Tegmark’s Vast Math

I recently had a surprise chance to meet Max Tegmark, and so I first quickly read his enjoyable new book The Mathematical Universe. It covers many foundations of physics topics that he correctly says are unfairly neglected. Since I’ve collected many opinions on foundation of physics over decades, I can’t resist mentioning the many ways I agree and disagree with him.

Let me start with what Tegmark presents as his main point, which is that the total universe is BIG, almost as big as it could possibly be. There’s a vast universe out there that we can’t see, and will never see. That is, not only does space extent far beyond our cosmological horizon, but out there are places where physics sits in very different equilibria of fundamental physics (e.g., has a different number of useful dimensions), and nearby are the different “many worlds” of quantum mechanics.

Furthermore, and this is Tegmark’s most unique point, there are whole different places “out there” completely causally (and spatially) disconnected from our universe, which follow completely different fundamental physics. In fact, all such mathematically describable places really exist, in the sense that any self-aware creatures there actually feel. Tegmark seems to stop short, however, of David Lewis, who said that all self-consistent possible worlds really exist.

Tegmark’s strongest argument for his distinctive claim, I think, is that we might find that the basic math of our physics is rare in allowing for intelligent life. In that case, the fact of our existence should make us suspect that many places with physics based on other maths are out there somewhere:

Why do the five arrows in the bottom panel conspire to allow any habitable range of Higgs properties? This could well be a fluke: five random arrows would allow some range with 19% probability, so we need only invoke a small amount of luck. … However, it’s perfectly plausible that further physics research could uncover more striking fine-tuning of this discrete type with, say, ten or more arrows conspiring to allow a habitable range for some physical parameter or parameters. And if this happens, then we can argue … that this is evidence for the existence … of other universes where the laws of physics are different, giving quite different requirements for life!

I accept that this consideration would modestly push us in this direction, just as the existence of physics parameters fine-tuned for life modestly pushes us to believe that other spatial regions out there have different values of those parameters. I accept these because I accept the self-indication principle in indexical inference, which says that the info that you exist should increase your belief in possible worlds where you might have existed, in proportion to the number of slots where you could have existed.

However, it isn’t clear that Tegmark accepts the self-indication principle, since he also says that you should believe in many worlds quantum mechanics if you survive many rounds of quantum suicide, which is where you randomly suicide depending on a quantum event. The self-indication principle does not say that seeing this should increase your belief in many worlds, as there aren’t substantially more slots for you to occupy in scenarios where quantum suicide attempts fail. Since both many worlds and stochastic quantum mechanics predict exactly the same conditional probabilities of observations given quantum suicide attempts, such data can’t distinguish them. So I wouldn’t see surviving quantum suicide as further evidence for many worlds.

Tegmark also argues that we should reject a description of reality that has math plus extra “baggage.” Apparently, if there are many possible math descriptions of universes, then it would be extra “baggage” for the universe to point to only one math and say “that’s the math that really exists.” This is pretty close to the argument of David Lewis. But Tegmark also seems to accept that there is some “measure” over all these maths, and that the reason to take some actions over others is to favorably change that measure. Yet this measure also seems to also be an extra “baggage” over and beyond the math descriptions themselves. The main difference I see between adding a “this exists” pointer and adding a measure over the space of maths is that the later approach has more slots for me to exist, and so is favored by self-indication.

Tegmark does say that we struggle to find reasonable measures for many physics problems, and as a result we don’t actually know what inflation implies. He suspects this is due to our over-eager embrace of the concept of infinity. But Tegmark also follows Everett in using infinity to derive the Born probability rule in quantum mechanics. Everett noted that the relative measure of Born-rule-deviant worlds approached zero after an infinity of measurements, and thus the final state is “as if” those worlds didn’t exist. To get the same effect, Tegmark instead considers a superposition of an infinity of spatial regions out there containing the same exact observer. But this trick wouldn’t work with a finite set of such regions, and I have my doubts that it is fair to treat such regions is if they hadn’t decohered. (My approach to deriving the Born rule in many worlds stays with strictly finite everything.)

Tegmark also seems to think that a similar approach explains the second law of thermodynamics, but that doesn’t at all make sense to me. The main thing to explain there is why our best measure for predicting the past is pretty much the inverse of our best measure for predicting the future. I didn’t see Tegmark addressing that at all. (Btw, Sean Carroll recently mostly rejected the solution he wrote a book elaborating.)

My last disagreement with Tegmark is that he sees the simulation argument as self-destructing because since we know nothing about the universe that might simulate us, it could also be simulated by another universe, which is a “a reductio ad absurdum.” But I think it quite reasonable to put substantial weight there being real costs of computation in the base universe, which means they would tend to limit the ability of universes they simulate to spawn yet further universes. Which makes it less likely that we are a second level simulation, relative to a first level one.

Let me conclude with some of the many ways I agree with the book:

• We reject or accept total theories, not places. If parallel universes are implied by our best theories, we should believe in them.

• It is vitally important to figure out good physics measures, and good physics explanations for them.

• I also prefer to reason about theories without infinities, when nearly as plausible on other grounds. Take a limit to infinity as the very last theory step, if possible.

• Our brains aren’t doing much quantum computing.

• Quantum mechanics seems random because we observers are splitting; the universe is deterministic.

• Foundations of physics gets unfairly neglected. People who do it have to hide it to save their careers.

• The world is hurting itself by not coordinating to reduce existential risk. We have a vast great potential future if we don’t kill ourselves.

• We don’t need a new understanding of consciousness to solve foundations of physics problems. “Consciousness is the way information feels when being processed in certain complex ways.” Saying that our math “exists” is pretty much the same as saying that the stuff that fits our math is the stuff that “feels” when arranged in certain ways.