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.

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.

A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.

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

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

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