Axiomatics and Formalism in Mathematics

Abstract

New terminology is defined in terms of old terminology. Any tract of logic contains only finitely many words. Therefore there must be a beginning, and that beginning cannot depend on anything that came previously. There must therefore be elementary terms that remain undefined. These terms are called“undefinable.”We strive to keep the number of these terms to a minimum. Since the terms are formally undefinable, we must describe them in a manner that is heuristically self-evident.

If every mathematician occasionally, perhaps only for an instant, feels an urge to move closer to reality, it is not because he believes that math ematics is lacking in meaning. He does not believe that mathematics consists in drawing brilliant conclusions from arbitrary axioms, of jug gling concepts devoid of pragmatic content, of playing a meaningless game. —Errett Bishop

The pursuit of knowledge is, I think, mainly actuated by love of power. —Bertrand Russell

Logic is founded upon suppositions which do not correspond to anything in the actual world—for example, the supposition of the equality of things, and that of the identity of the same thing at different times. —F.W. Nietzsche

The totality of thought is a picture of the world. —Ludwig Wittgenstein

Logic is the art of going wrong with confidence. —Joseph Wood Krutch

Logic is the art of convincing us of some truth. —lJean de La Bruyère

Brouwer, who has done more for constructive mathematics than anyone else, thought it necessary to introduce a revolutionary, semimystical theory of the continuum. —Errett Bishop

‘Can you do addition?’ the White Queen asked. ‘What’s one and one and one and one and one and one and one and one and one and one?’ ‘I don’t know,’ said Alice. ‘I lost count.’ —Lewis Carroll, Through the Looking Glass