Tegmark and the Engines of Mathematics

My belated reply to Scott Alexander on Tegmark’s Mathematical Universe

As strange as a clockwork cosmology

It appears that the theistic debates of yesteryear are back in fashion. Which is good news for me, because I didn’t have a newsletter twelve years ago and I have a lot of thoughts I’d like to share. For those who don’t me, I’m a pseudonymous AI researcher working on core modeling for Gemini at Google. I also have some background in computational complexity theory.

This time, what I’ll be writing is a slightly-belated response to Scott Alexander’s writing on Tegmark’s Mathematical Universe Hypothesis (a.k.a. the MUH), which is the hypothesis that “All mathematical objects exist.” The proposal is that there’s already an abstract sense in which mathematics “exists” long before anyone actually does the work of proving it exists, so what if that form of mathematical existence was enough to imply real existence. It’s basically a theory of everything, often (as in Scott’s case) presented as an alternative to theism. Scott’s piece is already nearly four months old, and a lot of responses have already been written, so I’ve probably missed the best of the debate. Still, I figure I’m better late than never.

Most of the responses to Scott unsurprisingly seem to have come from theists. I am not a theist, however. In fact, I’m deeply sympathetic to the mathematical worldview; my entire last piece was basically a full-throated mathematical defense of atheism. Even so, there’s always been a piece of the MUH that really bugged me, and no one ever seems to mention it: why are we all so okay with Tegmark's assertion that the output of a computation (or math problem) exists before you actually run the computation?

Or, to say it another way: computation and math are just movement of information, and information takes up physical space and uses physical energy. If the proponents of the MUH want to claim that all math exists, then they are necessarily either hypothesizing some larger universe where those resources exist, or they are inventing a whole new type of existence for which we have no evidence.

This might not be any sort of hard proof against the MUH (Tegmark undoubtedly considered this when he wrote his work), but I think an appreciation of this makes the hypothesis a much harder philosophical pill to swallow. There’s a very real sense in which, at least in this world, computation and mathematics only happen when something makes them happen.

But First: A Brief Defense of Tegmark’s Hypothesis

Full disclosure: I have never read Tegmark’s book. What I know of his hypothesis comes to me from discussions I’ve had with people who have, Wikipedia, and the writing of the two Scotts (Alexander and Aaronson - but when I write “Scott” without qualification I’ll mean Alexander). Still, I believe I have a pretty good idea of what’s in it, even if I may be missing a couple specifics. Something like the MUH can seem like a very natural conclusion from a certain type of computational worldview very similar to my own.

And I’d like to take a moment to defend that worldview, since my objection to the MUH stems from that very same perspective. The core of this worldview is physicalism, the idea that all of existence is the result of physical processes and physical processes alone. This includes not just matter and the universe but consciousness itself: it is the claim that our experiences are nothing more than byproducts of physics, a logically necessary outcome of the particles in our brains moving the way that they do.

If you are already determined not to believe in physicalism, then I won’t try to convince you here (maybe in another post). In fact, if you aren’t a physicalist, you probably already have your own collection of objections to the MUH - there was no shortage of responses from theists/non-physicalists to Scott’s piece. The most I’ll say in its defense is that mathematics and physics have so far given humanity immeasurably more insight into the world than any other approach, and physicalism is simply the wager that no other explanations are needed - that the universe is “physics all the way down” so to speak.

But if the universe is just physics, and computers can easily simulate physics and math, then one type of physicalist argument goes that if you had the right program (and the right computer) you could simulate the universe. And if you did simulate the universe, then you would also simulate everything it entailed, including consciousness.

If you start thinking about it too hard, the line between computation and physics starts to blur. Sometimes, it’s easy to see that there isn’t a difference between simulating something and being the thing itself. For instance, there’s no such thing as a “simulation of computing 2+2” - you are either doing the computation, or you aren’t. But most people probably wouldn’t make this claim for, say, the orbit of the Earth around the Sun; in that case we typically think about the Earth and Sun as physically existing, and when we simulate Earth’s orbit we do so knowing that it’s just an approximation of the real thing, and this would be true even if we simulated the earth down to the smallest molecule. Some things, like consciousness, are trickier. Still, one could argue that if consciousness is just a byproduct of computation, then that makes it a lot more like 2+2. Consciousness becomes something that can’t be simulated, it either exists or it doesn’t.

There are a couple directions you could go with this idea. You could say for instance that, since we understand the world only through our consciousness, maybe we don’t even need to have an actual planet Earth or an actual sun, maybe we’re all just in a computer simulating us. That gets you to Nick Bostrom’s famous Simulation Argument (or, if you prefer, the movie The Matrix). I assume most of my readers are familiar with this idea already.

But the thing is, it’s also possible to go in the other direction, to say maybe we were wrong earlier to differentiate between the Earth and the math that describes it. Everything is just physics, right, but what is physics? As far as our theories are concerned, it’s just math. Down to the very smallest particles, there is no aspect of the Earth that cannot be, in principle, described by math. So: maybe that’s all the Earth is. Maybe that’s literally all anything is, just math being math. And if so, if there is nothing out there but mathematics, then why would we think that the only existing math is the math that makes up our universe? From this perspective, limiting ourselves to just the math we can find in this particular universe seems arrbitrary. Wouldn’t it make far more sense to assume that all math exists in the same way that we exist?

And with that we have the Mathematical Universe Hypothesis. Whereas a physicalist believes that all existing things can be described by math, an MUH proponent believes the converse: all things described by math exist.

Whatever Happened to Parsimony?

There are a couple easy objections to this line of thinking. For instance, there was an old saying on the site LessWrong which I think applies here: “the map is not the territory.” The fact that mathematics gives us a perfect map of the world does not make it the same as being the world.

There’s also an unfalsifiability problem. Unfalsifiable things can still be true, of course, but they can never be proven true, which becomes more and more of a problem the more extreme your claim becomes.

I think falsifiability is probably the best argument against the MUH, better even than the one I’ll focus on here. It’s related to the main argument that Scott Aaronson presented 11 years ago: that maybe the MUH true, maybe it’s not, but either way the theory itself provides no actual insight; it’s just speculation making no meaningful predictions and supplying no methods for learning about the universe(s) we’re actually in. The fact that mathematics can be used to describe our existing universe isn’t really evidence of anything, because of course math describes all the physics we know - if there was something math couldn’t describe, we wouldn’t call it physics.¹

But I have a different objection, one which I haven’t personally seen discussed anywhere in this round of discussion. Which is that from my perspective, there is a huge difference between describing a computation and actually doing it.

Both of the pieces on Tegmark I’ve linked here ignore this particular issue. Each Scott, presumably channeling Tegmark, wrote something in their piece to the effect of “but why should we even need to run the simulation to say that the computation exists?” Scott Alexander wrote it cleanest: “The fact that [consciousness] exists in possibility-space is enough for the being to in fact be experiencing it.”

And I just don’t understand why we’re all so quick to let that assumption slide. What ever happened to Occam’s Razor? The difference between “possibility” and “actuality” is pretty damn big! If ever there were an extraordinary claim requiring extraordinary evidence, surely that is it, no?

Let’s take the exact example Scott used in his original piece, about Conway’s Game of Life, which is a simple deterministic game played on a grid. The grid starts with some cells filled, and then each “turn” a different set of cells wind up filled based on some deterministic set of rules. As Scott discusses, depending on where you start the game, you can run literally any possible computer program. There’s a sense, therefore, in which all of these configurations already exist - they’re baked directly into the rules of the game, just waiting to be discovered.

Conway’s Game of Life is played on a grid. The grid starts filled, and each turn the cells are updated based on a set of deterministic rules. Certain starting configurations can create very elaborate outcomes when run long enough. Image from playgameoflife.com and Wikipedia.

For instance, there is a Game of Life configuration that calculates the first ten million digits of pi - you simply fill out the grid just right, follow the rules, and eventually you get digits of pi. The fact “this configuration will give you the digits of pi” already existed long before any one discovered it, long before Conway even “discovered” the rules of his game. Even if the universe we live in never came to be, if no conscious entity ever graced the entire vastness of existence, it would still be true that if you filled a large enough grid with just the right squares and then played Conway’s Game of Life for whatever number of turns, you would wind up spitting out the first ten million digits of pi².

All this is true, I don’t deny it. But it’s also true that until someone or something actually does play the Game of Life with that configuration, there is nowhere in this universe where that grid actually exists. It is only the possibility of the grid that we can all agree exists everywhere all the time. To the extent that the grid does exist before someone actually plays the game, it is somewhere else, in some other form. The information that composes the grid literally cannot be found anywhere in the bounds of this universe.

And sure, there may be another “universe” where the mere possibility of the grid does cause it to exist, but it is a universe that bears absolutely no resemblance to our own. This is a claim far beyond anything presented in any other multiverse theory that I know of, which usually assumes at least some bounds on what can or cannot exist. Wherever “possibility space” exists, it’s not in the multiverse.

The Information View

I think many proponents of the MUH have heard similar objections before. I suspect the first retort to the previous point would be something along the lines of “obviously the math doesn’t exist in a physical location - it’s math, it’s just an abstraction. That doesn’t mean that the truth behind it isn’t real.”

But the point I’m making when I say something like “a particular configuration of the Game of Life doesn’t exist in this universe” is that the grid in the Game of Life is not just an abstraction. If you run the Game of Life on a computer, the information that composes the grid as it goes through each turn quite literally exists in a very physical sense. There would be a whole bunch of bits in the computer’s memory, and these bits would be located on physical hardware taking up physical space and passing physical electrical signals. And as the grid changed from one configuration to another, the bits would change too, and in a way that perfectly reflected the changes made to the grid.

The same would be true for the code choreographing the changes. It would be encoded as bits, or maybe pieces of it would be built into the structure of the computer’s core processor. The information may not be explicit - if I gave you a diagram of the computer’s circuitry you would certainly have to do quite a bit of work to figure out exactly how the relevant information was coded - yet whatever the code was, whatever mechanism was manipulating the game, it would still have to physically exist. The information behind it would be physical.

There’s a popular phrase in the field of Quantum Computing, “Information is Physical,” which is best explained in yet another many year old Scott Aaronson piece. I’ll let the interested reader comb through the more technical details there, but the essence is basically this: according to the laws of physics within this universe, information in any form must have some actual physical existence. You can’t “hide” information - if it exists, there is something in this universe that is in some way different because the information is there.³ The abstract computations in a computer all need physical bits and processors, the thoughts in your brain need neurons to send signals. And this applies to all of physics - the location and dynamics of all the universe’s many bodies and particles are a form of information.

Which is really just another way of saying everything I’ve already said. Whatever kind of existence this “possibility space” has, it’s qualitatively different than any other type of existence that we as humans have ever encountered.

And I know that’s not proof, I know that the proponents of the MUH are explicitly claiming that the way "possibility space” exists is valid and real. They would probably retort that requiring physical existence for math is getting it backwards - that it is in fact the math which creates the physical reality, not the other way around.

But I’m not ready to just accept such an extreme reversal of our current understanding of the universe - there isn’t any evidence that “possibility space” or anything in it has ever really existed. In fact within the bounds of our universe it very explicitly does not exist. Even God, if he did exist, would exist in a stronger, realer sense than “possibility space.”⁴ Without a compelling reason to believe in this alternate form of existence, I cannot help but defer to Occam and his Razor.⁵

And Now, Back to Your Regularly Scheduled Speculation

The MUH is a fascinating hypothesis. Even though I consider it implausible, even I can admit that there’s something incredibly appealing about it.

There’s a certain inevitability about it, it really is the culmination of a certain scientific viewpoint. the problem, though, is that it also plays right into a particular blindspot for the practical that’s relatively easy to develop when you’re mathematically minded.

Long division is a great example of what I mean. If you still remember how to do it, there are some problems that you could probably do without breaking too much of a sweat - 141 divided by 3, 774 divided by 9, etc. There isn’t really anything interesting to say about it: you know it, it’s there if you need, the algorithm “exists.”

But that doesn’t mean I could just ask you to compute 17,293,235,987,234,354,990 divided by 34,364. That would take you a while. A computer might seem to do it instantly, but actually it would still take several times longer than asking for 143 divided by 3, and it would still need to exert a fair bit of energy to move all those bits around. It wouldn’t be enough to know how division could be done, in order to actually get the answer, you would need to actually do the math.

But scientific pursuit, especially at the highly theoretical levels where individuals like Tegmark operate, never really involves solving the equivalent of really large division problems. A theoretical problem is generally solved the moment the algorithm is invented and proved correct, or when the underlying equation is discovered. Rarely does it ever need to be solved on real numbers, and to the extent that it is, it is either solved on paper with toy examples, or out-of-sight by a computer doing millions or billions of operations a second. It gets easy to forget - at a raw, unconscious level - that nature has not stopped at merely finding the equation, but is actually working actively and constantly to keep the motor of the world turning onward.

In the universal we actually have, the one from which we must draw all further conclusions, information really is physical. It only exists in one specific configuration at one specific time, and only when something else puts it there. To go beyond that, to say that there is a sense in which all information exists all the time, is as I’ve said an extraordinary claim.

I think it’s only because mathematics is already so abstract that we allow ourselves to even consider the MUH. You wouldn’t accept an argument that fairies exist simply because they are a possibility - that would be absurd. But because we have so routinely seen mathematics befuddle our intuitions in ways that seem magical, because it gives us things like imaginary numbers and computers that can play video games or store twenty-million-item sorted lists, we are more forgiving of the idea that there truly is magic there. But there isn’t. Underneath our pixelated screens exist a vast array of circuitry moving atoms from one place to another, creating and rearranging real, physical information.

I said I was a physicalist, and physicalism is the belief in the physical as all there is. Everything I’ve seen in the physical world suggests that mathematical truths emerge only through the grinding and transitioning of information.

But the Mathematical Universe Hypothesis posits more than just the physical. Maybe it’s correct, maybe mathematical truth does really in some sense exist, floating out there in the ether. Maybe underneath it all really is some endless unseen engine, made of mathematics but still infinitely beyond it, turning and churning forever as it spawns physical reality out of vast nothingness.

Maybe. But I wouldn’t bet on it.

1. There are also more technical objections around esoteric ideas like encodings and measures. All of these critiques have also been well-worn and addressed to some degree or other by Tegmark. Interestingly, one of those restrictions appears to be that the computations used to define the universe must be finite and decidable. Ironically, not only do I think such a restriction is basically an admission that the MUH can’t work, but I would also argue (unless I’m missing something - very possible) that they’re completely unnecessary. Why should the computer program that defines the universe ever halt?

2. In some appropriate encoding at some pre-designated region of the grid, obviously.

3. Exception: The hidden information could have no interaction whatsoever with the outside universe. “A comment in the source code of the universe” as Scott Aaronson puts it.

4. Also, why wouldn’t the MUH include God? I don’t really see any reason why God (or something very much like God) should be mathematically impossible. Arbitrarily more complex than a universe without him in it, sure, but not necessarily undefinable. As I understand from Scott’s writing the MUH does use simplicity as a sort of measure over universes, so that would indicate that our universe doesn’t have a God, but it would not rule out the physical existence of another universe that does.

5. And I’ll add that in my opinion consciousness is actually one of the most intuitive examples of why you need physical computation to get an output. If, as the physicalists claim, consciousness is really just a byproduct of the computations in the brain, then that would seem to imply that where there isn’t an actual brain processing away, there isn’t any consciousness.

Scott used a simulation of consciousness as his main argument for the MUH in his original piece, but to me consciousness is one of the most visceral examples of something that only exists because of an active process that causes it to exists.