What Are Mathematical Coincidences (and Why Does It Matter)? Marc Lange University of North Carolina at Chapel Hill mlange@email.unc.edu Although all mathematical truths are necessary, mathematicians take certain com- binations of mathematical truths to be ‘coincidental’, ‘accidental’, or ‘fortuitous’. The notion of a ‘mathematical coincidence’ has so far failed to receive sufficient attention from philosophers. I argue that a mathematical coincidence is not merely an unforeseen or surprising mathematical result, and that being a misleading com- bination of mathematical facts is neither necessary nor sufficient for qualifying as a mathematical coincidence. I argue that although the components of a mathematical coincidence may possess a common explainer, they have no common explanation; that two mathematical facts have a unified explanation makes their truth non-coincidental. I suggest that any motivation we may have for thinking that there are mathematical coincidences should also motivate us to think that there are mathematical explanations, since the notion of a mathematical coincidence can be understood only in terms of the notion of a mathematical explanation. I also argue that the notion of a mathematical coincidence plays an important role in scientific explanation. When two phenomenological laws of nature are similar, despite concerning physically distinct processes, it may be that any correct scientific explanation of their similarity proceeds by revealing their similarity to be no math- ematical coincidence. 1. The phenomenon to be saved All mathematical truths are necessary. Accordingly, there might seem to be no place in mathematics for genuine coincidences. 1 However, mathematicians and non-mathematicians alike sometimes encounter a pair (or more) of mathematical facts, F and G, about which they naturally ask, ‘Is it a coincidence that both F and G obtain? Or is it no coincidence?’ The fact that F and G both obtain, though necessary, 1 Except in the technical sense (employed in ‘coincidence theory’) where, for example, the coincidence set of two functions f(x) and g(x) is the set of x’s where f(x) 5 g(x), as when two curves intersect (‘coincide’). Mind, Vol. 119 . 474 . April 2010 ß Lange 2010 doi:10.1093/mind/fzq013 Advance Access publication 23 July 2010 at University of North Carolina at Chapel Hill on August 27, 2010 mind.oxfordjournals.org Downloaded from