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Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap

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After the great enjoyment and education I obtained from Potter's excellent new book SET THEORY and its PHILOSOPHY, I immediately bought a copy of this treatise, subtitled PHILOSOPHIES OF ARITHMETIC from KANT to CARNAP.

I am a mathematician, not a professional philosopher. I am very interested in the philosophical ideas that motivate, generate, justify or refute parts of mathematics. Yet, after reading this dedicated effort by Potter to explain and criticize the thinking of Kant, Frege, Wittgenstein and Carnap regarding the nature of arithmetic truths, I came away, regarding those four, agreeing even more with Gauss' disparagement of "the say-nothing word-wisdom of the metaphysicians" (letter to Taurinus in 1824). At least Potter in his Conclusion section, announces frankly that "all the accounts we have considered have turned out to be flawed."

Fortunately, the chapter on Dedekind reviewed what I had already enjoyed about his work in Potter's set theory book. The chapters on Hilbert and Gödel were outstanding, though much too brief. Potter's discussion of his version of Gödel's second incompleteness theorem, with a discussion of outer consistency and inner consistency and their formal equivalence in case the proof function is "well-presented" was especially fascinating. I wish he had supplied a reference or two (e.g., to Sol Feferman's work) for the technical details he skipped in his argument .

Just as he wrote a much better philosophy of set theory book the second time, I think Michael Potter is certainly capable of writing a much better philosophy of arithmetic book the second time, provided he focuses only on the seriously mathematical work in that subject, such as that by Hilbert, Gödel, Gentzen, Kleene, Tarski, Feferman, Matiyasevich, Davis and other more recent mathematical workers in the logic of arithmetic.

My advice to mathematicians is to avoid this book except for the chapters on Hilbert and Gödel. Mathematicians really don't worry, as Frege did, about whether or not Julius Caesar was a number.