Neo-Logicism and Russell’s Logicism

Certain advocates of the so-called "neo-logicist" movement in the philosophy of mathematics identify themselves as "neo-Fregeans" (e.g., Hale and Wright), presenting an updated and revised version of Frege's form of logicism.Russell's form of logicism is scarcely discussed in this literature and, when it is, often dismissed as not really logicism at all (in light of its assumption of axioms of inWnity, reducibility and so on).In this paper I have three aims: Wrstly, to identify more clearly the primary meta-ontological and methodological diTerences between Russell's logicism and the more recent forms; secondly, to argue that Russell's form of logicism oTers more elegant and satisfactory solutions to a variety of problems that continue to plague the neo-logicist movement (the bad company objection, the embarrassment of richness objection, worries about a bloated ontology, etc.); thirdly, to argue that neo-Russellian forms of logicism remain viable positions for current philosophers of mathematics.i. introduction I t was for a while fashionable to say that logicism was dead.Even now we see the sentiment repeated.1 As recently as 2010, a website developed for a popular logic textbook 2 claimed that "in the 1920's, the German mathematician and logician Kurt Gödel made an astonishing discovery, a discovery that eTectively ended the hopes of the logi-128 kevin c. klement 3 What is particularly strange about Herrick's way of putting the point is that he somehow thinks Gödel's results pose a greater problem for reducing mathematics to logic than to set theory, whereas of course Gödel's results equally show that no recursively axiomatized set-theoretic foundation for mathematics is complete.4 See especially Crispin Wright, Frege's Conception of Numbers as Objects (Aberdeen: Aberdeen U. P., 1983), and Bob Hale and Crispin Wright, The Reason's Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics (Oxford: Clarendon P., 2001).Neil Tennant and Stewart Shapiro have also done work in sympathy with their programme, though without the same kind of commitment to the project.Moreover, George Boolos, Richard Heck and Kit Fine have contributed in important ways, though none of them, I think, actually endorse the position in the end.cists", namely, that "there are truths of arithmetic that are in principle not provable", and "with tight deductive proofs that convinced everyone" (!), Gödel proved that "logicism is false", and indeed: Today, logicians agree that although mathematics can be derived from set theory, it cannot be derived from pure logic alone.Thus was an exciting new theory Wnally proven false.

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