I reviewed a book about the math foundations crisis for the New York Times

I thought the book had some interesting virtues but was also quite badly overwritten, and very confusing in its treatment of the mathematics. Here’s my review.

For length reasons I wasn’t able to fit in all the material I wrote, so here’s a section addressing the confusingness, which didn’t make it into the final piece. Only here at Quomodocumque do you get the director’s cut!

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This is not the book to read if you want a precise understanding of the mathematics Hilbert, Brouwer, and Russell were going to war over. Bardi’s over-the-top style is confusing as often as it’s entertaining. “Contemplate a never-ending, constantly evolving universe of single-grain galaxies. Think about an eternity of revelation in every single grain for every subsecond of existence. That’s starting to get at the difference between large and small infinity—an infinite number of sand buckets versus the infinite variation inside every single bucket. Big infinity is much bigger. But even then, you’re not close to defining the continuum.” Even for someone like me who knows the definition of the continuum, this is hard to parse. I can’t imagine what sense it makes to everybody else. Later, Bardi writes:

“The continuum hypothesis says cardinality of an infinite set is like the Highlander – there can be only one!”

This is not, in fact, what the continuum hypothesis says, and indeed it is false; the whole point of Cantor’s theory, as Bardi explains perfectly well elsewhere in the book, is that there are many cardinalities (roughly, sizes) of infinite sets. Bardi sacrifices clarity of exposition here for the sake of a verbal flourish, and not even much of one – a movie catchphrase reference that only a small age band of readers, including Bardi and me, is likely to recognize.