What can one say about a book on infinity that hasn’t been said before? An infinite number of things, presumably, but I’ll make this brief.
The book, Approaching Infinity, is by philosopher Michael Huemer. Perhaps you’ve heard of him — but why? If you’re a libertarian, but not a philosopher or “into philosophy,” it’s likely because of his well-received book, The Problem of Political Authority (2013).
If you’re a libertarian and, though not a philosopher, are into philosophy, you may also be aware of Huemer’s excellent online-available essays on the right to own a gun and the right to immigrate. (I imagine readers on both the Left and Right are now gnashing their teeth.)
Huemer, like Robert Nozick before him, is clearly better described as a philosopher who is a libertarian than as a “libertarian philosopher.”
But Huemer is nothing if not prolific. Libertarians who are really into philosophy may even be aware of his criticism of Ayn Rand, his argument that we sometimes have a duty to disregard the law, his argument that attorneys have a moral obligation not to defend unjust causes, his criticism of the US government’s War on Drugs, and his essay on why people are irrational about politics (also a TED talk!).
But — and this is the point I want to stress — even though he’s published much of interest to libertarians, Huemer, like Robert Nozick before him, is clearly a person better described as a philosopher who is a libertarian than as a “libertarian philosopher.” His first book, Skepticism and the Veil of Perception (2001), dealt with epistemology (the field of study that led to his hiring at University of Colorado, Boulder); his second, Ethical Intuitionism (2005), focused on ethics. Now, having covered epistemology, ethics, and politics, Huemer, in Approaching Infinity, turns to the philosophy of mathematics (with an occasional nod to some issues in the philosophy of science). Clearly a well-rounded guy, philosophically speaking.
Also an iconoclast:
- Although most philosophers since Descartes have opposed direct realism (the view that we are directly aware of real, physical objects), Huemer argues for just that point of view.
- Although most modern philosophers oppose ethical intuitionism, the view that we can have direct knowledge of objective moral truths, Huemer again argues for exactly that.
- Although most people readily accept political authority, and most philosophers are not anarchists, Huemer argues both against political authority and for a capitalist version of anarchy.
So it should surprise no one that Huemer, in analyzing some foundational issues in mathematics in order to solve various paradoxes of infinity, is willing to advance bold claims.
Almost everyone is familiar with at least some infinity paradoxes. We’ve all heard about Zeno and why that ball coming at you will never reach you, or why the hare can never catch the tortoise. Any you’re probably aware that strangeness results when even simple arithmetic is applied to infinity. E.g., ∞ = ∞ +1. Subtract ∞ from both sides: 0 = 1.
But I had no idea there were at least 17 different paradoxes associated with infinity. From Hilbert’s hotel to Gabriel’s horn . . . from Thomson’s lamp to Benardete’s paradox . . . from ancient Greek problems to dilemmas developed only in the past century . . . Huemer describes them all and then starts to evolve some background needed to solve the infinity paradoxes. There are discussions of actual and potential infinities, of Georg Cantor’s set theory, of the theory of numbers, of time and space, of both infinity and infinitesimals. Of the metaphysically impossible and the logically impossible. Of the principle of phenomenal conservatism (which Huemer introduced in his epistemology text), and even of the synthetic a priori.
Huemer, in analyzing some foundational issues in mathematics in order to solve various paradoxes of infinity, is willing to advance bold claims.
In building the background to handle the infinity paradoxes, Huemer argues that extensive infinities (including the cardinal numbers) can exist but not as specific magnitudes. Thus, the positive integers are infinite, in the sense that for any such number you can find higher positive integers, but not in the sense that there is a number “infinity” that is higher than all the positive integers. You cannot add and subtract “infinity” as I did in the previous paragraph. And he argues that while extensive magnitudes (time, space, volume) can sensibly approach infinity in this understanding, infinite intensive magnitudes (such as temperature, electrical resistance, attenuation coefficient, etc.) are metaphysically impossible. This distinction allows several paradoxes to be solved, or avoided.
A fascinating section of the text discusses various forms of impossibility. Sometimes philosophers note that X is physically impossible, given the laws of the universe as we now understand them, but nonetheless that it could be possible in a similar but slightly different possible world — say, with a slightly different Coulomb constant. But at other times X is deeply physically impossible. Consider these two alternatives described by Huemer:
Compare this pair of questions:
A. If I were to add a teaspoon of salt to this recipe, how would it taste?
B. If I were to add a teaspoon of salt to this recipe in an alternative possible world in which salt is a compound of plutonium and mercury and we are sea creatures who evolved living on kelp and plankton, how would it taste?
Huemer notes that it’s not merely that we have no idea about how to answer B but that, more importantly, even if we could answer B, answering it gives us no intuitions, is of no help in trying to figure out the answer to A. Though Huemer makes this point in the context of determining what counts as a solution to an infinity paradox, it also has direct application to various thought experiments in other areas of philosophy and to what counts as a helpful or unhelpful thought experiment. (On this see my own work, “Experiment THIS!: Libertarianism and Thought Experiments.”)
Related to the paradoxes of infinity are the problems of infinite regress. You may have heard of the problem of the regress of causes: asked what caused A, you explain that it was caused by B. But what caused B? C caused B. But … here is an infinite regress. Does this imply that we never really understand what caused A?
There are other interesting infinite regresses: of reasons, of truths, of resemblances, etc. Huemer offers helpful insights here as well, elaborating various factors that determine whether such infinite regresses are vicious or benign.
Did I mention that Huemer can be iconoclastic? Consider these passages from Approaching Infinity:
- “There are certain philosophical assumptions that tend to generate strong resistance to my views, and these assumptions are commonly accepted by those interested in issues connected with science and mathematics . . . I have in mind especially the assumptions of modern (twentieth-century) empiricism . . . the doctrine that it is impossible to attain any substantive knowledge of the world except on the basis of observation.”
- “In the original, core sense of the term ‘number,’ zero is not a number. . . . Why is zero not a number in the original sense? Because a number, in the primary sense, is a property that objects can have, whereas zero is not a property that objects can have.” Huemer extends the concept of number to include zero but explains why such an “extension” does not work for “infinity” as a number.
- “There are reasons to doubt that sets exist. No one seems to be able to explain what they are, they do not correspond to the ordinary notion of a collection, and core intuitions about sets, particularly the naive comprehension axiom, lead to contradictions.”
In his final chapter, Huemer, taking to heart Nozick’s concerns about coercive philosophy, offers readers his own thoughts about problems that remain: which of his answers leave him concerned or unsatisfied, arguments that are incomplete, areas for further exploration.
As in his earlier books on ethics, epistemology, and politics, Huemer’s style is as easy and enjoyable as his logic is rigorous. Intelligent laypeople who are interested in philosophy can follow his thoughts without difficulty. No Hegel here.
Because I have little background in the philosophy of mathematics, I approached Huemer’s latest effort with trepidation, despite having very much enjoyed his three earlier books. But now that I’ve read it, I highly recommend it. The best news: before finishing Approaching Infinity, you’ll have to read halfway through it, and before that one-quarter of the way, and before that one-eighth, and before that. . . . Yet despite this you can read it through to the very end, and be enthralled on every page.