A sane mind should not be guilty of a logical fallacy, yet there are very fine minds incapable of following mathematical demonstrations.

...the feeling of mathematical beauty, of the harmony of numbers and of forms, of geometric elegance. It is a genuinely aesthetic feeling, which all mathematicians know

When the logician has resolved each demonstration into a host of elementary operations, all of them correct, he will not yet be in possession of the whole reality, that indefinable something that constitutes the unity ... Now pure logic cannot give us this view of the whole; it is to intuition that we must look for it.

A small error in the former will produce an enormous error in the latter.

Geometry is not true, it is advantageous.

What is a good definition? For the philosopher or the scientist, it is a definition which applies to all the objects to be defined, and applies only to them; it is that which satisfies the rules of logic. But in education it is not that; it is one that can be understood by the pupils.

A reality completely independent of the spirit that conceives it, sees it, or feels it, is an impossibility. A world so external as that, even if it existed, would be forever inaccessible to us.