On Alexander On Tegmark and God

Recently, Scott Alexander of Astral Codex Ten (unofficially the favorite blog/column of most of a certain kind of philosopher) wrote a short piece on the interaction between arguments for theism and Max Tegmark’s “mathematical universe.” The headline? Tegmark’s view undermines the leading arguments for the existence of a God. Unfortunately, Alexander makes a number of mistakes. I am here to point them out. Bentham’s Bulldog (hereafter BB) also wrote a reply, and I am going to avoid reinventing as many of his wheels as I can (when I echo him, it’s where he simply said all that needed saying).

First: Tegmark. Max Tegmark defends the view that “every consistent mathematical structure” is physically realized. Intuitively, think of it like this: we can describe a lot of physical reality using mathematical structures. “Wave functions,” “Hilbert spaces,” other various fancy gadgets that are known mainly by theoretical physicists, mathematical physicists, and philosophers of physics. These gadgets create natural “spaces” with different options for descriptions of reality, and if we are constrained only by consistency there are a vast array of these spaces. Tegmark’s view is that each of these descriptions does describe something that is just as real as the thing we inhabit (itself a mathematical object). This places him in the philosophical tradition of plenitudinous ontology, which goes back at least as far as Plotinus and includes people like David Lewis and his modal realism as well as some multiverse theorists. Tegmark’s mathematical universe, however, is much bigger than any of these other theorists, since it includes, e.g., both ‘regions’ where the continuum hypothesis is true and ones in which it is false.

I haven’t studied Tegmark’s position carefully, but I am a little worried by the use of ‘consistent’ in its definition. Consistency is notoriously difficult to characterize. Gödel’s Incompleteness Theorems more-or-less scupper attempts to define consistency formally. This yields, depending on one’s disposition delightful or obnoxious, results such as the consistency of ZFC (the standard set theory) with NotConZFC (an axiom that says that ZFC is not consistent). Whether Tegmark allows a ZFC+NotConZFC region of his mathematical universe then raises a dilemma. If he does not, he is not in fact allowing all consistent structures to be realized. If he does, then whether one structure in his universe ‘ought to be included’ will itself be at issue: in the NotConZFC region, ZFC is out the way a geometry of square circles would be. There’s probably a decent answer to this on behalf of Tegmark (e.g. one with very careful metamathematics, maybe with reference to Joel David Hamkins’s Hyper-Platonism), so I wouldn’t hang the whole reply to Alexander on it, but it is a reason to worry beyond the typical parsimony/theoretical economy concerns that plenitudinous ontologies always raise.

In his attempt to show that Tegmark undermines the case for theism, Alexander lays out five theistic arguments he thinks are the strongest and then explains why they don’t work if there is a Tegmarkian mathematical universe. I will quote him on each and then comment, Fisk-style.

AFAICT, this obviates the top five classical arguments for God:

Spoiler: these are not the top 5 classical arguments. Minimally, the ontological argument belongs in there if we’re going for a ‘historical’ understanding of classical arguments, and if we’re taking classical to modify ‘God’ then there are a number of arguments for classical theism that command more attention/respect in contemporary philosophy of religion than some of these. BB made a nice list of arguments that Alexander misses, so I won’t linger on the point.

Cosmological: Why is there something rather than nothing? Because mathematical objects are logically necessary, and “existence” is just what it feels like to be a conscious observer on the inside of a mathematical object.

No one actually argues for theism on the basis of ‘why is there something rather than nothing.’ Not when they’re being careful, anyway. Theism does not even try to explain why there is a God (a something, to everyone but the most obtuse apophaticist). The argument that usually gets presented on the ‘why is there something rather than nothing’ headline is the argument from contingency, which observes that there could have been nothing contingent and asks for an explanation for this fact. I’m not sure how Tegmark defines modal operators in his universe; it may turn out that like Spinoza, Tegmark has no contingency in his universe. If so, this undermines the argument at the price of denying the data it relies on. The next step in the dialectic would be to ask which better explains our experience of contingency: real contingency + theism or a Tegmarkian Universe with contingency as an illusion? I’m not going to play the tape any further forward, but I’d be hesitant to bet on Tegmark. The idea that things could have been different is a pretty deep-seated one.

Now, there are various tricks to reintroduce contingency into a universe like Tegmark’s. The details aren’t important and will take us into a long excursus on modal metaphysics, but as soon as contingency comes back, we can again ask the question: what explains the contingency? Tegmark’s answer will depend on the details, but in either case you will end up with theism vs. some sort of ur-principle. But here’s the catch: theism vs. some sort of ur-principle was always the game in arguments from contingency. The jaunt through Tegmarkian plenitude is totally unnecessary if that’s where we ultimately end up. I’m actually not that keen on the argument from contingency, but I don’t see how Tegmark’s ontology changes the debate.

Fine-tuning: Why are the values of various cosmological constants exactly perfect for life? Because there are zillions of mathematical objects, but only the ones capable of hosting life do so. Therefore, a conscious observer inevitably finds themselves inside a mathematical object capable of hosting life.

This is just the multiverse response to the fine-tuning argument. Whether the multiverse response to the fine-tuning argument works is an open question. If it does, the Tegmarkian multiverse will do the trick but is overkill. If it does not, the Tegmarkian multiverse will not do the trick either.

Argument from comprehensibility: why is the universe so simple that we can understand it? Because in order for the set of all mathematical objects to be well-defined, we need a prior that favors simpler ones; therefore, the average conscious being exists in a universe close to the simplest one possible that can host conscious beings.

I wouldn’t cash ‘comprehensibility’ out as simplicity. Neither would folks who offer Comprehensibility arguments for theism. Probably the most detailed version of that argument is Robin Collins’s ‘fine tuning for discoverability’ argument. Collins has good reasons to think that his argument would not be undermined by a multiverse, so they would not be undermined by a Tegmarkian multiverse.

First cause argument: All things must have a cause. What is the cause of a cellular automaton’s starting position? There is none within the automaton itself. If a human is simulating the automaton on a computer, there’s some sense in which the cause is in the human’s world - it’s whatever made the human choose to simulate it from that starting position instead of another. But when you consider the automaton as a mathematical object, it doesn’t need a cause; you can start an automaton any way you want; they’re all just different mathematical objects. If we were selecting for simplicity, we would expect for most objects to start as a singularity and then explode outward (hmmmmm…)

Again, no one actually starts a theistic argument with ‘all things must have a cause.’ God, a thing, is causeless on theism. The best causal arguments rely on claims like ‘no causal chain (or no causal chain of a certain kind) can descend infinitely,’ and ‘no causal chain can begin without a self-moving starter.’ Principles like these are controversial, but if true would hold in a Tegmarkian universe too, and if false would not require a Tegmarkian universe to falsify. So once again the jaunt through Tegmarkland is not helpful.

Teleological argument: Why does the world have interesting structures like living things? There is no penalty for realized complexity, only for complexity of the starting laws. Given that we’re conscious beings, the world must be complex enough to contain conscious beings.

Usually, people fold both the cosmological fine-tuning argument and the biological design argument under the banner of ‘teleological,’ but I’m not here to quibble over terminology. This argument is really only significant as an object of historical interest. Arguments from biological design are basically dead in professional circles, and Darwin killed them (it’s philosophy so this is not a universal truth, but it’s pretty close to one).

I appreciate a lot of what Scott Alexander has to say, but someone needs to get him a copy of Two Dozen or So Arguments for the Existence of God or The Blackwell Companion to Natural Theology if he wants in on the God debate (yes these are both 5+ years old and are no longer cutting edge, but they’re still good jumping off points).