User talk:ZackMartin/Probability and Idealism

There are three basic views about the relationship between mind and matter:

• materialism: matter alone is fundamental, mind is derivative, mind is reducible to matter
• idealism: mind alone is fundamental, matter is derivative, matter is reducible to mind
• dualism: each of matter and mind are separately and independently fundamental, neither is derivative of the other, neither is reducible to the other

What is the prior probability? Now, I would say we do not know which of these are true. So we should assign a uniform prior, of around about a third.

But are any further views possible? Certainly one could devise some. One example is 'partial dualism' - mind is in part fundamental and irreducible to matter, in part non-fundamental and reducible to matter. If one tries, one can think of other possible positions. But, consider the following factors:

1. these alternative positions are generally more complex than the above three positions
2. to my knowledge, they have no known advocates or adherents
3. they are not fleshed out. For example, in 'partial dualism', how does one define the boundary between the reducible and irreducible parts of mind? Why is one part reducible and the other not?

For the above reasons, I think we should collectively assign these alternative views a prior close to zero. However, if someone came forward to advocate for one of them, and fleshed out the details, that would justify raising the probability, since it would address concerns (2) and (3) -- albeit concern (1) would remain.

Now, dualism has the particular problem of interactionism -- explaining how two separate, independent and mutually irreducible substances can interact -- which is a non-issue for materialism and idealism, since by reducing one substance to the other, their interaction is reduced to the interaction of one substance with itself. I think this justifies us assigning dualism a lower probability than materialism and idealism. A separate consideration is the fact that dualism is more complex than either materialism or idealism -- it posits two fundamental substances rather than one -- which may also justify reducing the probability.

So materialism and idealism have equal probability, dualism significantly less, and other views far less.

Materialists will say -- the success of neuroscience, etc., is evidence for materialism, and thus justifies considering idealism more likely. But -- there is nothing in the actual or potential future results of neuroscience that idealists should find threatening or objectionable. They can agree completely on the science -- they just give the same science a differing metaphysical interpretation. (And metaphysical interpretations of science are not actually part of science, they are hangers-on.) You are attached to some kind of brain imaging machine, such that when you think of things, you see consistent patterns which correspond to those thoughts appear on the machine's display. A materialist interprets this as proof that mind is fundamentally derivative of matter. But to an idealist, they are observing a pattern in experience, a correlation between two experiences of mind -- an internal mental experience of thinking a particular thought, an external sensory experience of seeing certain patterns on a computer screen.

Neuroscience threatens dualists because of the problem of interactionism. Many of them feel the need to believe that somewhere in the brain something 'magical' is taking place, the meeting of mind and matter. But, the further neuroscience advances, the less room is left in the brain for this mysterious magical interaction, which of course can only exist in the brain as long as it is a mystery to us. So, the success of neuroscience threatens in the long run to discredit dualism. But, idealists see no such threat -- they do not believe there is anything magical or mysterious or unexplainable by science happening in the brain. They just differ from materialists in their metaphysical interpretation of those scientific results.

So, we are left with materialism and dualism having equal probability. Next I plan to present an argument based on simulation that I think implies idealism has greater probability than materialism. But, before I move on to that, I want to see if you agree or disagree with the above.

I am wondering where Tegmark's radical platonism fits into the above - is it a form of materialism or idealism, or is it one of the "other views" (If it is, what I said about the other views lacking advocates is untrue.) But I am unsure of how to classify it. --Maratrean (talk) 07:06, 3 March 2011 (UTC)

Ok, I'll take this piece by piece. First, I agree with your discussion of assigning priors; a uniform prior for idealism, materialism, and dualism is probably a good idea. The fact that dualism posits the existence of two substances may or may not increase it's complexity - as I discussed on this page, when we assign complexity penalties a system for complexity we should really look at the complexity of the mathematical laws that describe the system. A dualistic universe might be more complex in this sense because it may need additional laws to govern the interaction of matter and mind, but it is still possible for it to be less complex than an idealistic or materialistic universe, so I wouldn't assign a strict penalty based on this.

I also agree with most of what you've said about science's understanding of the brain threatening dualism, and I would go as far as to say that dualism, insofar as it is falsifiable, has been mostly falsified. Of course it is still possible for there to be some kind of mind-stuff out there, but we've shown that it probably won't have any consequences on the reality we observe.

This brings us to your discussion of materialism and idealism, which you define as philosophical interpretations of the evidence we have. This essentially entails that we cannot use any evidence to support our claims, only philosophical arguments. I'm fine with this, but it definitely puts the entire discussion on much shakier ground.

Your discussion of Tegmark's ideas are interesting, and I admit after reading his paper "Shut Up and Calculate" a few months ago I was irked by that fact that it is difficult to tell the difference between a fundamental physical entity and a fundamental mathematical one. To relate this to the matter at hand: I think we need to find some kind of criteria for distinguishing whether an entity is fundamentally mental or fundamentally physical. On the talk page of your essay, one of the things we disagreed about was the need for observers - could we say that the universe is ultimately idealistic if it requires observers, and ultimately materialistic if it does not?

In summary: I agree with most of what you've put forward except for what I've discussed in the paragraph above, which is a point I think we need to iron out before we proceed. But it is possible that I missed something, so we may have to revisit some of these ideas later. Tetronian ^{you're clueless} 20:59, 4 March 2011 (UTC)

I'd agree with the idea of assigning complexity on an algorithmic basis, save that algorithmic complexity is (in general) uncomputable. So it is a nice idea theoretically, but seemingly intractable in practice. On the other hand, trying to come up with a tractable definition of complexity is also very hard - it is easy to talk about at a vague high-level, but hard or impossible to produce a precise definition which is tractable.

On an aside -- one of the problems with Bayesianism is how to assign priors. Subjective Bayesianism says you pick whatever prior represents your own personal estimation of likelihood -- which doesn't really work for discourse, since each side will naturally pick a greater prior for their favoured view than for others. Objective Bayesianism claims some priors are objectively more correct than others -- but, although considering simple cases, one gets a strong and interpersonally consistent intuition that a certain prior is objectively correct, in more complex/obscure cases intuition is going to break down, and its difficult to formulate rules which apply to all cases -- and, if two people disagree about which rule to apply, there seems to be no way to resolve that dispute. Frequentism does not have these problems; on the other hand, it can't be applied to problems like this.

We were discussing earlier how to determine the prior of infinite claims. If our rules for assigning priors are justified by agreement with our intuitions -- well, the mathematics of infinity is highly counterintuitive, so it may not be proper to apply rules justified by our intuitions to infinite scenarios. Which may well leave us with the position that we can't assign a prior, since we have multiple competing ideas of what prior we should assign, and no way to decide between them.

About Tegmark's view -- the issue relevant here is whether it represents a fourth choice alongside materialism/dualism/idealism, or if it is a variant of one of those three choices? If we are assigning uniform priors, that make quite a difference.

One argument against Tegmark's view is that it implies every possible universe is actual. Arguably, such a view has an infinite complexity, the maximal possible complexity. If more complexity justifies assigning a lower prior, that would suggest infinite complexity would justify assigning a prior of zero. But, if we try to define 'complexity' in terms of algorithmic complexity theory, to my knowledge that theory only applies to finite or countably infinite strings. But the multiverse of this theory cannot be described by a countably infinite string -- so, we can't apply algorithmic complexity theory to calculate that complexity.

Consider a series of infinite random strings, of increasing cardinality (e.g. aleph-null, aleph-one, etc). Arguably, those of higher cardinality are more "complex" than those of lower cardinality. But, if we assign the aleph-null string a probability of zero, how can we assign a lower probability to the aleph-one string? Maybe that could be resolved if one allowed infinitesimal probabilities, but usually we define probability in terms of the real interval [0,1], so this would violate the Archimedian property of the reals.

What is the cardinality of Tegmark's multiverse? Ah, the multiverse of this theory is essentially a universal set (the set of all sets, the set of everything) or a universal category. But then we run into Russell's paradox. The usual way to avoid this paradox is to not allow such collections. So, Tegmark's theory cannot be expressed in mathematical terms it seems, since mathematics can't talk about the whole of itself without running into Russell's paradox.

If Tegmark's theory is actually a fourth choice, rather than one of the three, I think these considerations justify assigning Tegmark's theory a prior of zero, to the extent we can assign it a prior at all. --Maratrean (talk) 10:57, 8 March 2011 (UTC)

I am what E. T. Jaynes calls a "subjectively objective" Bayesian: I understand that priors are subjective in that they depend on incomplete information, but they much be assigned objectively using mathematical rules. That is, though a prior reflects my state of belief, I don't have the freedom to make this state of belief whatever I want. In regard to algorithmic complexity: Solomonoff Induction is uncomputable if you don't restrict the space of allowed programs, but if you limit yourself to a certain set of programs you can get a computable probability distribution as a result. See also MML.

For the purpose of our discussion, I'm willing to leave Tegmark's universe by the wayside and focus only on idealism and materialism. I'm not entirely sure if Tegmark's idea of fundamental mathematical structures is even coherent, and if it is I'm not sure how it would differ from materialism or idealism, as I discussed above. While I don't agree with all of your arguments, I do agree that there are problems with computing its complexity.

That said, I'm ready to hear your arguments for idealism. The only issue I'd like to see resolved is this (quoted from above): "Could we say that the universe is ultimately idealistic if it requires observers, and ultimately materialistic if it does not?" I think we need to carefully define the difference between a fundamentally mental and a fundamentally physical structure as we continue on with this discussion. Tetronian ^{you're clueless} 03:04, 10 March 2011 (UTC)

"Could we say that the universe is ultimately idealistic if it requires observers, and ultimately materialistic if it does not?"

Idealism implies that an observerless universe is impossible. Materialism implies that an observerless universe is possible.

In an idealist universe, things observed exist. Things never observed neither exist nor do not exist, they are in a sort of limbo - it is like reading a novel or watching a movie, then asking what happens next? Of course, some "what happens next" are more likely than others given what actually happens - but none of them are actual, they simple exist in different degrees of potential. For a materialist universe, unobserved things exist and have particular properties, even if no observer ever observes or comes to know of their existence. (I propose we ignore the Copenhagen interpretation of quantum mechanics in this discussion - it is in many ways non-materialist - unless, if you happen to favour it, we can discuss it.)

I am not sure an idealist vs. a materialist universe look different "from the inside". But "from the outside", they look quite different -- i.e. they differ in those two factors I identified above. --Maratrean (talk) 09:39, 10 March 2011 (UTC)

I agree with all of that, and I'm ready to hear your arguments. Tetronian ^{you're clueless} 04:18, 12 March 2011 (UTC)

Simulation Hierarchy Argument

Okay, firstly, let us define "universe" as something which appears to be a universe from the perspective of an observer internal to it, whatever that universe is ultimately. So, from a materialist viewpoint, there might be one actual universe, and zero or more simulated universes which are simulated directly or indirectly in that actual universe. But we will call them both "universes" - from the perspective of an internal observer, they are usually going to be indistinguishable.

Now, one universe can be simulated in another. Let us define a relation sim between two universes u and v - u sim v - universe u is simulated in v; so, there exists some computer (or hypercomputer) in v which simulates u. Let us restrict sim to direct simulation only (as opposed to nested simulation), but indirect simulation is the transitive closure of sim, sim*.

Taking a universe u, we can define the simulation hierarchy H(u) as the set {v|u sim* v}. Let us define H'(u) as u union H(u). The relation sim constitutes an ordering over the set H'(u).

Now, as we have already discussed, from a purely mathematical perspective, H(u) can be finite or infinite - there could be an infinite nesting of simulations. But let us define some properties that u could have in terms of the shape of its H(u):

Firstly, linearity - u is linear if there is no a in H'(u) such that there exists b, c such that a sim b and a sim c. Conversely, if there is such an a, then u is non-linear. This is upward non-linearity, not downward non-linearity - in downward non-linearity, a sim c and b sim c - in upward non-linearity, a sim b and a sim c. Also, let us restrict non-linearity to the ultimately non-linear: a sim b and a sim c, where there is no d such that b sim* d and c sim* d.

Secondly, circularity - u is circular if there is an a in H'(u) such that a sim* a. Conversely, if there is no such a, then u is non-circular. The simplest form of circularity is the self-simulating universe, u sim u.

Thirdly, hierarchy-finitude - u is hierarchy-finite if the set H(u) has finite cardinality; hierarchy-infinite if the set H(u) has infinite cardinality.

Fourthly, hierarchy-Archimedianity: Let us define the hierarchy range between a and b, where a sim* b, is the set of all c, where c equals neither a nor b, where a sim* c and c sim* b. Hence, u is hierarchy-Archimedian if there exists no a,b elements of H'(u), such that the hierarchy range of a and b is infinite. (In other words, the hierarchy is Archimedian if sim has the Archimedian property with respect to the elements of the hierarchy.) Obviously, to be non-Archimedian the hierarchy must be infinite, yet many infinite hierarchies will still be Archimedian.

Note that while circularity implies in a sense an infinite regress of universes, it would be a regress composed of only a finite number of distinct universes - provided the hierarchy is Archimedian.

Now, if the simulation hierarchy is non-linear, it would have different branches, and thus could be finite non-circular in some branches, and infinite or circular in others. Branches themselves could branch, for multiple levels of non-linearity.

Now, to sketch the argument from here: Materialism assumes the simulation hierarchy of our universe must be linear, non-circular and finite. If any of these do not hold, materialism has grave difficulties. Whereas, for idealism, any of those possibilities is a non-issue. So we can use that to inform the conditional probability of materialism and idealism with respect to these possibilities. If non-linearity, circularity or infinity are true, then materialism is almost certainly, or even certainly, false. Whereas, if they are true, that has no direct impact on the probability idealism - although, if they make materialism unlikely or impossible, and if idealism is the most likely alternative to materialism, their truth would make idealism likely.

This argument requires us to assign priors to the propositions "The simulation hierarchy of our universe is non-linear", "The simulation hierarchy of our universe is circular", "The simulation hierarchy of our universe is infinite". Since we don't know anything about the simulation hierarchy of our universe, we should use some kind of uniform prior - although, we would need to understand the internal relations between these propositions to determine exactly what that prior should be.

Anyway, the use of a uniform prior would produce a significant probability that one of those three propositions is true, from which would we draw the conclusion that idealism is more likely to be true than materialism. --Maratrean (talk) 05:21, 13 March 2011 (UTC)

I will accept all of these definitions, and I very much like the way you've set this up - it's very organized and thorough. Now then, let's get to the things I do disagree with, which I'll break up into sections.

First, I'd like to address a minor flaw I noticed: you seem to be arguing that non-linearity is a problem for materialism, but I don't think this is so. Let's break the problem down further: as you point out, there are two kinds of non-linearity, upward and downward. We can immediately discard the case of upward non-linearity, or "a sim b and a sim c", because this doesn't threaten materialism in any way. In the case of downward non-linearity, or "b sim a and c sim a", I think we need to examine what's going on more carefully. Specifically, what do we really mean by ∃a? If ∃a means, "there exist a universe 'a' that is described by the bitstring i_a", then when we say "b sim a and c sim a" what we are really saying is "b sim i_a and c sim i_a", or in plain English, "universes b and c are simulating some universe described by the bitstring i_a". Honestly, I'm not entirely sure what to make of this case. Does this mean that there exist two identical universes described by i_a, or is there only one? In either case, I don't see this as undermining materialism, since either case allows for the existence of some kind of fundamental physical matter in universes b or c.

But even if you are correct and downwards non-linearity would be evidence against materialism, I don't think it significantly affects your argument, and here's why: you are arguing that it is more likely that at least one of the following statements is true:

1. "The simulation hierarchy of our universe is non-linear"
2. "The simulation hierarchy of our universe is circular"
3. "The simulation hierarchy of our universe is infinite"

If we assigned uniform priors to each of these, then p(1) = p(2) = p(3) = .5. Thus, for all of them to be false we would have: p((~1)*(~2)*(~3)) = .125, or 1/8. (If I am correct about non-linearity, then we would have p((~2)*(~3)) = 1/4). As a result, you argue, we should update on this as follows: p(idealism|sim argument) = (p(idealism)*p(sim argument|idealism)) / (p(idealism)*p(sim argument|idealism)+p(materialism)*p(sim argument|materialism)) = (.5*.875) / (.5*.875 + .5*.125) = .875. Thus, according to your argument, p(idealism) shifts from .5 to .875.

I have two problems with this: first, this is by no means a strong argument. We are using uniform priors and only the tiniest bit of information, and though the inference itself is indeed valid it is extremely inexact. Also, .875 is not exactly a doozy of a posterior probability - that is, you've invoked a lot of abstract reasoning for a relatively weak result. However, math is math, so I won't argue too strongly on that front. Second, we aren't using all of the information we have in this argument - there are more details we can include to make our probability assignments more exact. Specifically, we need to be looking at the average probability that any given universe in our set of all possible universes will create any simulation. Call this probability s. Now, s is obviously going to have to be less than 1 because there are most likely some universes in our set that will not produce simulations (e.g. empty ones). Thus, the prior probability of statement (3) must be close to zero because in order for (3) to be true, it means that an infinite number of universes must have simulations. The probability of this being the case is the limit of p(s)^∞, which of course is 0. This drastically changes the value of p((~1)*(~2)*(~3)) and makes it close to 1/4 (or even 1/2 if I am correct about non-linearity).

This discussion of s also affects the probabilities of (2) in that it necessarily lowers p(2) because it requires that an "adjacent" series of universes contain simulations. Thus the probability of any particular "circle" of universes must be directly based on p(s)^n, where n is the average number of universes involved in a circle. (It's actually a lot more complex than this, but it has to start with this value.) Obviously n could be small, but it may not be. Of course it is possible for p(s)^n to be >= .5 if n is small and s is large, but I think n is most likely 2 or 3 (I find n=1, or a sim a, to be rather implausible from a computational perspective) and s is probably relatively small. This also serves to lower p((~1)*(~2)*(~3)) by an amount that will depend on the values of n, s, and a number of other factors.

I also have some additional comments about the probability of (2), but I'll save these for later as I've given you a lot to argue with already. Tetronian ^{you're clueless} 03:43, 14 March 2011 (UTC)

I suppose in regards to non-linearity, I think you are right that, if we admit parallel material universes, two parallel material universes could separately and independently simulate exactly the same simulated universe.

In terms of the infinite case, I don't agree with your reasoning that we can use limits with respect to probability. Suppose I pick a natural number completely at random (if we accept the axiom of choice, then doing so is logically possible, even though it is not going to be physical possible). Let us call that number m; but I have not told you what it is. Now, you take some other natural number n. What is the probability that P(m=n)? You don't know what m is, so P(m=n)=0. But consider all the propositions p_n = "m=n". Clearly P(p_n) = 0 for all n. Yet, the infinite conjunction of all p_n has a probability of 1. The same argument can be made with say geometric points - if you pick a point totally at random, the probability of any given point being the point you picked is 0. But the infinite conjunction of those propositions with respect to every point has the probability 1. So we see, there are infinite sequences of probability, each of which is zero probability, but their infinite conjunction is non-zero probability. This shows that we can't determine the probability of an infinite conjunction by multiplication of the probabilities of the conjuncts, nor by using an infinite product.

Another argument: take the claim "The universe is infinite". Prima facie, we know nothing about whether the universe is finite or infinite, so we'd think if we adopted a uniform prior, we'd conclude that the claim has a probability of 0.5. But if we follow your line of reasoning, we should conclude that "The universe is infinite" has a probability of 0, and "The universe is finite" has a probability of 1. But, surely, we don't know that one claim is vastly more likely than the other; so, that suggests a problem with your reasoning.

Another argument: Let us order possible universes by their cardinality. Firstly, let us group together all the finite possible universes. Then, all those of cardinality aleph-null. Then, an every increasing succession of classes of possible universes having ever greater cardinality. Now, compare the quantity of universes in the first element of the sequence to those collectively in the subsequent elements of the sequence. It is vastly smaller - in fact, it is impossible to say how small it is - suffice to say, it is as small as we like - zero if our proportions will be reals, if we permit infinitesimals then as small an infinitesimal as we like. Hence, if we don't know what universe we are in, we should conclude that the probability of us existing in a finite universe is zero, and the probability of us existing in a transfinite universe is 1. (In fact, for any given cardinal, the probability that our universe has that cardinality is zero, and the probability that the cardinality of our universe is greater than that is 1.) Which is the opposite conclusion to the conclusion you reach. (I am not necessarily saying this argument is correct - I am simply using it as an example of how difficult it is to reason about probabilities of infinite objects... multiple arguments, each individually seemingly reasonable, can reach opposite conclusions - which, goes back to my original contention, we should give the infinity hierarchy proposition a prior probability of 0.5 not 0).

For the circular case, consider Quines - computer programs which generate themselves as output. These are well known to exist (there is a theorem in computability theory which shows that in any Turing equivalent language Quines exist). There are also programs which are "Quines plus", their own code is embedded in their output string, but there is also other content; or programs which output some transformation of themselves. In which case, is it really inconceivable that a universe could simulate itself? i.e. a program whose output represents a universe which contains a computer which executes the very same program? --Maratrean (talk) 07:46, 15 March 2011 (UTC)

Another argument: Consider the claim "The volume of the universe is no larger than V", for some finite V. Let us only consider Vs sufficiently large that current observational evidence does not conflict the resulting proposition. Now, consider all possible universes - how many of them in which V is true? Well, the number in which it is true is infinitely smaller than the number in which it is false. Let me put it this way - what proportion of all possible spheres have a volume less than one cubic metre? An infinitely small proportion. So, we conclude, the probability of the proposition "The volume of the universe is no larger than V" is 0, and that of its negation is 1. But this is true for all possible V. No matter how large a universe we are considering, there will always be an infinitely larger number of observers in larger universes; so any observer almost surely exists in a larger universe than the size we are considering. The conclusion seems to be, that the probability the universe is infinite is 1, and the probability it is finite is 0. Which is the opposite of your conclusion - that infinity has probability 1 and finitude probability 0.

Another version: Rather than ordering universes by cardinality, we could order them by ordinality. So first come the finite universes, then the infinite Archimedian universes, then non-Archimedian infinite universes... so, there will be vastly more infinite possible universes than finite one (and hence vastly more observers), and vastly more non-Archimedian universes than Archimedian ones. So, if we don't know what kind of universe we are in, there is a probability of 1 that we are in an infinite non-Archimedian one, and a probability of zero that we are in a finite or Archimedian infinite one. (I came up with this version, since I'm not really sure what it means to assert the existence of universes with ever greater cardinalities - ever greater ordinalities I can more intuitively conceive of.) --Maratrean (talk) 12:54, 15 March 2011 (UTC)

edit break

After thinking about this some more and seeing your arguments, my argument against infinite hierarchies is indeed wrong, though not entirely for the reasons you wrote here. I need to analyze the problem a little more thoroughly, after which I'll tell you what I think about your counterarguments. Tetronian ^{you're clueless} 01:01, 16 March 2011 (UTC)

Ok, my fully general response to the infinite hierarchies question can be summed up with these two pictures (Sorry for the blur - I don't have a good camera. Click on them to see the bigger version.).

The probability that an infinite chain would exist is the middle formula in the second picture (just set n equal to infinity). I'm reasonably sure that the formulas are correct, I made them based on the diagram. But let me know if you disagree. If these equations are correct, then we should be able to use our knowledge of a and P(s) to shed some light on this question (at last, something remotely empirical!). What are your thoughts on the values of these variables?

Also, I'm still mulling over your other replies to my "limit of p(simulation) = 0" argument, but I'll respond to those soon. One idea stands out to me, though: even if we have an infinite hierarchy of simulations, that doesn't mean that we don't have a non-simulated universe, as the picture shows: it is possible for it to "bottom out" (or in the case of the picture, "hit a ceiling") somewhere. Though the probability that we are in that basement universe would be infinitely small, it would still exist. Thus, I don't think the mere existence of an infinite hierarchy is an argument against materialism - you would need a hierarchy that extends infinitely in both directions. How we would assign probability to this scenario vs. the "infinite but with a bottom" scenario drawn above is the problem I'm currently working on. Tetronian ^{you're clueless} 02:26, 17 March 2011 (UTC)

Before I forget, some quick thoughts on your arguments:

Another argument: take the claim "The universe is infinite". Prima facie, we know nothing about whether the universe is finite or infinite, so we'd think if we adopted a uniform prior, we'd conclude that the claim has a probability of 0.5. But if we follow your line of reasoning, we should conclude that "The universe is infinite" has a probability of 0, and "The universe is finite" has a probability of 1. But, surely, we don't know that one claim is vastly more likely than the other; so, that suggests a problem with your reasoning. No, we shouldn't stick to our uniform prior if we have additional information we can incorporate into the problem - again, see Nick Bostrom's "Anthropic Bias" - virtually every thought-experiment in that book from Chapter 2 onwards is an example of how to do this. So we can indeed have non-uniform probability distributions a priori.

For the circular case, consider Quines - computer programs which generate themselves as output. These are well known to exist (there is a theorem in computability theory which shows that in any Turing equivalent language Quines exist). There are also programs which are "Quines plus", their own code is embedded in their output string, but there is also other content; or programs which output some transformation of themselves. In which case, is it really inconceivable that a universe could simulate itself? i.e. a program whose output represents a universe which contains a computer which executes the very same program? Correct. But the argument about two universes simulating a universe with the exact same bitstring description applies here as well. The universe in question is really just simulating an identical copy of itself, not itself. This is just a specific example of the general case I laid out in my objection to non-linearity.

So, if we don't know what kind of universe we are in, there is a probability of 1 that we are in an infinite non-Archimedian one, and a probability of zero that we are in a finite or Archimedian infinite one. I disagree. In general, we don't really need to consider observers here, since all materialism requires is that we have a "basement" universe that is fundamentally physical. It doesn't matter if we're in it or not.

I'll address the rest of the arguments later - I'm tired and I need to study. Tetronian ^{you're clueless} 02:37, 17 March 2011 (UTC)

I am interested in what you present in your photographed paper, but am struggling to follow them a bit. I will leave you some time - take your time - to write it up more clearly, and then we can continue the discussion. Write it up with whatever you wish (e.g. (La)TeX or Microsoft Word equation editor), then upload it as a PDF... --(((Zack Martin)))^™ 11:37, 17 March 2011 (UTC)

Or, you could just use the <math> tag (which is a subset of TeX anyway)... --(((Zack Martin)))^™ 11:56, 17 March 2011 (UTC)

Ok, I will do so. Tetronian ^{you're clueless} 12:22, 17 March 2011 (UTC)

Here it is, with some additional details for clarity's sake:

$m=total\ number\ of\ expected\ universes$

$P(s)=probability\ that\ at\ least\ one\ simulation\ will\ be\ created$

$a=average\ number\ of\ simulations\ created$

$d=stack\ depth\ (number\ of\ nestings)$

$k=number\ of\ universes\ at\ d=0$

$e_{d}=number\ of\ expected\ nonterminating\ chains\ of\ length\ d$

$f_{d}=number\ of\ expected\ terminating\ chains\ of\ length\ d$

$P(c_{d})=probability\ that\ at\ least\ one\ nonterminating\ chain\ of\ length\ d\ exists$

And here's the sample diagram (made in Word):

From these definitions and the diagram we can derive:

$e=a^{d}s^{d}k\$

$f=k(1-P(s))\sum _{d=0}^{n}a^{d-1}s^{d-1}$

$P(c_{d})={e \over e+f}={a^{d}s^{d} \over a^{d}s^{d}+(1-P(s))\sum _{d=0}^{n}a^{d-1}s^{d-1}}$

(This equation for P(c) is a better way of writing what was in the picture - the original version was actually slightly incorrect, and this one cancels out k)

$m=k\sum _{d=0}^{n}a^{d}$

That's everything. Thoughts? Tetronian ^{you're clueless} 16:23, 17 March 2011 (UTC)

OK, so the red circle at d=0, that is (say) our universe?

And what is the direction of the arrow between d=0 and d=1?

I mean, is the universe at d=1 simulated in the universe at d=0, or vice versa?

I am planning to write up a detailed explanation of what I am trying to say, but you probably won't have it for another 12-24 hours from the time of writing. --(((Zack Martin)))^™ 08:26, 18 March 2011 (UTC)

Sorry for the confusion - the universe at d = 0 is simulating the one at d = 1. Therefore the direction of the arrows shows the "direction" of the simulation - all of the d = 1 universes are simulated inside the d = 0 one, and so on. Tetronian ^{you're clueless} 12:12, 18 March 2011 (UTC)

Right, well that is actually back to front from what I am talking about. Reading your diagram downward, you have a downward infinite regress. But if you went the other way, upwards, you'd have an upward infinite regress. Not an infinite hierarchy of universes simulated on top of a universe, but a universe simulated on top of an infinite hierarchy of universes. I think, upward infinity is the issue for materialism - if materialism is true, there must be a basement universe; if the hierarchy is upwardly infinite, it is possibly basementless; but if it is basementless, then how can materialism be true? Or in other words, if we knew the probability of our universe being basementless, we can use that to discount the probability of materialism; but the probability of idealism can't be so discounted, since idealism doesn't care if our universe is basemented or not. --(((Zack Martin)))^™ 12:23, 18 March 2011 (UTC)

That would probably mean a few minor tweaks to the math; I'll see what I can do. Tetronian ^{you're clueless} 12:41, 18 March 2011 (UTC)

The latest incarnation

see here --(((Zack Martin)))^™ 06:01, 19 March 2011 (UTC)

Wow, that's very extensive - I'll reply as soon as I can. Tetronian ^{you're clueless} 14:39, 19 March 2011 (UTC)