Frege on knowing the foundation

Abstract

The paper scrutinizes Frege’s Euclideanism—his view of arithmetic and geometry as resting on a small number of self-evident axioms from which nonself-evident theorems can be proved. Frege’s notions of self-evidence and axiom are discussed in some detail. Elements in Frege’s position that are in apparent tension with his Euclideanism are considered—his introduction of axioms in The Basic Laws of Arithmetic through argument, his fallibilism about mathematical understanding, and his view that understanding is closely associated with inferential abilities. The resolution of the tensions indicates that Frege maintained a sophisticated and challenging form of rationalism, one relevant to current epistemology and parts of the philosophy of mathematics. From the start of his career Frege motivated his logicism epistemologically. He saw arithmetical judgments as resting on a foundation of logical principles, and he saw the discovery of this foundation as a discovery of the nature and structure of the justification of arithmetical truths and judgments. Frege provides no focused and sustained account of the foundation, or of our epistemic relation to it. It is clear from numerous remarks, however, that he saw the foundation as consisting of primitive logical truths, which may be used as axioms and which are self-evident. He thought that they are in need of no justification from any other principles. Logical and arithmetical principles other than the self-evident ones are justified by being provable from self-evident axioms together with selfevident definitions and self-evident rules of inference. At first glance, Frege may seem to be a prototypical representative of the Euclidean, rationalist tradition in the epistemology of logic and mathematics. But further scrutiny reveals a more complex situation. Although Frege regards the axioms as self-evident, he expressed a sophisticated modern awareness of the fact that what can seem obvious may turn out not even to be true. (The distinction between self-evidence and obviousness will be discussed.) Ironically, this awareness led to at least dim anticipations of the problem with his fifth axiom, a problem that decimated his logicist project. Frege was aware that principles that he put forward as axiomatic—even some that, unlike Axiom V, have endured as basic principles of logic—were not found to be obvious by his peers. In arguing for his logic he made use of methods that were explicitly pragmatic and contextualist, in senses that will be discussed. The relations between these