Approaching Infinity by Michael Huemer, by Bryan Caplan

My favorite philosopher has just published a new book on metaphysics and philosophy of mathematics, entitled Approaching Infinity . 



While I have little intrinsic interest in infinity, I can't imagine a better book on the topic.  I devoured the whole thing this weekend.  Mike begins by cataloging six forms of "infinite regress" and seventeen puzzles about infinity.  He then carefully reviews and critiques earlier theories of infinity, from ancient Greek philosophy to modern set theory.   (If you suffered through set theory in Ph.D. microeconomics, you'll especially enjoy the latter discussion).  Finally, he presents his own theory of infinity, beginning with the key distinction between logical impossibility and metaphysical impossibility.  Then he uses it to distinguish "virtuous" from "vicious" infinite regresses, and solve his seventeen puzzles.

My favorite part of the book: Mike's reply to the "anti-foundational" view that our beliefs can be justified through an infinite series of reasons rather than resting on foundational premises.  Mike:

Perhaps there could be a being with an infinite series of reasons for one of its beliefs; perhaps not.  Be that as it may, it is extremely implausible that humans are such beings.  Even among those few philosophers who defend the possibility of an infinite series of reasons, none has provided any examples to show how such a series would go.

In fairness, publisher-imposed length limits prevent a philosopher from stating an infinitely long argument.  If, however, someone were to state even the first fifty steps in the infinitely long chain of reasons that justifies the proposition [I exist], this would go a long way toward convincing me that there might be such an infinite claim.  No one has done anything like that; indeed, no one seems able to provide even the first ten steps.

This extreme implausibility applies to most (perhaps all) other infinite regresses that involve human beings.  For instance, suppose one held that in order for any choice to be truly free, the agent must freely choose the motives for which the original choice was made.  This leads to an infinite regress of choices, and whether or not it is metaphysically possible, it is extremely implausible that any human being ever performs such an infinite series of choices.

I doubt I'll ever read another book steeped in the philosophy of mathematics.  But Mike is such a singular intellect I'm grateful to read his thoughts about anything.  If you're going to read one book about infinity, choose Approaching Infinity.

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