Approaching Infinity by Michael Huemer
“Infinity” is a concept that, if you’re not careful, can really bite you in the ass. In his latest book, Approaching Infinity, the philosopher Michael Huemer attempts to sharpen our idea of infinity to address two areas of concern.
One is the nature of infinite regresses. A famous one is the Regress of Causes, where one event needs to be caused by another event, but that event needs to be caused, etc all the way down the line. Thomists’ attempt to to address this regress is to stipulate an uncaused “unmoved mover” that starts the whole process going. Note that this was a regress that people thought they needed to solve; Huemer calls it thus a “viscous regress”. Other regresses are classified as “benign” though, like starting from the postulate that some proposition P is true. Then it’s true that P is true. It’s also true that it is true that P is true. And so on. Nobody really complains about that regress.
The other area is a number of famous “paradoxes of the infinite”. Examples include Zeno’s paradox, Thomson’s lamp, Galileo’s paradox, and Hilbert’s hotel (there are 17 paradoxes discussed in all). In each case, there’s a paradox that occurs when we assume an infinity is involved. Take Galileo’s paradox (please!): which are great in extent, the natural numbers (1, 2, 3, 4, 5, …) or the perfect squares (1, 4, 9, 16, 25, …)? At first it seems like there should be more natural numbers, since for any finite list of numbers from 1 to n there will be both perfect square and non-squares like 7 or 18. But you can map every natural number to a square (just square the number!) so (1 ↔ 1), (2 ↔ 4), (3 ↔ 9), (4 ↔ 16), and so on. Since every spot on that ladder is filled, it looks as though there are just as many perfect squares as natural numbers. A paradox!
Huemer goes over two classical accounts of the infinite, that of Aristotle and that of Georg Cantor, and finds both wanting in various ways. There are multiple chapters of the philosophy of numbers, sets, and geometrical points that I think fairly present the views of the usual Cantorian orthodoxy before poking holes in them. Even if you ultimately disagree with Huemer’s account, I think it’s a very readable and enjoyable introduction to issues in the philosophy of mathematics.
We then come to Huemer’s own account: extrinsic infinities are allowable or at least possible, whereas intrinsic infinities are not. An extrinsic property is one that changes when you change the “size” of the object in question. So things like size itself, volume, mass, energy content, etc. If you double the size of a block of wood, you double its mass. An intrinsic quantity is one that is comparably scale-invariant; things like temperature, speed, color, etc. If you imagine one cup of boiling water and then bring another cup of boiling water together with it, the temperature of the water does not change.
With this theory of the infinite in hand, Huemer is able to (mostly) resolve the 17 paradoxes and give an account of viscous and benign regresses. For the case of the Regress of Causes, and infinitude of causes going back in time is an extrinsic one, and so in principle non-problematic. The Thomist proposal of an “unmoved mover” who is infinitely powerful fails on this account, though, since such an entity would involve infinite intrinsic magnitudes.
If you’re interested in understanding infinite quantities which if you’ve done any work in the STEM fields you’ll have come across, I wholeheartedly recommend Approaching Infinity.