The Department of Logic & Philosophy of Science presents
"Frege: Logical Objects by Abstraction and their Criteria of Identity"
with Matthias Schirn, Munich Center of Mathematical Philosophy, University of Munich
Friday, March 22, 2013
Social Science Tower, Room 777 (LPS Conference Room)
In Grundlagen, Frege suggests the following strategy for the envisaged introduction of the real and complex numbers as logical objects: it should proceed along the lines of his introduction of the cardinals, namely by starting with a tentative contextual definition of a suitable number operator in terms of an abstraction principle whose right-hand side (a second-order or higher-order equivalence relation) was couched in purely logical terms, and by finally defining these numbers as equivalence classes of the relevant equivalence relation. In this talk, Schirn will discuss Frege’s introduction of the cardinals as well as his projected introduction of the “higher” numbers with special emphasis on the so-called Julius Caesar problem. Moreover, he will analyze the variant of this problem that Frege faces in Grundgesetze when he comes to introduce his prototype of logical objects, namely courses-of-values, by means of a stipulation later to be enshrined in Basic Law V. The problem is now clad in formal garb. A second main issue, closely related to the first, is the nature and role of Frege’s criteria of identity for logical objects. Schirn will argue, among other things, (i) that pursuing the above strategy for the introduction of the higher numbers would inevitably lead to a whole family of possibly intractable Caesar problems; (ii) that in Grundlagen Frege considers the domain of the first-order variables to be all-encompassing, while in Grundgesetze there is a serious conflict between (a) his practice of taking the first-order domain to be all-inclusive when he elucidates and defines first-level functions and (b) the assumption underlying his proof of referentiality in §31 that the domain is restricted to the two truth-values and courses-of-values; (iii) that the range of applicability of the criteria of identity inherent in Hume’s Principle and in Axiom V — equinumerosity and coextensiveness of first-level concepts (functions) — lack the requisite unbounded generality even after having undergone considerable extension; (iv) that regarding the Caesar problem for cardinal numbers and its alleged solution the situation in Grundgesetze differs significantly from that in Grundlagen. If time allows, Schirn will also respond to the hypothetical question whether under certain conditions and especially in the aftermath of Russell’s Paradox Frege could have introduced the real numbers via an appropriate abstraction principle without offending against his logicist credo.
For further details, please contact Patty Jones, firstname.lastname@example.org or 949-824-1520.