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

mathematical explanation: why it matters 139as specific arguments for realism about certain mathematical entities.¹ Iaminterested in articulating a possible parallel between uses of the indispensabilityof mathematics in science as described by Baker and Colyvan and the case ofmathematics. Perhaps the best argument one can get here is one foreshadowedin Feferman (1964). Discussing Gödel’s claim that the postulation of theCantorian sets was just as justified as that of physical bodies in order toobtain a satisfactory theory of sense perceptions, Feferman claimed that thedevelopment of mathematics strongly supported the following interpretationof the argument:Abstraction and generalization are constantly pursued as the means to reachreally satisfactory explanations which account for scattered individual results. Inparticular, extensive developments in algebra and analysis seem necessary to givereal insight into the behavior of the natural numbers. Thus we are able to realizecertain results, whose instances can be finitistically checked, only by a detourvia objects (such as ideals, analytic functions) which are much more ‘abstract’than those with which we are finally concerned. The argument is less forcefulwhen it is read as justifying some particular conceptions and assumptions, namelythose of impredicative set theory, as formally necessary to infer the arithmeticaldata of mathematics. It is well known that a number of algebraic and analyticarguments can be systematically recast into a form which can be subsumed under(elementary) first order number theory. (Feferman, 1964, p.3)²Feferman seems to think that a persuasive form of the Gödelian argument goesas follows:1. There are scattered results in one branch of mathematics (the data,this might be finitistically verifiable propositions concerning the naturalnumbers) which call for an explanation.2. Such an explanation is obtained by appealing to more abstract entities(say ideals and analytic functions).3. We thus have good reason to postulate such abstract entities and tobelieve in their existence.¹ This is different from, although related to, the issue of objectivity discussed in Leng (2005),pp. 172–173. She discusses, following Waismann and Steiner, an example concerning how theexplanation of a known fact about the real numbers can be used to ‘to support the non-arbitrarinessof our extension of the number system to complex numbers’ (p. 172) or of our ‘representations ofcomplex numbers’ (p. 173). But the issue here is not that of realism about the mathematical objects.² Feferman’s work showed that predicative analysis could not be formally sufficient to obtain all thearithmetical consequences of impredicative mathematics. However, he also claimed until recently thatpredicative mathematics was sufficient to prove all of the arithmetical consequences of mathematicalinterest. He now agrees that certain arithmetical results (such as the modified form of finite Kruskal’stheorem (FKT ∗ ); see Feferman (2004) and Hellman (2004))) which cannot be proved predicatively areof mathematical interest.