An attempt to reconcile my materialist tendencies with my strong belief in the primacy of mathematics.
I'm a dualist, sort of. Here's the deal. For some time, I called myself a monist, since I believe that all we need to think is a physical nervous system and whatever that nervous system needs to function. In particular, I don't think we need supernatural help, like souls. What, then, is a mind? Clearly a mind isn't a nervous system, nor any other physical object, but my initial assumption precludes the possibility that it's supernatural. Then if there is such a thing as a mind at all (and not everybody agrees that there is (Wittgenstein, for one, would object), but that's another story), it has to be an abstract entity, like evil or the number 2. It follows that the reason we say I have a mind is not because I have certain body parts but because I behave in a certain way. And it follows from that that there's nothing about our concept of mind that prevents, say, a computer program from having a mind.
Now suppose that I open a new Emacs buffer and write a Perl program that has my mind, then run this program, then jump off a cliff. My remains are summarily gobbled up by a bunch of seagulls. Technically, I'm dead. But to say that I no longer exist as a person is preposterous, because the Perl program that has my mind, and hence, for all intents and purposes, is me, is still running. It can finish this essay and post it on my website under my name and truthfully say that it isn't impersonating anybody.
All this is nothing new, merely what writers like Daniel Dennet and Douglas Hofstadter have long believed. But let me also add how my thinking about a certain class of other abstract entities—namely, mathematical facts—changed during my first few years of college. I once believed that mathematics was invented, not discovered, and was just as arbitrary and the product of human whimsy as unicorns and God. Now I've taken a more Platonic perspective. What makes mathematics—and, more generally, all tautologies—special is that it constrains reality. What happens to be the case is a subset of what logically could be the case. For this reason, it seems proper to regard the number 2 not as unreal or only partially real, but as more real than physical things like matter and energy. "The square root of 2 is irrational." is a stronger statement than "The earth revolves around the sun.", because the former ultimately has greater consequences for our understanding of the material world. In short, it seems most correct to say that as imaginary as they are, mathematical objects really exist. What's more, a mathematical object's existence is independent of our knowing about the object. The square root of 2 was irrational long before the Greeks realized it; similarly, the Fibonacci numbers, the Platonic solids, and the dihedral group of order 8 all predate humankind.
Now suppose a certain mind—say, mine—was rigorously defined as a mathematical object. We've already accepted that it could be expressed as a Perl program, so this has to be possible. Then my mind is technically independent of my body. It existed long before I was born, and it will keep existing after I'm dead. Killing me doesn't destroy my mind any more than burning all the world's analysis textbooks destroys (−∞, +∞). And so, in a weird sense, I'm actually a dualist.
When I first wrote the above essay, my friend Will Cerbone responded to the passage
For this reason, it seems proper to regard the number 2 not as unreal or only partially real, but as more real than physical things like matter and energy. "The square root of 2 is irrational." is a stronger statement than "The earth revolves around the sun.", because the former ultimately has greater consequences for our understanding of the material world.
You have a perfectly good argument based on the fundamentality of the fact (that math is what we use to describe the substrata of reality), and you're tossing it away for a dubious consequentiality? Stand by your field, man!
In the spirit of the scientific method, I can only justify my belief that a description is valuable with the description's predictive power. If mathematics described reality but didn't tell us anything new, it would be a mere religion, no matter how often we referred to it while talking about the world. I mean, suppose I invented a concept of something called a cosmic muffin, and proceeded to explain all physical phenomena in terms of cosmic muffins. Unless my explanations actually included predictions, and these predictions in turn resisted falsification, it would be preposterous for me to claim that cosmic muffins are more real than matter or energy.
So I see mathematics as causally but not epistemically primary.
A later conversation, in 2015, made me realize that the idea of treating the whole of mathematics as real on empirical grounds doesn't make sense. In particular, the infinite can't be distinguished from the finite empirically, so there can never be empirical facts with any relevance for infinite objects in distinction from their finite counterparts. And the finite is only an infinitesimal part of mathematics, like a mote of dust adrift in space. It follows that empirical facts, even all of them together, are neutral to the bulk of mathematics. So in order to consistently believe that all well-defined things exist, I have to assume it outright. Which I'll do, 'cause that's how I roll.