Aristotelian realism

An article I read in Aeon Magazine by James Franklin gives me a good springboard for some of my own thoughts about the philosophy of mathematics. The author (who has a book on the subject) essentially opposes two extreme positions, the one nominalist and the other Platonist. The nominalist seems to say that mathematics doesn't study any real objects; it is merely a language, a series of tautologies that has great instrumental value but has no content on its own. The Platonist says that, on the contrary, mathematical objects exist in their own realm, and that the human mind has access to that realm through contemplation and logical reasoning.

The problem with the first view is that to any mathematician, it seems fairly straightforward to assert that we actually discover something--not just logical relationships between symbols, but actual content. The problem with the second view is that the world of mathematical concepts seems remote; how can we physical beings have access to it?

The alternative is Aristotelian realism, which asserts that mathematical objects inhere in nature. Our minds have access to them initially through observation, then through abstraction and logical reasoning.

This alternative is very attractive for at least two reasons. One reason is that it makes sense of applications far more easily than either Platonism or nominalism. Why should mathematical models be so good at describing real world phenomena? Under the Platonist view, there's not much reason even to wonder about it, since mathematical objects are eternal and inherently separate from the contingent world we live in. Under the nominalist view, the puzzle is why a mere language would be so effective in discovering things about the universe before they are even observed (think about the mathematical development of general relativity). Realism has a simple explanation: we draw mathematical concepts out of the real world, so it's natural that we should use them to explain how it works.

Another reason is that it's satisfying from the point of view of a practicing mathematician. Platonism also has that trait, in that it elevates the objects of mathematical study themselves. But Aristotelian realism allows us to assert that mathematics has real content without divorcing it from common experience. I find this accords well with my own practice of mathematics, both in research and teaching. I always emphasize to my students that common sense should be the starting point for thinking about any mathematical problem. Of course we have to take a long journey out from that starting point, but ultimately each step is grounded in reasoning that any flesh and blood human being can understand.

For me there's a third, more theological reason to appreciate Aristotelian realism. Franklin alludes to theological import himself:

Aristotelian realism stands in a difficult relationship with naturalism, the project of showing that all of the world and human knowledge can be explained in terms of physics, biology and neuroscience. If mathematical properties are realised in the physical world and capable of being perceived, then mathematics can seem no more inexplicable than colour perception, which surely can be explained in naturalist terms. On the other hand, Aristotelians agree with Platonists that the mathematical grasp of necessities is mysterious. What is necessary is true in all possible worlds, but how can perception see into other possible worlds? The scholastics, the Aristotelian Catholic philosophers of the Middle Ages, were so impressed with the mind’s grasp of necessary truths as to conclude that the intellect was immaterial and immortal. If today’s naturalists do not wish to agree with that, there is a challenge for them. ‘Don’t tell me, show me’: build an artificial intelligence system that imitates genuine mathematical insight. There seem to be no promising plans on the drawing board.

This paragraph is delightfully provocative. I suspect many proponents of artifical intelligence believe they are not so far off as Franklin believes, but I can neither confirm nor deny such claims. In any case, artificial intelligence is not what interests me most. Instead, I tend to fixate on this question, "What is necessary is true in all possible worlds, but how can perception see into other possible worlds?"

To me the advantage Aristotelian realism has over Platonism is that it lets us see the eternal, even the sacred, in all things. Whereas the Platonist sees objects in the world as mere shadows on the wall, as it were, the Aristotelian sees them as sources of truth in themselves. For this reason I think Aristotelianism can affirm creation in a way that Platonism can't.

It is common for applied mathematicians to point out that their models are only approximations of reality, and that real life, unlike beautiful mathematical theories, is "messy." And I think that both for the nominalist and the Platonist, there is a sense in which one must choose between the beautiful realm of theory and the messy realm of facts. I reject this dualism by taking the radical position that eternal, necessary truths are inherent in real objects. I do not thereby deny the contingency of the universe; of course it could have been different from the way it is. Yet every object reveals necessary truths; paradoxically, we find the infinite and the eternal in the finite and temporary.

To put it in starkly theological terms, I would compare Platonism to gnosticism and nominalism to idolatry. The one would have discovery be a way of escaping the created order; the other would have discovery be entirely about finite, contingent reality. Instead, I think discovery involves an interlocking of the temporal and the eternal. From real world objects we discover eternal, necessary truths; in return, we can use these eternal truths to understand--and also care for--the world we inhabit.

Indeed, is it not the mystery of whether physical laws are truly necessary that drives so much of theoretical physics? One encounters mathematical relations between objects with fundamental constants which can be measured empirically, and it is natural to wonder whether such constants could actually be deduced from some deeper principle. Or whether the laws of physics themselves are actually corollaries of some more fundamental Law. Could the universe have "come into being" through some means other than what we call the "big bang"? Such questions magnify the interlocking of the eternal and the temporal, the necessary and the contingent. God's glory shines in all things, to such an extent that it is difficult to see where his invisible glory ends and the more visible nature of things begins.

As a corollary, I see mathematics not so much as a way of escaping into abstract truths in a higher realm, nor as a mere tool of the sciences, but rather as a humble servant of empirical investigation. We study mathematics not only to understand what the world is like but also how it must be, and in that sense it gives some of the deepest insight of any science. Yet the inspiration for its progress is not so much a desire to ascend toward heaven as to see the heavenly on earth. Whose heart can be so cold as to resist finding the beauty in Euler's formula? Yet if we never saw such things as oscillations in common experience, I'm sure we never would have seen such a beautiful equation.

The science of theology

In attempting to explain the Scientia aspect of theology, Vanhoozer says the following about science (The Drama of Doctrine, Ch. 8, p. 247):

While impersonal empirical procedures may be the mark of modern science, they are not that of Aristotle, Aquinas, or Barth. For Aristotle, a science proceeds from the "first principles" that are intrinsic to some aspect or area of reality. These principles are ontological (real) before they are theoretical (propositional). To be scientific is to approach a subject matter in a way that is appropriate with its first principles. ... For both Aquinas and Barth, Christian theology is the science that approaches its subject matter, God, on the basis of first principles that themselves proceed from God: "For sacred doctrine, those first principles are...scripture and Christ."

I don't think the classical approach gets off on the right foot. The theoretical does precede the ontological, because all language, including the word ontological itself, is theory-laden. Theory is more than a package of propositions. It is something like a "procedure," as Vanhoozer suspects, but more pertinently it is an interpretive matrix. Critical to the whole discussion of epistemology is that a theory is an evolving interpretive matrix. Ontology, for us feeble creatures, is really a bit of a fantasy. We could learn nothing if we didn't already have a theory about how things work. It is only because our theories are constantly being bombarded with corrections that we are able to learn more.

It is telling that Vanhoozer has such a gaping hole in his definition of knowledge, here (next page):

Theories of knowledge come and go, and theology would frankly be ill advised wholly to invest in any single account. Some minimal account of knowledge is, however, needed; hence the following provisional definition: knowledge is the product of a disciplined approach to a particular subject matter.

The irony of this statement is that he gets the first part right: theories come and go, precisely because we are ever-evolving creatures. But his provisional definition of knowledge obviously lacks all accounting for spontaneous increase in knowledge, including those wonderful discoveries which occur mostly by accident. Granted, his main concern in this chapter is discipleship. Yet I don't think it is a trivial matter to completely overlook the way in which knowledge grows more spontaneously than we often believe. Perhaps Vanhoozer ought to remember Barth's words: theology is a free science.

What's in an idea?

About a week ago I wrote a weird little post about "unity and continuity in creation." Just as I said, that post was meant to be continued, and I think here's where I get to what I'm really thinking about.

I was listening to a talk given by Tony Jones called, "Smackdown--Plato versus Aristotle." That's almost nerdier than something a mathematician would come up with. I like it.

The reason he was talking about Plato and Aristotle is that these ancient Greeks disagreed fundamentally on what knowledge and thought really are. Jones argued that in our Western culture, we have inherited most of our vocabulary about thought from Plato.

I've noticed that I usually either consciously or subconsciously think that whenever I'm talking or thinking about big issues, my mind is ascending into some "thought land," some Platonic heaven where the Truth can be ascertained directly, if only we train our minds to see.

I think this is reinforced by the way we are taught in our culture from an early age. We go into a classroom to talk about those abstract truths we're required to learn. Then we leave the classroom to do what we actually want to do with our lives.

This pattern reinforces a Platonic separation between the "higher" and "lower" forms of knowledge, a distinction I think is false. I've never read the book Shop Class as Soulcraft, but I hear that it tries to combat this distinction, and for good reason.

I talked about in my previous post how we're completely created beings. Now I want to say that from this, we know that our thoughts and ideas are also completely created, and thus part of the continuum of creation.

What I really mean is this. If you separate knowledge into "higher" and "lower," then you get some people trying to reach for a higher plane of existence through their learning, while others just leave all that "fancy book-learning" to the weirdos in the academy.

In relation to what I do, mathematics, this is what I hear constantly: "How is this useful? How is this relevant to me? What can you do with that?" People want something they can hold in their hands, at least metaphorically (but often literally) speaking.

But in the academy, often knowledge for knowledge's sake is still a virtue. So you can see in the separation between different groups (almost different classes) of people how this Platonic theory of knowledge is played out.

What concerns me is that you can see this not just in an academic setting, but in the church, as well. Protestantism has actually prided itself on stressing orthodoxy rather than orthopraxy. This comes from a theological debate over justification by faith versus works, but I really wonder if what we're really doing is being Platonists.

Think about it: it's like those who have faith have some higher knowledge. We've sought the knowledge truly worth having, as opposed to something common people can "hold in their hands," so to speak.

Isn't that what lines like, "You just have to believe!" are really all about? You just have to believe because this kind of knowledge only comes to you when your mind escapes creation and enters the "spiritual" world. (Thus, things like scientific evidence will not help you.)

Whether it's the church obsessed with orthodoxy or the academy obsessed with, well, itself, what we get is this radical separation between different kinds of knowledge, and so people are given a black or white decision. Either seek the higher knowledge, or seek the lower knowledge.

This rather divisive choice is presented to people precisely because of our theory of knowledge, which coincides with Plato's desire to escape the world with his mind and find some purer truth.

But what happens when we say that actually all thought is part of the continuum of creation? I confess I don't fully know the answer, because generally speaking, our society doesn't think this way yet. I do have some ideas, but right now I think I'll just mention one.

It may sound cliche, but I think one thing we could use more of is togetherness. When we acknowledge that our minds are created things, and that therefore our thoughts all remain right here on earth no matter how sophisticated they are, we realize what is really going on when we talk about ideas that matter to us.

What we're really doing is what I talked about before: we're housekeeping. We're bringing some sort of order to God's created world. And because we're all part of that creation, we're all in this together. Ideas aren't meant to lift people's minds up out of the world, but rather to make the world better to live in.

I hear people say something like, "It's not about ideas, but about relationships." They say that because they're reacting to the old way of thinking about ideas. What we should be saying is that ideas are about relationships--our relationship to every created thing.

This is probably already being said by someone else much smarter and more famous than I am, but, it's nice to be able to say it the way I want to say it.