Logicism Revisited*

I was led to revisit logicism by an historical riddle, so I will begin with that. Mathematics always played a key role in the philosophical battle between empiricists and rationalists or intellectualists. The empiricists always had trouble with mathematics: some (like Locke) said it consisted of 'trifling' or 'verbal' propositions, others (like Mill) said it consisted of empirical truths (Hume vacillated between these two as regards geometry). Neither account seemed plausible. The intellectualists, on the other hand, derived their chief comfort and inspiration from mathematics. Anyone who denied that a priori reasoning could issue in genuine knowledge was met with the triumphant question 'What about Euclid's geometry?'. Russell describes the situation well (in his [1897], p. 1):

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