Arithmetic has began to totter.

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

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.

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

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.

Arithmetic has began to totter.

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