The Mathematical Realism of Thomas Aquinas
/Reviewed by Dr. Timothy Kearns, Professor of Western Civilization and Writing at Legion of Christ College of Humanities. His philosophy articles include “Derived Quantity and Quantity as Such—Notes toward a Thomistic Account of Modern and Classical Mathematics.”
Rioux, Jean W. Thomas Aquinas’ Mathematical Realism. Cham: Springer International Publishing, 2023. 280pp; hardback: $129; eBook $99.
Jean Rioux has written an admirable synthesis of the main debates in the philosophy of mathematics between those of a largely classical perspective, particularly as in Plato, Aristotle, and Thomas Aquinas, and the whole modern mathematical project begun especially by Cantor. The book weaves together three main threads: a sympathetic concern for what is commonly known or accepted in our time by educated readers, a careful interpretation of the thought of Aristotle and Thomas Aquinas, and an engagement and historical summary of the key developments that have made modern mathematics different from its classical predecessors. The best quality of the book is precisely Rioux’s interweaving of the threads of his investigation. A dogmatic Thomist might be disappointed to find modern concerns taken so seriously, just as a dogmatic modernist might be frustrated at Rioux’s efforts to bring both sides together in his inquiry with a clear sense also of what educated opinion in our time would maintain. To be sure, Rioux is a Thomist, and even one who comes down to agreeing with Aquinas as against his modern critics and against contemporary philosophers of mathematics. But that does not keep Rioux from taking those criticisms or the more general modern efforts seriously and sympathetically. The interested Thomist may like to know up front that Rioux does not agree with authors like Maurer and Svoboda and Sousedik that Aquinas’s views can accommodate and integrate modern mathematics as such, in a more or less straightforward way. Rioux’s peculiar claim on this issue is that modern mathematics is best understood from Aquinas’s perspective as a part of the art of mathematics, a sort of quantitatively-reasoned creative art. I will return to this below.
The basic structure of the work is historical. Rioux begins, perhaps surprisingly, with Plato. He outlines both the major views of Plato that prepared the way for the kind of synthesis he sees in Aristotle and also the ways in which Platonism—or extreme realism in mathematics—can be seen to fail, both from an Aristotelian perspective and also from that of a more general perspective of our time. Again, one of the strengths of the book is Rioux’s inclusion of just this latter kind of perspective. This treatment of Plato, although not one of the longer parts of the book, also shows a connection that Rioux will use later and helps to unify the conclusion of the book: Rioux draws on a few modern authors who have sought to explain how mathematical objects for Plato occupy an intermediate place (14–16). To summarize, if we do not calculate with twoness or with just the two of a kind we see before us, then the numbers we do use must somehow be intermediate between the Forms (like twoness) and concrete things. Rioux echoes that the key here is “in how mathematics is done” (15), which at least points us toward the kind of awareness of mathematics as an art that Rioux will draw on later.
Rioux outlines Aristotle’s views in Chapters 3–5, covering Aristotle on the objects of mathematics, mathematics and the middle sciences, and on abstraction and intelligible matter. Rioux draws on much of the relevant literature here from the contemporary scholarship on Aristotle; so, this section is an important one to study for Thomists interested in these questions. But one topic I did not see him address, which would been easily done, is the persistent question and objection on hears from modern Aristotelian scholars like Jonathan Barnes and others that Aristotle’s syllogistic is so poor that it cannot even capture the kind of reasoning one finds in Euclid’s Elements. The response to this objection is just that the figures referred to in the Elements are not singulars but representations of universals and thus reasoning about them is universal and categorical. (Note that the Elements as a text can be read without reference to accompanying figures, as in the 9th century manuscript of Euclid in the Vatican Library [Vat.gr.190.pt.1,2], or with different realizations of those figures as in Thomas Heath’s translation and earlier renaissance editions.) Including this issue would have added one more to the list of uses of this already useful book.
The whole of Part II (Chapters 6–9) is devoted to a careful elucidation of Aquinas’s views on mathematics, from mathematics as a science to mathematics as an art. Much of this will be familiar to Thomists who are interested in mathematics, but Rioux has provided a great service to Thomists by bringing everything together in one place, putting it into order, and engaging with the relevant modern scholarship on Aquinas. Rioux’s interpretation of Aquinas accords well with the standard account among Thomists. Yet throughout, Rioux has kept a certain dialectical distance in the form of providing the perspective of an educated contemporary reader and how he might react to some of what is being argued for, especially if it might be outside the mainstream of contemporary thought. This gives Rioux’s book an added edge that is often lacking in the writings of Thomistic philosophers. It also makes his case all the more powerfully when he does argue—convincingly—that Aquinas’s views are the right ones.
A word about Rioux’s central Chapter 7 (“To Be Virtually”), on virtual existence and mathematics. Much of this chapter is argumentatively engaged not with the scholarship on Aquinas (for on this relevant issue, sources are sparse), but with contemporary philosophies of mathematics like intuitionism, through which Rioux lays the groundwork for his later claim that modern mathematicals are object of the art of mathematics but not the science of mathematics. For Aquinas, Rioux says, on any question within the order of quantity,
either a given statement or its contradictory is true even before it has been shown to be so by reduction to first principles . . . Though we cannot see everything contained in mathematical first principles, still, what is contained therein is determinately contained, in a manner sufficient also to determine our minds when drawing a conclusion. As for the conjecture [whether there is a greatest pair of primes that differ by two], given the nature of the mathematical unit, the property of being greatest is either compatible in part or universally incompatible with [the] twin prime [conjecture]. Given our intellectual frailty—that we cannot see in a glance what the principles require in this instance—what remains is to discover those connections required to reach a conclusion. Yet that is a merely contingent event. It could occur now, or never. (120)
The point is that the arithmetical unit and the three dimensional magnitude (or body, as Rioux says) contain virtually all the respective quantities and therefore all the truths about them. I found this point especially worth making since I had not heard it made elsewhere and it seems to be right, and importantly so.
After his articulation of Aquinas’s views, not in a vacuum but engaging with those of contemporary mathematics, Rioux proceeds to consider other the major schools of contemporary mathematics, including the Aristotelian realism of the Sidney School, e.g. in James Franklin’s book, An Aristotelian-Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure (Palgrave-MacMillan, 2014). Much of what Rioux does is show the problems that each of the major schools face either in their own terms or as posed by other schools against them or even just from the perspective of the educated reader. He also criticizes them from the classical perspective of Aquinas. One strength is that, even in this the most argumentative part of the book, Rioux never loses sympathy for (and hopefully with) his interlocutors. Rioux dismisses nothing out of hand, but is able, perhaps from his long study of the relevant texts and authors, to consider each and all in a way that cannot be described as unfair. Even readers already familiar with Rioux’s other work and with classical criticisms of contemporary mathematics will find here many texts quoted at some length from original authors on just the kinds of issues Thomists would want to see addressed.
Rioux does make a proposal from a Thomistic perspective for how best to account for what modern mathematical discoveries really are and what they are about. As I mentioned above, Rioux maintains that the best way to understand them is as products of mathematical art. Here is a summary of the different aspects he sees in mathematics and how overemphasizing one or the other can lead to error:
Choosing to emphasize its scientific (and so realist) character, one might be inclined to see its objects—all its objects—as having a place in either the immaterial or material realms; such an emphasis might see the use of constructive definitions as one step toward fictionalism. On the other hand, overemphasizing its artistic character might lead to an exaggerated emphasis on free constructability among the mathematicals; other considerations (such as compatibility with mathematical principles) might then be regarded as merely limiting. Finally, focusing upon the real benefits of mathematica utens—the middle sciences, for example—could well lead someone to consider success the sole, or at least the overwhelmingly important, criterion of its value; the more practically minded among us might then tend to regard the real as irrelevant. (132)
Rioux’s final question in the book and his answer reveal the center of his own proposal for understanding modern mathematics: “[W]hat of mathematical models which are assumed false, yet which produce consistent results—the counterpart to applying the episteme of mathematics to physical problems?” (275) Regarding the application of fictitious mathematical constructions and models, Rioux says,
We call such a use heuristic, which is to say one is employing a method toward a determinate result, either prescinding from the question whether it is or is not a legitimate method, or even knowing that it is illegitimate but adequate to the short-term goal. . . . We mentioned i, above, as an instance. The tedium of constantly reducing i to its nonimaginary roots—especially if its use seems not to compromise the truth of the matter under consideration—is a high price to pay. It is likewise possible to render balance sheets without including negative numbers in the calculations, or to recast stress tests in terms of ever-changing units so as to avoid fractional numbers. Still, what a high price to pay, especially if we do not thereby compromise the truth. (Ibid.)
Here I confess that I remain unconvinced. A better approach, it seems to me, emerges from Rioux’s chapter on virtual being together with his critiques of some of the kinds of supposed discoveries in the last century: perhaps many modern mathematicals (but not all since some Rioux shows to be contradictory) are virtually contained within the unit for discrete quantities, within three-dimensional magnitude for continuous ones—somehow.