Philosophy of Mathematics

Frege's Theorem and Foundations for Arithmetic 1. The Second-Order Predicate Calculus and Theory of Concepts In this section, we describe the language and logic of the second-order predicate calculus. We then extend this calculus with the classical comprehension principle for concepts and we introduce and explain λ-notation, which allows one to turn open formulas into complex names of concepts. 1.1 The Language The language of the second-order predicate calculus starts with the following lists of simple terms: object names: a, b, …object variables: x, y, …n-place relation names: Pn, Qn, … (n ≥ 1)n-place relation variables: Fn, Gn, … (n ≥ 1) The object names and variables denote, or take values in, a domain of objects and the n-place relation names and variables denote, or take values in, a domain of n-place relations. From these simple terms, one can define the formulas of the language as follows: φ & ψ =df ¬(φ → ¬ψ) φ ∨ ψ =df ¬φ → ψ φ ≡ ψ =df (φ→ψ) & (ψ→φ) ∃αφ =df ¬∀α¬φ If ν1 and ν2 are any object terms, ν1 = ν2 is a formula. to

Gottlob Frege First published Thu Sep 14, 1995; substantive revision Mon Oct 22, 2012 Friedrich Ludwig Gottlob Frege (b. 1848, d. 1925) was a German mathematician, logician, and philosopher who worked at the University of Jena. Frege essentially reconceived the discipline of logic by constructing a formal system which, in effect, constituted the first ‘predicate calculus’. In this formal system, Frege developed an analysis of quantified statements and formalized the notion of a ‘proof’ in terms that are still accepted today. Frege then demonstrated that one could use his system to resolve theoretical mathematical statements in terms of simpler logical and mathematical notions. 1. According to the curriculum vitae that the 26-year old Frege filed in 1874 with his Habilitationsschrift, he was born on November 8, 1848 in Wismar, a town then in Mecklenburg-Schwerin but now in Mecklenburg-Vorpommern. Frege never fully recovered from the fatal flaw discovered in the foundations of his Grundgesetze. 2.

Logicism and Neologicism 1. Historical background Kant had held that both arithmetic and (Euclidean) geometry were synthetic a priori, just as—for him—metaphysics was. Indeed, this was to explain the special status of both mathematics and metaphysics, so that the latter could enjoy the exalted status of the former. We might, indeed, at first suppose that the proposition 7 + 5 = 12 is a merely analytical proposition, and follows by the principle of contradiction from the concept of a sum of 7 and 5. Kant's search for conceptual containments is confined to those that he might be able to find among just the explicit constituents of the proposition concerned, unmediated by any connections with related concepts that do not themselves occur within the proposition. only the general features of succession and iteration in time can guarantee the existence and uniqueness of the sum of 7 and 5 …; only the unboundedness of temporal succession can guarantee the infinity of the number series, and so on ….[4] 1.1 Dedekind ν γ γ′