Gottlob Frege (1848-1925) Wittgenstein: Foundations of Arithmetics is a paragon of philosophical analysis. Michael Dummett: Frege is the grandfather of analytic philosophy. Dummett: Frege – Philosophy of Language (1974)

Mathematician with strong philosophical interests and somewhat limited knowledge. Born in Wismar, „Pfarrersohn”, studied and taught in Jena. Never reached the position of an ordinarius (full professor). PhD in Göttingen, attended lectures of Hermann Lotze about the philosophy of religion. 1879 – Begriffsschrift (Conceptual Notation). First logical language that is rich enough to display the structure of every consequence in mathematics – this language is called conceptual notation. + a deductively complete logical theory (= every semantically correct inference can be justified by deduction). Basic principle: the Leibnitian idea of a lingua characteristica: the syntax should make logical connections totally transparent. Goal: the reduction of the arithmetics to logic (Leibniz again).

1884 – Grundlagen der Arithmetik (Foundations of Arithmetics) Philosophical program: „anti-psychologism”, anti-empirism, absolute truths. Critical analysis: what numbers are not – they are neither physical nor mental. Distinction between „concepts” (properties) and objects(individuals). Distinctions between concepts of different „grade” (order) – (properties of individuals, properties of first-grade properties, …) Two basic results of the critical analysis: Cardinality propositions (like ‘I have two hands’ , ‘ There were twelve apostles’ are about predicates („concepts”). The expressions ‘there are two’, ‘there are twelve’ and the like express concepts of second grade (they are second order predicates) – as well as the expression ‘there are’ or ‘there exists’. (Hume principle:) Two concepts have the same cardinality iff there is a one-to-one mapping between the objecs falling under them. Husserl’s criticism (Philosophie der Arithmetik, 1891) The informal exposition of the construction of natural numbers from logical concepts. General definition of cardinality (including infinite cardinalities). The cardinality of the concept ‘natural number’ is infinite (= different from any natural number).

1891-92 – three semantical studies of basic importance (Function and Concept, Concept and Object, On Sense and Reference [denotatum, nominatum, …]). Truth-values (True, False) as objects. Expressions have two sorts of semantical value: sense (Sinn) and referent (Bedeutung). „Frege triangle” : Expression – sense – referent. Three-component semantical notions are widespread from Aristotle to Frege’s contemporaries. There are two–component theories saying that the referent identifies the sense (medieval nominalists, J. St. Mill) – Frege seems to refute their views in On Sense and Reference. But the real novelties are these: The sense (the capacity of a sign to participate in information transfer) is a non-mental property. Sentences are the fundamental or prior sense-bearers. A dangerous further novelty: the concept of value range (Werthverlauf; a generalization of extension).

1893-1903: Grundgesetze der Arithmetik (Basic Laws of Arithmetics) I-II. 1st vol.: New version of the conceptual notation, based on the results of the Foudations and the semantical studies. The language presupposes that every function has a value range. Axiom V. gives the identification criterion for value ranges Formalizes the construction of natural numbers, following the sketch given in the Foundations (with some minor corrections). No informal analysis and critical part. 2nd vol: Finishes the theory of natural numbers, begins a theory of real numbers. The ideas are not really new and the critical passages are at least doubtful. Postscript : Russell’s letter and paradox. Frege’s reaction: not every function has a value range, Ax. V. should be restricted. Proposes a solution that proves to be wrong (see Quine: On Frege’s Way Out). Total collapse of his career and oeuvre.

The logicist program and Frege’s results Leibniz, Bolzano: arithmetics is nothing but logic developed further. Kant: logic is analytic, arithmetical truths are synthetic a priori. To refute Kant you should produce: Definitions of the basic notions of arithmetics (0, successorship, operations) from purely logical notions; Deductions of the axioms of arithmetics from basic principles of logic. This makes sense only if you give a clear and tenable delimitation of the scope of logic. Above that, you should guarantee on some way that nothing else gets used. The formalization of logic serves both these aims. A new branch in philosophy: philosophy of mathematics. Claims about mathematics that can be investigated by mathematical tools. A new branch in mathematics: foundational research. Research to answer questions about final (?) foundations of mathematics, mathematical existence, certainty of mathematics etc.

Frege’s philosophical legacy „Anti-Cartesian turn”: not the individual mind, but language and communication. Little known by his contemporaries (but: Husserl, Hilbert, Russell, Wittgenstein). One single famous student: Rudolf Carnap (logic only). Rediscovery around 1950: metaphysical turn in Anglo-saxon philosophy. From the 1980’s: neo-fregeanism (neo-logicism) in the philosophy of mathematics. (Arithmetics can be (consistently) founded on second-order logic and Hume’s principle.)