## Thursday, February 26, 2009

### How Many Regions of Spacetime Actually Exist?

I’ve been wondering how many regions of spacetime there are in the world. As often happens with philosophy, after thinking about the question I have no very firm opinion. But I think I can see what the plausible options are. So I thought I’d throw out a few of them, and commenters can tell me whether I’ve missed any important ones. I’d also be interested in what people think the right answer to the question is (I’ve indicated below what I’m guessing the majority answer will be.) For my purposes here I’m prepared to count both spatial regions and temporal regions as spatio-temporal regions, so if you believe in space and not time, for example, you may still believe in a lot of spatiotemporal regions in this generous sense.

Here’s what seems to me to be the main plausible options:

Zero: You’re likely to answer zero if you think that relationalism about spatiotemporality is true: that is, there are no pieces of space and time, just spatial and temporal relations between things. I suppose you might also answer zero if you think that ultimately science will reductively eliminate space and time in terms of some sub-spatiotemporal structure in the world.

One: I think this is probably the least plausible answer on this list. You might think the answer is one if you think spacetime is an _ens totum_ in a way that doesn’t even allow for sub-regions. Or you might think the answer is one if you’re a certain kind of monist.

A Large Finite Number: This covers a lot of options! When trying to work out which large finite number, the following considerations seem to be relevant:

1) Are there scattered sub-regions? I believe in unrestricted fusion for regions, but some people might not, if you think that the regions must all be connected, or have to partially be all at the same time, or something else. If you do believe in unrestricted fusion for regions, then that will put a constraint on the cardinality of regions: whenever the cardinality of jointly non-overlapping regions is N, the cardinality of regions altogether will be at least 2^N-1.

2) What is the size of the smallest regions? I think science is pointing us towards thinking the smallest regions of spacetime are a planck-length across and a planck-time long, which puts about 10^42 of them in each metre. But if you think the minimum size is bigger, or smaller, that’ll obviously make a difference to your final count.

3) How big space is and how long time is. If space or time was infinite in the large, then even if there were minimum-sized regions, the total number of them would be infinite. At least unless presentism or something were true - if space was finite, but “time was infinite” as the presentist understands that expression, we still wouldn’t have extra regions in the past or the future. Even if some sort of growing-block view were true, we could ignore an infinite future in our calculations, and just have to worry about past and present spatio-temporal regions.

A week ago I would have guessed the total number of space-time regions in the actual world was something less than 2^10^100. More even than the national debt of the USA, but significantly less than one googolplex.

Continuum-Many: If spacetime regions have a minimum, non-zero, non-infinitesimal size, then if spacetime is infinite in the large in the standard way (i.e. aleph0 metres or seconds), and we have unrestricted composition for regions, then the total number of spacetime regions will be 2^aleph0, which is the cardinality of the continuum. If forced to guess, this would be my current guess. You might also think this if you believe there are a continuum of distinct space-time regions in a small space-time region, but you do not think that unrestricted composition for regions is true.

Beth2: That is, 2^continuum many. I think this by far the most orthodox answer: if you think that space and time are made up of points in the usual way, or even gunk in the usual way, and you believe in unrestricted composition, then this seems to be the obvious way to go. If you have those common assumptions, it probably won’t matter to you for these purposes whether there are an infinite number of seconds in the universe, or an infinite number of metres. I predict this is the popular favourite.

Greater than Beth2, smaller than the first strongly inaccessible cardinal: This option doesn’t get a lot of love, but I don’t know why not. Why are the people who are so sure that there are points of space and time so sure that there is only a continuum of them in a metre or a second? There is a parsimony argument for picking the lowest infinity that will do the job, and I like those kinds of arguments, but does everyone else?

(Why stop at the first strongly inaccessible cardinal? No very good reason - though you’ll think it itself isn’t a great candidate, if you think the cardinality of regions is 2^(cardinality of points))

Proper class many: Maybe the spatial continuum has a greater cardinality than any set. Pierce thought something like that, though I don’t know anyone alive who does. Doing measure theory will be awkward if there are this many.

“More than proper class many”: if you think there are proper-class many points and you believe in unrestricted composition of regions, and that any fusion of points determines a region, you end up here. You might want to deny there’s a cardinality of regions at all if you think this, hence the scare quotes.

I’ve left out options for potential infinities, ontological indeterminacies and Meinongianism - this post is already too long!

1. Zero imo. (See Anaxagoras, Zeno, Melissos, Aristotle, Hobbes, Descartes, Leibniz, Maupertuis, and Kant).

2. I am extraordinarily interested to see what Jon Schaffer has to say about this post as it relates to super-substantivalism and monism.

*crosses fingers*

3. An option along the same lines as "more than proper class many" is that quantification over all regions of space-time is incoherent, in the same way that some say quantification over all sets is incoherent. I've been playing with a couple arguments for the view that this is at least a possibility.

4. Anonymous 11:20,

I'm not sure I see the relevance of either priority monism (PM) or super-substantivalism (SS). I suppose that (SS) gets you the claim that there are at least as many regions as there are material objects. But, I don't think it gives any more guidance than that. (PM) doesn't give us anything about the answer to the question in the post, does it? Though, I suppose it might be relevant to answering questions about how many fundamental or non-derivative regions there are.

5. Great question, Daniel! I'm afraid I'm going to disappoint Anon 11:20, but I'd bet on Beth2 regions, for exactly the reasons you provide:

1. There is an empirical question as to the structure of spacetime. Our best theories (e.g. General Relativity) treat spacetime as pointy. Maybe if Quantum Loop Gravity proves to succeed we'll get a lattice-like picture, but for now I think the money has to be on a pointy 4d manifold. So that gets us to continuum-many.

2. Then there is the metaphysical question of how composition works for regions. I like unrestricted composition generally, but I think that even those who don't like unrestricted compositions in some domains (e.g. material objects) don't find it nearly so bad for spacetime.

Of the rest: all the greater than Beth2 options strike me as possible but without any empirical support (that I know of). The zero and one options have something to be said for them, depending on one's background metaphysics. But anything between 2 and continuum-many strikes me as equally without empirical support.

6. I go for 5 regions.Well, why not?If JoScha is avails himself of sending philosophy to holidays and let the guys from the physics department do the job for him and Nolan seconds him in that, then why not ask your local magician? I did so and the answer I got was 5.Believe it or not.
I am actually quite surprised by Schaffer´s answer.Given the Hawthorne/Arntzenius position
on the matter, I would have thought the question of the existence of points would belong to the a priori realm once more.
What´s more, I am stunned by the opinion that gunk theory couldn´t propose an empirically adequate reconstruction of General Relativity
in kosher terms.
Even more unargued for seems to be Nolan´s position that there is empirically evidence for
Planck´s length.I wonder how physics is thought to establish that these regions can´t be divided.
I know that you have both papers where you argue for this claims.But I am not convinced in the least.
Nolan does a fine job in putting all the other options on the table.But just as in Hypergunk etc. there is not the slightest idea of which
of the options to prefer.If you the discount the "argument from parsimony",that is.But such
an argument is nonsense. So I declare it a mystery, which means that I don´t have a clue.
P.S.:
Jonathan Lowe, put your hat in the ring and go for One.

7. My inclination, for what little it's worth, is to say either 0 or 1. (Failing that, I suppose I'd think continuum-many. I'm not an unrestricted-composition sort of guy.)

8. Daniel: I wonder how you would do physics in a spacetime finite volumes of which contain more than continuum many points? The physical theories of spacetime I'm familiar with are all based in ordinary differential geometry, which is about finite-dimensional manifolds, which by definition are locally isomorphic to R^n. I don't know how you'd begin to define, e.g., the notion of the gradient of a scalar field, if you were trying to work in something bigger.

9. Daniel: a combination of curiosity and ignorance makes me ask: Why would you pose the question? What else hangs on it?

In any case, here's an obvious way to address it:

(1) First ask the question is: What is the topology of spacetime? Here we ask physicists. Mavs41, we don't ask magicians. Physicists are better than magicians at telling us about spacetime.
(2) Then ask: What is a region? I'm not sure there are any illuminating answers here, except fiat. (Maybe this is because I don't understand what hangs on the question.) One good answer is: a region is any open set in the topology.
(3) A bit of maths now answers our question.

This approach probably gives the answer "Continuum many" for reasons other than those suggested by Nolan. If spacetime is thought of as R^4, then there are only continuum-many regions (i.e. open subsets) of R^4. (Of course, if for some philosophical reasons you don't like points, you like, you can rework that thought in terms of a pointless topology.)

Final point. Absent any version of the continuum-hypothesis, there's an incommensurability between answers in terms of the continuum (e.g. beth^2) and answers in other terms (e.g. more than the first strongly inaccessible cardinal; more than proper-class-many, etc.) I don't see how you'll resolve those without answering the question: Is the (generalised) continuum hypothesis true?

10. There's been lots of interesting discussion so far. I'll come back to some of the earlier questions in a bit. In response to Tim, I wasn't asking the question in order to get at some other interesting question, though it seems to me that there are plenty of interesting questions connected with it.

That's an interesting suggestion that regions are all the open sets of the topology. I suspect that's only a prima facie tempting answer for some topologies - in the usual sort of discrete space that would have the result that there were no regions, I think, which would be odd.

I'm afraid I'm not seeing why a lack of an answer about the generalised continuum hypothesis leaves us with incommensurability between answers in terms of beths and answers in terms of inaccessible cardinals and proper classes. As far as I can see, the powerset axiom is enough to guarantee that the cardinality of the continuum is set-sized (and so less than proper-class many). It's also clear that the cardinality of the continuum is not strongly inaccessible, even without any continuum hypothesis.

Without the generalised continuum hypothesis, we will have an indeterminacy in whether many answers stated in alephs are larger or smaller than answers stated in terms of beths. But I'm not seeing how that causes trouble for comparing answers in the terms I'm using.

There are axiomatisations of set theory that leave it open whether continuum-many is proper-class many, I know. (I think ZF minus the power set axiom is one of them, but I haven't checked.) Of course if we were using such a weakened set theory, we would need something extra to make the relationships between the various answers I considered above clear. I was assuming we had at least ZF plus urelements (since I wanted sets of non-sets). That's obviously open to challenge too.

11. Daniel: point absolutely taken about CH; my bad. I guess what I had in mind was Cohen's (sometime) view that the classical continuum is bigger than any aleph. With Choice, that's an odd thing to say. One way to understand it is using a predicative set-theory, so that the "full power set" of omega is proper-class-sized. But if the full power set of omega is indeed a set: yes.

Regarding: "in the usual sort of discrete space [claiming that regions are the open sets] would have the result that there were no regions". Do you have a particular space in mind? It's certainly not true generally, e.g.: in the Discrete Topology (proper name) any arbitrary set of points will be a region.

12. Tim: you're right about Discrete Topology. About ten minutes after posting I started thinking that the thing I was calling the "usual sort" might not have been usual at all!

What I had in mind were models where there is zero distance between adjacent minima, and the minima are all the same size. I was thinking that there wouldn't be any open regions on such a space since the adjacent minima would be as close to each other as they are to themselves. (Luckily "closeness" isn't transitive in such models!)

I was of course wrong to say that that way of thinking about things is "usual". Probably more usual is the sort of thing where each minimum is at least the minimum (non-zero) distance from any other. Then any arbitrary set of minima will be open. In talking about distance I'm going beyond purely topological notions, of course, but I hope it's clear enough what's implied about the topology.

Now that I think about it, even the thought that the idiosyncratic model I described above would lack open regions is mistaken - the whole space would be open, and the null region too (if there is one). So there'd be one or two regions in such a space.

Out of curiosity, why do you think Choice makes Cohen's sometime view about the continuum odd, or any odder than it would be already?

13. It just occurred to me - another way to get more regions in the model I described is if you have disconnected chunks of spacetime - each maximal disconnected chunk will count as open. We could quibble about whether we are dealing with one space anymore if we allow disconnected chunks, but as long as the chunks all co-existed this option could be relevant to the question I started with in the post.

14. Daniel: on the topic of discrete geometries, you might be interested in a paper by Forrest in Synthese 1995, which presents a very tractable geometry for discrete space. But discrete space is a hot topic in philosophy of physics; I think there are *lots* of proposals floating around right now.

Some quick thoughts on allowing spacetime to be disconnected:

(1) How do we know about disconnected regions? I'm not a verificationist, but if there's no path from us to them...

(2) If there can be disconnected regions, then we can give arbitrarily large answers to your question; just postulate arbitrarily many disconnected regions. This probably makes your question less interesting. (Though maybe not: it depends why you asked the question!)

(3) Talking about actual disconnect spatiotemporal regions relates intriguingly to Lewis' understanding of a possible world; for him, disconnected spatiotemporal regions are separate possible worlds.

[Re: the tangential question about Cohen and Choice. I just meant that, if the continuum is larger than any aleph, it's larger than any well-orderable set so, given Choice, the continuum is larger than *any* set. Despite the Gricean implicature, I didn't stop to think about what happens when Choice fails!]

15. Tim: I'm a fan of that Forrest paper, and you're right that discrete space is a topic of current interest in physics - I think the smart money should be on spacetime turning out to be discrete, though there's plenty of smart money still on the continuous option (See Jonathan Schaffer's comment, above).

In response to the quick thoughts:
(1) It would be tricky. I can imagine coming up with e.g. symmetry laws that predicted disconnected spacetime regions, or even plain old induction - Bigelow and Pargetter's paper "Beyond the Blank Stare" has some nice thought experiments along the latter lines.

(2) There are _possible_ (or at least feasible) answers of all sorts of large sizes if there can be disconnected regions, but I'm not sure why that would make the question less interesting.

(3) Yes. (Well, almost yes, Lewis leaves it open that there could be spatiotemporally disconnected pieces of one world that were related by certain sorts of relations that would be analagous to spatio-temporal ones.) I take it Lewis is wrong about what is involved in things being worldmates, though obviously if he is correct then we can only have (spatiotemporally) disconnected spacetimes if they are connected in one of the "analagously" spatiotemporal relations he talks about.

16. Sorry if I haven't read the thread carefully enough, but I didn't see an answer to Tim's question: why is "how many regions are there?" an interesting question? You replied: "I wasn't asking the question in order to get at some other interesting question", but didn't give any positive motivation for the question (as far as I've seen).

From a purely mathematical point of view there's of course some interest in asking what kinds of geometrical structures can be defined, as you've been discussing. And, independently, there's interest in asking what mathematical structures are needed in order to work with General Relativity or any other given (or speculative) physical theory. But I don't presently understand what interest there is in the question of "how many regions of spacetime there are in the world". Could you explain this a little further?

Thanks.

17. Not wanting to speak for Daniel, but I can see interest in the question.

First, you might just find the question intrinsically interesting. Either there's a fact of the matter about how many regions of space-time there are, or there isn't, or the question is ill-formed and needs to be reposed. The latter two options look interesting in themselves (e.g. what exactly is wrong with the question in the last case, given it seems to be formed from words we understand; what is this sense of "no fact of the matter" in the second one). If there is an answer to the question, what's the reason *not* to be interested in it? (I don't mean: devoting one's life work to it, but I guess no-one is proposing to do that)

But I don't think that intrinsic interest exhausts the reasons you might have for asking the question. If there are only finitely many regions of space-time, that poses problems for some positions e.g. in the philosophy of maths, that presuppose that there are infinitely many. (Some of Field's work on conservativeness of mathematics presupposes we've got infinitely many regions, IIRC; some of his later work on determinacy gives similar partial results, on the basis of assumptions about space-time). If there are proper class many regions, then certain interesting theses in set theory with urrelemente look bad (e.g. Vann McGee's urrelemente axiom, that the urrelemente form a set, looks threatened). On the same assumed answer, model-theoretic semantics typically assumes that the domain of quantification is a set, so if there's a model-theoretic "intended interpretation" of English, then not only do our most unrestricted quantifications miss out on some sets, they also miss out on some concreta. Implicit methodology might also be revealed by asking the question: e.g. if we think that beth2 is the right answer, but we could find equivalent theories with more points and so more regions (maybe in the sort of way Daniel sketches in the later post) then it looks very much like we're relying on something like quantitative parsimony. Finding reason to believe that quantitative parsimony is operative in our thinking about such things is potentially significant as we might then feel on more solid ground appealing to it in other areas.

Even if you don't find those questions pressing, they illustrate, I think, how answers to questions like this can provide constraints on theoretical space in a host of other domains. So you might find the asking the question interesting in order to store up ammunition for later debates.

18. Maybe just asking "why not be interested?" was too quick and flip. I guess I was thinking that working out the consequences of best physical theory + your favourite metaphysics looked prima facie interesting to me; it then looks fair to ask whether there are positive reasons for thinking that the question is uninteresting.

For my own part, I'm kinda tempted to eliminate anything that smells like mereology from metaphysics, so that might leave me with points as the only real regions, with general "region-talk" being ditched in favour of plural talk about points (and subregion talk being replaced with "is among" ideology). Presumably there'd have to be interesting plural relations among pluralities of points to give you the required structure.

That still leaves open the question of how many points there are, of course.

19. Under the beth2 section you said: "if you think that space and time are made up of points in the usual way, or even gunk in the usual way, and you believe in unrestricted composition, then this seems to be the obvious way to go".

I thought the usual way of thinking of gunk was the algebra of regular open sets in R^n (of which there are only continuum many... I think. A countable basis would be the spheres of rational radius and rational centre.)

I think the continuum answer will actually be much more popular than you suggest because of gunk lovers...

20. Andrew: You're right about gunky spaces - that was a silly mistake of mine. More power to the continuum answer! I should have twigged when Tim mentioned only counting open sets of points as regions, above, since that's the classic model in pointy space of gunky space.

Stuart: I'm not sure how to answer your question about why my question is interesting, because there seem to be a number of ways of interpreting it. One is a question about the history of my being interested in it - what caused the interest? Another is a request that I make it interesting to you, if I can, though without knowing much about what you are interested in that could be difficult. Of course, sometimes "why is this interesting?" is used to suggest that it's not interesting, or to convey that the speaker isn't interested and doesn't want to talk about it. I assume that's not what was going on here, if only because it seems a silly response to a blog entry when you're not under any obligation to read it, let alone comment on it.

Despite my worry that I don't know where the question is coming from well enough to offer a satisfactory answer, I'll have a quick go. I am interested in the question for its own sake, as Robbie Williams suggests: I suspect a lot of metaphysicians find large-scale questions about fairly fundamental features of our universe to be often interesting in themselves.

I also think that the question is connected to a lot of interesting metaphysical and methodological questions about spacetime and how to investigat it. For example, whether it is infinitely divisible and if so how; what role infinities higher than the continuum should play in physical theories, if any; what mereological principles hold for space and time; and a number of others that have already come up just in this comments thread.

I might be able to say some more in the way of "positive motivation" if I had some idea of what you were interested in and/or thought people should be interested in.

Oh, one last issue that might be in play here - some people (maybe Stuart, maybe Tim) might be wondering whether I'm asking a metaphysical question, or one relevant to metaphysicians qua metaphysicians. (This is a metaphysics blog, after all.) I think I am and it is, but I won't try to defend that here, except by pointing out that it seems tangled up in a number of more traditional metaphysical questions.

21. When I asked "why ask the question?" I just wondered what impact the question might have on other areas of metaphysics. It seemed like an odd question to ask, just for interest's sake, because I expect the answer you want to give will drop out immediately from other views you have, namely (as I said in my first comment):
(1) what is the topology of spacetime?
(2) what do you count as a region?
And I wondered: why not ask those questions directly? (Of those questions, I happen to think that the first is very interesting, and the second isn't. But maybe that's just me.)

Regarding Robbie's comments as to why the question is interesting. People like Field certainly have a problem with the Axiom of Infinity. Fair cop. But I don't see the immediate problem with saying "there are proper-class many regions". One obvious way to make sense of that is using three theories:

(S) a theory of the pure sets that can also make sense of talk about proper classes; perhaps NBG or MK.
(P) a mathematized physical theory which talks about regions of spacetime.
(M) a model-theory, say ZFCU, which takes the regions of spacetime as urelements.

So McGee's axiom holds in M; and in M, we can consider the "intended" models of P and S, and assert that there is a bijection between a proper class of S and the points of spacetime. Finally, quantifying over all the regions of spacetime---a sentence of P---is given a semantics in M.

22. Hi Tim,

So the idea is that there are more than S-set many regions, but there is an M-set of regions? I guess I get it, but I wonder about what reasons we'd have to believe in S-sets. After all, presumably if you have M-sets around, we'd be able to fairly charitably interpret S-set and class talk as talk about a certain initial segment of the M-sets (though it depends on exactly what theory S is).

But maybe your point is that we can have a model theory for a language on which "there are more than proper class many regions" comes out true---by interpreting set-talk as not ranging over all the sets used in the model theory (which presumably is enforced anyway if the domain of interpretation is set-sized). Is that what you were thinking of?

23. That should have been "there are proper class many regions", not "there are more than proper class many regions". Sorry

24. Hi Robbie: I had in mind the following (although I think there are plenty of ways to cut this cake):

S gets interpreted in M. This probably means that the S-sets are an initial *inner* segment of the M-hierarchy. (An inner segment, because we want our model of S to consist only of pure sets, but the M-hierarchy is a hierarchy with urelements.)

The question "why believe in S-sets?" now comes out as "why believe in a pure subcollection of an initial segment of the M-set hierarchy?". The answer to that will be "it is a theorem of M that that collection exists".

25. Hi Tim,

Right, I get it. So in the object language we can't truly talk of sets of anything other than other sets (I guess not plausible as a hypothesis about English set/function-talk, but we can run with it). Maybe with second order resources we can still make sense of the claim that there are more than set-many regions. And from the metalinguistic point of view, there's an M-set of all regions.

Just to bring this out: maybe S could be a set theory without axioms of infinity, receiving an interpretation about the hierarchically finite M-sets. Then to say "there are more than set many regions" in the object language is true if there are infinitely many regions. The proposal is kinda like that, but with bigger infinities involved.

Of course, one could then wonder about whether there might not be more than *M-set* many regions flying around. Obviously there can't then be an M-set of all regions; the McGee axiom would fail for M; and if you're doing model theory with M-sets, the intended interpretation of the object-language can only contain some of the sets (e.g. an initial inner segment, as per your description) and only some of the regions (M-set many of them).

I can imagine using these sorts of claims about number of concreta (or maybe modalized versions of them) to argue for the possibility of quantification over more-than-M-set-many items (assuming in particular the pure sets are necessary existants). That's potentially interesting, I think, because we'd then have to give up on the idea that the interpretation of every language has to have a (M-)set-sized domain. And so one obstacle to the idea of unrestricted quantification over all sets would be undermined. But I agree that if all we had was more than S-set many regions, where S is from an external point of view just an initial inner segment of the M-sets, this wouldn't get started.