EDIT: See the reply to this comment by Amicus; I think their interpretation of Tegmark is correct. See also the SEP page for structural realism, where Tegmark's view is discussed: https://plato.stanford.edu/entries/structural-realism/
I feel like I don't understand Tegmark's view. It seems like he has to be saying one of two things, and both are really implausible.
(1) The first interpretation is that the universe and its constituents *are mathematical entities*. But this seems obviously false: mathematical entities are abstract (i.e. non-spatial, non-temporal, causally effete) objects, whereas we are obviously concrete objects. Tegmark might say that we were wrong to think of mathematical objects as abstracta, but then his claim ceases to be interesting: he's gained the ability to say the words "we are mathematical entities" with a straight face, but only at the cost of completely redefining the relevant terms.
(2) The second interpretation is that the various concrete objects in the world somehow *depend* on mathematical objects. In other words, our concrete universe exists *because* it can be mathematically modelled in some way. But what exactly is this dependence relation supposed to be? Is it causation (surely not)? Is it grounding? Maybe Tegmark could give up the claim that there is a strict dependence relation here, but then I don't see how the view differs from modal realism: it would just be the claim that all of the possible universes exist, coupled with a specific account of possibility as something like "mathematical consistency."
On either interpretation, it just smells like Lewisian metaphysics (the problems of which it thus inherits) coupled with some implausible claims about abstracta.
Also, regarding interpretation (1): Pruss once suggested that Platonic dualists should consider the view that the objects of our experience are abstracta (he even explicitly connects this to Tegmark's view). But he raised this option only for the sake of remarking on its absurdity! https://alexanderpruss.blogspot.com/2015/01/if-you-going-to-be-platonist-dualist.html
> The first interpretation is that the universe and its constituents *are mathematical entities*. But this seems obviously false: mathematical entities are abstract (i.e. non-spatial, non-temporal, causally effete) objects, whereas we are obviously concrete objects. Tegmark might say that we were wrong to think of mathematical objects as abstracta, but then his claim ceases to be interesting: he's gained the ability to say the words "we are mathematical entities" with a straight face, but only at the cost of completely redefining the relevant terms.
Tegmark's position is that there is no such distinction - a particularly radical form of ontic structural realism.
Your friend made another interesting point in favor of theism that Scott hasn't dealt with: theism is more elegant, and posits a less ugly, disunified picture of fundamental reality, bc it makes it so that there are intrinsic connections between the basic categories of things in the world: consciousness, moral facts, abstract objects, physical things, etc. All these things have reasons for being there given that God exists, because he grounds them; whereas on naturalism the universe is filled with a lot of arbitrary detail that didn't have to be there, but just happens to be. Theism more strongly predicts a world where there are specifically things like consciousness/moral facts/psychophysical harmony/etc., whereas naturalism could be true of a world whether or not those things existed (meaning it expects them less strongly).
Moral facts are facts about the nature of perfection, which is what God is by the "Perfect God" hypothesis's own definition; math facts are ideas entertained within an infinite intellect, explaining why they're necessary and transcend physical examples* (see footnote); consciousness exists because a perfect being would want to realize things of value, and you can't have love/moral growth/virtue/good stories without conscious beings; physical things exist because they are also useful for telling a great story (even if you doubt this, I'm not sure why idealism isn't an appealing proposition; I've always thought consciousness was more likely to be fundamental than physical things anyway).
Also, don't beautiful scientific theories usually get points for being more beautiful than others? Theism deserves credit if so: what could be more beautiful than the idea that perfection exists, and the reason the world exists is to tell the greatest (most beautiful) story ever told?
--- Footnote: About math needing God ---
It seems like if the physical world stopped existing (even on Tegmark's view, right?), it wouldn't stop being true that 2 + 2 = 4. But if 2 + 2 = 4 is true regardless of whether there is a physical world, it's hard to see how mathematical facts can *just exist* (Where? How?), without occupying anything or having any substrate or location.
But it’s very easy to see how they can exist as ideas in a mind. The trouble is, there haven’t always been human minds, and it seems bizarre to think that 2 + 2 = 4 would stop being true at the moment humans stop existing for contingent historical reasons.
If an infinite metaphysically necessary and all knowing mind that created everything else exists, though, it’s very easy to see how abstract (math) objects could exist necessarily--they exist within a necessarily existing divine intellect for ever and ever.
What's at stake in these debates about god's existence? As I understand it, the minimal versions of theism that are supported by philosophical arguments are just deism. Does this have any implications at all for how to live?
(1) The truth of minimal theism is intrinsically interesting, just as e.g. the truth of Platonism would be.
(2) As the other commentor (lumpy boi) notes, having a high credence in theism should raise your priors for the various theistic religions. This is especially true once you realize that if God exists, then you probably have various obligations to him, which you are failing to live up to. This might make you more receptive to the idea that God has provided a means of atoning for your sins.
(3) If minimal theism is true, then various actions became rational for you which would be irrational if theism were false (e.g. praying, asking God to forgive your sins, and so on). The Enlightenment deists who subscribed to "natural religion" often (sometimes?) did these things, so there's precedent for minimal theists acting this way. One might also argue that theism has special moral implications (e.g. Richard Swinburne argues that theism makes suicide intrinsically wrong, since if your life is gift from a loving God, then suicide basically amounts to throwing away a loved one's gift once it has ceased to please you).
I'm with you on it being intrinsically interesting, but I was asking about implications for how to live. You get at that in points (2) and (3), but the kinds of implications you point to don't follow at all from minimal theism.
Why on earth would minimal theism make it likely that you have obligations to this god? Or that there are such a thing as sins? Or that this god cares about praying? The conception of god that seems to be at play in your response is very much that of religion, but there's no reason to assume that the god of minimal theism is anything like that very specific conception. That seems to me like assuming that aliens look specifically like those grey humanoid figures we see in popular fiction.
It doesn't seem to me that accepting in minimal theism should increase your credence in theistic religions, because whenever I've engaged with those religions they've been riddled with internal inconsistencies that make them logically impossible to be true.
Btw, I worry that I'm sounding combative here when that's not my goal at all! I appreciate you taking the time to answer my question.
[any claim about God, including theistic religions or that you have an obligation to God, etc.] => [God exists]
So P([any claim about God, including theistic religions or that you have an obligation to God, etc.] | [God exists]) > P([any claim about God, including theistic religions or that you have an obligation to God, etc.])
So unless you did a really weird update of a shape like "this argument proves that God probably exists AND that we owe God nothing" you gotta increase your credence in [any claim about God, including theistic religions or that you have an obligation to God, etc.] as you increase your credence in [God exists]
Well, even minimal theism says that you were created by a perfectly good and loving being. It seems plausible that even if you knew nothing else about this being, you should still think that you have some obligations to it (e.g. gratitude for having been created), simply in virtue of what it's done for you.
Regarding theistic religions, you should have a credence of much less than 1 that those religions are in-fact internally inconsistent (if only because lots of really smart people who've thought hard about the relevant issues deny that this is so). And so long as your credence in this claim is less than 1, then finding out that an entailment of these religions (in this case, minimal theism) is true should raise your credence in those religions, for the reasons that I.M.J. McInnis lays out.
Also, I'm not sure what you mean when you refer to there being a "very specific conception" of God in the theistic religions. The philosophical traditions of Christianity, Judaism, and Islam all think of God as a perfect or maximally great being. Of course, they make lots of claims about what this being has done in history, but that's not about their model of God per se.
Here's an analogy: suppose you want to know if a particular event in a given cornfield was a UFO crash or not. You then find out (via independent evidence) that there are, in-fact, aliens of exactly the sort which the UFO theory posits concerning the cornfield incident. This should raise your confidence that the incident was, in-fact, a UFO crash. The same goes for e.g. God and the resurrection of Jesus.
Analogy: if I told you "I have a brother" and you're not sure whether you should believe me, but you learned that I have a sibling, you should increase your believe that I have a brother
Firstly, if the god proposed by the religion is internally inconsistent such that it cannot exist, then proving a necessary condition for that god shouldn’t increase your credence in it.
Secondly, whilst your probabilistic logic is sound, isn’t it the case that for any given claim about god, proof of god’s existence would equally increase the probability that the opposite claim about god is true? For example, “you have no obligations to god”. So why would the existence of god increase the relative credence you have about any claim about god?
The first is a good clarification! If I told you "I have a flarfnarf, which is a type of sibling both green and colorless," then learning I had a sibling should not increase your credence that I actually have a flarfnarf.
(At first I thought this objection was irrelevant. As they say, "zero and one are not probabilities," and when you think "the god proposed by this religion is logically inconsistent; it has zero probability" you should instead thing "either the god proposed by this religion is logically inconsistent, or something about my reasoning is broken." and you should never have probability 0 on "something about my reasoning is broken." But if I have reasoned "Hinduism in particular is totally contradictory---not for anything about god in general, but about Vishnu in particular"---then believing minimal theism shouldn't make me increase my "something about my reasoning is broken" credence. (Though I do think you should be very very hesitant to say "I am certain modulo Cromwell" about anything about which there's vociferous disagreement!))
For the second point, I think you're misunderstanding the meaning of "opposite." (It's a forgivable mistake, W.v.O. Quine made it too, when he argued for single-sorted logic.)
Let $xOy$ denote "$x$ has obligations to $y$". Let $Gz$ denote "$z$ is God." Let $u$ be the constant symbol denoting you. Suppose that God is unique (if extant), i.e. $\forall x,y(Gx \wedge Gy \implies x=y)$.
"You have obligations to God" is $\exists y ( Gy \wedge uOy)$.
"You do not have obligations to God" can be interpreted as $\not\exists y (Gy \wedge uOy)$, but I dislike this. It munges two things: God doesn't exist, and God exists but you bear no obligations toward God.
The correct partitioning of possibilities is:
God exists, You have obligations to God $\exists y ( Gy \wedge uOy)$.
God exists, You have no obligations to God $\exists y (Gy \wedge \not uOy)$
God does not exist, so it's meaningless to ask whether you have obligations or not $\not \exists y (Gy)$
So eliminating the third option increases credence in the first two options.
The historical sidebar here is: Quine thought you should still be able to say "well, I have no obligations to God because God doesn't exist" just like you should be able to say "the number 17 is not hungry, of course not, it's not a number." But it's provable that the system of single-sorted logic in which you stipulate that 17 is not hungry (nor is anything else that can't be hungry) is Morita equivalent to one in which you stipulate that 17 *is* hungry (and so is everything else that can't be "actively unhungry"). Which is bad! So you have to preserve the "true/false/meaningless" distinction if you want to preserve the "true/false" distinction in a nice way for predicates.
If God exists, he might want something. If he stands behind the moral law, might he want you, personally, to be moral? Will he make any demands of you: and if he does, how could his demands be resisted? A God, even a deistic God, is not a tame lion. He is not safe.
Lewis wrote about this, and how when he became a theist (not a Christian, not yet anyway) he slowly came to realize what the existence of God meant for him personally:
"I distinguished this philosophical “God” very sharply (or so I said) from “the God of popular religion”. There was, I explained, no possibility of being in a personal relation with Him. For I thought He projected us as a dramatist projects his characters, and I could no more “meet” Him, than Hamlet could meet Shakespeare. I didn’t call Him “God” either; I called Him “Spirit”. One fights for one’s remaining comforts.
...
"The real terror was that if you seriously believed in even such a “God” or “Spirit” as I admitted, a wholly new situation developed. As the dry bones shook and came together in that dreadful valley of Ezekiel’s, so now a philosophical theorem, cerebrally entertained, began to stir and heave and throw off its gravecloths, and stood upright
and became a living presence. I was to be allowed to play at philosophy no longer. It might, as I say, still be true that my “Spirit” differed in some way from “the God of popular religion”. My Adversary waived the point. It sank into utter unimportance. He would not argue about it. He only said, “I am the Lord”; “I am that I am”; “I am”.
"People who are naturally religious find difficulty in understanding the horror of such a revelation. Amiable agnostics will talk cheerfully about “man’s search for God”. To me, as I then was, they might as well have talked about the mouse’s search for the cat.
...
"Remember, I had always wanted, above all things, not to be “interfered with”. I had wanted (mad wish) “to call my soul my own”. I had been far more anxious to avoid suffering than to achieve delight. I had always aimed at limited liabilities. The supernatural itself had been to me, first, an illicit dram, and then, as by a drunkard’s reaction, nauseous. Even my recent attempt to live my philosophy had secretly (I now knew) been hedged round by all sorts of reservations. I had pretty well known that my ideal of virtue would never be allowed to lead me into anything intolerably painful; I would be “reasonable”. But now what had been an ideal became a command; and what might not be expected of one? Doubtless, by definition, God was Reason itself. But would He also be “reasonable” in that other, more comfortable, sense? Not the slightest assurance on that score was offered me. Total surrender, the absolute leap in the dark, were demanded. The reality with which no treaty can be made was upon me. The demand was not even “All or nothing”....Now, the demand was simply “All”."
Thanks for taking the time to send this. Can any of these specific things that god might want of you be known? If not, it doesn’t seem decision relevant
With all sincerity, you could ask him. If I was a deist, that would be my first line of inquiry. Prayer is a practice that has been believed in and relied upon for the vast majority of humankind for the majority of human history, and if a God of some type exists he just might answer.
There are many arguments that can get you to theism, but theism is not the same as deism. The deist believes that God exists, but he doesn't do anything anymore. He doesn't answer prayers, give prophets visions, work miracles, etc. A God like that would be impossible to know, for if God does not reach out to us we have no way of reaching to him. Yet deism is just as much a choice as Christianity, or Hinduism, or pantheism, or any of the other places you might land after you had concluded to your satisfaction that a god of some type exists. What reason would you have to prefer deism to the alternatives: except, perhaps, in that it requires the least of us. If the God of deism is real, it would affect our lives very little. The philosophical arguments for God may lead us to believe there is a lion at large: it is comforting to believe he is caged, but do we have reason to do so?
Lewis writes a bit on this in Mere Christianity, though in reference to a general belief in an impersonal "Life Force" which is similar enough to deism for our purposes:
"When you are feeling fit and the sun is shining and you do not want to believe that the whole universe is a mere mechanical dance of atoms, it is nice to be able to think of this great mysterious Force rolling on through the centuries and carrying you on its crest. If, on the other hand, you want to do something rather shabby, the Life-Force, being only a blind force, with no morals and no mind, will never interfere with you like that troublesome God we learned about when we were children. The Life-Force is a sort of tame God. You can switch it on when you want, but it will not bother you. All the thrills of religion and none of the cost."
The concept of God has often been tied to morality: to act as a ground of moral law, a judge, and the source of our conscience. If this is true then it has significant implications for our life as well. Lewis writes a bit on this in Mere Christianity as well, expanding on his own experience in becoming a theist:
“We have not yet got as far as the God of any actual religion, still less the God of that particular religion called Christianity. We have only got as far as a Somebody or Something behind the oral Law. We are not taking anything from the Bible or the Churches, we are trying to see what we can find out about this Somebody on our own steam. And I want to make it quite clear that what we find out on our own steam is something that gives us a shock. We have two bits of evidence about the Somebody. One is the universe He has made. If we used that as our only clue, then I think we should have to conclude that He was a great artist (for the universe is a very beautiful place), but also that He is quite merciless and no friend to man (for the universe is a very dangerous and terrifying place). The other bit of evidence is that Moral Law which He has put into our minds. And this is a better bit of evidence than the other, because it is inside information. You find out more about God from the Moral Law than from the universe in general just as you find out more about a man by listening to his conversation than by looking at a house he has built. Now, from this second bit of evidence we conclude that the Being behind the universe is intensely interested in right conduct — in fair play, unselfishness, courage, good faith, honesty and truthfulness. In that sense we should agree with the account given by Christianity and some other religions, that God is "good." But do not let us go too fast here. The Moral Law does not give us any grounds for thinking that God is "good" in the sense of being indulgent, or soft, or sympathetic. There is nothing indulgent about the Moral Law. It is as hard as nails. It tells you to do the straight thing and it does not seem to care how painful, or dangerous, or difficult it is to do. If God is like the Moral Law, then He is not soft. It is no use, at this stage, saying that what you mean by a "good" God is a God who can forgive. You are going too quickly. Only a Person can forgive. And we have not yet got as far as a personal God — only as far as a power, behind the Moral Law, and more like a mind than it is like anything else. But it may still be very unlike a Person. If it is pure impersonal mind, there may be no sense in asking it to make allowances for you or let you off, just as there is no sense in asking the multiplication table to let you off when you do your sums wrong. You are bound to get the wrong answer. And it is no use either saying that if there is a God of that sort — an impersonal absolute goodness — then you do not like Him and are not going to bother about Him.”
if you think the arguments for theism/deism are good then that increases your priors for the explanations of some miracles. i.e. if atheism is true then it's hard to run swinburne's bayesian arg in the resurrection of god incarnate. if theism is true then explanations like those are salient/more plausible.
I enjoy what you are writing very much as a committed theist (and Catholic), but your use of probabilities on infinite sets oscillates between careless and nonsensical - you are not alone in this issue of course, as Scott has the same problem.
So, as a mathematician, let me give you a basic crash course on probabilities :
So here is the thing : the basic formula of probabilities,
P(A)=card(A)/card(Ω),
only works if the universe set Ω is a finite set.
It doesn't work if Ω is infinite, especially if A is infinite itself. What you need is a probability measure m on your set Ω. A measure is a generalization of the notion of length (invented by Lebesgue for his theory of integration). A probability measure m is simply a measure such that the measure of the whole set Ω is equal to 1.
Then you can simply define the probability of some subset A by
P(A)=m(A) - no need for a division as the denominator, p(Ω), would simply be 1.
On a given set you can define multiple measures. The most standard one on the set of all real numbers is the Lebesgue measure L, which has the nice property that L([a,b])=b-a. You can't directly use it as a probability measure on the set of all real numbers (because L(R)=infinity), but you can use it on the segment [0,1]. But there are many other possible measures. For example the Dirac measure defined by D(A)=1 if 0 is in A and D(A)=0 if 0 is not in A is a perfectly defined probability measure on [0,1] too.
However note that the singleton {0} has measure 1 for the Dirac measure, so P({0})=P([0,1]). A finite set has the same probability as an infinite set !
And if you take the Lebesgue measure, you can find sets that have the same cardinality as [0,1], but have Lebesgue measure 0, and thus probability 0 for the Lebesgue measure.
As a summary : you need to define your probability measure, or any talk about probabilities does not make any sense.
> Every particular person is in the scenario ideal for their flourishing: that probably doesn’t involve being deceived.
The “Probably” dooms your argument. You have to be able to say that the number of people who are in worlds where induction fails is of a *smaller* cardinality then the number of people where induction holds (I think all this discussion of cardiologies of infinity is nonsense, but setting that aside…). There’s no way you can do that. If all the genocides, animal torture, and wild animal suffering in this world could be justified, then surely a world where everything is great but where induction is fake is justifiable. Maybe there are beings that really hate predictability (some humans probably do) That world would be great for them!
If even 1/Graham’s number people are in worlds without induction, then you can make a 1:1 correspondence between people who are in such worlds and people who are not… induction is thus fatally undermined, since we have no way to know whether we are in a world with it or one without.
This is mistaken. The point of the argument is that you *don't* have to be able to say that the non-deceived people are fewer in number than the deceived people: all you have to be able to say is that for each individual person, considered in isolation, it is probable that they would not be deceived.
This is the whole point of the Hilbert's Hotel dice roll example. When trying to figure out what you should expect to roll, you don't look at the *number* of people who get various outcomes, since then you'd get absurd results (e.g. that you're just as likely to roll a 6 as to roll a 1-5). Rather, you consider the individual case in isolation, where you can easily see that, given the relevant objects and properties, you have e.g. a 1/6 chance of rolling 6.
> When trying to figure out what you should expect to role, you don't look at the cardinality, since then you'd get absurd results.
A. I interpreted the dice example as being about whether you think individuals lack induction. But my point was that someone cannot simultaneously believe that induction is a brute fact and that the number of people for which induction fails is equal to the number of people for which induction works.
If I misunderstood that argument…
B. The dice rolling example is also misapplied. Your situation is not of infinitely many people randomly choosing among a finite list of outcomes. It’s infinite people choosing one of an infinite list of outcomes. If you were in Hilbert’s hotel and rolled a dice with one face for every world where induction holds and one face for every world where induction fails, your odds would be 50/50.
C. There are “absurd” results everywhere here. All of Matthew’s theodicies are facially absurd. Comparing infinities like this is absurd. Talking about “unsetly” many things like that isn’t a made up gibberish word is… absurd. Not sure why this absurdity is the bridge too far.
(A) I don't understand what you're saying here. Matthew doesn't think induction is "a brute fact" (he thinks that God puts people in induction-friendly circumstances).
(B) Sure, there are infinitely many ways *to be deceived*, but there's basically only one way to be non-deceived (i.e. if your faculties and perceptions are generally reliable), and so all we need to worry about is whether or not that one non-deceived option obtains. I guess one could say that there's lots of ways for your faculties to be generally reliable (e.g. you could have 20/20 vision, or 20/21, 20/22, etc.), but that seems too fine-grained to me, akin to saying that there's lots of ways to roll a 6 (since after all, the die might bounce this way rather than that). All we really care about is the coarse-grained "reliable or not-reliable" issue. Also, even if there were an equal number of deceived and non-deceived possibilities, the theist shouldn't take their odds to be 50/50, since theism predicts that the non-deceived options are more likely.
(C) I don't agree that the theodicies are absurd, but I'm a bit more worried by your seeming opposition to comparing infinities. Is your view that contemporary mathematics is gibberish, since it talks about differently-sized infinities? If so, don't you think it's a bit more likely that you're misunderstanding things than that contemporary mathematics has taken a fundamentally wrong turn (which you, a layman, have managed to diagnose)?
> Matthew doesn't think induction is "a brute fact"
Matthew once deleted me in a live argument with me about this, so I would respectfully disagree. My understanding is that “God does it” is a means to create a unifying explanation for this fact, since discarding it would entail skepticism (and 95% of the basis for Matthew’s theism is a deep rooted belief that we’re not in a skeptical scenario.)
Doesn’t really matter anyways. Maybe he’s changed his position, or was just adopting it for the sake of argument.
> (B) Sure, there are infinitely many ways *to be deceived*, but there's only one way to be non-deceived (i.e. if your faculties and perceptions are generally reliable)
1. Not sure what the implication of this is
2. I view this as a question of people who are being deceived versus not being deceived, not “ways” of those things happening. Unless those are the same thing?
3. There are many ways to not be deceived. The laws of physics could all be fake which hold by pure chance, and we just get very lucky.
4. Wouldn’t this enhance my argument, since the proverbial dice would have many faces saying “deceived”, but only one saying “not deceived”
> even if there were an equal number of deceived and non-deceived possibilities, the theist shouldn't take their odds to be 50/50, since theism predicts that the non-deceived options are more likely.
If the number of people in two situations are equal, then you are equally likely to be in either one. Theism may predict that there are more situations where people are not deceived, but it predicts equal numbers of people in both. Mathew expressly concedes this.
Given that who “you” are is a random selection from the people God creates, it’s equal!
You cannot make these distinctions between infinities of the same cardinality.
> I'm a bit more worried by your seeming opposition to comparing infinities. Is your view that contemporary mathematics is gibberish, since it talks about differently-sized infinities?
1. I don’t think you can do Bayesian probability over different infinities.
2. Even if you could, for probabilities to escape being 50/50, you need to have an infinity of a larger cardinality on one side, in which case it would dominate and make the probability go to 100 or 0. This is what mathematics does when it “compares differently sized infinities”.
3. Practically speaking, is infinity useful at all for modern mathematics? Is there any practical application where you have an uncancelled infinity? I don’t know. If the answer is “no there are not”, then a maybe just throw it out. If there are, then I believe they are all between cardinalities, which still defeats your argument.
(A) But if the explanation is "God did it," then it isn't a brute fact.
(B) You say that "If the number of people in two situations are equal, then you are equally likely to be in either one." But that's false, as the Hilbert Hotel dice example shows us: the number of people who roll 6 is equal to the number of people who roll 1-5. But clearly the probability that you will roll 1-5 is greater (five times greater, to be precise) than the probability that you will roll 6.
Also, no, your argument would not be enhanced, since the theist's claim isn't that there are more *ways* of being non-deceived: the claim is that God puts his thumb on the proverbial scale. Think of it like this: it's not that God adds more sides to the die; rather, he uses a loaded die, which is loaded towards non-deceptive possibilities :)
(C) Yes, infinity is of extreme (even fundamental) importance to much modern mathematics. We can also just prove that there greater and smaller infinities. Cantor did this for the real and natural numbers back in 1874. I think you'll find this helpful: https://en.wikipedia.org/wiki/Cantor's_diagonal_argument
Based on your response to point C, I am forced to conclude that you are either not actually reading my comments or are trolling. I’m not going to engage any further.
I always love to see some good back and forth! That said, I think there’s a couple of features of the simplicity-based arguments that you’re underrating.
Regarding 2.3, one of Scott’s points is that a metric of simplicity isn’t something that just gets tacked onto Tegmark’s theory—it’s an unavoidable part of it. You can’t construct a uniform probability distribution over a countably infinite set, and this situation is analogous. (What’s the probability that a number chosen from a uniform distribution between 0 and infinity is less than 1? The answer isn’t even 0, it’s undefined!) However, you *can* select elements with well-defined probabilities if you use a non-uniform distribution. As Scott argues, since picking a distribution is mandatory, the most natural option is something based on simplicity. I think this avoids the epicycles problem more elegantly than something like God being unlimited goodness—although *goodness* (as opposed to some other trait) is arguably hard to define and picked out over other traits a posteriori, mathematical simplicity is not only something that can be precisely defined, but also something so critical to Tegmark’s theory that it doesn’t make mathematical sense without it.
I think it also addresses skepticism. Universes where everything turns into a pile of jello ten seconds from now would have very complicated laws of physics compared to our current ones. This is double true given special relativity and related issues—I don’t know whether it’s even possible to write down a version of the standard model where a physical constant changes at a certain moment in time, since picking out a single instant would throw Lorentz invariance straight out the window and maybe blow up the entire mathematical framework of quantum field theory. The end result could be surprisingly complicated compared to our known laws of physics, making them exponentially less likely.
>although *goodness* (as opposed to some other trait) is arguably hard to define and picked out over other traits a posteriori, mathematical simplicity is not only something that can be precisely defined, but also something so critical to Tegmark’s theory that it doesn’t make mathematical sense without it.
Yeah, I think it should be stressed that the mathematical universe hypothesis is not just a competitor in simplicity to theism, it overwhelmingly beats it on that score. As you point out, "goodness" is probably not a great candidate for a primitive concept. But even if one is a moral realist who disagrees with that claim, there's still a question left over as to which collection of net-good worlds God instantiates, since the theist here doesn't think he instantiates all of them (because they say that would vitiate induction). Whatever process God uses to pick, it would probably be unspeakably complex, at least in terms of description compressibility. He'll have lots of "multiverse selection algorithms" to choose from, which might compensate somewhat for the complexity of each individual "algorithm," but it doesn't look very good. If I proposed some mathematical theory of physics that depended on (say) some non-principal ultrafilter that could only be constructed via the Axiom of Choice, and the resulting physics depended in some important way on the specific *choice* of ultrafilter (i.e., in a way that the resulting behavior of matter would be non-isomorphic across different choices, whatever "isomorphism" is here), I'd consider that a hideous theory indeed.
You can find contrived encodings of hypotheses according to which non-inductive hypotheses are simpler. This is the whole grue issue. It seems you need to measure the simplicity of a theory's ontology (see here https://jesseclifton.substack.com/p/notes-on-occam-via-solomonoff-vs), not clear how that works on Tegmark's view.
As you're probably aware, the encodings are supposed to be with respect to some ideal language whose basic terms are considered privileged/primitive. This would rule out perverse grue-type hypotheses. Of course, as you're also probably aware, the natural objection is to say that the privileged language in question is arbitrary in contrast to all the more grue-like ones. But as a sympathizer to subjective Bayesianism, I don't think this really needs to be a big deal. I just *do* think some concepts are more appealing than others they're technically interdefinable with and don't feel a big pull to justify it.
I don't think the simplicity of a theory's ontology is a suitable replacement for the computability-theoretic approach (problematic as it may be). Surely you can add tons of epicycles to any theory to preserve that there's only one kind of thing, or a small absolute number of concrete instantiations of that kind of thing. But the epicycles will be bad, and it's not clear how to make sense of an auxiliary hypothesis being an epicycle or why that's a bad thing without appealing to something intuitively similar to minimum description length.
> respect to some ideal language whose basic terms are considered privileged/primitive
Sure, but these basic terms aren't part of Tegmark's theory, which contradicts what I thought was the thrust of the comment I originally responded to, which is that Tegmark + Bayes is enough to get you out of skepticism.
It seems like it would be a benefit of theism relative to Tegmark if it didn't have to say "...because humans just happen to want to use the language with such-and-such primitive terms".
>Sure, but these basic terms aren't part of Tegmark's theory, which contradicts what I thought was the thrust of the comment I originally responded to, which is that Tegmark + Bayes is enough to get you out of skepticism.
AFAIK Tegmark doesn't say anything about the nature of simplicity, at least in this context; Scott is combining the former's theory with his own views about simplicity, not attributing everything to Tegmark. But maybe you're not suggesting otherwise and I'm misunderstanding your comment here.
>It seems like it would be a benefit of theism relative to Tegmark if it didn't have to say "...because humans just happen to want to use the language with such-and-such primitive terms".
Given that the theist does in fact have to say something about the nature of simplicity, since they have to use simplicity considerations in more ordinary practical or scientific explanatory contexts, it's not clear that they can avoid having to ultimately say similar things. So if the atheist has simplicity-as-low-Kolmogorov-complexity or whatever as a general problem (i.e., he has to worry about issues like choice of a privileged reference Turing machine), he's not doing much worse than his theist counterpart.
> 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.
I find this to be a very shaky line of reasoning. We do not appear to exist in a world that is optimal for our flourishing at all - this is the basic problem of evil, which I won't elaborate on since you are familiar with it. So the theist must be committed to the fact that the evil in our world must be necessary to achieve some greater good. Fine. But the same line of reasoning can be applied to induction - in fact it's much easier to conceive of a world where induction is not necessary for flourishing than it is to conceive of a world where the evil that we see is necessary. So when it comes to induction I don't see theism as having any advantage here.
2.2. Moral Facts seems easily addressed by naturalist forms of Moral Realism, like the one defended in Peter Railton's Moral Realism. It's a really popular metaethical view (and not the only one to grapple with this)!
The paper you link for moral knowledge seems to fall apart in the first paragraph.
"We also know of many bad things. We know suffering is bad. And we know that acts like racial discrimination and harming the innocent are morally wrong."
But we don't know those things, or, at least, we didn't for a long time. Racial discrimination was not just 'not wrong', it was accepted as right for most of human history. In fact, I would argue that the fact that we see morals evolve, not just over 1000's of years but within a few generations, would support a naturalistic view of moral knowledge rather than God-provided moral knowledge. They try to draw a distinction between what's "really" right or wrong vs what we "think" is right or wrong, but that assumes that what we currently "think" is right or wrong is what's "really" right or wrong, which a quick glance at history suggests isn't an assumption we should make.
To use your example, the fact that there is so much written trying to figure out morals, much of it conflicting, would suggest that we have a *very* unreliable view of the table in front of us (if it even is a table), and it's only through generations upon generations of testing and shared knowledge that we're able to even approach what it is we're looking at. Previous civilizations thought they were looking at chair, the fools. (Fast-forward 1,000 years: "They thought they were looking at a table? Ha!")
The way the authors describe the naturalistic view of moral knowledge as a happy accident would be analogous to calling human evolution a happy accident, rather than a series of systems where more successful one outcompete less successful ones. I don't think you'd find a naturalist who describes evolution as a happy accident.
I'm willing to accept that there may be a moral knowledge argument for God, but I don't think this paper does a good job defending that idea.
The fact we thought something for a while doesn't mean that it's true. Our views about science have equally changed. And as I explain, the argument works just as much for other kinds of knowledge, rather than just moral.
Even right now, our views are not fully moral. Many people still aren't vegans (including myself, though I would eagerly vote to ban factory farming in any referendum--though sadly many people won't), and many people still support banning child sex dolls/robots.
"There are infinite people" means that there are some people with the property of being infinite. You should be saying there are "infinitely many people."
(This is a substantive nitpick! "There are infinite sets" and "there are infinitely many sets" are very different statements! It's true that there are infinitely many integers, but it is not true that there are infinite integers.)
Also "unsetly" doesn't appear to be a word? Maybe I missed a coinage but you might mean "proper class": a class of things that's too big to be a set.
Quick answer just to point to a reference in case you might be interested. I don't think Scott's article goes in depth enough to decide whether your objections are answered or not, but here's a very detailed presentation of Tegmark's mathematical universe, which goes into many of the details of how it explains the world we see, including all these characteristics of fine tuning and intelligibility.
By the way, as far as unsetly many people etc goes -- it's been a while since I took math, but all sets of discrete objects are by definition only countably infinite.
We seem to live in a world with at most countably infinite people: you can draw a box around each person and say that it has no other people in it. If there are uncountably many people in all of everything, then we'd observe a discrete self with probability zero. The vast, vast, vast majority of people that exist would exist in a non-discrete form where this is impossible.
You've advanced the notion a few times that there must be more than countably infinitely many people/consciousnesses, but as far as I can tell, that either requires a nonstandard notion of consciousnessness within our universe, or else accepting that we live in an infinitely improbable universe.
>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.
From this example, it seems like you agree with the general idea that when the mathematics of anthropic principles (or the principle of indifference, or whatever) fails to deliver a meaningful answer due to pathologies with infinity, it's rationally licit to fall back to other principles, e.g., going with (what you take to be) local physical chance in this case. So, since the math of anthropics in the Tegmark multiverse is similarly intractable, why is it so much worse for Scott to fall back to something like Kolmogorov complexity to decide credences?
In fact, it seems like you have to do this as well. On your view, there are infinitely many people in vastly more complex universes than our own, but you still prefer simpler theories of the natural world over more complex ones (e.g., the motion of planets around the sun being described by general relativity over Ptolemaism with a gazillion epicycles). You don't feel bound at all to use anthropics here, because you can't! And you also can't just chalk this up to God's "protection of induction," because the very thing we're trying to decide in this context is how to induce!
For those of you who may be interested, I made a post doubting the truth of the mathematical universe hypothesis based on our subjective experience of time and emotional valence:
Regarding point 2.3, I think this objection is less serious than you make it out to be. IIRC it's impossible to assign a uniform probability distribution on the set of possible worlds for the same reason it's impossible to do so on a countably infinite set or the real number line so your hand is forced anyways.
In addition--correct me if I'm wrong--it seems that any distribution would converge on giving increasingly complex sets of laws less and less measure, by any metric of complexity (more precisely, notions of complexity defined by the character length/byte size of the set of laws within some specific arbitrary programming language.)
In general, it's not true that the only way to compare the sizes of infinite sets is with cardinality. There are various ways of putting measures on infinite sets that give you the intuitive result that, eg, the size of the set (0, 1/2) is twice the size of the set (0, 1/4). Granted, the set if possible worlds is unstructured enough that there isn't a y obvious or natural way to put a measure on it, but nor is it obvious that there's no good way to do so:
I'm giving Lebesgue measure as an example of a measure that let's us compare sets with the same cardinality, not saying it's the relevant one here. My point is that your claims about cardinality are kinda beside the point. When we're thinking about the probability of selecting a member of some subset of some infinite set in other contexts, relevant questions concern measure, not cardinality, of sets in question. So question for scott is whether there's a way of putting a measure on the set of possible worlds with the features that Scott wants.
I thought he wanted a measure over worlds, after which you can worry about who you are in the world (which is only a problem if there are infinitely many people you might be, which there's no guarantee of).
I'm confused. On most pictures, possible worlds are not arranged in space. If you're imagining a giant multiverse in which each possible world is located in a particular region, then you might put a measure over the worlds that basically looks at how much space they take up. But that is clearly not Scott's picture, or tegmark's. He's thinking of worlds as corresponding to something like abstract mathematical descriptions (eg, laws plus initial conditions) that fix what happens. You can try to put a measure on such descriptions, but the descriptions are not arranged in space; no reason to think it will be anything like a lebesgue measure.
EDIT: See the reply to this comment by Amicus; I think their interpretation of Tegmark is correct. See also the SEP page for structural realism, where Tegmark's view is discussed: https://plato.stanford.edu/entries/structural-realism/
I feel like I don't understand Tegmark's view. It seems like he has to be saying one of two things, and both are really implausible.
(1) The first interpretation is that the universe and its constituents *are mathematical entities*. But this seems obviously false: mathematical entities are abstract (i.e. non-spatial, non-temporal, causally effete) objects, whereas we are obviously concrete objects. Tegmark might say that we were wrong to think of mathematical objects as abstracta, but then his claim ceases to be interesting: he's gained the ability to say the words "we are mathematical entities" with a straight face, but only at the cost of completely redefining the relevant terms.
(2) The second interpretation is that the various concrete objects in the world somehow *depend* on mathematical objects. In other words, our concrete universe exists *because* it can be mathematically modelled in some way. But what exactly is this dependence relation supposed to be? Is it causation (surely not)? Is it grounding? Maybe Tegmark could give up the claim that there is a strict dependence relation here, but then I don't see how the view differs from modal realism: it would just be the claim that all of the possible universes exist, coupled with a specific account of possibility as something like "mathematical consistency."
On either interpretation, it just smells like Lewisian metaphysics (the problems of which it thus inherits) coupled with some implausible claims about abstracta.
Also, regarding interpretation (1): Pruss once suggested that Platonic dualists should consider the view that the objects of our experience are abstracta (he even explicitly connects this to Tegmark's view). But he raised this option only for the sake of remarking on its absurdity! https://alexanderpruss.blogspot.com/2015/01/if-you-going-to-be-platonist-dualist.html
> The first interpretation is that the universe and its constituents *are mathematical entities*. But this seems obviously false: mathematical entities are abstract (i.e. non-spatial, non-temporal, causally effete) objects, whereas we are obviously concrete objects. Tegmark might say that we were wrong to think of mathematical objects as abstracta, but then his claim ceases to be interesting: he's gained the ability to say the words "we are mathematical entities" with a straight face, but only at the cost of completely redefining the relevant terms.
Tegmark's position is that there is no such distinction - a particularly radical form of ontic structural realism.
That's very helpful, thanks! Now that you say that, I see that Tegmark's view is discussed on the SEP page for structural realism.
Your friend made another interesting point in favor of theism that Scott hasn't dealt with: theism is more elegant, and posits a less ugly, disunified picture of fundamental reality, bc it makes it so that there are intrinsic connections between the basic categories of things in the world: consciousness, moral facts, abstract objects, physical things, etc. All these things have reasons for being there given that God exists, because he grounds them; whereas on naturalism the universe is filled with a lot of arbitrary detail that didn't have to be there, but just happens to be. Theism more strongly predicts a world where there are specifically things like consciousness/moral facts/psychophysical harmony/etc., whereas naturalism could be true of a world whether or not those things existed (meaning it expects them less strongly).
Moral facts are facts about the nature of perfection, which is what God is by the "Perfect God" hypothesis's own definition; math facts are ideas entertained within an infinite intellect, explaining why they're necessary and transcend physical examples* (see footnote); consciousness exists because a perfect being would want to realize things of value, and you can't have love/moral growth/virtue/good stories without conscious beings; physical things exist because they are also useful for telling a great story (even if you doubt this, I'm not sure why idealism isn't an appealing proposition; I've always thought consciousness was more likely to be fundamental than physical things anyway).
Also, don't beautiful scientific theories usually get points for being more beautiful than others? Theism deserves credit if so: what could be more beautiful than the idea that perfection exists, and the reason the world exists is to tell the greatest (most beautiful) story ever told?
--- Footnote: About math needing God ---
It seems like if the physical world stopped existing (even on Tegmark's view, right?), it wouldn't stop being true that 2 + 2 = 4. But if 2 + 2 = 4 is true regardless of whether there is a physical world, it's hard to see how mathematical facts can *just exist* (Where? How?), without occupying anything or having any substrate or location.
But it’s very easy to see how they can exist as ideas in a mind. The trouble is, there haven’t always been human minds, and it seems bizarre to think that 2 + 2 = 4 would stop being true at the moment humans stop existing for contingent historical reasons.
If an infinite metaphysically necessary and all knowing mind that created everything else exists, though, it’s very easy to see how abstract (math) objects could exist necessarily--they exist within a necessarily existing divine intellect for ever and ever.
What's at stake in these debates about god's existence? As I understand it, the minimal versions of theism that are supported by philosophical arguments are just deism. Does this have any implications at all for how to live?
(1) The truth of minimal theism is intrinsically interesting, just as e.g. the truth of Platonism would be.
(2) As the other commentor (lumpy boi) notes, having a high credence in theism should raise your priors for the various theistic religions. This is especially true once you realize that if God exists, then you probably have various obligations to him, which you are failing to live up to. This might make you more receptive to the idea that God has provided a means of atoning for your sins.
(3) If minimal theism is true, then various actions became rational for you which would be irrational if theism were false (e.g. praying, asking God to forgive your sins, and so on). The Enlightenment deists who subscribed to "natural religion" often (sometimes?) did these things, so there's precedent for minimal theists acting this way. One might also argue that theism has special moral implications (e.g. Richard Swinburne argues that theism makes suicide intrinsically wrong, since if your life is gift from a loving God, then suicide basically amounts to throwing away a loved one's gift once it has ceased to please you).
I'm with you on it being intrinsically interesting, but I was asking about implications for how to live. You get at that in points (2) and (3), but the kinds of implications you point to don't follow at all from minimal theism.
Why on earth would minimal theism make it likely that you have obligations to this god? Or that there are such a thing as sins? Or that this god cares about praying? The conception of god that seems to be at play in your response is very much that of religion, but there's no reason to assume that the god of minimal theism is anything like that very specific conception. That seems to me like assuming that aliens look specifically like those grey humanoid figures we see in popular fiction.
It doesn't seem to me that accepting in minimal theism should increase your credence in theistic religions, because whenever I've engaged with those religions they've been riddled with internal inconsistencies that make them logically impossible to be true.
Btw, I worry that I'm sounding combative here when that's not my goal at all! I appreciate you taking the time to answer my question.
[any claim about God, including theistic religions or that you have an obligation to God, etc.] => [God exists]
So P([any claim about God, including theistic religions or that you have an obligation to God, etc.] | [God exists]) > P([any claim about God, including theistic religions or that you have an obligation to God, etc.])
So unless you did a really weird update of a shape like "this argument proves that God probably exists AND that we owe God nothing" you gotta increase your credence in [any claim about God, including theistic religions or that you have an obligation to God, etc.] as you increase your credence in [God exists]
Well, even minimal theism says that you were created by a perfectly good and loving being. It seems plausible that even if you knew nothing else about this being, you should still think that you have some obligations to it (e.g. gratitude for having been created), simply in virtue of what it's done for you.
Regarding theistic religions, you should have a credence of much less than 1 that those religions are in-fact internally inconsistent (if only because lots of really smart people who've thought hard about the relevant issues deny that this is so). And so long as your credence in this claim is less than 1, then finding out that an entailment of these religions (in this case, minimal theism) is true should raise your credence in those religions, for the reasons that I.M.J. McInnis lays out.
Also, I'm not sure what you mean when you refer to there being a "very specific conception" of God in the theistic religions. The philosophical traditions of Christianity, Judaism, and Islam all think of God as a perfect or maximally great being. Of course, they make lots of claims about what this being has done in history, but that's not about their model of God per se.
Here's an analogy: suppose you want to know if a particular event in a given cornfield was a UFO crash or not. You then find out (via independent evidence) that there are, in-fact, aliens of exactly the sort which the UFO theory posits concerning the cornfield incident. This should raise your confidence that the incident was, in-fact, a UFO crash. The same goes for e.g. God and the resurrection of Jesus.
Hope that helps!
Analogy: if I told you "I have a brother" and you're not sure whether you should believe me, but you learned that I have a sibling, you should increase your believe that I have a brother
Firstly, if the god proposed by the religion is internally inconsistent such that it cannot exist, then proving a necessary condition for that god shouldn’t increase your credence in it.
Secondly, whilst your probabilistic logic is sound, isn’t it the case that for any given claim about god, proof of god’s existence would equally increase the probability that the opposite claim about god is true? For example, “you have no obligations to god”. So why would the existence of god increase the relative credence you have about any claim about god?
The first is a good clarification! If I told you "I have a flarfnarf, which is a type of sibling both green and colorless," then learning I had a sibling should not increase your credence that I actually have a flarfnarf.
(At first I thought this objection was irrelevant. As they say, "zero and one are not probabilities," and when you think "the god proposed by this religion is logically inconsistent; it has zero probability" you should instead thing "either the god proposed by this religion is logically inconsistent, or something about my reasoning is broken." and you should never have probability 0 on "something about my reasoning is broken." But if I have reasoned "Hinduism in particular is totally contradictory---not for anything about god in general, but about Vishnu in particular"---then believing minimal theism shouldn't make me increase my "something about my reasoning is broken" credence. (Though I do think you should be very very hesitant to say "I am certain modulo Cromwell" about anything about which there's vociferous disagreement!))
For the second point, I think you're misunderstanding the meaning of "opposite." (It's a forgivable mistake, W.v.O. Quine made it too, when he argued for single-sorted logic.)
Let $xOy$ denote "$x$ has obligations to $y$". Let $Gz$ denote "$z$ is God." Let $u$ be the constant symbol denoting you. Suppose that God is unique (if extant), i.e. $\forall x,y(Gx \wedge Gy \implies x=y)$.
"You have obligations to God" is $\exists y ( Gy \wedge uOy)$.
"You do not have obligations to God" can be interpreted as $\not\exists y (Gy \wedge uOy)$, but I dislike this. It munges two things: God doesn't exist, and God exists but you bear no obligations toward God.
The correct partitioning of possibilities is:
God exists, You have obligations to God $\exists y ( Gy \wedge uOy)$.
God exists, You have no obligations to God $\exists y (Gy \wedge \not uOy)$
God does not exist, so it's meaningless to ask whether you have obligations or not $\not \exists y (Gy)$
So eliminating the third option increases credence in the first two options.
The historical sidebar here is: Quine thought you should still be able to say "well, I have no obligations to God because God doesn't exist" just like you should be able to say "the number 17 is not hungry, of course not, it's not a number." But it's provable that the system of single-sorted logic in which you stipulate that 17 is not hungry (nor is anything else that can't be hungry) is Morita equivalent to one in which you stipulate that 17 *is* hungry (and so is everything else that can't be "actively unhungry"). Which is bad! So you have to preserve the "true/false/meaningless" distinction if you want to preserve the "true/false" distinction in a nice way for predicates.
If God exists, he might want something. If he stands behind the moral law, might he want you, personally, to be moral? Will he make any demands of you: and if he does, how could his demands be resisted? A God, even a deistic God, is not a tame lion. He is not safe.
Lewis wrote about this, and how when he became a theist (not a Christian, not yet anyway) he slowly came to realize what the existence of God meant for him personally:
"I distinguished this philosophical “God” very sharply (or so I said) from “the God of popular religion”. There was, I explained, no possibility of being in a personal relation with Him. For I thought He projected us as a dramatist projects his characters, and I could no more “meet” Him, than Hamlet could meet Shakespeare. I didn’t call Him “God” either; I called Him “Spirit”. One fights for one’s remaining comforts.
...
"The real terror was that if you seriously believed in even such a “God” or “Spirit” as I admitted, a wholly new situation developed. As the dry bones shook and came together in that dreadful valley of Ezekiel’s, so now a philosophical theorem, cerebrally entertained, began to stir and heave and throw off its gravecloths, and stood upright
and became a living presence. I was to be allowed to play at philosophy no longer. It might, as I say, still be true that my “Spirit” differed in some way from “the God of popular religion”. My Adversary waived the point. It sank into utter unimportance. He would not argue about it. He only said, “I am the Lord”; “I am that I am”; “I am”.
"People who are naturally religious find difficulty in understanding the horror of such a revelation. Amiable agnostics will talk cheerfully about “man’s search for God”. To me, as I then was, they might as well have talked about the mouse’s search for the cat.
...
"Remember, I had always wanted, above all things, not to be “interfered with”. I had wanted (mad wish) “to call my soul my own”. I had been far more anxious to avoid suffering than to achieve delight. I had always aimed at limited liabilities. The supernatural itself had been to me, first, an illicit dram, and then, as by a drunkard’s reaction, nauseous. Even my recent attempt to live my philosophy had secretly (I now knew) been hedged round by all sorts of reservations. I had pretty well known that my ideal of virtue would never be allowed to lead me into anything intolerably painful; I would be “reasonable”. But now what had been an ideal became a command; and what might not be expected of one? Doubtless, by definition, God was Reason itself. But would He also be “reasonable” in that other, more comfortable, sense? Not the slightest assurance on that score was offered me. Total surrender, the absolute leap in the dark, were demanded. The reality with which no treaty can be made was upon me. The demand was not even “All or nothing”....Now, the demand was simply “All”."
Thanks for taking the time to send this. Can any of these specific things that god might want of you be known? If not, it doesn’t seem decision relevant
With all sincerity, you could ask him. If I was a deist, that would be my first line of inquiry. Prayer is a practice that has been believed in and relied upon for the vast majority of humankind for the majority of human history, and if a God of some type exists he just might answer.
There are many arguments that can get you to theism, but theism is not the same as deism. The deist believes that God exists, but he doesn't do anything anymore. He doesn't answer prayers, give prophets visions, work miracles, etc. A God like that would be impossible to know, for if God does not reach out to us we have no way of reaching to him. Yet deism is just as much a choice as Christianity, or Hinduism, or pantheism, or any of the other places you might land after you had concluded to your satisfaction that a god of some type exists. What reason would you have to prefer deism to the alternatives: except, perhaps, in that it requires the least of us. If the God of deism is real, it would affect our lives very little. The philosophical arguments for God may lead us to believe there is a lion at large: it is comforting to believe he is caged, but do we have reason to do so?
Lewis writes a bit on this in Mere Christianity, though in reference to a general belief in an impersonal "Life Force" which is similar enough to deism for our purposes:
"When you are feeling fit and the sun is shining and you do not want to believe that the whole universe is a mere mechanical dance of atoms, it is nice to be able to think of this great mysterious Force rolling on through the centuries and carrying you on its crest. If, on the other hand, you want to do something rather shabby, the Life-Force, being only a blind force, with no morals and no mind, will never interfere with you like that troublesome God we learned about when we were children. The Life-Force is a sort of tame God. You can switch it on when you want, but it will not bother you. All the thrills of religion and none of the cost."
The concept of God has often been tied to morality: to act as a ground of moral law, a judge, and the source of our conscience. If this is true then it has significant implications for our life as well. Lewis writes a bit on this in Mere Christianity as well, expanding on his own experience in becoming a theist:
“We have not yet got as far as the God of any actual religion, still less the God of that particular religion called Christianity. We have only got as far as a Somebody or Something behind the oral Law. We are not taking anything from the Bible or the Churches, we are trying to see what we can find out about this Somebody on our own steam. And I want to make it quite clear that what we find out on our own steam is something that gives us a shock. We have two bits of evidence about the Somebody. One is the universe He has made. If we used that as our only clue, then I think we should have to conclude that He was a great artist (for the universe is a very beautiful place), but also that He is quite merciless and no friend to man (for the universe is a very dangerous and terrifying place). The other bit of evidence is that Moral Law which He has put into our minds. And this is a better bit of evidence than the other, because it is inside information. You find out more about God from the Moral Law than from the universe in general just as you find out more about a man by listening to his conversation than by looking at a house he has built. Now, from this second bit of evidence we conclude that the Being behind the universe is intensely interested in right conduct — in fair play, unselfishness, courage, good faith, honesty and truthfulness. In that sense we should agree with the account given by Christianity and some other religions, that God is "good." But do not let us go too fast here. The Moral Law does not give us any grounds for thinking that God is "good" in the sense of being indulgent, or soft, or sympathetic. There is nothing indulgent about the Moral Law. It is as hard as nails. It tells you to do the straight thing and it does not seem to care how painful, or dangerous, or difficult it is to do. If God is like the Moral Law, then He is not soft. It is no use, at this stage, saying that what you mean by a "good" God is a God who can forgive. You are going too quickly. Only a Person can forgive. And we have not yet got as far as a personal God — only as far as a power, behind the Moral Law, and more like a mind than it is like anything else. But it may still be very unlike a Person. If it is pure impersonal mind, there may be no sense in asking it to make allowances for you or let you off, just as there is no sense in asking the multiplication table to let you off when you do your sums wrong. You are bound to get the wrong answer. And it is no use either saying that if there is a God of that sort — an impersonal absolute goodness — then you do not like Him and are not going to bother about Him.”
if you think the arguments for theism/deism are good then that increases your priors for the explanations of some miracles. i.e. if atheism is true then it's hard to run swinburne's bayesian arg in the resurrection of god incarnate. if theism is true then explanations like those are salient/more plausible.
I enjoy what you are writing very much as a committed theist (and Catholic), but your use of probabilities on infinite sets oscillates between careless and nonsensical - you are not alone in this issue of course, as Scott has the same problem.
So, as a mathematician, let me give you a basic crash course on probabilities :
So here is the thing : the basic formula of probabilities,
P(A)=card(A)/card(Ω),
only works if the universe set Ω is a finite set.
It doesn't work if Ω is infinite, especially if A is infinite itself. What you need is a probability measure m on your set Ω. A measure is a generalization of the notion of length (invented by Lebesgue for his theory of integration). A probability measure m is simply a measure such that the measure of the whole set Ω is equal to 1.
Then you can simply define the probability of some subset A by
P(A)=m(A) - no need for a division as the denominator, p(Ω), would simply be 1.
On a given set you can define multiple measures. The most standard one on the set of all real numbers is the Lebesgue measure L, which has the nice property that L([a,b])=b-a. You can't directly use it as a probability measure on the set of all real numbers (because L(R)=infinity), but you can use it on the segment [0,1]. But there are many other possible measures. For example the Dirac measure defined by D(A)=1 if 0 is in A and D(A)=0 if 0 is not in A is a perfectly defined probability measure on [0,1] too.
However note that the singleton {0} has measure 1 for the Dirac measure, so P({0})=P([0,1]). A finite set has the same probability as an infinite set !
And if you take the Lebesgue measure, you can find sets that have the same cardinality as [0,1], but have Lebesgue measure 0, and thus probability 0 for the Lebesgue measure.
As a summary : you need to define your probability measure, or any talk about probabilities does not make any sense.
> Every particular person is in the scenario ideal for their flourishing: that probably doesn’t involve being deceived.
The “Probably” dooms your argument. You have to be able to say that the number of people who are in worlds where induction fails is of a *smaller* cardinality then the number of people where induction holds (I think all this discussion of cardiologies of infinity is nonsense, but setting that aside…). There’s no way you can do that. If all the genocides, animal torture, and wild animal suffering in this world could be justified, then surely a world where everything is great but where induction is fake is justifiable. Maybe there are beings that really hate predictability (some humans probably do) That world would be great for them!
If even 1/Graham’s number people are in worlds without induction, then you can make a 1:1 correspondence between people who are in such worlds and people who are not… induction is thus fatally undermined, since we have no way to know whether we are in a world with it or one without.
This is mistaken. The point of the argument is that you *don't* have to be able to say that the non-deceived people are fewer in number than the deceived people: all you have to be able to say is that for each individual person, considered in isolation, it is probable that they would not be deceived.
This is the whole point of the Hilbert's Hotel dice roll example. When trying to figure out what you should expect to roll, you don't look at the *number* of people who get various outcomes, since then you'd get absurd results (e.g. that you're just as likely to roll a 6 as to roll a 1-5). Rather, you consider the individual case in isolation, where you can easily see that, given the relevant objects and properties, you have e.g. a 1/6 chance of rolling 6.
> When trying to figure out what you should expect to role, you don't look at the cardinality, since then you'd get absurd results.
A. I interpreted the dice example as being about whether you think individuals lack induction. But my point was that someone cannot simultaneously believe that induction is a brute fact and that the number of people for which induction fails is equal to the number of people for which induction works.
If I misunderstood that argument…
B. The dice rolling example is also misapplied. Your situation is not of infinitely many people randomly choosing among a finite list of outcomes. It’s infinite people choosing one of an infinite list of outcomes. If you were in Hilbert’s hotel and rolled a dice with one face for every world where induction holds and one face for every world where induction fails, your odds would be 50/50.
C. There are “absurd” results everywhere here. All of Matthew’s theodicies are facially absurd. Comparing infinities like this is absurd. Talking about “unsetly” many things like that isn’t a made up gibberish word is… absurd. Not sure why this absurdity is the bridge too far.
(A) I don't understand what you're saying here. Matthew doesn't think induction is "a brute fact" (he thinks that God puts people in induction-friendly circumstances).
(B) Sure, there are infinitely many ways *to be deceived*, but there's basically only one way to be non-deceived (i.e. if your faculties and perceptions are generally reliable), and so all we need to worry about is whether or not that one non-deceived option obtains. I guess one could say that there's lots of ways for your faculties to be generally reliable (e.g. you could have 20/20 vision, or 20/21, 20/22, etc.), but that seems too fine-grained to me, akin to saying that there's lots of ways to roll a 6 (since after all, the die might bounce this way rather than that). All we really care about is the coarse-grained "reliable or not-reliable" issue. Also, even if there were an equal number of deceived and non-deceived possibilities, the theist shouldn't take their odds to be 50/50, since theism predicts that the non-deceived options are more likely.
(C) I don't agree that the theodicies are absurd, but I'm a bit more worried by your seeming opposition to comparing infinities. Is your view that contemporary mathematics is gibberish, since it talks about differently-sized infinities? If so, don't you think it's a bit more likely that you're misunderstanding things than that contemporary mathematics has taken a fundamentally wrong turn (which you, a layman, have managed to diagnose)?
> Matthew doesn't think induction is "a brute fact"
Matthew once deleted me in a live argument with me about this, so I would respectfully disagree. My understanding is that “God does it” is a means to create a unifying explanation for this fact, since discarding it would entail skepticism (and 95% of the basis for Matthew’s theism is a deep rooted belief that we’re not in a skeptical scenario.)
Doesn’t really matter anyways. Maybe he’s changed his position, or was just adopting it for the sake of argument.
> (B) Sure, there are infinitely many ways *to be deceived*, but there's only one way to be non-deceived (i.e. if your faculties and perceptions are generally reliable)
1. Not sure what the implication of this is
2. I view this as a question of people who are being deceived versus not being deceived, not “ways” of those things happening. Unless those are the same thing?
3. There are many ways to not be deceived. The laws of physics could all be fake which hold by pure chance, and we just get very lucky.
4. Wouldn’t this enhance my argument, since the proverbial dice would have many faces saying “deceived”, but only one saying “not deceived”
> even if there were an equal number of deceived and non-deceived possibilities, the theist shouldn't take their odds to be 50/50, since theism predicts that the non-deceived options are more likely.
If the number of people in two situations are equal, then you are equally likely to be in either one. Theism may predict that there are more situations where people are not deceived, but it predicts equal numbers of people in both. Mathew expressly concedes this.
Given that who “you” are is a random selection from the people God creates, it’s equal!
You cannot make these distinctions between infinities of the same cardinality.
> I'm a bit more worried by your seeming opposition to comparing infinities. Is your view that contemporary mathematics is gibberish, since it talks about differently-sized infinities?
1. I don’t think you can do Bayesian probability over different infinities.
2. Even if you could, for probabilities to escape being 50/50, you need to have an infinity of a larger cardinality on one side, in which case it would dominate and make the probability go to 100 or 0. This is what mathematics does when it “compares differently sized infinities”.
3. Practically speaking, is infinity useful at all for modern mathematics? Is there any practical application where you have an uncancelled infinity? I don’t know. If the answer is “no there are not”, then a maybe just throw it out. If there are, then I believe they are all between cardinalities, which still defeats your argument.
(A) But if the explanation is "God did it," then it isn't a brute fact.
(B) You say that "If the number of people in two situations are equal, then you are equally likely to be in either one." But that's false, as the Hilbert Hotel dice example shows us: the number of people who roll 6 is equal to the number of people who roll 1-5. But clearly the probability that you will roll 1-5 is greater (five times greater, to be precise) than the probability that you will roll 6.
Also, no, your argument would not be enhanced, since the theist's claim isn't that there are more *ways* of being non-deceived: the claim is that God puts his thumb on the proverbial scale. Think of it like this: it's not that God adds more sides to the die; rather, he uses a loaded die, which is loaded towards non-deceptive possibilities :)
(C) Yes, infinity is of extreme (even fundamental) importance to much modern mathematics. We can also just prove that there greater and smaller infinities. Cantor did this for the real and natural numbers back in 1874. I think you'll find this helpful: https://en.wikipedia.org/wiki/Cantor's_diagonal_argument
Based on your response to point C, I am forced to conclude that you are either not actually reading my comments or are trolling. I’m not going to engage any further.
I always love to see some good back and forth! That said, I think there’s a couple of features of the simplicity-based arguments that you’re underrating.
Regarding 2.3, one of Scott’s points is that a metric of simplicity isn’t something that just gets tacked onto Tegmark’s theory—it’s an unavoidable part of it. You can’t construct a uniform probability distribution over a countably infinite set, and this situation is analogous. (What’s the probability that a number chosen from a uniform distribution between 0 and infinity is less than 1? The answer isn’t even 0, it’s undefined!) However, you *can* select elements with well-defined probabilities if you use a non-uniform distribution. As Scott argues, since picking a distribution is mandatory, the most natural option is something based on simplicity. I think this avoids the epicycles problem more elegantly than something like God being unlimited goodness—although *goodness* (as opposed to some other trait) is arguably hard to define and picked out over other traits a posteriori, mathematical simplicity is not only something that can be precisely defined, but also something so critical to Tegmark’s theory that it doesn’t make mathematical sense without it.
I think it also addresses skepticism. Universes where everything turns into a pile of jello ten seconds from now would have very complicated laws of physics compared to our current ones. This is double true given special relativity and related issues—I don’t know whether it’s even possible to write down a version of the standard model where a physical constant changes at a certain moment in time, since picking out a single instant would throw Lorentz invariance straight out the window and maybe blow up the entire mathematical framework of quantum field theory. The end result could be surprisingly complicated compared to our known laws of physics, making them exponentially less likely.
>although *goodness* (as opposed to some other trait) is arguably hard to define and picked out over other traits a posteriori, mathematical simplicity is not only something that can be precisely defined, but also something so critical to Tegmark’s theory that it doesn’t make mathematical sense without it.
Yeah, I think it should be stressed that the mathematical universe hypothesis is not just a competitor in simplicity to theism, it overwhelmingly beats it on that score. As you point out, "goodness" is probably not a great candidate for a primitive concept. But even if one is a moral realist who disagrees with that claim, there's still a question left over as to which collection of net-good worlds God instantiates, since the theist here doesn't think he instantiates all of them (because they say that would vitiate induction). Whatever process God uses to pick, it would probably be unspeakably complex, at least in terms of description compressibility. He'll have lots of "multiverse selection algorithms" to choose from, which might compensate somewhat for the complexity of each individual "algorithm," but it doesn't look very good. If I proposed some mathematical theory of physics that depended on (say) some non-principal ultrafilter that could only be constructed via the Axiom of Choice, and the resulting physics depended in some important way on the specific *choice* of ultrafilter (i.e., in a way that the resulting behavior of matter would be non-isomorphic across different choices, whatever "isomorphism" is here), I'd consider that a hideous theory indeed.
You can find contrived encodings of hypotheses according to which non-inductive hypotheses are simpler. This is the whole grue issue. It seems you need to measure the simplicity of a theory's ontology (see here https://jesseclifton.substack.com/p/notes-on-occam-via-solomonoff-vs), not clear how that works on Tegmark's view.
As you're probably aware, the encodings are supposed to be with respect to some ideal language whose basic terms are considered privileged/primitive. This would rule out perverse grue-type hypotheses. Of course, as you're also probably aware, the natural objection is to say that the privileged language in question is arbitrary in contrast to all the more grue-like ones. But as a sympathizer to subjective Bayesianism, I don't think this really needs to be a big deal. I just *do* think some concepts are more appealing than others they're technically interdefinable with and don't feel a big pull to justify it.
I don't think the simplicity of a theory's ontology is a suitable replacement for the computability-theoretic approach (problematic as it may be). Surely you can add tons of epicycles to any theory to preserve that there's only one kind of thing, or a small absolute number of concrete instantiations of that kind of thing. But the epicycles will be bad, and it's not clear how to make sense of an auxiliary hypothesis being an epicycle or why that's a bad thing without appealing to something intuitively similar to minimum description length.
> respect to some ideal language whose basic terms are considered privileged/primitive
Sure, but these basic terms aren't part of Tegmark's theory, which contradicts what I thought was the thrust of the comment I originally responded to, which is that Tegmark + Bayes is enough to get you out of skepticism.
It seems like it would be a benefit of theism relative to Tegmark if it didn't have to say "...because humans just happen to want to use the language with such-and-such primitive terms".
>Sure, but these basic terms aren't part of Tegmark's theory, which contradicts what I thought was the thrust of the comment I originally responded to, which is that Tegmark + Bayes is enough to get you out of skepticism.
AFAIK Tegmark doesn't say anything about the nature of simplicity, at least in this context; Scott is combining the former's theory with his own views about simplicity, not attributing everything to Tegmark. But maybe you're not suggesting otherwise and I'm misunderstanding your comment here.
>It seems like it would be a benefit of theism relative to Tegmark if it didn't have to say "...because humans just happen to want to use the language with such-and-such primitive terms".
Given that the theist does in fact have to say something about the nature of simplicity, since they have to use simplicity considerations in more ordinary practical or scientific explanatory contexts, it's not clear that they can avoid having to ultimately say similar things. So if the atheist has simplicity-as-low-Kolmogorov-complexity or whatever as a general problem (i.e., he has to worry about issues like choice of a privileged reference Turing machine), he's not doing much worse than his theist counterpart.
> 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.
I find this to be a very shaky line of reasoning. We do not appear to exist in a world that is optimal for our flourishing at all - this is the basic problem of evil, which I won't elaborate on since you are familiar with it. So the theist must be committed to the fact that the evil in our world must be necessary to achieve some greater good. Fine. But the same line of reasoning can be applied to induction - in fact it's much easier to conceive of a world where induction is not necessary for flourishing than it is to conceive of a world where the evil that we see is necessary. So when it comes to induction I don't see theism as having any advantage here.
2.2. Moral Facts seems easily addressed by naturalist forms of Moral Realism, like the one defended in Peter Railton's Moral Realism. It's a really popular metaethical view (and not the only one to grapple with this)!
The paper you link for moral knowledge seems to fall apart in the first paragraph.
"We also know of many bad things. We know suffering is bad. And we know that acts like racial discrimination and harming the innocent are morally wrong."
But we don't know those things, or, at least, we didn't for a long time. Racial discrimination was not just 'not wrong', it was accepted as right for most of human history. In fact, I would argue that the fact that we see morals evolve, not just over 1000's of years but within a few generations, would support a naturalistic view of moral knowledge rather than God-provided moral knowledge. They try to draw a distinction between what's "really" right or wrong vs what we "think" is right or wrong, but that assumes that what we currently "think" is right or wrong is what's "really" right or wrong, which a quick glance at history suggests isn't an assumption we should make.
To use your example, the fact that there is so much written trying to figure out morals, much of it conflicting, would suggest that we have a *very* unreliable view of the table in front of us (if it even is a table), and it's only through generations upon generations of testing and shared knowledge that we're able to even approach what it is we're looking at. Previous civilizations thought they were looking at chair, the fools. (Fast-forward 1,000 years: "They thought they were looking at a table? Ha!")
The way the authors describe the naturalistic view of moral knowledge as a happy accident would be analogous to calling human evolution a happy accident, rather than a series of systems where more successful one outcompete less successful ones. I don't think you'd find a naturalist who describes evolution as a happy accident.
I'm willing to accept that there may be a moral knowledge argument for God, but I don't think this paper does a good job defending that idea.
The fact we thought something for a while doesn't mean that it's true. Our views about science have equally changed. And as I explain, the argument works just as much for other kinds of knowledge, rather than just moral.
Even right now, our views are not fully moral. Many people still aren't vegans (including myself, though I would eagerly vote to ban factory farming in any referendum--though sadly many people won't), and many people still support banning child sex dolls/robots.
"There are infinite people" means that there are some people with the property of being infinite. You should be saying there are "infinitely many people."
(This is a substantive nitpick! "There are infinite sets" and "there are infinitely many sets" are very different statements! It's true that there are infinitely many integers, but it is not true that there are infinite integers.)
Also "unsetly" doesn't appear to be a word? Maybe I missed a coinage but you might mean "proper class": a class of things that's too big to be a set.
Quick answer just to point to a reference in case you might be interested. I don't think Scott's article goes in depth enough to decide whether your objections are answered or not, but here's a very detailed presentation of Tegmark's mathematical universe, which goes into many of the details of how it explains the world we see, including all these characteristics of fine tuning and intelligibility.
Here's the link, in case you want to have a read: https://alwaysasking.com/why-does-anything-exist/
By the way, as far as unsetly many people etc goes -- it's been a while since I took math, but all sets of discrete objects are by definition only countably infinite.
For instance, https://math.stackexchange.com/questions/4172014/why-discrete-set-must-be-countable
We seem to live in a world with at most countably infinite people: you can draw a box around each person and say that it has no other people in it. If there are uncountably many people in all of everything, then we'd observe a discrete self with probability zero. The vast, vast, vast majority of people that exist would exist in a non-discrete form where this is impossible.
You've advanced the notion a few times that there must be more than countably infinitely many people/consciousnesses, but as far as I can tell, that either requires a nonstandard notion of consciousnessness within our universe, or else accepting that we live in an infinitely improbable universe.
>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.
From this example, it seems like you agree with the general idea that when the mathematics of anthropic principles (or the principle of indifference, or whatever) fails to deliver a meaningful answer due to pathologies with infinity, it's rationally licit to fall back to other principles, e.g., going with (what you take to be) local physical chance in this case. So, since the math of anthropics in the Tegmark multiverse is similarly intractable, why is it so much worse for Scott to fall back to something like Kolmogorov complexity to decide credences?
In fact, it seems like you have to do this as well. On your view, there are infinitely many people in vastly more complex universes than our own, but you still prefer simpler theories of the natural world over more complex ones (e.g., the motion of planets around the sun being described by general relativity over Ptolemaism with a gazillion epicycles). You don't feel bound at all to use anthropics here, because you can't! And you also can't just chalk this up to God's "protection of induction," because the very thing we're trying to decide in this context is how to induce!
For those of you who may be interested, I made a post doubting the truth of the mathematical universe hypothesis based on our subjective experience of time and emotional valence:
https://perkeleperusing.substack.com/p/several-arguments-against-the-mathematical
Regarding point 2.3, I think this objection is less serious than you make it out to be. IIRC it's impossible to assign a uniform probability distribution on the set of possible worlds for the same reason it's impossible to do so on a countably infinite set or the real number line so your hand is forced anyways.
In addition--correct me if I'm wrong--it seems that any distribution would converge on giving increasingly complex sets of laws less and less measure, by any metric of complexity (more precisely, notions of complexity defined by the character length/byte size of the set of laws within some specific arbitrary programming language.)
In general, it's not true that the only way to compare the sizes of infinite sets is with cardinality. There are various ways of putting measures on infinite sets that give you the intuitive result that, eg, the size of the set (0, 1/2) is twice the size of the set (0, 1/4). Granted, the set if possible worlds is unstructured enough that there isn't a y obvious or natural way to put a measure on it, but nor is it obvious that there's no good way to do so:
https://en.m.wikipedia.org/wiki/Lebesgue_measure
Right but those will be changed by moving people around.
I'm giving Lebesgue measure as an example of a measure that let's us compare sets with the same cardinality, not saying it's the relevant one here. My point is that your claims about cardinality are kinda beside the point. When we're thinking about the probability of selecting a member of some subset of some infinite set in other contexts, relevant questions concern measure, not cardinality, of sets in question. So question for scott is whether there's a way of putting a measure on the set of possible worlds with the features that Scott wants.
But if you’re doing a measure over people it will be a function of their arrangement
I thought he wanted a measure over worlds, after which you can worry about who you are in the world (which is only a problem if there are infinitely many people you might be, which there's no guarantee of).
But any way to take a measure over worlds will be subject to their arrangements.
I'm confused. On most pictures, possible worlds are not arranged in space. If you're imagining a giant multiverse in which each possible world is located in a particular region, then you might put a measure over the worlds that basically looks at how much space they take up. But that is clearly not Scott's picture, or tegmark's. He's thinking of worlds as corresponding to something like abstract mathematical descriptions (eg, laws plus initial conditions) that fix what happens. You can try to put a measure on such descriptions, but the descriptions are not arranged in space; no reason to think it will be anything like a lebesgue measure.