The Number Systems

Abstract

It is in fact quite difficult to construct—from first principles—a system of natural numbers in which the multiplication operation is workable.

...there are good reasons to believe that nonstandard analysis, in some version or other, will be the analysis of the future.—Kurt Gödel

Sire, I have no need of that hypothesis.—Pierre Simon de Laplace

I must create a system or be enslaved by another man’s.—William Blake

We ought not to be over-anxious to encourage innovation in cases of doubtful improvement, for an old system must ever have two advantages over a new one: it is established, and it is understood.—Charles Caleb Colton

I frame no hypothesis; for whatever is not deduced from the phenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy.—Isaac Newton

Logical consequences are the scarecrows of fools and the beacons of wise men.—T.H. Huxley

Errors are not in the art but in the artificers.—Isaac Newton

Man is a credulous animal and must believe something. In the absence of good grounds for belief, he will be satisfied with bad ones.—Bertrand Russell

Logic is simply the architecture of human reason.—Evelyn Waugh