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Beyond the Axioms: The Question of Objectivity in Mathematics

I will be discussing the axiomatic conception of mathematics, the modern version of which is clearly due to Hilbert; but one aspect of it goes back to Cantor. I refer here to Hilbert’s view that the demands of mathematical existence and truth are entirely met by a suitable axiomatization. By whatever dialectical process we come to adopt the axioms, whatever ‘intuitions’ lead us to them, the question of mathematical truth or existence becomes welldefined only with the introduction of the axioms. Now Cantor is one of the last people, ever, to cite in support of the axiomatic method; but Hilbert’s point about mathematical existence and truth is a natural consequence of the discussion in Foundations of a General Theory of Manifolds [Cantor, 1883] of free mathematics and in particular of the limited obligation of free mathematics to establish only the immanent existence of its objects. Speaking of this in connection with his transfinite numbers, he wrote