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

356 colin mclartythen they could not use each other’s results without also verifying theequivalence of their chosen definitions. Any major paper uses scoresof structures, each of which has many different set theoretic reductions(compare homology in Section 13.3.1). Checking so many equivalences isinfeasible. Leaving them implicit would court disaster. A set theoreticspecification may be used in some step of a proof but those details mustbe rigorously irrelevant to the statement of the theorem (see discussion ofTheorem 4).• Methods must handle structures at every level from natural numbers tofunctors by comparable means that readily relate any two levels.These methods are used in practice, by necessity, with deductive rigor, evenif they violate some philosophical theories.⁵ Practice also dispels the idea thatstructuralism abstracts away from intuitive content. Structuralist tools give themost direct known path from ‘pure’ content to rigor.Section 13.1 extends Resnik’s structuralism by the standard practice ofidentifying some structures as parts of others. Certain injections S 1 ↣S 2 ofone structure S 1 into another S 2 are taken as identity preserving and thus asmaking S 1 a part of S 2 . We use textbook treatments of the real and complexnumbers to argue that such identity is not defined by any logical principles butby stipulation or tradition. This opens up a philosophical topic of explaininghow particular cases come to be accepted as identity preserving. Section 13.1.2argues that Shapiro’s distinction of systems and structures does not help tounderstand current practice.Section 13.2 pursues the original point of structuralist methods—definingstructures as themselves places in patterns of structures rather the way thatResnik describes in his later chapters. It takes polynomials as an example anddiscusses foundations. The ontology so far as it goes is far from philosophicallyclassical: The Leibniz law of identity fails as many individuals lack fullyindividuating properties. Nothing here prevents philosophers from goingbeyond practice and attributing a more classical ontology to mathematicalobjects. We know that mathematics can be re-interpreted in various waysto give it a classical ontology. But a further task faces anyone who claimsthat only classical ontology is conceptually sound: Either show that contraryto the appearances practice concretely does recognize such ontology; orexplain how conceptually unsound practice can succeed so well. Whateverposition one takes in debates over foundations and ‘structuralism,’ it remains⁵ See related arguments from practice against Quine’s ontology in Maddy (2005, p.450). CompareKrömer (2007, p. xv) on the way category theory became standard mathematics despite some clashwith official ZF foundations.