Objections and Comparisons
Relativism is distinguished from pluralism; mathematical theses which are true in some systems but not others are held to express different propositions which different truth values in each. Comparisons are drawn with if-thenism, postulationism, and deductivism. A number of objections are tackled: that proofs need not be formal, that mathematicians believe theses without having proofs, that axiom systems can be inadequate and incomplete, that any consistent sentence counts as a mathematical truth, since it is provable from some system. In response, further comparisons are drawn with neo-logicism, and the role of abstraction principles such as Hume's principle discussed. The superiority of neo-formalism is urged, in the treatment for example of the ‘Julius Caesar problem’ and the problem of pairwise inconsistent abstraction principles. Finally Gödel's incompleteness theorem and the problem of true, but unprovable, Gödel sentences, is raised. A quick response by the neo-formalist is rejected, and the difficulty set aside until the last chapters.
Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.
If you think you should have access to this title, please contact your librarian.