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

It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word.

Arithmetic has began to totter.

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

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.

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

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