Bryan Caplan  

Approaching Infinity by Michael Huemer

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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.

COMMENTS (10 to date)
Lawrence D'Anna writes:

They teach economists set theory? Oh my god that's hilarious. Why?

If you ever decide to read a second book on the philosophy of math, I'd suggest The Autonomy of Mathematical Knowledge by Curtis Franks.

Michael Huemer writes:

Thanks, Bryan! You are a speedy reader and very helpful reviewer.

David Jinkins writes:

Lawrence: Probability/Measure theory relies on set theory, and econometrics is based on probability. Rigorous PhD econometrics courses use this stuff to study the behavior of estimators. Applied economists then apply the estimators to data to recover things like the wage returns to an additional year of education, or the effect of the minimum wage on unemployment.

Keith writes:

Bryan: Steven Landsburg is not a philosopher, but his "The Big Questions" certainly delves into the philosophy of mathematics and was a fun read.

David Jinkins: I can see why set theory might be covered in a rigorous econometrics class, but why microeconomics?

Justin writes:

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HM writes:

Economics studies optimization, and set theory is pretty handy to formulate a lot of optimization problems. Reason is that you choose the best option among all the options you have, and "all the options you have" is of course a set that depends on your income and prices.

Once you think about it set-theoretically, it becomes pretty easy to formalize the notion that we are better of with lower prices etc etc

With a sharp intuition for economics, there are alternative ways of understanding this of course. But if you add some set theory to good economic intuition, it actually allows pretty efficient communication of economic ideas.

Roger Sweeny writes:

I read David Foster Wallace's Everything and More: A Compact History of Infinity and I gotta say, didn't get much out of it.

Anyone know how the two books compare?

Lawrence D'Anna writes:


OK, so we're talking about learning the language of sets, pairs, cartesian products, functions, power sets, maybe ordinals then? Not stuff like Cohen forcing, large cardinals and Martin's axiom?

That makes more sense.

Still I'm surprised. Measure theory? Like, econometrics needs to know about sigma-algebras and Lebesgue integration? There's a lot of practical probability math you can do without talking about measure theory. I'm fascinated. What kinds of questions in econometrics depend on this stuff?

Michael Huemer writes:

Roger: Yes, my book is better.

S Mack writes:

Infinitely better?

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