Contra Scott Alexander On Whether Tegmark's View Defeats Most Theistic Arguments

God>Math

1 Introduction

Come now, and let us reason together, saith the Lord

—Isaiah 1:18

Scott Alexander, whom a friend of mine once cited during a high school debate as 'the smartest person on the internet,' has been writing about God. He notes that the atheism vs. theism debates of the 2010s are back—people (including me) are writing long essays about why God exists, and others are replying with long essays about why God does not exist.

Scott’s latest volley in Nietzschean eternal back and forth between theists and the atheists on the internet claims that Tegmark's Mathematical Universe Defeats Most Proofs Of God's Existence. Tegmark’s basic view: every single possible mathematical structure is instantiated. Every consistent set of mathematical laws concretely exists somewhere, but the simpler ones exist in greater numbers. Scott thinks that most of the arguments for God are impotent against such a view.

2 The failure of Tegmark’s view

2.1 Consciousness

Scott left one major class of arguments off his list of theistic arguments that the Tegmark view supposedly explains: arguments from consciousness! In my view, arguments from consciousness are some of the best arguments for God. Specifically, there are two really good arguments:

1. On atheism, it’s surprising that consciousness exists at all.

2. On atheism, it’s surprising that there’s psychophysical harmony (I won’t try to explain what that means, but I recommend if you want to learn more see here, here, or here).

Merely instantiating a bunch of math doesn’t guarantee either of those things (note: as is explained in all the above links, this doesn’t actually assume dualism). Thus, the Tegmark view is completely impotent against one of the best arguments for God.

2.2 Moral knowledge

Another one of the best arguments for God is from moral knowledge. The moral facts aren’t the sorts of things that can move around atoms in your brain. Thus, if you have the right moral beliefs, it would be a huge coincidence. It would be like if you had a totally unreliable visual faculty, where the things that you saw had nothing to do with what was really in front of you, but happened to be right about what was in front of you.

In order to have your beliefs be justified, they have to be based on what’s true. If you know that you’re seeing a table because you’re hallucinating, then you have no justification for thinking there’s a table. I’ll explain the more precise version of this principle in a footnote.¹ But on atheism, by default, all moral knowledge is that way, so you have no basis for trusting it.

The same point, by the way, applies just as much to other facts about math, modality, logic, and what priors are reasonable. Thus, being a moral skeptic doesn’t get you out of the argument’s threatening tentacles. Here, Tegmark’s view is no help.

2.3 Epicycles

The simplest version of the Tegmark view would hold simply that all mathematical structures exist. But this implies that you’d probably be in a complex universe, because there are more of them than simple universes. To get around this, Tegmark has to add that the simpler universes exist in greater numbers. I’ll explain why this doesn’t work in section 3, but it’s clearly an epicycle! It’s an extra ad hoc assumption that cuts the cost of the theory.

2.4 Low probability

Even if the Tegmark view explains much of the same data as theism, the data points can still be evidence for theism. If, say, the fine-tuning argument just knocks out the non-Tegmark versions of atheism—which start out occupying the vast majority of atheism’s probability space—it’s still managed to be pretty successful.

2.5 Doesn’t explain the data

In his article, Scott lists a bunch of arguments that he thinks the Tegmark view explains as well as theism. I disagree with all of these. I’ll list the arguments as Scott phrases them in bold and then explain why it doesn’t explain it.

1. Cosmological: Why is there something rather than nothing? Scott claims that the Tegmark view explains this 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.” It’s true that it’s necessary that mathematical objects exist purely abstractly—there’s necessarily an equation that describes our laws. But it’s not logically necessary that these equations would be concretely instantiated, so that there’s a real world running these laws. (I’m not much moved by the cosmological argument, but if you are, I don’t think you should be much convinced by Scott’s response).

2. Fine-tuning: Why are the values of various cosmological constants exactly perfect for life? Scott proposes to explain fine-tuning by invoking an observer selection effect—every mathematical structure is instantiated, we only find ourselves in the worlds with observers. But this is no different from the traditional multiverse response, and has the associated problems (e.g. probably most multiverses produce mostly short lived brains that briefly fizz into existence in the recesses of outer space before quickly dying and this doesn’t explain the fact that the universe is finely-tuned for discoverability).

3. Argument from comprehensibility: why is the universe so simple that we can understand it? But this is not the argument from comprehensibility! It’s not about the world being simple enough to understand, it’s about the world having various surprising features, unrelated to simplicity, that make it so that we’re able to understand it.

4. First cause argument: All things must have a cause. Here Scott proposes that “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.” But this is precisely what proponents of this argument deny! They think a thing that exists in time needs a cause! (I don’t think the first cause argument is very good, but if a person is moved by it, the Tegmark view is no help.

Thus, I do not think that the view helps much with most theistic arguments.

3 Skepticism

One thing that Scott did not mention but could have is that the Tegmark view explains the anthropic data. On the Tegmark view, the number of people that exist would be the biggest number of people there could be! That gives you enough people to explain the fact that you exist (if, as I suggest, you’re likelier to exist if more people exist, and should thus think the number that exists is the most that it could be, the Tegmark view accommodates that). But I think the Tegmark view has various problems and cannot explain most of the evidence favoring theism.

The biggest problem for the view is that it collapses induction (a while ago Scott and I had a lengthy back and forth about this). On the Tegmark view, there are unsetly many people with every property: because there are infinite mathematically describable worlds like ours until one second but that turn to jello or a pile of beans one second from now. But there’s no reason to think we’re not in such a world. There are infinite in each case.

Now, the reply given by proponents of the Tegmark view is that the simpler worlds exist in great numbers (I’m about to plagiarize myself FYI—I’m funky like that!). The problem is that it doesn’t make much sense to talk about greater numbers of worlds unless one is a bigger cardinality than the other. The way infinities are measured is by their cardinality—that’s determined by whether you could put the members of the infinite set in one to one correspondence. If you have five apples, and I have five bananas, they’re sets of the same size, because you can pair them 1:1.

Often, infinities can be the same cardinality even if one seems bigger than the other. For instance, the set of all prime numbers is equal in size to the set of all natural numbers, because you can pair them one to one: you can pair 1 with the first prime, 2 with the second prime, 3 with the third prime, and so on.

Crucially, even if deceived people are rarer and non-deceived people are common, the number (measured by cardinality) of deceived people will be the same as the number of non-deceived people. To see this, suppose that there are infinite galaxies. Each galaxy has 10 billion people who are not deceived and just one person who is deceived. Intuitively you’d think that there are more non-deceived people than deceived people.

This is wrong! There are the same number. Suppose the galaxies are arranged from left to right, with a leftmost galaxy but no rightmost galaxy. Imagine having the deceived people from the first 100 trillion galaxies move to the first galaxy (containing 10 billion deceived people). Next, imagine having the next 100 trillion galaxies move to the second galaxy. Assuming you keep doing this for all the people, just by moving the people around, you can make each galaxy have 100 trillion people who are deceived and only 10 billion who aren’t deceived. So long as the number of deceived people is not a function of where the people are located, it’s impossible to hold that there are more deceived people than non-deceived people based on the fact that deceived people are rarer than non-deceived people. How rare deceived people are can be changed just by moving people around.

(Technically on the Tegmark view the number of people is too big to have a cardinality, but the same basic point applies).

Now you might worry: if God makes infinite worlds, won’t there still be infinite massively deceived people? So then shouldn’t this undermine induction? No, because for every particular person, they’re placed in a world optimal for their flourishing, which is unlikely to be a counterinductive world.

Here’s an analogy: suppose you’re in Hilbert’s hotel (that’s an infinitely big hotel). There are infinite copies of you in the hotel. You roll a six-sided die. What’s your credence in the die coming up 1-5? The answer is, of course, 5/6. But note: the cardinality of the people who get 1-5 is the same as the cardinality of people who get 6. In fact, by moving people around, you could make it so that every room has a hundred people who get 6 and only one who gets 1-5—or the opposite. This is a weird property of infinity—if you have two infinites of the same cardinality, you can match them up any which way. You can match ten members of one set to each member of the other set or do the opposite.

On theism, the situation is rather like that in Hilbert’s hotel when you roll dice. Every particular person is in the scenario ideal for their flourishing: that probably doesn’t involve being deceived. However, on non-theistic views, there’s no analogous process: there are simply infinite deceived and non-deceived people, and the infinites are the same cardinality. Thus, non-theistic views consistent with SIA probably result in skepticism.

4 Conclusion

Tegmark’s view is interesting and has some nice upsides. But, in my judgment, it is riddled with unresolvable problems and fails to explain much of the evidence that theism explains. Thus, theism wins overall.

At the end of his article, Scott suggests that nitpicking problems with Tegmark’s view is beside the point, writing:

But also, I think nitpicking specific holes misses the point. In Miles Donahue’s post on these arguments, he says he can’t really think of a great response to fine-tuning, but suspects that the terrain is too difficult and unexplored to give up and say God is the only answer. This answer was first proposed c. 2014. I only know about it because Tegmark writes about AI and x-risk enough that some of my friends are big fans. If it’s true, it’s true. But if it’s false, then the very fact that we waited this long to get it suggests that there are lots of possible godless explanations of the universe (that satisfy the supposed proofs of God’s existence) that we haven’t thought of yet. Instead of taking the proofs at their word that it’s God or nothing, we may fairly expect many undiscovered third alternatives.

I think this is wrong. Analogy: some creationists recently had a solution for explaining the apparent age of the universe using special relativity. This view had various mathematical problems, but was innovative and new. Should we say that “nitpicking specific holes misses the point.” No! At some point, there are enough bits of evidence for a theory that it’s likelier than unknown explanations—especially when the kinds of evidence favoring it are broad and diverse.

I also think the prospects for another view like Tegmark’s are rather dim. There are various views one can adopt that involve taking a fundamental thing and setting it to infinity—having it exist without bound. These include:

1. Theism: a mind totally without bound.

2. Modal realism: possible worlds instantiated without bound.

3. The Tegmark view: mathematical structure instantiated without bound.

But there aren’t that many fundamental things! So there are relatively few theories that just involve instantiating something fundamental without bound. Thus, there aren’t many theories with similar simplicity and nonarbitrariness to theism. And the ones that do exist—Tegmark’s view, modal realism—have tremendous problems and don’t even explain the many lines of data that theism does.

1. Let a component of reasoning be an experience providing non-inferential justification, pattern non-inferentially connecting experiences to beliefs, or belief.

Define the content of a component as in the case of a belief, the content of the belief, in the case of a pattern connecting experiences non-inferentially to beliefs, the legitimacy of the connective pattern, and in the case of an experience that justifies a belief, the belief justified by the experience.

The principle: every component of reasoning must be explained by truth of the content of the component of follow from a chain of components of reasoning that ultimately bottoms out in components of reasoning that are explained by their contents.