Elsewhere: God, cosmology, complexity

Spot the godOver on top cosmologist Sean Carroll’s blog, there’s a guest post by his fellow top cosmologist Don Page, who is a Christian. Page was responding to Carroll’s debate with William Lane Craig. Page does not find Craig’s Kalam Cosmological Argument persuasive, but has his own reasons for being a Christian, which you can read about over there (spoilers: maybe God is the simplest explanation for the fact that the universe is orderly; also the Resurrection happened).

The comment thread beneath the post is huge and goes off in all sorts of interesting directions. Page makes use of Bayes’ Theorem in his arguments. There are some people who use in their day jobs (rather than just reading Less Wrong and bullshitting, as I do) who respond to him, notably Bill Jefferys, staring here.

I’ve been commenting on and off. I reconstructed the threads I got involved in as the lack of threaded commenting over there makes it difficult to follow. I’ve been reading Peter Boghossian’s “A Manual For Creating Atheists” (which I hope to post about at some point) so I was trying for some Socratic dialogue and questioning of “faith” as a means of knowing. See how I got on:

Mathematicians wanted

I was interested in Daniel Kerr’s comments (for example, here, here, and finally here, in response to one of mine). He says that simplicity depends on a choice of mathematical language, but I thought this was just a constant factor. However, the comments rapidly go off into model theory and stuff about the Axiom of Choice, so I got lost. Can anyone comment on what he’s saying and whether he’s right?

6 thoughts on “Elsewhere: God, cosmology, complexity”

  1. I don’t know enough of the maths to truly follow what he’s saying, but it looks like he’s just talking about how to measure “simple” in the context of occam’s razor, that what it really means is “according to a shared intuition amongst scientists it’s hard to formalise”, but which approximates to the length of the theory expressed in some language (some combination of natural language and mathematical notation).

    Which fortunately usually agree, but unfortunately, often disagree in the interesting cases we’re not certain of.

    Like, lots of theories would be a lot more complicated if we didn’t have a symbol for “pi” and had to write it out as an infinite sum. But hopefully we’d still come to the same conclusions vis-a-vis occam’s razor.

    I _think_ all the maths is just examples of that sort of thing, not actually central to the point, but I’m not sure.

    I didn’t entirely agree with the examples of what constitutes “simple” but I agreed that that was the problem with Page’s post. I agree that the universe usually exhibits a bias towards mathematical elegance (that’s basically occam’s razor) and toward having life in it (it must do, because it does), but I’m not sure adding “God” simplifies the theory in a useful way. Like, no-one ever deduces the fundamental forces of the universe from a description of God, which suggests that adding God doesn’t let you leave anything _out_ so it doesn’t really make it simpler. Also, all the other things you might expect from a universe with God in seem to NOT be true — eg. the universe seems aggressively indifferent to human moral intuitions, whereas physics, biology, etc were tuned to make not just life, but more pleasant, more moral life, I’d agree that WAS an argument for God.

  2. Well, I *used* to be a mathematician. I got as far as

    “since the set of natural numbers requires a lot of complex information to specify in say mereology (with the appropriate choice of axioms) or the game of life. Both are Turing complete languages capable of expressing predicate logic, set theory, and thus all mathematical theories”

    before going “huh?” That appears to parse as “the set of natural numbers [is Turing complete language] capable of expressing…” which I think is bollocks. The natural numbers aren’t a language, any more than the alphabet or a collection of words is a language. They need a structure on top of them to make a language.

  3. I think the suggestion is that mereology (or some mereological theory) is, or at least includes, a Turing-complete language. Not that the natural numbers are one.

  4. So I think Daniel Kerr’s point about bits is something like this: If we know that the universe is complex enough to need something Turing-complete to describe it, but simple enough that anything Turing-complete can describe it, then indeed its descriptions in two Turing-complete languages differ by at most an absolute constant depending only on the languages. (Assuming both encoded in binary strings and assuming both have a way to treat arbitrary binary strings as data.) But (1) this constant might be large and (2) we aren’t actually entitled to assume that the universe is completely describable in computable terms. In the latter case, we don’t really know anything about how the length of its description might depend on how we describe it.

    It’s mostly #2 that he’s saying, though in the absence of any good reason to think our minds are more powerful than Turing machines I think we need only concern ourselves with computable theories and am accordingly more concerned with #1.

    His comment about the Löwenheim-Skolem theorem is wrong, I think. L-S says that (e.g.) there are countable models of the theory of the real numbers — but it doesn’t say that there are models in which the reals are countable. Indeed, in any model powerful enough to express the relevant notions the reals are uncountable; if you appeal to L-S, what you get is a countable model for which there are (e.g.) bijections between R and N outside the model. In any case, I don’t see why “there is a bijection between X and Y” should imply “X and Y are equally complex”.

    I’m also not sure what he’s saying about “modelling ZFC in a base 2 modular arithmetic”; something like Peano arithmetic doesn’t have any machinery for talking about sets, and doesn’t have any such thing as infinite strings of 0s and 1s, and in fact PA is a substantially weaker theory than ZFC in, e.g., the sense that there are statements about numbers that can be proved when you implement numbers in ZFC and use the axioms of ZFC but that can’t be proved using the axioms of PA. (A famous example is Goodstein’s theorem.)

    One of his comments, I think, was responding to Don Page’s argument that it is possible for God+nature to be simpler than nature alone, in (say) the sense that any minimally complex theory that fully describes nature has to have something that looks like God in it. I think Don is absolutely correct about this in principle, but it seems monstrously implausible to me that he’s correct in practice.

    1. But (1) this constant might be large and (2) we aren’t actually entitled to assume that the universe is completely describable in computable terms.

      OK, so his point is that maybe the universe is doing something that isn’t efficiently realisable on our hardware (by which I mean both the programs we’d run on computers and the brains which would interpret the results, since it’s possible to trade difficulty in one against difficulty in the other). So perhaps by preferring programs which are compact to implement and whose output natural for humans to interpret, we’re getting it wrong?

      there are countable models of the theory of the real numbers — but it doesn’t say that there are models in which the reals are countable.

      Hastily reading up on model theory, getting a headache: so those models are effectively of things which aren’t the reals?

      I think Don is absolutely correct about this in principle, but it seems monstrously implausible to me that he’s correct in practice.

      Yes, it seems like it’d have to mean that God was in fact a really simple algorithm which produced both the universe and the varieties of behaviour that people attribute to the Christian God. There are simple algorithms that produce complex behaviour, I guess, but not, as far as we know, complex people.

  5. A model is a model not of a thing but of a theory. There are countable models of any first-order theory, including (e.g.) ZFC set theory. (Assuming ZFC is consistent; otherwise it doesn’t have any models at all.) We can consider “the” real numbers in some countable model of ZFC; i.e., within that model we construct the natural numbers, the rationals, and the real numbers in some standard way. I put the word “the” in quotation marks because there are actually lots of ways to do those constructions, and they produce provably isomorphic results. So, the result of all this is a countable model of ZFC, containing a model of the real numbers.

    But that model of the real numbers isn’t countable within your model of ZFC. Of course it’s countable “from the outside” because any subset of a countable set is countable, but it’s not countable “from the inside”: there are no functions in your model of ZFC that establish a bijection between its copy of the real numbers and its copy of the natural numbers. (Because the usual proof of the uncountability of the real numbers applies just as well to that model as to any other.)

    Technical note: if you actually want to construct those countable models, you’ll run into a problem: if you could construct a model (countable or not) of ZFC you’d have proved that ZFC is consistent (because any theory with a model is consistent), and you can’t do that in ZFC by Goedel’s second incompleteness theorem. But what you can do is to construct a model for any finite subset of the axioms of ZFC, which is usually enough (e.g., a lot of relative consistency proofs work that way).

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