'Facts, facts, facts,' cries the scientist if he wants to emphasize the necessity of a firm foundation for science. What is a fact? A fact is a thought that is true. But the scientist will surely not recognize something which depends on men's varying states of mind to be the firm foundation of science.

Having visual impressions is, of course, necessary for seeing things, but it is not sufficient. What must be added is not anything sensible. And it is precisely this that unlocks the outer world for us; for without this non-sensible something, each of us would remain locked up in his inner world.

Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build my arithmetic.

It is possible, of course, to operate with figures mechanically, just as it is possible to speak like a parrot: but that hardly deserves the names of thought. It only becomes possible at all after the mathematical notation has, as a result of genuine thought, been so developed that it does the thinking for us, so to speak.

The thought: A logical inquiry

The aim of scientific work is truth. While we internally recognise something as true, we judge, and while we utter judgements, we assert.

I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems while the root drives into the depths.