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

402 colin mclartyrams. The equation expresses an isomorphism of sets#(dogs in this sheep pen) = #(rams){dogs in this sheep pen} ∼ = {rams}and the real interest is the specific correspondence: at this moment my dogs arefacing one ram each and that is all the rams. Corfield gives practically importantexamples from combinatorics and topology, as our Sections 14.2.1 and 14.2.2raised equations of Betti numbers into isomorphisms of homology groups.Turning every equation into an isomorphism is just the same thing as turningevery isomorphism into an equivalence—an ‘isomorphism up to isomorphism’.It is not clear how far this can be taken. We have seen that mathematicianscannot entirely dispense with isomorphism in favor of equivalence and socannot entirely dispense with equality in favor of isomorphism. On theother hand, as Corfield says, philosophers have missed the real importanceof equivalence as a kind of sameness of structure (2005, p.76). Mathematicalphysicist John Baez has taken this viewpoint very far, originally using n-categoriesor higher dimensional algebra as a revealing approach to quantum gravity, but alsolooking at it conceptually all across mathematics (Baez and Dolan, 2001).Philosophers are right that structural mathematics raises issues such as: howare purely structural definitions possible? And what is the role of identity versusstructural isomorphism? But let us take Resnik’s point that ‘mathematicianshave emphasized that mathematics is concerned with structures involving mathematicalobjects and not with the ‘‘internal’’ nature of the objects themselves’(Resnik, 1981, p.529). This is already the rigorous practice of mathematics.That practice offers working answers with powerful and beautiful results.Acknowledgments. I thank Karine Chemla, José Ferreirós, and Jeremy Grayfor discussions of the topics here, and William Lawvere for extensive critiqueof the article.BibliographyAlexandroff,Paul(1932), Einfachste Grundbegriffe der Topologie (Berlin: Julius Springer).Translated as Elementary Concepts of Topology (New York: Dover, 1962).Alexandroff, Paul and Hopf, Heinz (1935), Topologie (Berlin: Julius Springer).Reprinted 1965 (New York: Chelsea Publishing).