Defining Measures Over Spaces Richer Than The Continuum

Cian Dorr asked an interesting question in the comments of my previous post:

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.

I'm probably not comfortable enough with the maths to come up with elegant treatments of mathematics with higher cardinalities than the continuum, but here's one way of doing it, though it is a bit kludgy. Let us start with a space that has 2^continuum many points. Instead of defining the fields, measures etc. over points, define them over equivalence classes of points, where the equivalence classes contain 2^continuum-many points each. For example, the distance measure needed to get the right predictions in the physics we do doesn't treat as different all the points "around" a given point. (I’ll talk in this entry about “space” though the remarks will carry over straightforwardly to spacetime.)

You might wonder what right these so-called points have to be called "points" if e.g. a distance metric does not distinguish them. Why aren’t the equivalence classes better candidates to be identified as the "points"? But there are a few answers available: maybe there's also a more discerning and more natural function F from points to somethings that does distinguish the points inside our equivalence classes, and our distance measure is a crude abstraction from F that is good enough for practical purposes. Or maybe the natural relation in the area is not a function on equivalence classes, but a relation between points that is uniform across these equivalence classes: that is, when we have two equivalence classes C and D, then if any member of C stands in R to any member of D, then every member of C stands in R to every member of D. Or it should be easy enough to come up with other marks of distinction that the members of the equivalence classes have that make them better candidates to count as points than the classes - maybe the members are the ultimate parts of spacetime, for example, or maybe we have general reasons for thinking classes can’t be points.

So the members of the equivalence classes might be the genuine points, and arbitrary fusions of them might have a good claim to be regions, thus giving us more than Beth2 regions. But the physical theories need not operate very differently - it just turns out that where we treated our mathematical physics as defining fields, measures, etc. over points, it should instead be treated as defining those quantities over equivalence classes of points. Of course, we are left with the question of what the fundamental physical relationships are that we are modelling with our functions, but I hope I’ve said enough to indicate that there are a number of options here.

If the model I have just given works, then it will be trivial to carry out a similar procedure to generate models of larger spaces: simply ensure that the equivalence classes contain N points, where N can be any cardinal you like. It does not work as smoothly once each equivalence class has more than set-many points, though that raises quite different sorts of problems.

These models mimic standard physics, and for some purposes we might want to see what sorts of models of higher-cardinality spaces we could come up with that exploit the extra structure of those spaces to produce more complicated “physical” structures. But the sort of model I described should be enough to raise an epistemic question I alluded to in my previous post. If physics as we do it would work just as well in these richer spaces, why be so sure that we are not in one of these spaces? I think some appeal to simplicity or parsimony is good enough to favour believing we are not in such a higher-cardinality space. But I seem to be more of a fan of parsimony than a lot of people. This case might be another one to support the view that many physicists implicitly employ parsimony considerations in theory choice, perhaps even considerations of quantitative parsimony!

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!