David Liggins

Much recent discussion in the philosophy of mathematics has concerned the indispensability argument—an argument which aims to establish the existence of abstract mathematical objects through appealing to the role that mathematics plays in empirical science. The indispensability argument is standardly attributed to W. V. Quine and Hilary Putnam. In this paper, I show that this attribution is mistaken. Quine's argument for the existence of abstract mathematical objects differs from the argument which many philosophers of mathematics ascribe to him. Contrary to appearances, Putnam did not argue for the existence of abstract mathematical objects at all. I close by suggesting that attention to Quine and Putnam's writings reveals some neglected arguments for platonism which may be superior to the indispensability argument.

Many philosophers posit abstract entities – where something is abstract if it is acausal and lacks spatio-temporal location. Theories, types, characteristics, meanings, values and responsibilities are all good candidates for abstractness. Such things raise an epistemological puzzle: if they are abstract, then how can we have any epistemic access to how they are? If they are invisible, intangible and never make anything happen, then how can we ever discover anything about them? In this article, I critically examine epistemological objections to belief in abstract objects offered by Paul Benacerraf, Colin Cheyne and Hartry Field.