If you hear a convincing argument, you should update your belief in the direction of the belief the argument argues for. If you update in the other direction (“come out stronger”), then either it’s not a convincing argument (by definition), or you’re doing it wrong.
I didn’t mean that your initial beliefs should come out stronger. I meant that having updated for good arguments, and by incorporating them, your beliefs will be more complete, better thought-out, and more sustainable for the future.
Well, one example of such a thing might be the Simulation Argument, which I believe has been mentioned to you. It’s an argument for the possible existence of something which might be called a “god” or “gods” (though that’s usually inadvisable due to semantic baggage). Our view of what exists and what could exist certainly incorporates an understanding of the possibility that we’re living in a simulation.
Theistic arguments per se, however, are generally bad.
The Simulation Argument is certainly quite an interesting one, since it was invented by an atheist (Nick Bostrom), and as far as I can tell is only taken remotely seriously by other atheists. Many of them (including me) think it is a rather better argument for some sort of “god” or “gods” than anything theists themselves ever came up with.
For other interesting quasi-theistic arguments invented by atheists, you might want to consider Tegmark’s Level 4 multiverse. Since any “god” which is logically possible can be represented by some sort of mathematical structure, it exists somewhere within the Level 4 multiverse. David Lewis’ modal realism has a similar feature.
All these arguments tend to produce massively polytheistic rather than monotheistic conclusions (and also they imply that Santa, the Tooth Fairy, Harry Potter and Captain Kirk exist somewhere or in some simulation or other).
If you want a fun monotheistic argument invented by atheists, try this one, which was published by Robert Meyer and attributed to Hilary Putnam. It’s a clever use of the Axiom of Choice and Zorn’s Lemma.
If you want a fun monotheistic argument invented by atheists, try this one, which was published by Robert Meyer and attributed to Hilary Putnam. It’s a clever use of the Axiom of Choice and Zorn’s Lemma.
Isn’t that just the First Cause argument, wrapped up in set-theory language?
Well yes, but it “addresses” one of the really basic responses to the First Cause argument, that there might—for all we know—be an infinite chain of causes of causes, extending infinitely far into the past. One of the premises of Meyer’s argument is that any such chain itself has a cause (i.e. something supporting the whole chain). That cause might in turn have a cause and so on. However, by an application of Zorn’s Lemma you can show that there must be an uncaused cause somewhere in the system.
If you don’t assume the Axiom of Choice you don’t have Zorn’s Lemma, so the argument doesn’t work. Conversely, if God exists, then—being omnipotent—he can pick one element from every non-empty set in any collection of sets, which is the Axiom of Choice. So God is logically equivalent to the Axiom of Choice,
He also defines away the causal-loop, or time travel, response, leaving only the uncaused cause; and then arbitrarily defines any uncaused cause as God. It looks like a good argument on the surface, but when I look at it carefully it’s not so great; it’s basically defining away any possible disagreement.
I should also mention that it’s not really a monotheistic argument. It only argues for the existence of at least one God. It doesn’t argue for the non-existence of fifty million more.
It’s reasonably fun as a tongue-in-cheek argument, but I wouldn’t want to use it seriously.
He also defines away the causal-loop, or time travel, response, leaving only the uncaused cause
Well I think premise 2 just assumes there aren’t any causal loops, since if there were, the constructed relation ⇐ would not be a partial order (let alone an inductive order).
There are probably ways of patching that if you want to explicitly consider loops. Consider that if A causes B cause C causes A, then there is some infinite sequence whereby every entry in the sequence is caused by the next entry in the sequence. So this looks a bit like an infinite descending chain.
The arguer could then tweak premise 2 so it states that any such generalised infinite chain (one allowing repeated elements) still has a lower bound (some strict cause outside the whole chain) and apply an adapted version of Zorn’s Lemma to still get an uncaused cause in the whole system.
The intuition being used there is still that any infinite sequence of causes of causes must have some explanation for why the whole sequence exists at all. For instance if there is an infinite sequence of horses, each of which arises from parent horses, we still want an explanation for why there are any horses at all (and not unicorns, say). Even if a pregnant horse if sent back in time to become the ancestor of all horses, then again we still want an explanation for why there are any horses at all.
The weakness of the intuition is that the “explanation” in such a weird case might well not be a causal one, so maybe there is no further cause outside the chain, or loop. (But even then, there is a patch: the arguer could claim that the whole chain or loop should count as a combined “entity” with no cause, ie there is still some sort of uncaused cause in the system).
I agree with you that the really weak part is just defining the uncaused cause to be “God”. Apart from confusing people, why do that?
And thanks for spotting the non-uniqueness by the way… the argument as it stands does allow for multiple uncaused causes. To patch that, the arguer could perhaps define a super-entity which contains all these uncaused causes as its “parts”. Or else add an additional “common cause” premise, whereby for any two entities a, b, either a is a cause of b, or b is a cause of a, or there is some c which is a cause of both of them.
The arguer could then tweak premise 2 so it states that any such generalised infinite chain (one allowing repeated elements) still has a lower bound (some strict cause outside the whole chain) and apply an adapted version of Zorn’s Lemma to still get an uncaused cause in the whole system.
That’s just assuming the result you want. I don’t think it makes a strong argument.
(But even then, there is a patch: the arguer could claim that the whole chain or loop should count as a combined “entity” with no cause, ie there is still some sort of uncaused cause in the system).
Counting a loop as a combined entity, on the other hand, could be very useful. The combined-entity loop would be caused by everything that causes any element in the loop, and would cause anything that is caused by any element in the loop. Do this to all loops, and the end result will be to eliminate loops (at the cost of having a few extremely complex entities).
This seems fine as long as there are only a few, causally independent loops. However, if there are multiple loops that affect each other (e.g. something in loop A causes something in loop B, and something in loop B causes something in loop A) then this simply results in a different set of loops. These loops, of course, can also be combined into a single entity; but if the causality graph is sufficiently well connected, and if there is a large enough loop, the end result of this process might be that all entities end up folding into one giant super-entity, containing and consisting of everything that ever happens.
I have heard the theory before that the universe is a part of God, backed by a different argument.
I agree with you that the really weak part is just defining the uncaused cause to be “God”. Apart from confusing people, why do that?
It honestly looks like a case of writing down the conclusion at the bottom of the page and then back-filling the reasoning. He can’t justify that part, so he defines it quickly and hopes no-one pays too much attention to that line.
And thanks for spotting the non-uniqueness by the way… the argument as it stands does allow for multiple uncaused causes. To patch that
Why do you want to patch that? A quick patch looks like (again) writing the conclusion first and then filling in the reasoning afterwards.
OK, I think we both agree this is not at all a strong argument, that the bottom line is being written first, and then the premises are being chosen to get to that bottom line and so on. However, I still think it is fun to examine and play with the argument structure.
Basically, what we have here is a recipe:
Take some intuitions.
Encode them in some formal premises.
Stir with some fancy set theory.
Extract the desired conclusion : namely that there is an “uncaused cause”
It’s certainly interesting to see how weak you can make the ingredients (in step 1) before the recipe fails. Also, the process of then translating them into premises (step 2) looks interesting, as at least it helps decide whether the intuitions were even coherent in the first place. Finally, if the desired conclusion wasn’t quite strong enough for the arguer’s taste (hmm, missing that true monotheistic kick), it’s fun to work out what extra ingredient should be inserted in to the mix (let’s put in a bit of paprika)
That’s basically where I’m coming from in all this..
Ah… I think I get it. You want to play with intuitions, and see which premises would have to be proved in order to end up with monotheism via set theory.
I don’t think it would be possible to get around the point of defining God in terms of set theory. Once you have a definition, you can see if it turns up; if God is not defined, then you don’t know what you’re looking for. Looked at from that point of view, the definition of God as a first cause is probably one of the better options.
Loops can still be a problem...
The arguer could then tweak premise 2 so it states that any such generalised infinite chain (one allowing repeated elements) still has a lower bound (some strict cause outside the whole chain) and apply an adapted version of Zorn’s Lemma to still get an uncaused cause in the whole system.
This can still fail in the case where two loops have their external causes in each other. (I think. Or would that simply translate into an alternate set of loops? …I think I could figure out a set of looped entities, such that each loop has at lest one cause outside that loop, that has no first cause).
To patch that, the arguer could perhaps define a super-entity which contains all these uncaused causes as its “parts”. Or else add an additional “common cause” premise, whereby for any two entities a, b, either a is a cause of b, or b is a cause of a, or there is some c which is a cause of both of them.
Either of those would be sufficient; though the first seems to fit more possible sets.
This can still fail in the case where two loops have their external causes in each other. (I think. Or would that simply translate into an alternate set of loops? …I think I could figure out a set of looped entities, such that each loop has at lest one cause outside that loop, that has no first cause)
I think if two loops were caused by each other, then there would be a super-loop which included all the elements from both of them, and then you could look for the cause of the super-loop. The Axiom of Choice would still be needed to show that this process stops somewhere.
Finally, I rather liked your thought that causality may be so loopy that everything is a cause of everything else. The only way to get a first cause out of that mess is to treat the entire “super-duper-loop” of all things as a single uncaused entity, and if you insist on calling that “God”, you’re a pantheist.
Let’s consider loops A->B->C->A->B->C and D->E->F->D->E->F.
Let’s say, further, that B is a cause of E and D is a cause of A. Then each loop has an external cause.
Then there are also a few other loops possible:
A->B->E->F->D->A->B->E->F->D (external cause: C)
A->B->E->F->D->A->B->C->A->B->E->F->D… huh. That includes all of them, in a sort of double-loop with no external cause. I guess that would be the super-loop.
Finally, I rather liked your thought that causality may be so loopy that everything is a cause of everything else. The only way to get a first cause out of that mess is to treat the entire “super-duper-loop” of all things as a single uncaused entity, and if you insist on calling that “God”, you’re a pantheist.
Better yet; no matter what causality looks like, you can still always combine everything into a single giant, uncaused entity. You don’t need to assume away loops or infinite chains without external causes if you do that.
I’ve been doing a bit more “stir in fancy set theory” over the weekend, and believe I have an improved recipe! This builds on the idea to treat chains and loops as a single “entity” and look for a cause of that entity. It is a lot subtler than just throwing every entity together into one super-duper-entity.
Here are a bunch of premises that I think will do the trick:
A1. The collection of all entities is a set E, with two relations C and P on E, such that: x C y if and only if x is a cause of y; x P y if and only if x is a part of y.
Note: This ensures we can apply Zorn’s Lemma when considering chains in E, but is not as strong as the full Axiom of Choice. If the set E is finite or countable, for instance, then A2 applies automatically.
A3. If x C y and x P z then z C y.
Informally, “anything caused by a part is caused by the whole”.
Definitions: We define ⇐ such that x ⇐ y if and only if x = y or there are finitely many entities x1, …, xn such that x1 = x, xn = y and xi is a cause of xi+1 for i=1.. n-1. Say that a set S is a “chain” in E iff for any x, y in S we have x ⇐ y or y ⇐ x. Say that such an S is an “endless chain” iff for any x in S there is some y not equal to x in S with y ⇐ x. Say that an entity y is “uncaused” if and only if there is no z distinct from y with z ⇐ y. Also say that x is a “proper part” of y iff x is not equal to y but x P y.
Note: These definitions ensure that ⇐ is a pre-order on E. Note that an endless chain may be an infinite chain of distinct elements, or a causal loop.
A4. Let S be any endless chain in E. Then there is some z in E such that every x in S is a proper part of z.
Lemma 1: For any chain S in E, there is an element x of E with x ⇐ y for every y in S.
Proof: Suppose S has an end (not endless). Then there is some x in S such that for no other y in S is y ⇐ x. By the chain property we must have x ⇐ y for every member y of S. Alternatively, suppose that S is endless, then by A4, there is some z in E such that every x in S is a part of z. Now consider any y in S. There is some x not equal to y in S with x ⇐ y, so there are x = x1… xn = y with each xi C xi+1 for i=1..n-1. Further, by A3, as x C x2, we have z C x2 and hence z ⇐ y.
Lemma 2: For any x in E, there is some y in E such that: y ⇐ x, and for any z ⇐ y, y ⇐ z.
Theorem 3: For any x in E, there is some uncaused y in E such that y ⇐ x.
Proof: Take a y as given by Lemma 2 and consider the set S = {s: s ⇐ y}. By Lemma 2, y ⇐ s for every member of S, and if S has more than one element, then S is an endless chain. So by A4 there is some z of which every s in S is a proper part, which implies that z is not in S. But by the proof of Lemma 1, z ⇐ y, which implies z is in S: a contradiction. So it follows that S = {y}, which completes the proof.
I’ve also got some premises for aggregating multiple uncaused entities into a single entity. This gives another approach to “uniqueness”. More on my next comment, if you’re interested.
For uniqueness, we build on the idea of all uncaused causes being part of a whole. The following premises look interesting here:
B1. If x P y and y P z then x P z; x = y if and only if x P y and y P x.
This states that P is a partial order, which is reasonable for the “part of” relation.
B2. If S is any chain of parts, such that for any x, y in S we have x P y or y P x, then there is some z in E of which all members of S are parts.
This states that E is inductively ordered by the “part of” relation.
B3. If x C z and y P z then x C y.
Informally, “a cause of the whole is a cause of any part”.
B4. Suppose that y ⇐ x and z ⇐ x and both y, z are uncaused. Then y P z or z P y, or there is some w of which both y and z are proper parts.
Informally, two uncaused y and z can’t independently conspire to cause x unless they are parts of a common entity.
Definition: Say that entities x and y are causally-connected if and only if x = y, or there are entities x=x1,..,xn=y with either xi C xi+1 or xi+1 C xi for each i=1..n-1.
B5. Any two entities in E are causally-connected.
Informally, E doesn’t “come apart” into completely disconnected components, such as a bunch of isolated universes.
Theorem 4: For any x in E, there is a unique entity f(x) in E such that: f(x) is uncaused, f(x) ⇐ x, and any other uncaused y with y ⇐ x satisfies y P f(x).
Proof: For any x, define a subset E’ = {y in E: y ⇐ x, y is uncaused}. Consider any chain of parts S in E’ with at least two elements. By B2 there is some z in E of which all members of S are parts. By B3, z must be uncaused (or else some w C z would also be a cause of all the members of S, which would require them all to be equal to w, so S would be a singleton), and by A3, z ⇐ x. So z is also a member of E’. By application of Zorn’s Lemma to E’, there is a P-maximal element f in E’ such that there is no other y in E’ with f P y. But then, by B4, for any y in E’ we must have y P f; this makes f unique.
Theorem 5: For any x, y in E, f(x) = f(y) if and only if x and y are causally-connected.
Proof: It is clear that if f(x) = f(y) then x is causally-connected to y (just build a path backwards from x to f(x) and then forward again to y). Conversely, suppose that x C y, then f(x) is uncaused and satisfies f(x) ⇐ y so we have f(x) P f(y). This implies f(x) = f(y). By a simple induction on n we have that if x is causally-connected to y, then f(x) = f(y).
Corollary 6: There is a single entity g in E such that f(x) = g for every entity x in E.
(Huh. One of the ancestors to this comment—several levels up—has been downvoted enough to require a karma penalty. I wonder if there should be some statute of limitations on that; whether, say, ten levels of positive-karma posts can protect against a higher-level negative-karma post?)
A4. Let S be any endless chain in E. Then there is some z in E such that every x in S is a proper part of z.
An interesting assumption. Necessary for theorem 3, but I suspect that it’ll mean that the original cause described in theorem 3 will then very probably be an entity z that is the earliest cause.
I also note that, while z consists of all the parts in the endless chain, there is no guarantee that any of the elements in the chain, even those that cause other elements in the chain, is in any way a cause of z. In fact, the way that z is defined, z may well be causeless (or, then again, z may have a cause). While I can’t actually find anything technically invalid in theorem 3, or in assumption 4, I get the general feeling of wool being pulled over my eyes in some way.
When I consider B3, it becomes even more important to note that z as a whole is not necessarily caused by any element that is a proper part of z. The cause of a part may or may not be the cause of the whole.
Hmmm… B4 appears to be pretty much just shoehorning monotheism in. It seems a questionable assumption; if I decide to get into my car and drive, and you decide to get into your car and drive, and we drive into each other, then we both are causes of the resultant accident but we are not the same. (We are not causeless, either, so it’s not quite a counterexample, just an explanation of why I don’t think B4 is justified,) B5 is unsupported, but I can prove that all entities that I will ever observe evidence of are causally connected (i.e. they are connected to the effects on my actions of having observed them) so it will look true whether it is or not.
Though I can raise questions about your assumptions, I can’t find anything wrong with your logic from then on. So congratulations; you have a very convincing argument! …as long as you can persuade the other person to accept your assumptions, of course.
Ah… I think I get it. You want to play with intuitions, and see which premises would have to be proved in order to >end up with monotheism via set theory.
I don’t think it would be possible to get around the point of defining God in terms of set theory.
Well now, here’s a devious approach, which would probably appeal to me if I ever needed to make a career as a philosopher of religion.
Let’s suppose a theist wants to “prove” that God—by his favourite definition—exists. For instance he could define a type G, whereby an entity g is of type G if and only if g is omnipotent, omniscient, perfectly good and so on, and has all those characteristics essentially and necessarily. Something like that. Then the theist finds a set of premises P, with some intuitive support, such that P ⇒ There is an uncaused cause.
And then he adds one other premise “Every entity that is not of type G has a cause” into the recipe to form a new set P’. He cranks the handle, and then P’ ⇒ There is an entity of type G. Job done!
Just in case someone accuses him of “begging the question” or “assuming what he set out to prove” he then pulls out the modal trick. He just claims that it is possible that P’ is true. This leads to the conclusion that “It is possible that there is an entity of type G”. And then, remembering he’s defined G so it includes necessary existence (if such a being is possible at all, it must exist), he can still conclude
“There is a being of type G”. Job done even better!
The modal trick reminds me of Descarte’s approach… God is definitionally perfectly good, which implies existence (since something good that doesn’t exist isn’t as good as something good that does), therefore God exists.
Huh. That modal trick is devious. But it doesn’t work. I can assume an entity that does something easily measurable (e.g. gives Christmas present to children worldwide), and then slap on a necessary existence clause; but that doesn’t necessarily mean that I can expect Santa later this year.
I think the ‘necessary existence’ clause requires a better justification in orderto be Templeton-worthy.
On reflection, the fact that an atheist would be able to come up with an argument for a god that’s more persuasive to atheists is unsurprising, especially when you consider the fact that most religious people don’t become religious via being persuaded by arguments. It’s definitely still amusing, though.
I’m definitely aware of Tegmark’s theory, though I admit I hadn’t considered it as an argument for any kind of theism. That seems like an awfully parochial and boring application of the ultimate ensemble, although you’re right that it can have that sort of application… although, if we define “supernatural” entities to mean “ontologically basic mental entities” a la Richard Carrier, would it really be the case that Tegmark’s multiverse implies the existence of such? I’m not sure it does.
Meyer’s argument begins with premises that are hilariously absurd. Defining entities as being able to be causes of themselves? Having “entities” even able to be “causes”? What? And all this without the slightest discussion of what kinds of things an “entity” can even be, or what it means to “exist”? No, this is nonsense.
Meyer’s argument begins with premises that are hilariously absurd. Defining entities as being able to be causes of themselves? Having “entities” even able to be “causes”?
I think this is mostly a presentational issue. The purpose of the argument was to construct a non-strict partial order “<=” out of the causal relation, and that requires x<=x. This is just to enable the application of Zorn’s Lemma.
To avoid the hilarity of things being causes of themselves, we could easily adjust the definition of ⇐ so that “x<=y” if and only if “x=y or x is a cause of y”. Or the argument could be presented using a strict partial order <, under which nothing will be a cause of itself. The argument doesn’t need to analyse “entity” or “exists” since such an analysis is inessential to the premises.
And finally, please remember that the whole thing was not meant to be taken seriously; though rather amusingly, Alexander Pruss (whose site I linked to) apparently has been treating it as a serious argument. Oh dear.
FWIW, the probability I place on the Simulation Argument being true is only a little higher than the probability I place on traditional theistic gods existing. Could be just me, though.
Well, traditional theistic gods tend to be incoherent as well as improbable. (Or one might say, improbable only to the extent that they are coherent, which is not very much.) So, I’m not sure how we’d integrate that into a probability estimate.
God-wise, I’ve never seen any evidence for anything remotely supernatural, and plenty of evidence for natural things. I know that throughout human history, many phenomena traditionally attributed to gods (f.ex. lightning) have later been demonstrated to occur by natural means; the reverse has never happened. These facts, combined with the internal (as well as mutual) inconsistencies inherent in most major religions, serve to drive the probability down into negligibility.
As for the Simulation Argument, once again, I’ve never seen any evidence of it, or any Matrix Lords, etc. Until I do, it’s simply not parsimonious for me to behave as though the argument was true. However, unlike some forms of theism, the Simulation Argument is at least internally consistent. In additions, I’ve seen computers before and I know how they can be used to run simulations, which constitutes a small amount of circumstantial evidence toward the Argument.
EDIT: I should mention that the prior for both claims is already very low, due to their complexity.
Epsilon is not a number, it’s a cop-out. Unless you put a number you are reasonably confident in on your prior, how would you update it in light of potential new evidence?
Well, so far, I have received zero evidence for the existence of either gods or Matrix Lords. This leaves me with, at best, just the original prior. I said “at best”, because some of the observations I’d received could be interpreted as weak evidence against gods (or Matrix Lords), but I’m willing to ignore that for now.
If I’m using some measure of algorithmic complexity for the prior, what values should I arrive at ? Both the gods and the Matrix Lords are intelligent in some general way, which is already pretty complex; probably as complex as we humans are, at the very least. Both of them are supremely powerful, which translates into more complexity. In case of the Matrix Lords, their hardware ought to more complex than our entire Universe (or possibly Multiverse). Some flavors of gods are infinitely powerful, whereas others are “merely” on par with the Matrix Lords.
I could keep listing properties here, but hopefully this is enough for you to decide whether I’m on the right track. Given even the basics that I’d listed above, I find myself hard-pressed to come up with anything other than “epsilon” for my prior.
Theistic arguments per se, however, are generally bad.
Why would we expect there to be good arguments for the wrong answer?
We here at Less Wrong have seen many arguments for the existence of God...All of those arguments are wrong.
Thank you for being unambiguous, this is exactly the sort of thing I wanted to see if this community actually believed. Personally I think it reflects poorly on anyone’s intellectual openness for them to believe the other side literally has no decent arguments.
Then you must believe the same with respect to homeopathic remedies, the flat earth society, and those who believe they can use their spiritual energy in the martial arts. Give us some good arguments for those.
There’s a lot of stuff out there for which it seems to me there is no good argument. I mean really, let’s try to maintain some sense of perspective here. The belief that everyone has a decent argument is, I think, pretty much demonstrably false. You presumably want us to believe that you’re in the same category as people who ought to be taken seriously, but I don’t really see how a belief in God is any more worthy of that than a belief in homeopathic remedies. At least, not based on your argument that all positions ought to be considered to have good arguments. If you’re trying to make a general argument, you’re going to get lumped in with them.
An argument can be “decent” without being right. If you want an example, and can follow it Kurt Godel’s ontological argument looks pretty decent. Consider that:
A) It is a logically valid argument
B) The premises sound fairly plausible (we can on the face of it imagine some sense of a “positive property” which would satisfy the premises)
C) It is not immediately obvious what is wrong with the premises
The wrongness can eventually be seen by carefully inspecting the premises, and checking which would go wrong in a null world (a possible world with no entities at all). Axiom 1 implies that if an impossible property is positive, then so is its negation (since an impossible property logically entails its negation). Axiom 2 says that can’t be true—a property and its negation can’t both be positive. So together these are a coded way of saying that all positive properties are possible properties. And then Axiom 5 (Neccessary existence is a positive property) goes wrong, because necessary existence is not a possible property in the null world. So it is not a positive property. Axiom 5 is inconsistent with Axioms 1 and 2.
There are arguments for the existence of God that are good in the sense that they raise my estimate of the likelihood of the existence of God by a substantial factor.
They aren’t sufficient to raise the odds to an overall appreciable level.
Sometimes, the issues really are cut-and-dried, though. To use a rather trivial example, consider the debate about the shape of the Earth. There are still some people who believe it’s flat. They don’t have any good arguments. We’ve been to space, we know the Earth is round, it’s going to be next to impossible to beat that.
Why would we expect there to be good arguments for the wrong answer?
I meant this as the rhetorical “we”, not “we, Less Wrong”.
And in general, you shouldn’t take me, or any other commenter in particular (even Eliezer), to represent all of Less Wrong. This is a community blog, after all.
Personally I think it reflects poorly on anyone’s intellectual openness for them to believe the other side literally has no decent arguments.
Edit: Sorry, I see that you quoted from that comment, so presumably you did read it. That said, I’m not sure that what I said was clear, given your subsequent comments...
I didn’t mean that your initial beliefs should come out stronger. I meant that having updated for good arguments, and by incorporating them, your beliefs will be more complete, better thought-out, and more sustainable for the future.
That is what many people here have done regarding theism. Seen the best arguments, and decided that they fail utterly. Eliezer quoted above talks about Modern Orthodox Judaism allowing doubt as a ritual, but not doubt as a practice leading to a result. You would have us listen to arguments as ritual, but not actually come to a conclusion that some of them are wrong.
I didn’t mean that your initial beliefs should come out stronger. I meant that having updated for good arguments, and by incorporating them, your beliefs will be more complete, better thought-out, and more sustainable for the future.
Well, one example of such a thing might be the Simulation Argument, which I believe has been mentioned to you. It’s an argument for the possible existence of something which might be called a “god” or “gods” (though that’s usually inadvisable due to semantic baggage). Our view of what exists and what could exist certainly incorporates an understanding of the possibility that we’re living in a simulation.
Theistic arguments per se, however, are generally bad.
The Simulation Argument is certainly quite an interesting one, since it was invented by an atheist (Nick Bostrom), and as far as I can tell is only taken remotely seriously by other atheists. Many of them (including me) think it is a rather better argument for some sort of “god” or “gods” than anything theists themselves ever came up with.
For other interesting quasi-theistic arguments invented by atheists, you might want to consider Tegmark’s Level 4 multiverse. Since any “god” which is logically possible can be represented by some sort of mathematical structure, it exists somewhere within the Level 4 multiverse. David Lewis’ modal realism has a similar feature.
All these arguments tend to produce massively polytheistic rather than monotheistic conclusions (and also they imply that Santa, the Tooth Fairy, Harry Potter and Captain Kirk exist somewhere or in some simulation or other).
If you want a fun monotheistic argument invented by atheists, try this one, which was published by Robert Meyer and attributed to Hilary Putnam. It’s a clever use of the Axiom of Choice and Zorn’s Lemma.
Isn’t that just the First Cause argument, wrapped up in set-theory language?
Well yes, but it “addresses” one of the really basic responses to the First Cause argument, that there might—for all we know—be an infinite chain of causes of causes, extending infinitely far into the past. One of the premises of Meyer’s argument is that any such chain itself has a cause (i.e. something supporting the whole chain). That cause might in turn have a cause and so on. However, by an application of Zorn’s Lemma you can show that there must be an uncaused cause somewhere in the system.
If you don’t assume the Axiom of Choice you don’t have Zorn’s Lemma, so the argument doesn’t work. Conversely, if God exists, then—being omnipotent—he can pick one element from every non-empty set in any collection of sets, which is the Axiom of Choice. So God is logically equivalent to the Axiom of Choice,
All totally tongue-in-cheek and rather fun.
He also defines away the causal-loop, or time travel, response, leaving only the uncaused cause; and then arbitrarily defines any uncaused cause as God. It looks like a good argument on the surface, but when I look at it carefully it’s not so great; it’s basically defining away any possible disagreement.
I should also mention that it’s not really a monotheistic argument. It only argues for the existence of at least one God. It doesn’t argue for the non-existence of fifty million more.
It’s reasonably fun as a tongue-in-cheek argument, but I wouldn’t want to use it seriously.
Well I think premise 2 just assumes there aren’t any causal loops, since if there were, the constructed relation ⇐ would not be a partial order (let alone an inductive order).
There are probably ways of patching that if you want to explicitly consider loops. Consider that if A causes B cause C causes A, then there is some infinite sequence whereby every entry in the sequence is caused by the next entry in the sequence. So this looks a bit like an infinite descending chain.
The arguer could then tweak premise 2 so it states that any such generalised infinite chain (one allowing repeated elements) still has a lower bound (some strict cause outside the whole chain) and apply an adapted version of Zorn’s Lemma to still get an uncaused cause in the whole system.
The intuition being used there is still that any infinite sequence of causes of causes must have some explanation for why the whole sequence exists at all. For instance if there is an infinite sequence of horses, each of which arises from parent horses, we still want an explanation for why there are any horses at all (and not unicorns, say). Even if a pregnant horse if sent back in time to become the ancestor of all horses, then again we still want an explanation for why there are any horses at all.
The weakness of the intuition is that the “explanation” in such a weird case might well not be a causal one, so maybe there is no further cause outside the chain, or loop. (But even then, there is a patch: the arguer could claim that the whole chain or loop should count as a combined “entity” with no cause, ie there is still some sort of uncaused cause in the system).
I agree with you that the really weak part is just defining the uncaused cause to be “God”. Apart from confusing people, why do that?
And thanks for spotting the non-uniqueness by the way… the argument as it stands does allow for multiple uncaused causes. To patch that, the arguer could perhaps define a super-entity which contains all these uncaused causes as its “parts”. Or else add an additional “common cause” premise, whereby for any two entities a, b, either a is a cause of b, or b is a cause of a, or there is some c which is a cause of both of them.
That’s just assuming the result you want. I don’t think it makes a strong argument.
Counting a loop as a combined entity, on the other hand, could be very useful. The combined-entity loop would be caused by everything that causes any element in the loop, and would cause anything that is caused by any element in the loop. Do this to all loops, and the end result will be to eliminate loops (at the cost of having a few extremely complex entities).
This seems fine as long as there are only a few, causally independent loops. However, if there are multiple loops that affect each other (e.g. something in loop A causes something in loop B, and something in loop B causes something in loop A) then this simply results in a different set of loops. These loops, of course, can also be combined into a single entity; but if the causality graph is sufficiently well connected, and if there is a large enough loop, the end result of this process might be that all entities end up folding into one giant super-entity, containing and consisting of everything that ever happens.
I have heard the theory before that the universe is a part of God, backed by a different argument.
It honestly looks like a case of writing down the conclusion at the bottom of the page and then back-filling the reasoning. He can’t justify that part, so he defines it quickly and hopes no-one pays too much attention to that line.
Why do you want to patch that? A quick patch looks like (again) writing the conclusion first and then filling in the reasoning afterwards.
OK, I think we both agree this is not at all a strong argument, that the bottom line is being written first, and then the premises are being chosen to get to that bottom line and so on. However, I still think it is fun to examine and play with the argument structure.
Basically, what we have here is a recipe:
Take some intuitions.
Encode them in some formal premises.
Stir with some fancy set theory.
Extract the desired conclusion : namely that there is an “uncaused cause”
It’s certainly interesting to see how weak you can make the ingredients (in step 1) before the recipe fails. Also, the process of then translating them into premises (step 2) looks interesting, as at least it helps decide whether the intuitions were even coherent in the first place. Finally, if the desired conclusion wasn’t quite strong enough for the arguer’s taste (hmm, missing that true monotheistic kick), it’s fun to work out what extra ingredient should be inserted in to the mix (let’s put in a bit of paprika)
That’s basically where I’m coming from in all this..
Ah… I think I get it. You want to play with intuitions, and see which premises would have to be proved in order to end up with monotheism via set theory.
I don’t think it would be possible to get around the point of defining God in terms of set theory. Once you have a definition, you can see if it turns up; if God is not defined, then you don’t know what you’re looking for. Looked at from that point of view, the definition of God as a first cause is probably one of the better options.
Loops can still be a problem...
This can still fail in the case where two loops have their external causes in each other. (I think. Or would that simply translate into an alternate set of loops? …I think I could figure out a set of looped entities, such that each loop has at lest one cause outside that loop, that has no first cause).
Either of those would be sufficient; though the first seems to fit more possible sets.
I think if two loops were caused by each other, then there would be a super-loop which included all the elements from both of them, and then you could look for the cause of the super-loop. The Axiom of Choice would still be needed to show that this process stops somewhere.
Finally, I rather liked your thought that causality may be so loopy that everything is a cause of everything else. The only way to get a first cause out of that mess is to treat the entire “super-duper-loop” of all things as a single uncaused entity, and if you insist on calling that “God”, you’re a pantheist.
Let’s consider loops A->B->C->A->B->C and D->E->F->D->E->F.
Let’s say, further, that B is a cause of E and D is a cause of A. Then each loop has an external cause.
Then there are also a few other loops possible:
A->B->E->F->D->A->B->E->F->D (external cause: C) A->B->E->F->D->A->B->C->A->B->E->F->D… huh. That includes all of them, in a sort of double-loop with no external cause. I guess that would be the super-loop.
Better yet; no matter what causality looks like, you can still always combine everything into a single giant, uncaused entity. You don’t need to assume away loops or infinite chains without external causes if you do that.
I’ve been doing a bit more “stir in fancy set theory” over the weekend, and believe I have an improved recipe! This builds on the idea to treat chains and loops as a single “entity” and look for a cause of that entity. It is a lot subtler than just throwing every entity together into one super-duper-entity.
Here are a bunch of premises that I think will do the trick:
A1. The collection of all entities is a set E, with two relations C and P on E, such that: x C y if and only if x is a cause of y; x P y if and only if x is a part of y.
A2. The set E can be well-ordered
Note: This ensures we can apply Zorn’s Lemma when considering chains in E, but is not as strong as the full Axiom of Choice. If the set E is finite or countable, for instance, then A2 applies automatically.
A3. If x C y and x P z then z C y.
Informally, “anything caused by a part is caused by the whole”.
Definitions: We define ⇐ such that x ⇐ y if and only if x = y or there are finitely many entities x1, …, xn such that x1 = x, xn = y and xi is a cause of xi+1 for i=1.. n-1. Say that a set S is a “chain” in E iff for any x, y in S we have x ⇐ y or y ⇐ x. Say that such an S is an “endless chain” iff for any x in S there is some y not equal to x in S with y ⇐ x. Say that an entity y is “uncaused” if and only if there is no z distinct from y with z ⇐ y. Also say that x is a “proper part” of y iff x is not equal to y but x P y.
Note: These definitions ensure that ⇐ is a pre-order on E. Note that an endless chain may be an infinite chain of distinct elements, or a causal loop.
A4. Let S be any endless chain in E. Then there is some z in E such that every x in S is a proper part of z.
Lemma 1: For any chain S in E, there is an element x of E with x ⇐ y for every y in S.
Proof: Suppose S has an end (not endless). Then there is some x in S such that for no other y in S is y ⇐ x. By the chain property we must have x ⇐ y for every member y of S. Alternatively, suppose that S is endless, then by A4, there is some z in E such that every x in S is a part of z. Now consider any y in S. There is some x not equal to y in S with x ⇐ y, so there are x = x1… xn = y with each xi C xi+1 for i=1..n-1. Further, by A3, as x C x2, we have z C x2 and hence z ⇐ y.
Lemma 2: For any x in E, there is some y in E such that: y ⇐ x, and for any z ⇐ y, y ⇐ z.
Proof: This is the version of Zorn’s Lemma applied to pre-orders.
Theorem 3: For any x in E, there is some uncaused y in E such that y ⇐ x.
Proof: Take a y as given by Lemma 2 and consider the set S = {s: s ⇐ y}. By Lemma 2, y ⇐ s for every member of S, and if S has more than one element, then S is an endless chain. So by A4 there is some z of which every s in S is a proper part, which implies that z is not in S. But by the proof of Lemma 1, z ⇐ y, which implies z is in S: a contradiction. So it follows that S = {y}, which completes the proof.
I’ve also got some premises for aggregating multiple uncaused entities into a single entity. This gives another approach to “uniqueness”. More on my next comment, if you’re interested.
For uniqueness, we build on the idea of all uncaused causes being part of a whole. The following premises look interesting here:
B1. If x P y and y P z then x P z; x = y if and only if x P y and y P x.
This states that P is a partial order, which is reasonable for the “part of” relation.
B2. If S is any chain of parts, such that for any x, y in S we have x P y or y P x, then there is some z in E of which all members of S are parts.
This states that E is inductively ordered by the “part of” relation.
B3. If x C z and y P z then x C y.
Informally, “a cause of the whole is a cause of any part”.
B4. Suppose that y ⇐ x and z ⇐ x and both y, z are uncaused. Then y P z or z P y, or there is some w of which both y and z are proper parts.
Informally, two uncaused y and z can’t independently conspire to cause x unless they are parts of a common entity.
Definition: Say that entities x and y are causally-connected if and only if x = y, or there are entities x=x1,..,xn=y with either xi C xi+1 or xi+1 C xi for each i=1..n-1.
B5. Any two entities in E are causally-connected.
Informally, E doesn’t “come apart” into completely disconnected components, such as a bunch of isolated universes.
Theorem 4: For any x in E, there is a unique entity f(x) in E such that: f(x) is uncaused, f(x) ⇐ x, and any other uncaused y with y ⇐ x satisfies y P f(x).
Proof: For any x, define a subset E’ = {y in E: y ⇐ x, y is uncaused}. Consider any chain of parts S in E’ with at least two elements. By B2 there is some z in E of which all members of S are parts. By B3, z must be uncaused (or else some w C z would also be a cause of all the members of S, which would require them all to be equal to w, so S would be a singleton), and by A3, z ⇐ x. So z is also a member of E’. By application of Zorn’s Lemma to E’, there is a P-maximal element f in E’ such that there is no other y in E’ with f P y. But then, by B4, for any y in E’ we must have y P f; this makes f unique.
Theorem 5: For any x, y in E, f(x) = f(y) if and only if x and y are causally-connected.
Proof: It is clear that if f(x) = f(y) then x is causally-connected to y (just build a path backwards from x to f(x) and then forward again to y). Conversely, suppose that x C y, then f(x) is uncaused and satisfies f(x) ⇐ y so we have f(x) P f(y). This implies f(x) = f(y). By a simple induction on n we have that if x is causally-connected to y, then f(x) = f(y).
Corollary 6: There is a single entity g in E such that f(x) = g for every entity x in E.
Proof: This follows from Theorem 5 and B5.
Done!
(Huh. One of the ancestors to this comment—several levels up—has been downvoted enough to require a karma penalty. I wonder if there should be some statute of limitations on that; whether, say, ten levels of positive-karma posts can protect against a higher-level negative-karma post?)
An interesting assumption. Necessary for theorem 3, but I suspect that it’ll mean that the original cause described in theorem 3 will then very probably be an entity z that is the earliest cause.
I also note that, while z consists of all the parts in the endless chain, there is no guarantee that any of the elements in the chain, even those that cause other elements in the chain, is in any way a cause of z. In fact, the way that z is defined, z may well be causeless (or, then again, z may have a cause). While I can’t actually find anything technically invalid in theorem 3, or in assumption 4, I get the general feeling of wool being pulled over my eyes in some way.
When I consider B3, it becomes even more important to note that z as a whole is not necessarily caused by any element that is a proper part of z. The cause of a part may or may not be the cause of the whole.
Hmmm… B4 appears to be pretty much just shoehorning monotheism in. It seems a questionable assumption; if I decide to get into my car and drive, and you decide to get into your car and drive, and we drive into each other, then we both are causes of the resultant accident but we are not the same. (We are not causeless, either, so it’s not quite a counterexample, just an explanation of why I don’t think B4 is justified,) B5 is unsupported, but I can prove that all entities that I will ever observe evidence of are causally connected (i.e. they are connected to the effects on my actions of having observed them) so it will look true whether it is or not.
Though I can raise questions about your assumptions, I can’t find anything wrong with your logic from then on. So congratulations; you have a very convincing argument! …as long as you can persuade the other person to accept your assumptions, of course.
Well now, here’s a devious approach, which would probably appeal to me if I ever needed to make a career as a philosopher of religion.
Let’s suppose a theist wants to “prove” that God—by his favourite definition—exists. For instance he could define a type G, whereby an entity g is of type G if and only if g is omnipotent, omniscient, perfectly good and so on, and has all those characteristics essentially and necessarily. Something like that. Then the theist finds a set of premises P, with some intuitive support, such that P ⇒ There is an uncaused cause.
And then he adds one other premise “Every entity that is not of type G has a cause” into the recipe to form a new set P’. He cranks the handle, and then P’ ⇒ There is an entity of type G. Job done!
Just in case someone accuses him of “begging the question” or “assuming what he set out to prove” he then pulls out the modal trick. He just claims that it is possible that P’ is true. This leads to the conclusion that “It is possible that there is an entity of type G”. And then, remembering he’s defined G so it includes necessary existence (if such a being is possible at all, it must exist), he can still conclude “There is a being of type G”. Job done even better!
Can I have the Templeton Prize now please?
The modal trick reminds me of Descarte’s approach… God is definitionally perfectly good, which implies existence (since something good that doesn’t exist isn’t as good as something good that does), therefore God exists.
ZZZZzzzzzz....
A closer parallel is Plantinga’s “victorious ontological argument”.
That one was from the late 20th century, not the 17th. All rather sad really...
Huh. That modal trick is devious. But it doesn’t work. I can assume an entity that does something easily measurable (e.g. gives Christmas present to children worldwide), and then slap on a necessary existence clause; but that doesn’t necessarily mean that I can expect Santa later this year.
I think the ‘necessary existence’ clause requires a better justification in orderto be Templeton-worthy.
And here I always thought God corresponded to an inaccessible cardinal axiom.
On reflection, the fact that an atheist would be able to come up with an argument for a god that’s more persuasive to atheists is unsurprising, especially when you consider the fact that most religious people don’t become religious via being persuaded by arguments. It’s definitely still amusing, though.
I’m definitely aware of Tegmark’s theory, though I admit I hadn’t considered it as an argument for any kind of theism. That seems like an awfully parochial and boring application of the ultimate ensemble, although you’re right that it can have that sort of application… although, if we define “supernatural” entities to mean “ontologically basic mental entities” a la Richard Carrier, would it really be the case that Tegmark’s multiverse implies the existence of such? I’m not sure it does.
Meyer’s argument begins with premises that are hilariously absurd. Defining entities as being able to be causes of themselves? Having “entities” even able to be “causes”? What? And all this without the slightest discussion of what kinds of things an “entity” can even be, or what it means to “exist”? No, this is nonsense.
I think this is mostly a presentational issue. The purpose of the argument was to construct a non-strict partial order “<=” out of the causal relation, and that requires x<=x. This is just to enable the application of Zorn’s Lemma.
To avoid the hilarity of things being causes of themselves, we could easily adjust the definition of ⇐ so that “x<=y” if and only if “x=y or x is a cause of y”. Or the argument could be presented using a strict partial order <, under which nothing will be a cause of itself. The argument doesn’t need to analyse “entity” or “exists” since such an analysis is inessential to the premises.
And finally, please remember that the whole thing was not meant to be taken seriously; though rather amusingly, Alexander Pruss (whose site I linked to) apparently has been treating it as a serious argument. Oh dear.
FWIW, the probability I place on the Simulation Argument being true is only a little higher than the probability I place on traditional theistic gods existing. Could be just me, though.
Well, traditional theistic gods tend to be incoherent as well as improbable. (Or one might say, improbable only to the extent that they are coherent, which is not very much.) So, I’m not sure how we’d integrate that into a probability estimate.
Agreed; but this doesn’t apply to lesser gods such as Zeus or Odin or whomever.
What are the values for these probabilities and how have you estimated them?
Both of the values are somewhere around epsilon.
God-wise, I’ve never seen any evidence for anything remotely supernatural, and plenty of evidence for natural things. I know that throughout human history, many phenomena traditionally attributed to gods (f.ex. lightning) have later been demonstrated to occur by natural means; the reverse has never happened. These facts, combined with the internal (as well as mutual) inconsistencies inherent in most major religions, serve to drive the probability down into negligibility.
As for the Simulation Argument, once again, I’ve never seen any evidence of it, or any Matrix Lords, etc. Until I do, it’s simply not parsimonious for me to behave as though the argument was true. However, unlike some forms of theism, the Simulation Argument is at least internally consistent. In additions, I’ve seen computers before and I know how they can be used to run simulations, which constitutes a small amount of circumstantial evidence toward the Argument.
EDIT: I should mention that the prior for both claims is already very low, due to their complexity.
Epsilon is not a number, it’s a cop-out. Unless you put a number you are reasonably confident in on your prior, how would you update it in light of potential new evidence?
Well, so far, I have received zero evidence for the existence of either gods or Matrix Lords. This leaves me with, at best, just the original prior. I said “at best”, because some of the observations I’d received could be interpreted as weak evidence against gods (or Matrix Lords), but I’m willing to ignore that for now.
If I’m using some measure of algorithmic complexity for the prior, what values should I arrive at ? Both the gods and the Matrix Lords are intelligent in some general way, which is already pretty complex; probably as complex as we humans are, at the very least. Both of them are supremely powerful, which translates into more complexity. In case of the Matrix Lords, their hardware ought to more complex than our entire Universe (or possibly Multiverse). Some flavors of gods are infinitely powerful, whereas others are “merely” on par with the Matrix Lords.
I could keep listing properties here, but hopefully this is enough for you to decide whether I’m on the right track. Given even the basics that I’d listed above, I find myself hard-pressed to come up with anything other than “epsilon” for my prior.
Thank you for being unambiguous, this is exactly the sort of thing I wanted to see if this community actually believed. Personally I think it reflects poorly on anyone’s intellectual openness for them to believe the other side literally has no decent arguments.
Then you must believe the same with respect to homeopathic remedies, the flat earth society, and those who believe they can use their spiritual energy in the martial arts. Give us some good arguments for those.
There’s a lot of stuff out there for which it seems to me there is no good argument. I mean really, let’s try to maintain some sense of perspective here. The belief that everyone has a decent argument is, I think, pretty much demonstrably false. You presumably want us to believe that you’re in the same category as people who ought to be taken seriously, but I don’t really see how a belief in God is any more worthy of that than a belief in homeopathic remedies. At least, not based on your argument that all positions ought to be considered to have good arguments. If you’re trying to make a general argument, you’re going to get lumped in with them.
An argument can be “decent” without being right. If you want an example, and can follow it Kurt Godel’s ontological argument looks pretty decent. Consider that:
A) It is a logically valid argument
B) The premises sound fairly plausible (we can on the face of it imagine some sense of a “positive property” which would satisfy the premises)
C) It is not immediately obvious what is wrong with the premises
The wrongness can eventually be seen by carefully inspecting the premises, and checking which would go wrong in a null world (a possible world with no entities at all). Axiom 1 implies that if an impossible property is positive, then so is its negation (since an impossible property logically entails its negation). Axiom 2 says that can’t be true—a property and its negation can’t both be positive. So together these are a coded way of saying that all positive properties are possible properties. And then Axiom 5 (Neccessary existence is a positive property) goes wrong, because necessary existence is not a possible property in the null world. So it is not a positive property. Axiom 5 is inconsistent with Axioms 1 and 2.
There are arguments for the existence of God that are good in the sense that they raise my estimate of the likelihood of the existence of God by a substantial factor.
They aren’t sufficient to raise the odds to an overall appreciable level.
Sometimes, the issues really are cut-and-dried, though. To use a rather trivial example, consider the debate about the shape of the Earth. There are still some people who believe it’s flat. They don’t have any good arguments. We’ve been to space, we know the Earth is round, it’s going to be next to impossible to beat that.
I should clarify that when I said:
I meant this as the rhetorical “we”, not “we, Less Wrong”.
And in general, you shouldn’t take me, or any other commenter in particular (even Eliezer), to represent all of Less Wrong. This is a community blog, after all.
Did you read what I wrote about what makes arguments good or bad...?
Edit: Sorry, I see that you quoted from that comment, so presumably you did read it. That said, I’m not sure that what I said was clear, given your subsequent comments...
That is what many people here have done regarding theism. Seen the best arguments, and decided that they fail utterly. Eliezer quoted above talks about Modern Orthodox Judaism allowing doubt as a ritual, but not doubt as a practice leading to a result. You would have us listen to arguments as ritual, but not actually come to a conclusion that some of them are wrong.