Review: The Autonomy of Mathematical Knowledge

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Review
Curtis Franks' (full disclosure: he's is a friend of mine) PhD-turned-book Autonomy of Mathematics: Hilbert's Program Revisited is an exciting new look at an overlooked aspect of early twentieth century mathematics. Franks' writing is crisp and engaging, as he paints the picture of a man and his philosophy that so many have spurned.

The face of a brilliant mathematician or of a sun-hat enthusiast.
(Photo: Wikimedia)

Commentary
In the 1920's, Hilbert launched a program that was ostensibly aimed at solving the foundational crisis of mathematics-- the issues of paradoxes (e.g., Russell's paradox). The traditional understanding is that Hilbert's program failed because Gödel's incompleteness theorems threw a monkey wrench into any sufficiently sophisticated system that tried to prove itself.

Franks' thesis is that this is a narrow understanding of Hilbert's goals. While Gödel's results did complicate certain endeavors, Franks' suggests that Hilbert was really trying to take back mathematics. That is, certain other endeavors were trying to resolve the foundational crisis by rooting mathematics in some other discipline (e.g., philosophy). Hilbert's goal was to keep mathematics strictly within the realm of mathematics-- a unique feature of the discipline.

The Autonomy of Mathematical Knowledge is admittedly not for everyone (perhaps not even for me)-- about 20% of the book involves theorems I faithfully assume describe what the surrounding text tells me they do. Yet, about 80% of the book is eminently accessible-- the historical context, the epistemic issues, and the attempt to reconstruct a neglected approach combine for a great read.

See Also

• Russell's paradox for our discussion of that issue.
• Computational Theory for Lawyers for our discussion of the issues of intensionality (discussed in the book; not to be confused with intentionality).