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Michael Huemer, Approaching Infinity, 2016

There are many good introductory books on the infinite, some of which have reached enormous popularity. Two notable examples are Rudy Rucker’s Infinity and the Mind and John Barrow’s The Infinite Book. Huemer’s book differs from books of this sort in one main aspect: it is essentially a philosophy book written by an academic philosopher, who aims to investigate the kind of reasoning involving the infinite rather than the infinite tout court. In fact, in the book, Huemer makes several references to appearances of the infinite outside philosophy, for instance, in mathematics, physics or theology, but the book’s main focus remains philosophical throughout.

Glaring incarnations of the infinite in philosophy (logic) are the paradoxes. In fact, it is traditionally assumed that the infinite is, in itself, an endless source of paradoxes. Therefore, the book starts with a list of 17 paradoxes. These include some notorious and well-explored arguments such as Zeno’s or Galileo’s paradoxes, as well as others less known to the general public, such as Gabriel’s horn in geometry, or the Martingale betting system in probability theory. Another bunch of difficulties traditionally associated with the infinite is that arising from ‘infinite regresses’. Take, for instance, the Kalam cosmological argument: since everything that is in the physical world is caused by something else, there must be a first cause, otherwise there would be an endless (infinite) sequence of causes. Another famous regress is that embodied in the argument known as the Third Man in Plato’s Parmenides: any form F0 is also F0, but the relationship between the form F0 and itself is, again, one further form F1, and this process can be continued indefinitely.

Section I is entirely devoted to illustrating such arguments. Section II introduces the main historical conceptions of the infinite, which should also help overcome the main difficulties associated with it. Among the doctrines examined, Cantor’s conceptions and methods, which, as is known, gave rise to contemporary set theory, feature prominently. However, the book also addresses, among other things, the dichotomy between actualism and potentialism (mainly due to Aristotle) and Kant’s famous antinomies of pure reason.

All these conceptions represent equally legitimate, if mutually alternative, ways to handle the paradoxes and, in general, the issues associated with the infinite. However, Huemer wants to present his own account of the infinite which, in his view, fares better than the accounts reviewed in Section II. This task he accomplishes in Section III.

Huemer’s overall strategy is twofold. On the one hand, he denies that the actualism/potentialism dichotomy, as historically articulated, is relevant to grasping the essence of the infinite. As he aims to show, for instance, Aristotle’s potentialist account is not entirely appropriate: Huemer is keen on adopting significant portions of Aristotelian potentialism, but he is not willing to adopt in full its logical consequence, the idea that the actual infinite is impossible. On the other hand, Huemer thinks that Aristotle was entirely right in assuming that there may not be such infinite entities as infinite magnitudes or numbers: precisely such things give rise to the sort of paradoxes we still discuss.

In sum, Huemer’s account may be viewed as a semi-Aristotelian, finitistic (in a sense, even intuitionistic) view which, however, does not deny the completeness (better, ‘completability’) of such collections as series of numbers, variables, causes, phenomenal properties etc. More precisely, Huemer distinguishes between ‘extensive’ and ‘intensive’ infinite magnitudes (p. 150): the former may be legitimately viewed as sums of their components, whereas the latter may not. Only the use of extensive magnitudes is fully sanctioned. Now, infinite cardinalities, such as ℵ0, are intensive magnitudes, and are, thus, illegitimate. In general, all those magnitudes which violate the intuitively given properties of finite numbers may not be seen as legitimate quantities (this extends to other quantities like infinitesimals). The main purpose of Huemer’s account of the infinite, as said above, is to ackle the paradoxes and the problem of infinite regress. In Section III the ‘old theory of the infinite’ is replaced by a ‘new theory of the infinite’, in essence, Huemer’s semi-Aristotelian view. For instance, paradoxes like the fact that, from 1 + ∞ = ∞ one may derive that 1 = 0 are defeated immediately, once one recognises that there are no infinite quantities which may be added to finite numbers in the same way as finite numbers are added to other finite numbers. Similarly, paradoxes like Galileo’s paradox disappear: it is just not true that there are as many square numbers as numbers, simply because the relationship ‘greater than’ ought not to apply to infinite collections, if such collections are incorrectly construed as numbers. Huemer’s solving strategy also extends to the paradoxes of geometry, such as that of measure: if points measure 0, and line segments are made of points, then line segments should measure 0. However, line segments are not made of points, hence the measure of a line segment is not simply the sum of its components (it is not an extensive magnitude). Finally, using the same conceptual strategy, one can also successfully cope with regresses. The Kalam cosmological argument must ultimately be rejected, as one may conceive of an infinite, complete series of causes, while denying that such causes are, altogether, an infinite cardinality.

The book has two main assets: first, it offers a coherent, well-thought strategy to address (and solve) the difficulties traditionally associated with the infinite. Secondly, it presents and discusses a plethora of arguments/paradoxes/ideas relating to or involving the infinite. By combining a deft analytic approach and a mastery of the scholarship of the infinite, the book really constitutes a good introduction to central themes in the philosophy of the infinite. However, there are respects in which Huemer’s book is clearly lacking.

First of all, Huemer presents his own account of the infinite as fully original and innovative, but one may legitimately question the correctness of this claim: Huemer mostly relies on Aristotle’s and other finitists’ ideas and conceptions. The clear novelty of his account is the fact that, although essentially finitist, it is also, in a way, actualist (although the author would deny that this is true from his standpoint). More precisely, Huemer is a finitist mathematically and metaphysically, and an actualist (although a special kind of actualist) logically, so he can consistently think that reality is not made of points or of infinite cardinalities, while, nonetheless, holding the view that there is no reason to give up the idea that actually (‘real’) infinite processes may be conceived of without any contradiction. However, what is presented to us as the main novelty in the book, that is the combination of finitism and actualism, is by no means argued for in depth. We are given some arguments – better, some hints at arguments – but it seems that Huemer is more interested in convincing his readers that the difficulties with the infinite may be solved by using his own account than in developing his ideas philosophically to a sufficient extent. As is clear, this is far from being satisfactory for anyone expecting discernible philosophical advances.

The book is also oversimplistic in many respects. Its discussions of sets, and of the view that set theory really is the foundation of mathematics, are slightly superficial, and there are too many intricate, but crucial, philosophical issues that the book simply does not take into consideration. In sum, while one may praise the author for providing us with a tidy, confident and profitable overview of the main problems, paradoxes, arguments and modes of reasoning associated with the infinite, one may not be equally happy about the book’s level of philosophical originality and analytic depth. There are surely valid reasons to read this book, such as its compelling style, the level of competence on the infinite it shows, and the interesting philosophical arguments and mathematics it discusses. However, there are equally strong reasons for disagreement and dissatisfaction with much of what it states (or purports to state), both because its main theses are not sufficiently argued for, and because they are so controversial that they would deserve another (and different) book.

Claudio Ternullo
University of Vienna

Reprinted from ESSSAT News and Reviews, n. 28:1 (March 2018), pp. 40-43.