by Charlie Huenemann

An early Euclidean manuscript

Materialism is the view that everything that exists is made of matter. What’s matter? It’s hard to say with both precision and completeness, but it can’t be far off to think of matter as whatever can engage causally with the known forces of nature: gravity, electromagnetism, and atomic forces (strong and weak). If a thing responds to any of those forces, that thing is material. Of course, maybe there are some unknown forces of nature, and we’ll have to revise as they become known, but right now, this seems to be an adequate criterion for judging what counts as matter.

But I don’t think materialism is true, and it’s not because I believe in spirits or love or imagination or magic. It’s because of math. Math is a science of form: it explores the possible forms or properties or systems that are possible. Some of these possible structures, of course, describe the real systems we come across in our world, which is neat, and makes physics possible. But there are many, many more possibilities than are actual. It doesn’t take many beers before a gang of interested mathematicians will start describing all sorts of things that could never come to exist in our puny world because they are too big or complicated.

As the gang of mathematicians start describing these otherworldly possibilities, they are not just making stuff up. They can get things right, and get them wrong. It’s really hard to do math, because it’s all about proofs, which slide easily from validity into invalidity, or from coherence into incoherence, with a move as subtle as the fall of an eyelash. But communities of mathematicians keep one another in check, and what they do is as rigorous as any human endeavor can be.

So what about all these non-actual possibilities? If they are not just make-believe, what makes them genuine as possibilities? The best answer, I think, is that there are truths about structure, truths about form, which outrun all of the truths about matter. Our material world is the real-life version of a relatively small set of possible structures, but there is a much bigger world in existence, which is the one that math describes.

A next-best answer – one that I would take if it really were an option – is that math is just the consequence of a set of rules mathematicians play by as they do their work. In this view, math is like chess or Go: each game has a huge number of possible ways to play out, but all of the ways are generated by the set of permissible moves, or the rules of the game. But math is not like this. Gödel proved that there are many mathematical truths that cannot be reached by generating truths from a finite set of rules. The truths outrun the rules. So math isn’t just whatever follows from a set of rules.

A next-worse answer – one that I won’t take so long as I can believe in truths about form – is that there are many, many material universes, each as real as ours, and every possible mathematical structure shows up as real in some universe or other. In this view, mathematicians are physicists, except they are describing all possible material universes, instead of just this one. 

I hate this view. The idea of many possible universes is fine, I guess, but I hate using that idea to explain why math is true. It seems to me that the arrow of explanation has to run the other way: namely, that what makes other universes possible in the first place is the fact that they can be coherently described through math. The reverse view, that the truths of math are true because they accurately describe what’s going on in other universes, just seems bonkers to me. How could any of us know what is going on in other universes? We can’t, except insofar as we know truths about possible structures and forms. And once we have this knowledge, who needs other possible universes? (Quantum physicists, maybe, but certainly not for explaining why the truths of mathematics are true.)

So, if math is true, materialism is false. The objects of mathematical knowledge, namely possible forms and structures, don’t engage causally with the known forces of nature. (If they did, mathematicians would need particle accelerators.) They are different sorts of beasts. How we can have any knowledge of them is an interesting question, and one that both Aristotle and Descartes agreed can only be answered by positing an immaterial intellect. I think that’s just an attempt to solve one puzzle with another (namely, the puzzle of what the heck an immaterial intellect is). All we can reasonably say is that our brains and our language and our interactions with other brains and languages somehow deliver unto us some knowledge of form and structure, and we use this knowledge to leverage our way into more knowledge of form and structure. 

But that’s just an outline of an answer. A substantive explanation of how human beings can have knowledge of form would require, I think, a monumental revision of how we conceive both reality and mind. This would not be just gluing on extra magical immaterial bits; it would be a big overhaul of metaphysics. We might follow Plato’s lead and take truths of form to be basic, and then try to explain matter in terms of form. Or we might follow Kant’s lead and think of form as intrinsic to any sort of representation whatsoever; which would take representation as basic. Either approach – as well as any other approach – would be a huge philosophical shift away from the naturalistic metaphysics that seems plausible only so long as we take math for granted.

I’ll say a bit more about each of these ideas. On the platonic view, there is an infinite domain of possible structures or forms. What do I mean? Think of an infinite library of blueprints, or possible configurations of energy. These forms are possible ways for things to exist. Add energy to one of these possible structures, and what you get is a material thing, like a hydrogen atom or a tulip. Through our abilities to use symbols and language, we can discover, describe, and navigate through these structures. Maybe we can do this because we ourselves are possible structures, infused with energy. Our own structure enables us to generate further structures, and that’s how we are able to do math. But the range of structures we can generate is sharply limited, as there are infinitely many other possible structures we’ll never be able to discover or describe. It’s hard to travel much further down this line without sounding like some Renaissance Neoplatonist like Giordano Bruno or, come to think of it, Johannes Kepler.

The Kantian view, on the other hand, refuses to say what ultimate reality is. Whatever it is, by the time it turns up in our experience it has been formatted in some math-friendly way. In this sense we are analogous to our laptops: there’s a world wide web out there with zillions of zeros and ones, but by the time they show up on our screens, they’ve been transformed into emails and GIFs and the page you are reading now. In some analogous way, we impose form onto our experience, and whatever part of us that does the imposing – the human operating system – also enables us to explore other formal possibilities beyond the ones we happen to run into as we browse through our experience (and thus math). Many good questions can be raised here about what “part” of us does this “imposing” and how it’s able to do it. If we’re not careful, trying out answers to these questions will lead us into the Mirkwood forests of German Idealism.

One thing’s for sure, though: matter isn’t all there is. And thinking about what else there has to be raises some dramatic and exciting philosophical possibilities.

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