Against Pointillisme about Space & Time — Part 1

0:00 Oh, right, the stick. Excuse me. Okay. Well, thank you. Yes, well, I feel very lucky indeed to be in Oxford, but I feel especially lucky and grateful to be here this week. So, thank you. And to Friedrich also, and all the organizers in the society, thank you very much for the invitation. So, it's true that this talk will be along the lines that Michael mentioned. I want to combine philosophy of physics and analytic metaphysics, and I'm going to just wonder a little bit how to turn this on. In fact, I would like today, if I can with the time, to say more than the abstract discussed. I would like to combat pointillism, not just as regards the structure of space and time, but also in classical mechanics. And this is part of a wider campaign, which I will begin by telling you about. The wider campaign is to deny pointillism. What is pointillism? Well, it is in effect there written, the doctrine, that a physical theory can be written down in such a way that the fundamental quantities are all of them defined on or associated with points of either space or space-time or point-sized bits of matter, and they represent intrinsic properties of those points of space or space-time or point-sized bits of matter. So the idea will be that a solution of the theory or in philosophers' terms a possible scenario or a complete history of the system can be represented as an uncountably long conjunction, a continuously long conjunction of all the intrinsic properties of all these infinitely many points, all point-sized bits of matter. And so I especially want to target today pointillism about the structure of space and time,

2:30 and my target here especially is Phil Bricker in a 1993 paper, and quantilism about, in mechanics, the concept of velocity, and my target is Michael Tooley, 1988, Dennis Robinson, 1989, and David Lewis, 1999. And all of these authors, all four papers, have what I will call a common vice, namely they want to defend quantilism against the apparent extrinsicality of the fundamental physical quantities by saying we need to give a heterodox construal of the physical quantity. So these are all four of them metaphysicians who, in effect, want to revise a little bit the foundations of geometry or mechanics. And I would like to say that it's not necessary to make the revision because we don't need to be quantilist. The fact is that their heterodox proposals are interesting, and I don't mean to be rude about them, but I am no pointillist, and although their proposals are interesting, in the sense that they are motivated by pointillism, I do not share these authors' motivations. So there are going to be three parts to this. I'll discuss the wider campaign, then I'll talk about space-time, then I'll talk about mechanics. So let me begin with the wider campaign. First of all, I have problems of philosophical method. I am trying to connect physics and metaphysics this is dangerous but I would want to say that although we live in an apparently quantum world if you are involved in the philosophy of nature you should have an eye on the deliverances of empirical inquiry

5:00 In particular, the natural sciences, physics. And it is also worthwhile, this goes back to Michael's kind opening remarks, to take notice of the structures of classical physics. Because they are more subtle, more problematic, and they underpin the quantum picture of reality in ways that philosophers tend to ignore. So my talk will be disregarding the quantum today. In fact, my talk can also disregard the technicalities of classical physics, in particular classical mechanics. Fortunately, more important, it can disregard an ongoing philosophy controversy that some people here will be involved in, which is how to analyze or to understand a distinction that some philosophers feel very important. It is the one I mentioned, intrinsic versus extrinsic as a distinction among properties with a corresponding distinction among relations, which we will come to. So this distinction is controversial. But there is a distinction which almost all of my campaign can use, which is much more straightforward. It is the clearer distinction between what are sometimes called the positive extrinsics and all the rest. And a positive extrinsic property is a property which has the implication that its instance is not the only thing in the universe. Now some of you who are into analytic metaphysics will recognize this topic from David Lewis's famous 1983 critique of a proposal by Kim. Kim suggested that an intrinsic property is a property which is compatible with the instance being the only contingently existing object in the universe. For example, is spherical. Well, setting aside the existence of spatial points or space-time points, there surely could be a spherical object alone in space.

7:30 contingently existing. So spherical was intrinsic, said Kim. Lewis criticized this by saying surely the predicate is the only contingently existing object in the universe. That predicate surely represents an extrinsic property, but it is compatible with being. Indeed, it implies being the only object in the universe. the phrase accompanied for is not the only contingently existing object in the universe and use the phrase is lonely to be is not accompanied. So Lewis's point was that loneliness and accompaniment are clearly extrinsic, but that Kim's analysis was that to be intrinsic was to be compatible with loneliness, and that just didn't seem to work. All I care about, really, today, we can just take the positive extrinsics, which is implying that the instance is accompanied, versus the rest. That is at least a clear distinction. It's worth saying at the beginning something which... It's worth saying at the beginning that both the philosopher's distinctions, the unclear one and the clearer one, are different from three distinctions which come naturally to the mind of a person who studies maths, especially differential geometry or even just elementary calculus. So the three distinctions that you meet in geometry are intrinsic versus extrinsic, mathematician's words for the notion of the geometry of a surface considered as embedded in a larger space one speaks of its extrinsic geometry but considered in itself one talks of its intrinsic geometry well that's not the same as the philosopher's distinction another distinction which comes up is between scalar and vector and other more complex mathematical objects like tensor or connection

10:00 Now, we will find that the metaphysicians we discuss often assume that a scalar quantity is, in fact, intrinsic to the point that has that value of the scalar quantity. This is actually a mistake, I think, but it is a common assumption of, for example, Bricker's paper and other metaphysics literature. and the third notion that you find in the calculus and in geometry is the idea of what is often called a local property namely that it's determined by an arbitrarily small neighborhood around the point in question but that you do need to look a little bit beyond the point in question but you can choose how mean you want to be little beyond as you so this third one will be the one that is most important for us the second one will be a little bit important, the first one number one, the mathematician's use of intrinsic and extrinsic that won't be an issue in this discussion so the I want to tell you all my big messages at the beginning so that you know where I'm going here are two widespread claims in the metaphysics literature I think that classical mechanics supports quantilism for one thing classical mechanics is free of various kinds of holism which seem to occur in quantum Then you have a qualification. Classical mechanics supports quantilism as regards space. Roughly, this means it avoids what I call spatial extrinsicality. That is, when there are extrinsic properties, there are implications for the world beyond the instance. In particular, I have emphasized accompaniment, other contingently existing things. spatial extrinsicality is my word for when these implications concern matters of contingent fact

12:30 elsewhere in space rather than at other times. This is just my jargon. So there's a general idea, I think, in metaphysics that quantilism as regards space is supported by classical mechanics. but there is a concession not as regards space time there is an admission that classical mechanics does involve in its property attributions implications concerning other times and the reason is there is a metaphysical debate many of you will know between two ways to conceive identity across time there is what you might call the common sense fully fledgedly and completely and wholly one and the same object at two times for example a rock at noon and a rock at 1201 it is the self same rock Aristotle maybe in our own day and this was my joke to show my culture from my local neighborhood Wiggins is an Oxford philosopher but the alternative view in Hume certainly in Quine and David Lewis is that you should think of there being stages or temporal parts of objects so that there is the rock at noon or the rock from noon to noon plus five seconds and then there's the rock at one minute past noon or from one minute past noon to one minute plus five seconds there are these short lived temporal parts and it's really then that have the transient properties like being hot at noon or being cold at 12.01. And I'll be using the words, these are Princeton words, endurance for the traditional view, perjurance for the weird Lewis view, and persistence for the neutral word that covers identity over time, matter how you wish to theorize about it as a metaphysician okay so persistence is the neutral word and there's a famous argument which you find I'm happy to say all you know in philosophy and

15:00 in science almost everything comes out of Germany and Austria places like Oxford and Cambridge very peripheral, but in particular Leibniz in 1698 gives this argument, but it was reinvented by Broad in Cambridge in the 20s, and it was reinvented by Kripke and Armstrong in Princeton and Sydney in the mid-70s. It's an argument that in the classical mechanics of truly continuous matter which is, so to speak, like butter, naively understood, a matter which fills space on every length scale, no matter how small you wish to conceive it. It is not made of atoms. There are no gaps. It's butter, butter, all the way down to the tiniest length scale you care to consider. butterfield, but butter. So imagine this continuum, or or something. If you imagine classical continuous matter so conceived Kripke, Lewis, Leibniz Broad, they say there is an argument concerning such matter that in order to understand what it is for it to persist over time you must believe in endurance. Perjurantism cannot make sense of the persistence kind of matter. The reason is that there are two scenarios which are surely distinct which the perjurantist, the advocate of perjurance, cannot make sense of. One scenario is, as it might be, a disk which is not rotating and it's rigid. So there it is, just sitting there. In a space-time diagram, the world lines of the continuous matter, the world lines of the point-sized bits of matter, will go straight up the page. In a space-time diagram, time goes up the page, and so the world lines will go straight up. It's just sitting there. The other scenario is that the disk is spinning, and therefore the world

17:30 lines will be helical they'll form a helix and it looks as if, never mind the details, the perjurantist cannot explain that difference, he cannot describe that difference so there will be in that sense temporal extrinsicality that's the widespread claims what I say is that classical mechanics pointeist and it can be perdurantist points really are written down here I want to say against pointeism that classical mechanics in fact denies pointeism as regards space not just as regards space time it needs to use spatially extrinsic properties more than is generally realized and today I'll be concerned with needing them for geometry but in another place I would argue that the balance laws of classical continuum mechanics require also spatial extrinsicality and I actually elsewhere say I won't discuss this a lot today that classical mechanics of continuum compatible with perjurance and the way to overcome the rotating disc argument is to give up having only instantaneous temporal parts as the ingredients for your account of persistence over time if you allow yourself a small dose of temporal extrinsicality which is roughly speaking like the mathematics notion of local, if you allow yourself a small dose of temporal extrinsicality, then you as a perjurantist can accommodate classical continual. So, that's the end of the wider campaign. That's the end of the big picture. Now I'm going to settle down and produce a

20:00 of space and time. So the fact is, I now have to confess, it's less interesting than I have made it appear. Because I have said I have my targets, who are pointillists. And it's not so interesting as that, that there are these people going on saying, ah, pointillism definitely. There is a concession by the metaphysicians something non-puntilist to talk about geometry. It would be rather ridiculous to think you could talk about the geometry of space wholly in terms of intrinsic properties of individual points, without relations between them. But what is, I think, interesting and worth doing is to ask exactly what is needed. what are the minimum necessary non-quantilist components in the description of the structure of space and time. Okay, clearly geometry and mechanics use vectors and indeed tensors and connections, but I will specialize to the simplest case of vectors. And we will therefore be facing the question, Can a property of a point that is represented by a vector be intrinsic to the point? Or for short, can a vectorial property be intrinsic? So vectorial property, that's just my jargon for a property that's mathematically represented by a vector. And the pointillists, like my four authors, all say, apparently not. we agree vectors have length and direction and surely directionality suggests extrinsicality and then they tend to say well we will nevertheless argue for a yes answer and I will of course be saying no need for this their yes answer comes at a price namely heterodoxy about some mathematical theories quantities

22:30 and I'm going to say no need let's let's have a look at the art contemporary exponent the great David Lewis many of you will know his doctrine of Humey and supervenience is at the center of his metaphysical system and I think it's best for me not to try explain all the background notions of possible world and perfectly natural property and duplicate and so on which he would discuss this and I'll assume that but I'd just like to bring out here that these notions are at play in this famous quotation from a 1994 paper in which he states the doctrine and I will point out that he is admitting at the beginning that there are spatio-temporal relations of distance and a relation of occupancy between point-sized bits of matter and spatial or spatio-temporal those relations, he admits, are needed, and they are not, on the face of it, intrinsic to a point. So we're going to come back to this doctrine as time goes on. But now let's ask what then do we, in fact, need to talk about length. Length of a spatial interval, for example. Well, I said that all would admit you couldn't really do geometry wholly in terms of the intrinsic properties of points. And this can be made vivid by the schoolchild paradox of the length of a line. If you like, it's the joke with which your modern course in measure theory might begin as an undergraduate mathematician. it's also of course the joke with which your discussion of Zeno in your ancient philosophy class might begin because this is a philosophical conundrum or if you like a brain teaser

25:00 or we would also call it a chestnut in English you know we have many metaphors let us ask how would you of a line. Well, the length of a line should be the sum of the lengths of its individual non-overlapping parts. Those parts are points. Each of them is of length zero. We now know thanks to Cantor that there are uncountably many of them, but surely uncountably many zeros should be like finitely many zeros. It should add up to zero. So the length of the line is zero. Well, there is a philosophical and a technical reply to this sophistry, which is really familiar to us all. The philosophical reply is the one that matters more right now. It's to do with the understanding of length as, in fact, not thus summed up. The measure theory reply will simply say something like, you shouldn't accept the uncountable additivity It's only for, as it might be, countable or perhaps even finite collections that you should accept the additivity. In any case, this situation triggers a metaphysician to consider the application of intrinsic and extrinsic as a dichotomy to relations. and Lewis proposes the following trichotomy admittedly you see on the slide here 1 and 2 and you'll say well where is Butterfield's 3 but this will be obvious when we get to 2 there are some relations which supervene or are determined by the intrinsic natures of the relata the obvious of similarity or difference in an intrinsic respect so if you think that height is an intrinsic property then being taller than would be an internal relation on the other hand there are relations that are not internal but which intuitively are determined by the nature of the composite

27:30 of the two relata taken together now at this point there's a little bit more familiar philosophical machinery which Lewis and many analytic metaphysicians now would use, namely the theory of parts or myriology or the theory of fusions. It's in a way a suspect idea. They will say that if you take two objects, as it might be Michael and me, then unnoticed in everyday life but ontologically important and yet a free lunch at no extra ontological cost, there's a third object, the fusion of Michael and me. So Lewis's idea is that if you take, as it might be, a classical proton and a classical electron that form a hydrogen atom, and the electron is a certain distance, as it might be, a centimeter from the proton, and you take what he would call a duplicate of that system, the two together the duplicate that is to say something matching in all intrinsic respects must have its proton and its electron of course but they must also be one centimeter only apart so Lewis's vision is that once you consider fusions of relata duplicates or intrinsic replicas if you like of the fusion must match as regards the between the parts. So his example, his paradigm example of a perfectly natural external relation is spatio-temporal relations. And I have to admit, as a kind of sympathetic outsider to Princeton Australian metaphysics, that, don't worry, I think by the end of this talk but it's going to take me until the last slide to get them fluently into position so I would say that it's reasonable to say that okay let an interval be a fusion of its points and let us say with Lewis that if two intervals match in intrinsic properties then they are congruent so we're going to have this meshing between the notion of fusion and the treatment of temporal relations and then indeed I would say it follows from that that the length of a curve is

30:00 in Lewis's sense an external relation among its points and therefore if the curve is in fact straight we would simplify to what we call a metric space rather than the Gauss-Riemann conception of length as given primarily on arbitrary curves and you could say that it it would be a relationship between just the two endpoints. So far, so good for the Lewisite, or the Pointillist, really. The Gauss-Riemann conception of length as a line integral of a certain infinitesimal contribution along the curve seems to fit Pointillism, and everything is more or less fine. But there is trouble, or what I've called a devil in the details, meaning a little trouble that is in fact going to be big. To do this in maths, you need to attribute a tangent vector, and indeed something more complicated, a metric tensor, to a point in the space. And therefore, we are back at the question I announced would be central. Can a vectorial property be intrinsic to a point? and at this point at this stage I've become more specific now and tell you a tiny bit about Bricker Bricker has three claims he in his paper does adopt a Lewisian neo-Humian framework of worlds and perfectly natural properties and intrinsic properties understood in Lewis's way in terms of natural properties and duplicates is just the buzzword for two things that completely match in their intrinsic properties and i've given his three main claims ugly mnemonic labels in brackets these are not his labels in fact he doesn't distinguish these claims it's just that you can see as his paper proceeds that there are these claims first of all says the metric tensor as orthodoxly conceived represents an extrinsic property of a point but he admits that it is a local property it is determined by an arbitrarily small neighborhood

32:30 around the point and lewis's humian supervenience said that all the perfectly natural properties are intrinsic to points, right? So there is a conflict between metra extra and the metric being perfectly natural. The natural way to resolve this conflict is to say, never mind, some things that science needs, or some perfectly natural properties, are extrinsic to points. The metric tensor is one of them. It's perfectly natural. Science needs it. And it's extrinsic. All is well. A little bit of anti-pointilism. A small dose of spatial extrinsicality. That would be the natural thing to say. A little bit of localness. Instead of ruthless, demanding pointilism requiring intrinsicness to the points. But, says Bricker, this is no good, this response. Because I, being a neo-Humian about modality, want to have possible worlds composed in a principle of combination-like way. This is not on the slide, but this is a Lewisian doctrine about building possibilities by cut and paste from other possibilities. You can take distinct existences and put them together in a new possibility. As Bricker describes in some detail, and this is not on a slide, we won't go into it, it is in fact true that this proposal makes necessary connections between distinct existences, like a point and arbitrarily near ones, which Bricker regards as bad news. So bad for him, and this is where my philosophical judgment diverges from Bricker, so bad that he would rather say the metric actually represents an intrinsic property of an infinitesimal neighborhood, which we must now understand as a genuine mathematical object. And in order to do this, please let us go to Abraham Robinson's non-standard analysis

35:00 and understand neighborhoods as genuine objects in Robinson's way. a façon de parler for the normal calculus notion of a limit or as when we speak of arbitrarily small neighbourhoods along the lines of for all there exists for all. It's not that kind of façon de parler. Okay, so this is Bricker of course you already know my reply. I've said it but here it is again. I accept metra extra I accept that the metrics represents an extrinsic property but the argument against the natural proposal neo-Humian argument using possibilities that came in the middle of the last slide it depends on the quantiest assumption that two space time points with intrinsic nature's being given by the values of scalars are distinct existences and can be put together and in particular there's the idea of a scalar the scalar curvature which is explicit in Bricker for a few lines and it represents exactly in effect the clash between his philosophy and mine because as I said earlier not every scalar represents an intrinsic property of a point endemic in philosophy in particular Bricker to think of scalars as doing so but there is this animal thanks to Riemann the scalar curvature it's a scalar but it does code local nearby behavior of the geometry right so Bricker also makes I think another mistake he tries to support his heterodoxy by saying well another is two paragraphs One is conventional and it's fine, the second one I object to. First he says, if each neighborhood of a point in classical continuum mechanics has a finite mass and a volume, then indeed, and you have done this in high school calculus, the mass density at a point can be taken as the limit of the ratio of mass to volume as you consider smaller and smaller volumes around the point in question,

37:30 a tower of ever smaller volumes and indeed says Bricker mass density is in that sense extrinsic but says Bricker it is customary in physics to instead take the mass density to be a primitive scalar field there is then an intrinsic mass density at each point and the mass of a finite region no matter how small is to be obtained by integration of this intrinsic property of mass density which is primitive now my objection is that it is not customary in physics to do so right it is in fact customary to proceed according to the first paragraph and if you were to look at a modern measure theory treatment of rigorous continuum mechanics for example in Clifford Truesdale you would find it proceeding in the first way so this is my my objection and I think for perfectly good reasons ok so why should one do that well the reason is that if you postulate basic quantities at a point, you are liable to have conundrums facing you. Some of you have seen this slide before, so I apologize if you have. Here is a conundrum. Imagine a massive unit square expands fourfold in the way shown in the picture. It's just a straightforward linear expansion by this function f, which sends the point two-thirds comma one-third to the point four-thirds comma two-thirds. Now, in classical continuum mechanics on a plane, you imagine a mass density, and you can imagine that the unit square is of uniform mass density, resulting four times larger square is also of uniform mass density, but the mass is preserved, and therefore the mass density goes down at every point by a factor of four. So if it was as it might be, one kilogram per square centimeter initially,

40:00 it will then later be 0.25 kilogram per square centimeter, because I've made a four-fold expansion. if mass density rho is basic there is a question to Bricker how does so to speak rho know how to decrease under this expansion but if you state the mass conservation finitely then it follows mathematically from the nature of f that it will go down well this is if I may enough I think about geometry I won't go on about that I would like to move to my third stage and I think I have 10 minutes 10 to 15 so I want to talk about velocity in particular in mechanics so velocity as orthodoxly understood in the sense we discussed at the beginning. It's determined by the trajectory of the object in an arbitrarily small temporal neighborhood. For what goes on now, we can talk about point particles in a void. You may as well think of a point particle in a void, or if you prefer, an object small and rigid enough to be treated as a point particle. the spatial extent of the object for the rest of the talk will not matter so you can imagine a point particle well because it is local I would say that it is hardly extrinsic so what do I mean by that well first of all I do admit to the pointillist or to the metaphysician in general yes velocity is extrinsic it does presuppose the notion of persistence. So if I'm talking with a perjurantist such as Lewis, for example, I have to admit that when I speak about the velocity of the object or the point particle or the rock at noon, that presupposes that it exists at other times,

42:30 which for Lewis or a perjurantist would mean there are other temporal parts, So there are other contingently existing objects, namely these other parts. And the attribution of velocity, even instantaneous velocity at noon, in that sense, implies accompaniment. It implies the existence of other contingently existing things, and it is extrinsic. It's even, in my phrase, positive extrinsic. it's unproblematically extrinsic in the sense of implying accompaniment and yet these implications are not exactly enormous, in fact they're mild they are only that the object should exist for some open interval perhaps very very small an open interval of times around the time in question Well there's also another implication. There's an implication if it's instantaneous velocity that there's a certain common limit for the average velocities that you define either from later in time getting ever smaller or from behind earlier in time getting ever smaller. So there's an implication there about the limitingly good behavior of average velocities as we consider smaller and smaller temporal intervals around the time, say, noon in question. So it's a small dose of temporal extrinsicality. And similarly, for acceleration and higher derivatives of position, there's a small dose of temporal extrinsicality. But now that's the conventional position which I would endorse. Now there's the heterodox position that I want to target. So Thule in 1988 says, I propose that orthodox velocity should be reconstrued. I propose that we will adopt the famous techniques of Frank Ramsey and David Lewis. David Lewis in his paper How to Define Theoretical Terms of 1970 explains the Ramsey idea of simultaneously defining

45:00 from a single body of doctrine several, if you like, many theoretical terms by the requirement that they should all be the unique realizers of a functional role which you can syntactically extract from the body of doctrine. So for each of the terms there is a functional role which you extract from the theory and for each term you say it is defined as the unique realiser of this functional role. So velocity is to be the intrinsic property hereby postulated by me too-ly that there is such a thing as this intrinsic property which is the unique realiser of this role why say that Thule criticises the orthodox view on a variety of what I consider weak grounds but he has a positive case which is that he wants to say again this is