What are numbers? What is the nature of arithmetical truth?

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

There is more danger of numerical sequences continued indefinitely than of trees growing up to heaven. Each will some time reach its greatest height.

Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician.

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.

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.

...one can hardly deny that mankind has a common store of thoughts which is transmitted from one generation to another.