Realism vs. Anti-Realism I: Introduction — Guest Post by G. Rodrigues

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[1]?

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[2]:

  1. We should believe the theory which best accounts for our sense experience.
  2. If we believe a theory, we must believe in its ontological commitments.
  3. The ontological commitments of any theory are the objects over which that theory first-order quantifies.
  4. The theory which best accounts for our sense experience first-order quantifies over mathematical objects.
  5. 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[3]. 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[4], 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[5], 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[6].

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[7].

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[8].

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[9], 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[10], 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.



[1] 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.

[2] 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.

[3] 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.

[4] I will have occasion to return to this point later on.

[5] 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.

[6] Further bibliographical references will be scattered throughout the posts for those interested in pursuing these matters.

[7] This is war; the air is burning, shrapnel will be flying everywhere. Better get down on the floor.

[8] Recommended reading: Mortimer Adler, Ten philosophical mistakes.

[9] Such Golden Age proclamations usually betray a singular lack of historical sense.

[10] 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.


  1. This is like reading from a menu in a 3 star restaurant, you know it is going to be excellent, but the portions are expected to be small….

  2. In following up on the link to the indispensability argument I found this:
    Standard readings of mathematical claims entail the existence of mathematical objects.  But, our best epistemic theories seem to debar any knowledge of mathematical objects.  Thus, the philosopher of mathematics faces a dilemma: either abandon standard readings of mathematical claims or give up our best epistemic theories.

    Further down we apparently find what the anthropology behind these so called best epistemic theories is:

    But, the rationalist’s claims appear incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies.

    Is this sort of self evident? Am I missing something here? Is this a counter argument or a colossal prejudice.

  3. To me it seems that the mathematical realists confuse the description of a thing with the thing itself.

    Mathematics can be used to describe the real world but so can words. So called mathematical objects are no more or less real than words are.

    In these terms, it is probably better to think of mathematics as a language for describing reality that is more pricise than traditional languages but a language non the less.

  4. But what in physical reality is meant by references to “conjoining and splitting topologies on a function space”? Radical nominalism has its own problems. A word like “dog” becomes a word precisely because there is something real in Fido, Rover, and Spot that is not particular to only one of them. That is why the realism of the universals is precisely the same as the realism of mathematics. The argument that language and mathematics are descriptors isn’t really an argument against realism.

  5. Ye Olde Statistician,

    Except that the description, even with the greater precision of mathematics, will always be incomplete, an aproximation of reality. You are confusing the description of reality with reality itself.

  6. And just to be clear, I am not arguing for or against either matmatical or linguistic realism. Mearly pointing out that every one of the arguments presented here for mathematical realism applies equaly well to all other forms of language. To the point that IF mathematical objects have reality beyond the mind, then words must as well.

    The arguments presented here offer no rational basis to distinguish mathematics from other languages.

  7. I wouldn’t mind (as a native Dutch speaker) a somewhat simpler text. Unless there’s going to be proof that neither the English, nor the Dutch, language exist. In which case the proof should in an as complicated kind of English as possible.

  8. @Matt
    To speak of words in the general sense you are doing does not make any sense to me in this context. There is a point in something corresponding to the word dog, which is immanent to all dogs, yet none such a correspondence exists for all Rovers, Spots and Fido’s in the world. But even with dogs, it is hypothetical, albeit real, what this immanent being of dogs is, at least to most of us I suppose. I experienced just how difficult it can get to come to terms with such things. Sander might well remember how I attempted to express the universal connected to the word house. I ran into an abstraction and failed. Yet with mathematics it is much easier. You can actually bring to expression what a circle or a triangle is. The definition and the construction coincide. Try that with dog!
    Therefore I do not agree with YOS that: the realism of the universals is precisely the same as the realism of mathematics. It could be true objectively, but subjectively the dog is much harder to apprehend than the circle.

  9. @rembie

    I am not specifically arguing that the realism of the universals is precisely the same as the realism of mathematics. I am mearly pointing out that as I read it the article above offer’s no basis to distinguish them.

    If you wish to convince me I am wrong you will have to show how the specific arguments in the article apply to one but not the other.

  10. @rembie

    What Matt is saying is that mathematics is a language. A mathematical description/definition of a circle may very well fully describe the concept of a circle but is not itself a circle. Also, the mathematical description of a circle fails to describe any circle that appears outside of the mind, say for instance, drawn on paper. So, it is precise only in the mind and nowhere else.

    Since the subject is words, here’s a dictionary definition of concept.

    con·cept [ kón sèpt ]

    1. something thought or imagined: something that somebody has thought up, or that somebody might be able to imagine
    2. broad principle affecting perception and behavior: a broad abstract idea or a guiding general principle, e.g. one that determines how a person or culture behaves, or how nature, reality, or events are perceived
    3. understanding or grasp: the most basic understanding of something
    4. way of doing or perceiving something: a method, plan, or type of product or design

    Synonyms: idea, notion, thought, perception, impression, conception, theory, model, hypothesis, view, belief.

    Note that one of its synonyms is model.

    Mathematics describes certain concepts but at no time are mathematical concepts any more real than any other concept.

  11. @Matt
    I do not have any wish to convince you of anything yet, I am myself searching for the right approach to this issue.
    What you are saying does not follow. Certainly the definition does not apply to a circle drawn on paper, But that does not justify the conclusion that the concept of a circle is just something in my mind. The definition in Dutch is in my mind, the definition in some other language in yours (I presume English). Yet the circle is in neither. There is no your circle and my circle. It is the interpretation of this fact that we should be talking about.

  12. @rembie

    The fact that mathematical concepts can be comunicated from one mind to another like any other knowledge does not remove those concepts from the realm of the mind and put them somewhere in the real world.

  13. @rembie

    Speaking of circles, we seem to be talking in one. The point is there is a distinction between descriptions and the whatever being described. You seem to have agreed with that when you said yet the circle is in neither. Mathematics is just another language. It may not be the same class as Dutch, German or English but a language nonetheless. The expression for a circle is just how a circle is described in math but, even there, it is not a circle — just a description.

    Whether circles or any other concept actually exist outside of the mind is perhaps an interesting question. I would like to see the argument for it. Even if true, that hardly qualifies their descriptions to also be so regardless of language choice.

  14. @Matt
    What do you mean by the real world? And what do you mean by communicated from one mind to the other? I apprehend what a circle is. I turn to someone next to me and I try to make him apprehend what a circle is too. It is not too different from pointing at something at the horizon. If the other person looks in that direction, he will simply see what I mean for himself.
    I would like to know in what way mathematics is a language to you. Do you mean it has symbols and grammar other than English or German? Or do you apply the term loosely because you find that mathematical objects are descriptive of things in the real world?

    I would distinguish between a couple of things. If we speak of a description of a circle I assume we mean the sequence of words by which we bring to expression what a circle is. That indeed is not itself a circle. Secondly we might have a mental picture of a circle. Surely you have yours and I have mine. I tend to favor very small circles, but that is just me.( Incidentally if we speak of dogs it is mostly on this level of mental pictures or actual sense experience). The mental picture is not the concept of the circle, it is simply the proxy of the circle we might otherwise be inclined to draw on paper as an aid. The pure concept cannot be pictured, neither does it consist of parts, like its description. Yet it is there.
    Through thinking we determine what a circle is, it is a dictate. A circle is indeed nowhere to be found, it must be created. The conundrum is that although such an object is seemingly made up by us, it is not arbitrary, but highly relevant with regard to the one world around us. Can you agree on this?

  15. . Do you mean it has symbols and grammar other than English or German?

    Yes. Not the field of mathematics but the expressions themselves.

  16. @rembie

    “What do you mean by the real world?”

    The real world is the realm of the physical. Things you can see, hear and touch. If you can’t distinguish these things from the things that are purely in your mind further discussion is pointless.

    “And what do you mean by communicated from one mind to the other?”

    That is what language (spoken, written, or anything else) is for. This is what we are doing at this blog. If you don’t understand this, what are you doing on this blog, throwing random symbols at a computer screen?

    “I turn to someone next to me and I try to make him apprehend what a circle is too. It is not too different from pointing at something at the horizon. If the other person looks in that direction, he will simply see what I mean for himself.”

    It is very different than what you describe. Without language and bi-directional comunication you can not assume that the other person knows which object you are pointing at. Additionaly you will find no perfect circles in nature to point to, so you absolutely can not communicate the mathematical concept of a circle in this manner.

  17. @Matt
    you will find no perfect circles in nature to point to, so you absolutely can not communicate the mathematical concept of a circle in this manner.

    Correct. But each and every one of us has to arrive at the concept of the circle by his own effort.

    The real world, the so called realm of the physical is real because we think about it. Without thoughts it would not be real to us (which is the only thing we need to consider). Thoughts are therefore an indispensible part of what we call real.

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