematical knowledge ... is, in fact, merely verbal knowledge. "3" means "2+1", and "4" means "3+1". Hence it follows (though the proof is long) that "4" means the same as "2+2". Thus mathematical knowledge ceases to be mysterious.

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.

I am obliged to interpolate some remarks on a very difficult subject: proof and its importance in mathematics. All physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.

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.

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