A couple weeks ago, my fellow blogger, Brett Salkeld, gave an interesting talk to a group of Catholic school teachers entitled There is no Neutral for Catholics. There is a lot going on here and I strongly recommend reading it. I am not in agreement with some of what he says, but Brett is a thoughtful theologian so he is worth engaging with. And in particular, I am very interested in his project. In this brief blog post, however, I want to engage with a specific argument that he is making about mathematics. I commented on his FB page and promised to say more, but I realized that I needed more space that a FB comment box comfortably allows. There have been some delays, but I hope my response is still timely.
The main thrust of Brett’s argument is that Catholic schools should teach all subjects from a distinctively Catholic point of view. I am generally sympathetic to this desire. However, he also goes on to argue that there is no “neutral” position from which to teach any subject. He has grounded his argument with the example of mathematics, arguing that
But can I say a word or two about math on the premise that, if even math can be taught from a Catholic point of view, anything can be taught from a Catholic point of view?
I will grant him this point, at least as a rhetorical strategy: if he is going to successfully write a book entitled Making Every Class Catholic, then mathematics is going to be one of his challenges. However, I found his argument about mathematics to be overly facile and, while I understand that this was being presented as a talk to school teachers, and is not an academic article, I still think it is problematic enough that I want to raise some points for him to consider before he gives this talk again, or incorporates this argument into his book.
There are two parts to his argument that I want to address: the ontological nature of mathematical objects–and whether mathematics can be taught in a “neutral” manner that avoids these ontological questions. First, on their ontology, he writes:
OK. Have you ever thought about what kind of thing a number is? In a way, it is not really a “thing” at all. The number 6, for instance, is not a thing like an orange or a puppy dog. You cannot ever point to it or put it in your pocket. You cannot smell it or taste it. There are not tall sixes and short sixes, yellow sixes and blue sixes. A 6 is not made of matter. We can represent it with a shape—written on paper or carved in stone—that we all agree means “six,” but we know that that shape is not itself 6; we know that another shape might have suited just as well.
But if a number is not a material thing, what does that say about reality? Is a number real? And I do not mean this in terms of the mathematical definitions of real and unreal numbers, but in the plainer, more everyday way of speaking: that is, “Do numbers really exist?” If they do, then immaterial things exist, even if it is obvious that they exist in a different way than material things.
Now, there are at least a couple of ways we could answer this question. We could say, on the one hand, that numbers are not real. We could say that they are just something we made up in our heads in order to be able to do useful things with material reality, which is, in fact, the only real thing. On the other hand, we could say that numbers are built into reality, that we did not make them up, but rather that, by carefully observing reality, we discovered numbers. That is, by paying attention to material reality with our senses, we could discern a deeper truth about a reality beyond the material with our minds.
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But let’s finish with math, shall we? A Catholic math class does not start from the premise that humans made up numbers because they are useful. And it does not start from the premise that it doesn’t matter what you think about the reality of numbers, as long as you can make them work properly. No, it starts from the premise that, as we hear in the Old Testament book of Wisdom, God has “arranged all thing by measure and number and weight.” Or, as Galileo who, despite his troubles, remained a faithful Catholic his whole life, is reported to have said, “Mathematics is the language in which God has written the universe.” These quotes, and you could find others as well, should be on the wall of every Catholic math teacher’s classroom.
In the third paragraph we have two common views about the nature of numbers and mathematical objects in general. The first I describe as naive constructivism: that mathematical objects have no reality outside the human mind, and are constructed by human beings based on material reality. I coin this term to distinguish this position from mathematical constructivism proper, which is a more narrow (though related) theory. The second might be termed naive Platonism, the idea that mathematical objects are real, though non-material, and exist in some ideal way. The devil is in the details, but this position has much in common with Plato’s theory of forms. Based on my own experience, I would say that the majority of mathematicians are naive Platonists, with a minority holding some explicitly constructivist position. If I had to choose a position among the two, I would call myself a naive Platonist: mathematical objects are real in some objective sense
I call both positions “naive,” however, because they are: they gloss over all manner of difficult and subtle ontological and metaphysical questions–questions that I have some appreciation of but as a professional mathematician (as opposed to philosopher of mathematics) have only a limited knowledge of. For instance, there is a significant gap between what Plato believed about mathematical objects and what Aristotle believed, driven by their very different views on metaphysics. For a brief overview of Aristotle’s position, see here. (The Stanford Encyclopedia of Philosophy does not appear to have a corresponding article on Plato.)
The real problem for me, however, comes when Brett strongly implies that the Catholic position must be naive Platonism. Immediately following the above description, he writes:
I hope it is clear that one of these attitudes towards numbers is Catholic and one is not.
The problems are manifold. First, I think the constructivist position is being caricatured: Brett elides between the idea that mathematical objects are constructed by human beings from their observation of the natural world and the assertion that the natural world is, a priori, all that there is (which is a position more properly described as materialism or philosophical naturalism). Now, philosophical naturalism may entail some form of constructivism, but I am not convinced that it is a necessary part of constructivism. Brett is welcome to make the case, but I don’t think the assertion can just be made without more argument.
Second, these are not the only two philosophical positions that one might adopt about the ontological status of numbers, and there is no reason to believe that of all of the philosophies of mathematics, only one of them is compatible with Catholic (or even Christian) thought. The German mathematician Kronecker is quoted as saying
God created the integers, all else is the work of man.
Though an aphorism is not a complete philosophical position, it does suggest that there are other ways to think of these questions. (It may or may not be relevant, but I note that Kronecker was born into a German Jewish family; according to Wikipedia he converted to Christianity late in life.) Therefore, I would suggest that there are a range of positions on the ontological status of the number six (and other, more exotic mathematical objects), not all of which will coincide nicely with the dichotomy constructed above. (Note, for instance, that this quote from Kronecker seems fully compatible with the verse from the Book of Wisdom that Brett cites.) And I question the assertion that only one of them is compatible with Catholic thought. So if Brett is going to argue that mathematics must be taught from a Catholic perspective, he needs to be clearer on what this means: what is to be allowed in the classroom, as it were, and what must be rejected out of hand.
But at this point, I want to turn to his second argument, that school teachers must take a stand: that one cannot be “neutral” on the subject of mathematical ontology. He states the matter in rather striking terms:
To answer this, let us consider that, not only is it possible to teach every subject from a Catholic point of view—it is impossible to teach any subject from a neutral point of view. There is no neutral. We will teach from a given perspective whether we are aware of this or not, whether it is the result of a conscious decision, or simply the unconscious replication of the perspective we ourselves have learned from others. And, as Catholic teachers, we should be very intentional about recognizing just what perspective we are teaching from.
The problem here is that, in my own lived experience as a professor of mathematics who has taught for close to 30 years, this is simply not the case. Unlike the majority of my colleagues, I have thought about the philosophical foundations of mathematics, and tried to discern what it means to assert that mathematical objects are “real.” But when I, a committed Catholic, teach calculus, or differential equations, or some advanced topic, what I teach and how I teach it will differ almost not at all from the same class taught by a colleague who is Jewish, or Muslim, or an uncertain agnostic, or perhaps even a militant atheist. The theorems will be the same, the proofs will be the same, the motivations for the results will be (more or less) the same. If you listen to our language for clues, you will be hard pressed, as I will refer “constructing” mathematical objects (implying that they did not exist until I strung the argument together, and that they exist only within our collective understanding), and others will refer to the “discovery” of this or that theorem (implying that it was pre-existent and part of reality waiting to be found).
From our graduate school days we are all, more or less, aware of the the great philosophical debates that revolve around mathematics; most of us know some details of Hilbert‘s ambitious project to formalize mathematics, and how that project ran aground on the work Godel. But only a fraction of us have ever actually read the proof of Godel’s incompleteness theorems, or delved into the deeper problems of mathematical philosophy. The reason is that, ultimately, it has remarkably little impact on what we do or teach. Some years ago there was one militantly atheist mathematician who published a book in which he denounced mathematicians for their naive Platonism, and advocated that the “only” logical stance was a thorough-going materialist constructivism. The response of the vast majority of professional mathematicians was “meh.”
The reason for this, and the mistake (or at least over-simplification) that Brett makes, is that he is conflating two different modes of thinking. To make this clear, I want to shift briefly from mathematics to the natural sciences, for example, physics. A physicist might, for philosophical reasons, be committed to materialism, and so his approach to his discipline will be a thorough going naturalism. On the other hand, a physicist who is a believer will approach his discipline in seemingly the same way, since her understanding of physics commits her to a pragmatic naturalism: physics, as a discipline, does not invoke the divine, and relies only on naturalist arguments and evidence. The net result is that the physics that is done is the same, even though in the background one physicist believes only in the material world and the other believes that the created world is a reflection of God’s glory.
I believe the same pragmatic/philosophical distinction is at work in mathematics. Mathematicians approach mathematical objects as abstract “things” which we have agreed to manipulate according to agreed upon logical rules, but which we often explain or explore using an intuition that draws upon real world patterns, but that also draws upon our own esoteric musing about how things ought to work in the “world” of mathematics. These are the “rules of the game”, as it were, and we have all bought into them for our research and teaching (perhaps with the rare exception of those like the materialist constructivist I mentioned above).
In light of all of this, I have to ask: why, precisely, does it matter, as Brett argues, that “as Catholic teachers, we should be very intentional about recognizing just what perspective we are teaching from”? To be clear, I am in favor of a philosophical introspection, by professional mathematicians, college faculty, as well as K-12 teachers, about the philosophical foundations of mathematics. However, it is not clear to me that the answers they arrive at must necessarily be “Catholic” in any exclusive sense. Indeed, as I noted above, it is not clear at all if there is a single “Catholic” answer to the question of the existence of mathematical objects.
Now, as I said, I believe that Brett is a thoughtful and sophisticated theologian–so I am relatively sure that he will have good answers to my questions and concerns. indeed, I worry that he is going to launch a counter-attack on my arguments that leaves me in (philosophical) shreds. But, to reach beyond the narrow concerns I have raised about the philosophy of mathematics, I am concerned that, unless he presents these ideas in a much more sophisticated fashion than given in this article (and to be fair, he is intended a whole book on the subject), his ideas will be reduced to the official Catholic(TM) stance on a whole number of disciplines, one which defines itself by exclusion as much as by what it actually contains. I can just imagine what certain parts of the Catholic home-schooling movement would do in these circumstances. A good cautionary note comes from looking at the imbroglio a couple years ago at the Franciscan University of Steubenville about teaching the book The Kingdom.
Let me finish now with an anecdote much beloved by mathematicians, which points to our own confused understanding of the perspective from which we teach. The Hungarian mathematician Paul Erdos, though a self-described atheist, insisted that God had a book in which was written the best and most elegant proofs of all mathematical theorems. He further said that, if you are a mathematician, “You don’t have to believe in God, but you should believe in The Book.” In his honor, two mathematicians have published a volume entitled Proofs From The Book, containing proofs of a number of results that are deemed to be from “The Book.” So an interesting challenge: go to a math conference, and in the hotel bar after the last talk, buy a couple rounds and ask the mathematicians gathered there about the “The Book”. Does it exist? How can we know that a proof is indeed from “The Book”? I doubt much will be resolved, but a good time should be had by all.
Coda: since I conclude by mentioning Paul Erdos, the question of my Erdos number will probably come up. According to the mathematical bibliographical database MathSciNet (which actually has a built in search function to compute this), my Erdos number is 3.
The featured image for this article are the Borromean Mobius rings, a mathematical sculpture commissioned by the Department of Mathematics at the University of Alabama. Image used by permission.
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