David Braddon-Mitchell and Kristie Miller have a fun new paper arguing that we can both believe in spatially extended mereological simples and agree that every extended object o has a part at each sub-region of the region which o occupies. The trick which is supposed to enable us to marry these two theses is to claim that space has smallest regions which have no sub-regions. (This is a position I've heard defended in conversation by others as well, so I'm coming round to the view that it ought to be taken seriously.)
For many of us, quantized space seems a very strange idea on the face of it, but Braddon-Mitchell and Miller argue that it must be taken seriously because of certain results in physics. They claim that 'physicists tell us that we cannot divide up space into any finer-grained regions than those constituted by Planck squares [i.e. areas of 10 to the power -66 centimetres squared]' and that physics 'tells us that talk of space breaks down altogether once we talk about regions smaller than the Planck square'. 'Hence', they conclude, 'we know that talking about something occupying a sub-region of a Planck square makes no sense: there is no such sub-region' (p. 224).
When they give a few details of what the physics actually shows, it turns out to be that 'there is nothing that could be taking place within these squares'. Braddon-Mitchell and Miller take it (and this is where I get puzzled) that this 'is to say that in principle, there cannot be anything that occupies the sub-regions of such a square' (p. 224, their emphasis).
For all I know, physicsists may well be claiming this sort of import for that sort of result, but I can't see how the transition could be that straightforward. What I can't understand is how any claim about what can or cannot take place with a Planck square - which is of course the sort of thing physicists can helpfully tell us about - could settle the question of whether such a square has sub-regions. Specifically, I wonder what sort of results are supposed to distinguish between the hypotheses:
1. that Planck squares have no sub-regions
and
2. that any Planck square (and therefore any sub-region of a Planck square) must, as a matter of nomic necessity, be uniformly filled.
The claim that nothing can be 'taking place' within a Planck square seems at best to get us as far as 2. What justifies the further step to 1? What sort of result from physics as it is currently practiced could possibly justify this further step?
(I ask these questions in self-confessed ignorance of the physics, but with a dose of scepticism as to whether we can get this much metaphysics out of it, whatever it is!)
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Hi
In writing the paper we were more concerned with a conditional claim. The that if physicists are right about space-time being an essentially macroscopic phenomenon that supervenes on properties at a scale where there is no space time, then this has the interesting consequences for the debate about extended simples etc that we describe.
The relatively brief comments about entropy and the planck square were not to actually give the argument from physics for the antecedent of the conditional, but to give the flavour of a more complete one. I don't think that argument is adequately spelled out in the philosophy of physics lit, though it is gestured at. The rough idea is that there are a number of features which are taken to be definitive of space-time, and most of these features are absent at very small scales. Hence the need, as in the paper to distinguish between the a priori idea of space that gives us the idea of scale, and the notion of space or space-time that is a theoretical term in physics.
So what disntnguishes (1) and (2) in your formulation is that whatever 'fills' the square does not have the features that play the theoretical role given to space-time in physics. Even if it does have what we toy with calling the 'Kantian' features in the paper. But spelling this out is something that needs doing...
Hi David,
Thanks very much for that. I guess I should read some of the relevant stuff before reasserting my scepticism ... (although at the moment I'm confident that it will survive the process)!
Actually, though, there seems to be another independently interesting step in the argument - where it is taken that whatever plays the theoretical role physicists assign to space (as opposed to the thing you're calling 'Kantian' space), is what determines the sense of sub-regionhood that should be inserted into the principle that extended objects have parts corresponding to the sub-regions of the space they occupy. If the principle that we're intuitively keen on is really one involving the Kantian notion, a position which saves a different principle doesn't solve all our problems.
What's to stop there being parts which are smaller than what for physicists' purposes is considered the smallest unit of space? Like the Planck square itself, these parts will be extended in the Kantian sense, although they are not extended in the physicists' sense. The parts won't be able to behave as objects ordinarily do, but do they need to? Perhaps some version of the principle that we're interested in *all* the parts, not just the non-arbitrary parts, could help support a negative answer to this question.
Just a quick question, from a position of utmost ignorance.
Sometimes when people talk about granular space-time, it sounds like they think of space like a piece of squared paper, where each minimal region of space is one of the squares.
But that's not directly motivated from the picture that events occupy regions of a certain minimal size, even if you add in the presumption that nothing could be a region unless an event can occupy it.
Intuitively, it's consistent with the constraints just mentioned that the minimal regions should (intuitively) "overlap": that A and B may be adjacent non-overlapping regions of space-time, and there exist a minimal region of space-time C that overlaps the right-hand half of A and the left-hand half of B.
Of course, we'd then owe some account of what "overlap" meant (it better not mean "share a subregion"). But there are options: maybe the "pseudo-regions" of the "Kantian space" mentioned could do as surrogates. Compatibly, maybe overlap could be understood mereologically. More spectulatively, maybe there's some way of cashing overlap out in terms of nomic necessities concerned the locatability of objects at the respective minimal regions: e.g. A and C overlap iff it is nomically necessary that nothing could wholly occupy A without partially occupying C.
So what I'd like to know is: is there a consensus among granular-space-time-lovers for either the overlapping or the non-overlapping conception of how space is "granulated"? And if so, is the reason for preferring one over the other based on physical considerations, or metaphysical considerations?
Great question Robbie, though as I'm also totally ignorant as to what the actual granular-space-time-lovers think I'll just hope that someone more knowledgeable might read this thread and answer it!
I'm not by any means an expert either, but this kind of thing has been discussed a fair bit around Oxford. I believe a kind of 'overlapping' conception is the most popular, because it's seen as the most physically plausible - any preference for it would certainly not be based on purely metaphysical considerations. Overlap would need to be explained, certainly, but I suspect that doing so is non-trivial - at this stage you'd need to make a connection with the measurement problem. Quantum effects can't really be discussed coherently except in terms of one interpretation or other.
Carrie, you seem to be asking how the move can be made from the physical discovery that there can be no differentiation within the Planck squares to the metaphysical claim that they have no subregions, which seems to be equivalent to the claim that as a matter of metaphysical necessity there can be no differentiation within the Planck squares. My guess is that this does seem to most physicists to follow, because most physicists don't draw clearly, or at all, the distinction between metaphysical and physical necessity.
Braddon-Mitchell is absolutely right to distinguish between the "the a priori idea of space that gives us the idea of scale, and the notion of space or space-time that is a theoretical term in physics." It's the latter which is of interest to physicists, the latter which they take to have discovered to be granular. It's not that the physicists take this to have conclusions for the former; it's just that they don't really care about the former at all.
Interesting paper, thanks for pointing it out.
Physicists don't (at least I'm not aware of any that do) make the claim that the plank scale is the smallest possible region of space-time. Given the limits of general relativity and quantum mechanics at present it is in principle not possible for us to measure a length smaller than a plank length because the tools we use to measure length (photons at this small scale) would have to have such large energies (greater the energy the shorter the wavelength) that the photons would then create small black holes and disappear into the black hole leaving us with no measurement. It is a practical problem of no tool to measure rather than a metaphysical limit to space-time.
Well, from the math/physics side, I can add some detail. The Planck length, 10 to the -33 cm, popped up in real-world measurements. Things like the relationship between the energy of a photon and its frequency. Nobody could explain why this number was what it was. It just made the equations work. For instance, if you assume that light can interact with things with ANY of energy, some of the equations "blow up," they give impossibly large results, sometimes infinity! These "un-physical" results, like a prediction of infinite energy in some experiment, disappear if we assume that energy (and light, and so forth) can only interact in multiples of some minimal unit. When you plug in the Planck length, the equations work! They predict what will happen in many real-world situations. And experiments have verified the existence of this fundamental length many, many times.
It's one of those things where, once you know to look for it, you find it everywhere, and it makes things work! Think of it this way: the Universe seems to stop keeping track of decimal places once you get to that length. It seems to be as closely as the Universal Calulator ever checks itself. Anything smaller than that gets completely "uncertain." Many physicists treat the Planck length as the basic unit of length, so what we normally think of as a centimeter is then 10 to the +33 Planck lengths. In fact, a lovely thing happens when you treat the Planck length as 1, and the speed of light as 1 - most of the math becomes MUCH simpler.
And just for reference, the Planck length is, in centimeters:
1/1,000,000,000,000,000,000,000,000,000,000,000 cm
And the speed of light, in units we are used to, is about:
670,000,000 miles per hour = 1,080,000,000 km per hour
Wow, hard to imagine! Hope this helps.
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