Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

mathematical explanation: why it matters 137the nymphal stage remains in the soil for a lengthy period, then the adultcicada emerges after 13 years or 17 years depending on the geographical area.Even more strikingly, this emergence is synchronized among the members ofa cicada species in any given area. The adults all emerge within the same fewdays, they mate, die a few weeks later and then the cycle repeats itself.’ (2005,p. 229) Biologists have raised several questions concerning this life-cycle butone of them in particular concerns the question of why the life-cycle periodsare prime. Baker proceeds then to a reconstruction of the explanation of suchfact to conclude that:The explanation makes use of specific ecological facts, general biological laws, andnumber theoretic result. My claim is that the purely mathematical component[prime periods minimize intersection (compared to non-prime periods)] is bothessential to the overall explanation and genuinely explanatory on its own right.In particular it explains why prime periods are evolutionary advantageous in thiscase. (Baker, 2005, p.233)It is of course not possible in this brief introduction to even summarize thereconstructed explanation and the additional arguments brought in supportof the claim that this is a genuinely mathematical explanation. Let us rathersummarize how such explanations give a new twist to the indispensabilityargument. The argument now runs as follows:(a) There are genuinely mathematical explanations of empirical phenomena;(b) We ought to be committed to the theoretical posits postulated by suchexplanations; thus,(c) We ought to be committed to the entities postulated by the mathematicsin question.The argument has not gone unchallenged. Indeed, Leng (2005) triestoresist the conclusion by blocking premise (b). She accepts (a) but questions theclaim that the role of mathematics in such explanations commits us to the realexistence (as opposed to a fictional one) of the posits. This, she argues, will begranted when one realizes that both Colyvan and Baker infer illegitimately fromthe existence of the mathematical explanation that the statements groundingthe explanation are true. She counters that mathematical explanations need nothave true explanans and consequently the objects posited by such explanationsneed not exist.Mathematical explanations of empirical facts have not been sufficientlystudied. We badly need detailed case studies in order to understand better thevariety of explanatory uses that mathematics can play in empirical contexts.The philosophical pay-offs might come from at least three different directions.