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

mathematical explanation: why it matters 145‘natural setting’. Finally, note that both Cournot and Mandelbrojt take it asa matter of course that mathematical explanations exist. Of course, the abovequotes can only be the beginning of the job. What is then the relationshipbetween explanation and generalization in mathematics? Such a major questioncannot be answered in this introduction but it is important to raise it sincethis web of relations between explanation, generality, and abstraction is arecurring one in all attempts to talk about explanation. Here I propose toindicate how the problem recurs in Kitcher’s writings. Kitcher is well knownas a defender of the theory of explanation as unification. Kitcher’s first articlewhere generalization and explanation are thematized is an article on Bolzano’sphilosophy of mathematics from 1975. As part of his analysis of Bolzano,Kitcher argued against the thesis that ‘a deductive argument is explanatory ifand only if its premises are at least as general as its conclusion.’ Against thethesis he raised the following objection:In any case, the generality criterion is ill-adapted to the case of mathematics.There is a very special difficulty with derivations in arithmetic, namely thatmany proofs use the method of mathematical induction. Suppose that I prove atheorem by induction, showing that all positive integers have property F. This isaccomplished by showinga) 1 has Fb) if all numbers less than n have F then n has F(Of course there are other versions of the method of induction). It would seemhard to deny that this is a genuine proof. [ ... ] Further, this type of proofdoes not controvert Bolzano’s claim that genuine proofs are explanatory; wefeel that the structure of the positive integers is exhibited by showing how1 has the property F and how F is inherited by successive positive integers;and, in uncovering this structure, the proof explains the theorem. But proofsby induction do violate the generality criterion[ ... ] Whatever account Bolzanogives and whatever generality it achieves for arithmetic, Bolzano would surely behard put to avoid the consequence that the proposition expressed by ‘1 has F’ isless general than that expressed by ‘Every number has F’. (Kitcher, 1975, p.266)I believe that this argument is too quick and that it suffers from anunfruitful attempt to use the complexity of the logical formulas as a measure ofgenerality. The first problem is that it is easy to reformulate the two premisesof induction as a single universal sentence, thereby eliminating ‘1 has F’ as anindependent premise. Secondly, not everybody agrees that proofs by inductionare explanatory (see Mancosu (2001, note 11)). Thus, I find Kitcher’s argumentto be neither here nor there.A related context in which Kitcher addresses the issue of generalizationis in his book TheNatureofMathematicalKnowledgefrom 1984. Thisisa