Denying Infinity: Pragmatism in Abraham Robinson’s Philosophy of Mathematics

Abstract

Abraham Robinson is well-known as the inventor of nonstandard analysis, which uses nonstandard models to give the notions of infinitesimal and infinitely large magnitudes a precise interpretation. Less discussed, although subtle and original–if ultimately flawed–is Robinson's work in the philosophy of mathematics. The foundational position he inherited from David Hilbert undermines not only the use of nonstandard analysis, but also Robinson's considerable corpus of pre-logic contributions to the field in such diverse areas as differential equations and aeronautics. This tension emerges from Robinson's disbelief in the existence of infinite totalities (any mention of them is ‘literally meaningless’) and the fact that much of his work involves them. I argue that he treats infinitary avenues of mathematics as useful tools to avoid this difficulty, but that this is not successful to the extent that these tools must be justified by a conservative extension from finitary mathematics. While Robinson provides a compelling and unorthodox pragmatic justification for the role of formal systems in mathematical practice despite their apparent infinitary presuppositions, he deflates mainstream mathematics to a collection of games that occasionally produces meaningful results. This amounts to giving up on a commitment to reconciling his finitism with his mathematical practice.