Mathematical realist Blaise Pascal
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
— Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”
These are the final words of E. Wigner’s famous essay, which you can find on-line here. Wigner uses the God-haunted word “miracle”, once in the quoted paragraph, several times in the whole essay. But His presence is more palpable when he speaks about the wonderful, unmerited gift, of mathematics; I do not know what was his relation with the Deity, but he may have found himself in the sad predicament of being thankful and having no one to thank for.
The sense of the miraculous, or more prosaically for those who distrust such intrusions of the numinous in our ordinary lives, the sense of wonder, is the beginning of all True Philosophy. There is a puzzle here and an answer must be found. Why is the universe orderly? Not only that, why is the orderliness of the universe of such a nature that it can be described and explained, at least in part (and I would add, only in part) by mathematics? And by supremely abstract mathematics, discovered and developed to solve purely mathematical problems, quite independently of its appropriateness for describing the real world?
A tentative answer is to adopt a realist stance and say that mathematical objects have an extra-mental reality just like the common ordinary objects of our experience like rocks, trees, persons and planets, and in studying them we are, somehow, somewhere, somewhen, discovering objective features of reality. This suggestion needs some considerable spelling out, but to see how it imposes itself forcefully, let us watch W. Quine make the so called indispensability argument:
- We should believe the theory which best accounts for our sense experience.
- If we believe a theory, we must believe in its ontological commitments.
- The ontological commitments of any theory are the objects over which that theory first-order quantifies.
- The theory which best accounts for our sense experience first-order quantifies over mathematical objects.
- We should believe that mathematical objects exist.
Lest there be any confusion, I hasten to add that I do not find this argument compelling. Premises 2 and 3 are particularly hard to swallow. If I invoke it here it is for two wholly different sorts of reasons: one, the delightful irony in borrowing a stick from your enemy (Quine was a confirmed naturalist) with which to beat your other enemies and two, to once again stress that, given how inimical mathematical realism is to naturalism, it is quite telling that Quine felt compelled to adopt it (and expansively redefine naturalism along the way). For otherwise, if mathematical objects are nothing but a product of the mind with no objective basis on reality (the mind itself being, on some accounts, the random material product of a highly contingent history), in the same way as fictions are, then what can possibly explain their appropriateness in the description of reality?
If that is indeed the case, and the Dirac operator of a spin manifold or the curvature of a connection are just as fictitious as novels or children stories, then are we not forced to retreat to the nigh-mystical position of Wigner? And if that is indeed the case, what is the principled distinction between say, the Dirac operator on a spin manifold, and quarks which because of QCD confinement cannot be observed free? Are we not committed to say that quarks are equally fictitious? And if that, then are we not obliged to conclude that science tells us nothing objective about the nature of the world? And if that, should not electrons or evolution by natural selection go the same way of quarks?
But enough of questions and on to the heart of the matter. Mathematical realism is just one among a cluster of related problems involving the ontological status of such seemingly abstract objects as universals, properties, propositions, relations, etc. All these notions are intimately related to one another. For the sake of simplicity, I will lump them all under one banner, although realists of various stripes will insist on all sorts of distinctions.
Here, I will defend not mathematical realism per se, but the closely related problem of realism with respect to universals. For those who have read The Last Superstition, you will know that Edward Feser addresses this problem in pages 39-49. Although I will return to some of the arguments made in those pages, here I will take a different route and follow J. P. Moreland’s Universals.
The plan for this series is then as follows: first I will present an account of the realist position with respect to universals, the phenomena that it purports to explain and the challenge it presents to anti-realists. Then I will survey some variants of anti-realism with respect to universals and point out, not only the problems they have, but how these problems typically recur in all anti-realist stances. Then I will respond to the more common anti-realist objections. I will wrap things up with a few words on why this seemingly abstruse and irrelevant problem is actually at the heart of many contemporary discussions by (very) briefly surveying one such example and along the way annoy to no end and give offense to those delicate, clueless, liberal souls. At least so I hope.
Some caveats are in order. My background is in mathematics, and to a lesser extent in physics. From this, two immediate corollaries follow:
1. As a general rule, mathematicians are notoriously bad expositors; over-abundance of technical detail in contrast to a dearth of understanding. In mathematics, this is somewhat inevitable, as knowing the technical details is more often than not what understanding amounts to. So while I cannot say that I have any of the qualities usually recognized in mathematicians like rigor, attention to detail, etc., you can surely expect that I amnot the exception to the general rule.
2. Of necessity, not being a philosopher, I will say nothing that is original. Or to put it in other words, the only originality I can claim is in my mistakes.
In 2, originality is used in the paltry sense of new or novel; but there is a deeper sense to it, related to the root word origin. To recognize it, it is probably best to step into the world of the arts, literature in particular. The greatness of an author like T. S. Eliot or James Joyce (if these examples offend your tastes, replace them by your own as similar remarks apply) lies in part in the fact that their genius has opened up a clearing in our common cultural heritage, where their voices rise and add up to the chorus (or cacophony: choose your preferred metaphor) of the voices of the Great and Magnificent Dead. There can be no understanding of Joyce, understanding in the deeper senses of literary criticism, without locating him within the total order of literature and clarifying his relationship with his predecessors, Shakespeare and Homer above all.
When we read Joyce in its strongest, say in the final pages of Finnegans Wake, we feel that a limit has been reached and that words have been found to express the hitherto inexpressible, what was always there since the origin, what is definitive of our nature of human beings qua human beings but that would not, and could not have been recognized unless it was first illumined to us. These illuminations then become the guiding lights in the inner theater of our imaginings. Something like this sense of continuity is lost in philosophy with the advent of the Cartesian revolution (and the Hobbesian revolution, and the Baconian revolution, and etc.) where the ties with Plato, Aristotle and their progeny, the Scholastics, were severed. The curt dismissal of a whole tradition without even dignifying to offer a semblance of criticism is not exactly the type of cultural continuity I am thinking of. In cycle after cycle, modern philosophers will raze to the ground the hard won wisdom of the past and build upon the ashes of their forefathers their own metaphysical edifices. But to borrow Kierkegaard’s charge against Hegel, peppered with some rhetorical flavor, no one, including their builders, wants to live in them because the darned things are so damn ugly.
The same Kierkegaard, no friend of Aquinas and co., in an intense little book called Repetition, proposes this term to replace the Platonic term of anamnesis or recollection. For Plato we have always known, but upon the shock of being dropped on the bucket of the world we have forgotten, and the travail of Wisdom is to recollect and reawaken what is origin-al within ourselves. Or as Francis Bacon puts it at the beginning of Essay LVIII Of Vicissitude of Things, in his very distinctive diction:
Solomon saith, “There is no new thing upon the earth”. So that as Plato had an imagination, “That all knowledge was but remembrance”; so Solomon giveth his sentence, “That all novelty is but oblivion”.
Kierkegaard means by repetition, not the stale, desiccated reiteration of old formulas, but a re-creation in the Apocalyptic terms of “Behold, I make all things new”. So the third corollary is a plea, and here I am following Feser again, for a renewed look upon the metaphysical tradition of the Scholastics. Not a return to some fabled Golden Age, that never existed anyway, but a development and elaboration upon the sound, realist metaphysics and philosophy of nature developed by those men and concomitantly, a gentle nudge to the reader, if any there be, to go search in more appropriate places for a more exhaustive explanation of the issues involved.
I will take as my fourth and last corollary, a warning and a dire one indeed. As the reader may have already observed, I am given, among other sins, to ramblings, digressions, asides, footnotes, parenthetical remarks, heavy doses of pedantry and (salutary) exaggerations. Add on top of this the fact that the series will drag itself through four more installments, and you may want to reserve your comments to future posts. Anyway, as the typical villain in a comic book would have it, “I am invincible!” (clenched fists, maniacal laughter), so feel free to Snipe, Snide and Snark; I may even respond in kind. Unless that is, someone bores me to death. Literally.
 This is not to deny that many developments in mathematics have indeed occurred in answer to problems posed by other disciplines, most notably physics. But the fact that such developments did come about that way, does not entail that they necessarily had to come about that way, this latter claim being patently absurd as even the most incipient knowledge of modern mathematics (starting about the beginning of the 19th century with the efforts to give calculus a firm and rigorous foundation) shows.
 H. Putnam, M. Resnik, etc. have advanced slightly different versions of the indispensability argument. See The Indispensability Argument in the Philosophy of Mathematics for more information.
 And then again, the modern strategy to defuse indispensability arguments, appropriately called dispensability arguments, involves rewriting the physical theories to avoid the quantification over mathematical objects. But such rewriting, even when successful, appeals to second-order logic or mereological axioms which are even more controversial and problematic. As Quine quipped, higher order logic is set theory in sheep’s clothing, so it is legitimate to wonder how successful the strategy is given that you avoid reference to mathematical objects (e.g. sets) only by introducing them surreptitiously and by the back door, suitably redressed. Once again I refer the reader to The Indispensability Argument in the Philosophy of Mathematics for more information and references.
 I will have occasion to return to this point later on.
 The reason why I will not be defending mathematical realism per se is in short, and to quote my lifelong intellectual hero Dr. Johnson, “Ignorance, madam, pure ignorance”. If I were a Platonist realist, I would have available a straightforward account of all of mathematics, but since I reject Platonism for reasons I will not delve into, this move is not available to me. I could still take the neo-Platonist route, which is indeed available, of saying that mathematical objects pre-exist in the mind of God from all eternity, but favoring an Aristotelian-Thomistic metaphysics something more is needed than this simple, stopgap answer. Mathematics is a fascinating and bizarre realm, in more than one sense, and as far as I know, Thomists have not written much about it, and what there is, it is hard to digest — at the level of PhD thesis 700 pages thick, couched in impenetrable jargon.
 Further bibliographical references will be scattered throughout the posts for those interested in pursuing these matters.
 This is war; the air is burning, shrapnel will be flying everywhere. Better get down on the floor.
 Recommended reading: Mortimer Adler, Ten philosophical mistakes.
 Such Golden Age proclamations usually betray a singular lack of historical sense.
 Here I confess I was really tempted to quote at length the endless ironies of J. Swift’s “A digression in praise of digressions. Just go read it.