Some variant of Tegmark 4 seems intuitively obvious to me. Indeed it's hard for me to see how any other model fits, once I realized that there really is Nothing Special About Us. The only universe-sets that make sense to me are the empty set and the "all possible universes" set. (with the exact criteria for possibility unknown)
I say this not because it answers the questions raised, but because it's been knocking around in my mind for years without having anyone I could say it to that would get the point.
Some interesting stuff about our conceptions of the world might fall apart if you adopt the mathematical universe. If you think that the entirety of mathematical structures exists in the same way, than it is hard to think what happens when you decide to do good to someone with the entire structure. The whole thing just "is there". Your decision could be thought of as a computational process that takes place in many different subsets. But the exact opposite decision still takes place where it takes place. Then you get something complicated in which your decision ends up conflates with self location in the near future. As if you deciding something doesn't change the whole, but only where in the whole are things of the "you" kind to be found.
And then, citing Lewis becomes helpful to find out about the minimal levels of complexity we are dealing with: As suggested above, let us call an individual which is wholly part of one world a possible individual." If a possible individual X is part of a trans-world individual Y, and X is not a proper part of any other possible individual that is part of Y, let us call X a stage of Y. The stages of a trans-world individual are its maximal possible parts; they are the intersections of it with the worlds which it overlaps. It has at most one stage per world, and it is the mereological sum of its stages. Sometimes one stage of a trans-world individual will be a counterpart of another. If all stages of a trans-world individual Y are counterparts of one another, let us call Y counterpart-interrelated. If Y is counterpart-interrelated, and not a proper part of any other counterpart-interrelated trans-world individual (that is, if Y is maximal counterpart-interrelated), then let us call Y a -possible individual. Given any predicate that applies to possible individuals, we can define a corresponding starred predicate that applies to -possible individuals relative to worlds. A -possible individual is a -man at W iff it has a stage at W that is a man; it -wins the presidency at W iff it has a stage at W that wins the presidency; it is a -ordinary individual at W iff it has a stage at W that is an ordinary individual. It -exists at world W iff it has a stage at W that exists; likewise it -exists in its entirety at world W iff it has a stage at W that exists its entirety, so - since any stage at any world does exist in its entirety - a -possible individual -exists in its entirety at any world where it -exists at all. (Even though it does not exist in its entirety at any world.) It -is not a trans-world individual at W iff it has a stage at W that is not a trans-world individual, so every -possible individual (although it is a trans-world individual) also -is not a trans-world individual at any world. It is a -possible individual at W iff it has a stage at W that is a possible individual, so something is a -possible individual simpliciter iff it is a -possible individual at every world where it -exists. Likewise for relations. One -possible individual -kicks another at world W iff a stage at W of the first kicks a stage at W of the second; two -possible individuals are -identical at W iff a stage at W of the first is identical to a stage at W of the second; and so on.
Notice [David Lewis's Possible Worlds] is distinct from the Mathematical/Ultimate in that there may be properties of non-mathematical kind.
I don't see why Tegmark 4 excludes those properties. Such a universe is still mathematically possible, in a sense akin to logically possible: it violates none of the Peano Axioms, for instance.
So the reason is that Tegmarks claim is that the the mathematical properties not only define the Multiverse, but also that they constitute the entire extension of it. If there were substances, properties, or objects, that behaved mathematically well, that would still falsify his claim.
It is less crazy than it sounds the more you study philosophy of physics I suppose. It basically depends on accepting or not that matter could be just relational properties, with nothing intrinsic.
It's a leap of faith to suppose that even our universe, never mind levels I-III, is exhausted by its mathematical properties, as opposed to simply mathematically describable. And I don't really see what it buys you. I suppose it's equally a leap of faith to suppose that our universe has more properties than that, but I just prefer not to leap at all.
What would it mean for our universe not to be exhausted by its mathematical properties? Isn't whether a property seems mathematical just a function of how precisely you've described it?
Let's start with an example: my length-in-meters, along the major axis, rounded to the nearest integer, is 2. In this statement "2", "rounded to nearest integer", and "major axis" are clearly mathematical; while "length-in-meters" and "my (me)" are not obviously mathematical. The question is how to cash out these terms or properties into mathematics.
We could try to find a mathematical feature that defines "length-in-meters", but how is that supposed to work? We could talk about the distance light travels in 1 / 299,792,458 seconds, but now we've introduced both "seconds" and "light". The problem (if you consider non-mathematical language a problem) just seems to be getting worse.
Additionally, if every apparently non-mathematical concept is just disguised mathematics, then for any given real world object, there is a mathematical structure that maps to that object and no other object. That seems implausible. Possibly analogous, in some way I can't put my finger on: the Ugly Duckling theorem.
I don't understand why restrict to "mathematically possible" universes, and arbitrarily exclude those which run, say, on cartoon physics, on magic, or completely randomly. These are far easier to imagine, so surely they are more likely to exist?
I don't understand why restrict to "mathematically possible" universes, and arbitrarily exclude those which run, say, on cartoon physics, on magic, or completely randomly.
In what sense are cartoon physics, magic systems, or random number generators not mathematically possible?
Can I do mathematically-impossible things like solve the Halting problem if you give me a copy of a Bugs Bunny cartoon or a random number generator?
How about a time-turner which solves the Halting problem (you exit the loop if and only if every algorithm halts)?
Sure, I can imagine a magic box which accepts any algorithm and tells you whether it halts. Therefore a universe with such a box has a right to exist.
Sure, I can imagine a magic box which accepts any algorithm and tells you whether it halts.
Doesn't that just mean your imagination is self-contradictory?
In what sense? It does not make me go mad or anything, it's just one of the many programs my brain runs.
I have very high confidence that the following question will not make your imagination explode. However, if you are at all worried about that, please stop reading.
What do you imagine the magic box will do if you feed it the algorithm: "Feed this algorithm to the magic box. If it says it halts, then go into an infinite loop. If it says it doesn't halt, then halt."
If you presuppose that the universe is not "mathematically possible", you can't really prove that it is possible for anyone to ask it that question. For that matter, it might just say "it halts" and be right. (It's a mathematically impossible world, and you're using math both when you're deciding what the algorithm will do and what the box should answer.)
By the way, the usual statement for the halting problem is that you can't make an algorithm that solves it, and by algorithm it usually means something a Turing machine, i.e. everything the algorithm does can be done by a Turing machine. In this case, assuming it makes sense to use math to reason about it, “Feed this algorithm to the magic box” is not actually something a Turing machine can do (it only has heads and tapes, no magic boxes). If you also give it the magic box it's no longer just a Turing machine, it's a [Turing machine + Turing oracle], which is something a bit different.
Imagine the magic box accepts algorithms expressed in Lisp (which theoretically allows unlimited memory). How do you express "feed something to the magic box" in Lisp?
By the way, does anyone know if it's proven impossible (or even if discussed) to build a limited halting-problem solving algorithm that works for all algorithms except those that contain (complete or limited) halting solver algorithms as subroutines, in which case they also halt but say something like "don't be an ass"?
Well, either your magic box can't cope with algorithms that talk about the magic box itself, or there's a contradiction going on.
Nothing's bad about it, but I don't think you can actually imagine the thing you said you could!
Maybe we have different definitions for the term imagine. As far as I'm concerned, by describing your question you imagined it. If you are worried about it being logically inconsistent in this particular universe, imagine a universe where an algorithm's halting behavior changes after it's been fed through the magic box in question. My universe - my rules. Or lack thereof.
Okay, at this point I think we have different definitions for "universe". The one you're describing can't be consistently described.
I think we have different definitions of the term "consistency". If you define it as "lack of contradiction in the classical first-order logic", then sure, but why be so restrictive?
Absolutely. It generates numbers at random and in one universe, it happens to always be right.
I don't see the relevance, though.
The relevance is that since our imagination runs on the Turing machine of our brains, whatever we can imagine is as likely to exist as any construct based on mathematical axioms, like Tegmark level 4.
Why are you jumping from some symbols being rearranged on a Turing machine to assuming unknown and arbitrarily complex instantations loosely resembling said symbols? Of course a brain 'imagining' something exists on level 4, but why credit any particular form of imagination as being coherent and also greater than mathematics? If you imagine a square triangle, how is that a refutation of Tegmark level 4, rather than, say, evidence that a brain can emit two words in succession which don't mean anything?
So basically, that's all that your point boils down to? "never mind the failure of millennia of imagination-based reasoning and the striking success of mathematical reasoning in those millennia, I'm just going to make imagination the arbiter of metaphysical possibility even if that means embracing contradictions and other such nonsense"? That's pretty lame.
So you refused to understand my original point and resorted to misrepresenting, strawmanning and eventually insults? Nice. Tapping out.
There are at least ten different conceptions of how the World can be made of many worlds.
But are those just definitional disputes? Or are they separate claims that can be evaluated. If they are distinct, in virtue of what are they distinct. Finally, do we have good grounds to care (morally) about those fine distinctions?
Max Tegmark's taxonomy is well known here.
Brian Greene's is less, and has 9, instead of four, kinds of multiverse, I'll risk conflating the Tegmark ones that are superclasses of these, feel free to correct me:
I don't understand branes well enough (or at all) to classify the others. The holographic one seems compatible with a multitude, if not all, previous ones.
Besides all those there is David Lewis's Possible Worlds in which all possible worlds exist (in whichever sense the word exist can be significantly applied, if any). For Lewis, when we call our World the Actual World, we think we mean the only one that is there, but what we mean is "the one to which we happen to belong". Notice it is distinct from the Mathematical/Ultimate in that there may be properties of non-mathematical kind.
So Actuallewis= Our world and Actualmost everyone else=Those that obtain, exist, or are real.
The trouble with existence, or reality, is that it is hard to pin down what it is pointing at. Eliezer writes:
and elsewhere
Now another interesting way of looking at existence or reality is
Reality=I should care about what takes place there
It is interesting because it is what is residually left after you abandon the all too stringent standard of "causally connected to me", which would leave few or none of the above, and cut the party short.
So Existenceyud and Existencemoral-concern are very different. Reality-fluid, or Measure, in quantum universes is also different, and sometimes described by some as the quantity of existence. Notice though that the Measure is always a ratio - say these universes here are 30% of the successors of that universe, the other 70% are those other ones - not an absolute quantity.
Which of the 10 kinds of multiverses, besides our own, have Existenceyud Existencemoral-concern and which can be split up in reality-fluid ratios?
That is left as an exercise, since I am very confused by the whole thing...