@1907russellRegressive proposes the following “regressive” method of determining the true axioms of mathematics: we begin with intuitions, or plausibility judgements, and then abduct to the best axiomatization of those judgments. This mirrors how laws of nature are determined in the sciences.
Related notes: Reflective equilibrium, after Rawls
Selected passages and quotations
- @1907russellRegressive: “The usual mathematical method of laying down certain premises and proceeding to deduce their consequences, though it is the right method of exposition, does not, except in the more advanced portions, give the order of knowledge. This has been concealed by the fact that the propositions traditionally taken as premises are for the most part very obvious, with the fortunate exception of the axiom of parallels. But when we push analysis farther, and get to more ultimate premises, the obviousness becomes less, and the analogy with the procedure of other sciences becomes more visible.”
- Godel (1990/1944): “[Russell] compares the axioms of … mathematics with the laws of nature and logical evidence with sense perception, so that the axioms need not … be evident in themselves, but rather their justification lies (exactly as in physics) in the fact that they make it possible for these ‘sense perceptions’ to be deduced… . I think that … this view has been largely justified by subsequent developments, and it is to be expected that it will be still more so in the future.”