Computers as a Source of A Posteriori Knowledge in Mathematics

ABSTRACT

Electronic computers form an integral part of modern mathematical practice. Several high-profile results have been proven with techniques where computer calculations form an essential part of the proof. In the traditional philosophical literature, such proofs have been taken to constitute a posteriori knowledge. However, this traditional stance has recently been challenged by Mark McEvoy, who claims that computer calculations can constitute a priori mathematical proofs, even in cases where the calculations made by the computer are too numerous to be surveyed by human agents. In this article we point out the deficits of the traditional literature that has called for McEvoy’s correction. We also explain why McEvoy’s defence of mathematical apriorism fails and we discuss how the debate over the epistemological status of computer-assisted mathematics contains several unfortunate conceptual reductions.

Acknowledgements

We wish to thank the anonymous reviewer of this journal for helpful and constructive comments on the first version of the manuscript.

Notes

1. It should be noted that Tymoczko's conception of the term ‘a priori’ is not clear, and seemingly he uses the term in several different senses. As pointed out by the anonymous reviewer of this paper, Tymoczko for instance expresses no fewer than three different senses of the term in this one sentence: ‘An a priori truth might be immediately evident, stipulated by convention, or, most common, known by reason independently of any experience beyond pure thought’ (Tymoczko 1979, 77). The community conception of the term however seems to be the operatively most important sense of the term in Tymoczko's argument.

2. As noted above there were earlier critics of the traditionalist stance, for example, Swart (1980) and Burge (1998). We have however chosen to focus on McEvoy here as his challenge is the most recent and the most general.

3. See Colton (2007) for examples of experimental mathematics and Sørensen (2010) for a definition and conceptual analysis of the field. Our own analysis of the use of computers in experimental mathematics will be given in section 5 below.