12th UK Foundations of Physics — Classical Mechanics & Kolmogorov-Sinai Entropy

0:00 Testing, one, two, three. Testing, one, two, three. Testing, one, two, three. the foundations of physics conference um just a couple of uh information uh matters i need to give you and then harvey's going to say a few words and then we'll start the session um coffee will be served at 9.30 each day, coffee and tea and biscuits. Please make use of them because we've all paid for them. If you want to come early before 9.30, please do. If you find yourself at the loose end and just want to sit around in the department and chat, it's usually open from at least, well, from 8 o'clock, 8.30. Coffee should be here, as I say, from 9.30. Lunch will be served at 12.30. Again, pile up the plates because we pay for this. And that will be again in the foyer on the main tables there. And we'll break for tea around 3 o'clock. There is a sign-up sheet for an end-of-conference dinner on Thursday night. No one so far has signed up. But if you would like to, I need to give the The Chinese restaurant, I need to give them the final figures by about 6 o'clock today. And the cost should be around £15, not including drinks. If you're interested in going out on the Thursday night to the Chinese restaurant, please sign up on the sheet. If anyone owes me money, please find me at some point, or a large gentleman will find you. With regard to the bursaries, I'm afraid the arrangement is we can waive the registration fee, but accommodation bursaries, money for accommodation, I'm afraid photographers have to pay us and then we get the bursary money from various disreputable entities like the Institute of Physics and the Analysis Trust. but they insist that we have to provide them with a full account before they give us the money so I'm afraid you have to pay us and then we can pay you back so

2:30 some people expect that's it there are any problems or concerns come and see me the program has changed slightly some people have dropped out but I hope you agree we have a very interesting set of talks ahead of us, and I'm going to hand it to Harvey for a few words. I'm sure that many of you knew Jeeva Anandam, and some of you may know that Jeeva died several weeks ago of modern neurone disease. Jeeva was, of course, a very distinguished theoretical physicist. There were several effects in physics named after him. I can think in particular of the Mandan force, electromagneticism, and the Oharnoff Mandan geometric phase, generalization of Barry's work, and quantum mechanics. But, of course, he was well known for his work on quantum phase effects and per space-time and many other things, theoretical physics. But, in the, Jima was a natural, natural philosopher, always interested in foundational issues. And in the 90s, the mid-90s, he was a regular, very distinguished visitor to Oxford and began to entertain the idea of doing formal graduate studies in the philosophy of physics in Oxford. eventually was accepted into the de-fill program, did indeed a de-fill on the roles of the symmetries and laws in geometry and modern physics, fine thesis. So his interesting foundations of physics principally were related to notions of symmetry and in particular his work on the physical foundations of geometry and physics. I think it's very, very beautiful. I learned an enormous amount from Jeeva's very special combination of mathematical virtuosity, but tremendous physical insight, physical intuition. So his loss will be keenly felt in the community foundations of physics, and he leaves behind him a son from his first marriage, who I think now is in his late teens or the 20s, and his wife, Pratchima, his second wife, Pratchima, and a four-year-old wife, too.

5:00 Incidentally, the diagnosis of Mildgarten Neuron Disease was not well known outside his immediate family and his department in South Carolina and the diagnosis was made within the last few moments. Thank you, Harvey. Our first speaker needs actually no introduction. Jeremy, from the University of Oxford, talking on the title, Classical Mechanics is Not Pointless and Can Be Perderantist. Thank you. Well, good morning everybody, and many people have been thinking, as Tony Sudbury told me he was, that it's a bit frustrating now, because no online dictionary that Tony Sudbury's students looked at defined either of the isms in the title. So a certain amount of explanation is more than usual. So this is actually a talk about, it's a project that I'm afraid only halfway through, about, in effect, the relationship between analytic metaphysics and continuum classical mechanics, that's the next focus. So this is my first foray, actually, to try to make philosophy out of fluid mechanics or continuum classical mechanics. And the idea is that analytic metaphysicians are attracted by a vague doctrine, quantism, which classical mechanics violates. don't know that, so there's a message to them in Africa. It's of some interest also for this community, I suppose, to hear that message. The second message is a kind of permission message that there's a doctrine, perjurantism, which the analytic metaphysicians have articulated, and then they say classical mechanics vetoes it for continuous bodies. However, I want

7:30 that once you give up the quantilism, which they are in Amadon, you have the permission to be a perjurantist. That's the idea. So it's not quantilism, but once you give up the quantilism, you can be perjurantist. So what are these doctrines? Well, quantilism is named after the art movement. And the art movement of Seurat draws a picture with dots. And the metaphysical doctrine is very roughly the doctrine that you can fully describe the world by going point by point through space, or space-time, and describing intrinsic properties of points. So for space, it would be something like the instantaneous state of the world is given by the point-by-point description of points' intrinsic properties, or perhaps the point-sized bits of matter's intrinsic properties. and in space-time context the doctrine would be that the whole history of the world is given by the point-by-point description of intrinsic properties this looks suspicious I think to most philosophers or foundations of physics people but it's attractive to analytic metaphysicians of a Humean stride not to everyone but those who have in the tradition of David Hume in analytic metaphysics now tend to be contracted by it. So perjuritism is a doctrine which is one side of a two-sided debate. This is the debate about the persistence of objects over time and how to understand that in philosophy. And there is what you might call the common sense view that an object persists over time by fully existing at two different times, say noon and 1205, it's wholly there, call the object X, it exists at noon, it exists at 1205, that is called endurance, fully-fledged identity of something over time. On the other hand, perjurance is the rival doctrine, not very commonsensical, but quite quite natural, I think, to physicists and mathematicians that you should talk of there being a stage, also called temporal part, as truly existing at noon, and a 1205 temporal

10:00 part of the stage exists. Now, when you first hear this, most people in physics or maths tend to think that this sounds like a distinction without difference. And there is, in fact, a kind of formal equivalence between these things, but I think that it As always in philosophy, at least in my opinion, you can get these formal results, but then when you go beyond that, you end up creating a climb. Various kinds of equivalents are plausibly broken by philosophical considerations of a multifaceted and endlessly controversial And so it's true that the doctrines of this talk will, I'm afraid, lead into that kind of mind, or run into the sands, or lead to this open sea of controversy. And I think it's crucial that I keep it short and allow time for discussion to get us back on track for a level of thought. So what I'm going to do, because it's now 10, 20, roughly three years ago, I'm sure that we can push these. Yeah, but I think what I'll actually, rather than bringing you the words on this slide, I'm talking about having a couple of words during a general discussion. There's some of this, and some of this, I think, is for this audience called Or due to my grandmother to . That is an expression for teaching people what they already know. If you want to know more about the different European language expressions of this idea, you need a batch of a little bit of attention. Okay, so... Well, it's a good thing I'm clever enough to change the font size, isn't it? Well, I couldn't, of course, kind of attack your Sufi and Robert Bishop, but that might be the idea of my difficult way of speaking. Thanks. Thanks. So just saying a bit more precisely, quantilism have broken down into a trio of doctrines

12:30 at which classical mechanics is widely taken to support. The fundamental quantities of the theory can or even should be and defined to be at the points of space or space-time, that as subs they represent intrinsic properties of those points and the models of the theory which a philosopher would call possible worlds according to the theory but a mathematician would call solutions, are fully defined by a specification of the quantities, values of all such points. Now, in a way, this doctrine, the exact words don't matter, of course. It's an ism. It's vague. And one of the aspects of controversy and constant matter of judgment for a philosopher to write about this kind of thing is what exactly you should say about bringing something pretty well-defined within physics, like a quantity of a specific physical theory into contact with a vague, broad, metaphysical idea like interesting property. So there are all kinds of issues of intellectual judgment in this game that I'm trying to play, and that's where differences will arise. In any case, what I admit is that classical mechanics does indeed proceed in terms of discussing extensionless bits of matter, and one thing was the Boscovich, in fact, this is already in Euler 10 years before, but normally attributed to Boscovich, the idea of a point particle. And the analysis of continual, by which I mean, of course, a body continuously filled by matter, not a swarm of point particles, but a truly continuous point, then only described in terms of the, as it might be in space-time terms, the world lines of the extensionless bits of matter forming a congruence of curves. And classical mechanics treats these extensionous bits of matter as enduring. You write, as it might be, qi of t for the position at time t of the i-th particle, and the label i seems to label once and for all the i-th particle, treated as one and the same throughout its existence. Okay. Well, I want to say, and this may not be used to many of you, but classical mechanics

15:00 has to take extended regions and their attributes as primitives, even though such regions overlap. And in that sense, the description is redundant. Now, there's an awful lot of aspects here in the sense that well, roughly speaking we typically think I think metaphysicians tend to think of holism as associated especially with quantum theory where there are special features and as Holger Lyle was explaining to me last night there are these things like homonomies in classical theories you can gloss as introducing a kind of non-quantilism within classical physics. But what I will be stressing is something more elementary, that when you do rigorous classical continual mechanics, and this is already in Euler in the 1770s, you need to attribute attributes, properties and relations, to extended spatial regions. But once you've rejected quantumism, this is getting into second gear. I mean, I'm still in the first gear, but it's getting into second gear. I'll get into second gear and shout louder, surely. So in any case, this remark is a remark about spatial extent being needed for classical mechanics. But once you deny quantism in that context, I want to argue that you can see the merits of anti-quantilism as regards time as well. And this gives you, as we'll see, the chance to be a perjuritist. So the background, again, one of the generalities to a general discussion, the background to this is that I think it's quite helpful to think of there being two pictures of classical mechanics. And these are my words, pictures that I think are worth articulating. in motion picture, which for present purposes is not problematic, and there's a

17:30 particles in motion picture which I deny. So the matter in motion picture is the idea, of course there's an enormous amount of discussion one could have about velocity of space and time, the notion of mass, the notion of force, but setting all that aside, for the present purposes, this unproblematic matter in motion picture is the idea that matter consists either of point, it's rather agnostic, it consists either of point particles, or it consists of extended bodies which are small enough and rigid enough that you will treat them as point particles, or it consists of extended bodies that are not treated as point particles, but they are rigid enough that you will describe them by manageably small finite number of coordinates, and in particular you will approximate them as rigid. So that view is agnostic about the micro constitution of extended bodies, and it goes along with a rather instrumentative philosophy of classical mechanics as a long enterprise of modeling. But who knows, even concerning a very small, apparently rigid body, what is in a constitution is. All we know is and the word particle in many, many mechanics texts is used in that sense. In particular, you finesse issues about what exactly happens when classical bodies contact one another, which is a topic, a rather deep and difficult topic in the foundations of mechanics. The more opinionated, and I think misleading picture, urges that if there are continuous bodies, not swarms of point particles, but truly continuous bodies, then one should analyze them in terms of their extension of the bodies. And that's what I would deny for continuous bodies. But I admit that these doctrines, quantumism and the particle-emotion picture, making it more precise, are plausible for whole particles. And so this is a little bit out of context here, but I will be arguing, along with this doctrine, that the perjuritist, not being a quantumist,

20:00 use a notion of instantaneous velocity or allied notions like momentum to account for persistence. That's something of a side remark about my observation. Then occurs on this slide a quick attack on free intellectual communities for forgetting two centuries of work. This will be no news I can't say it anyways to encourage a five-layer. So let's begin by attacking the people who are predominantly absent from expositions. I think that's the safe strategy. So modern philosophy of nature in analytic metaphysics tends to take these two pictures as unproblematic, and actually I want to urge the giants, Euler, Lagrange, Poisson, Hamilton, through to Hilbert, worked on this, worked on the problems associated with these pictures. That's why Hilbert's sixth problem says, looking back over the two centuries of work in foundations of classical mechanics, we've constructed the question of providing a realist formulation, we use it's expertized mechanics and probability, in a sense. These issues were largely sidelined in the quantum and relativity revolutions. But in the hands of people like Truesdell and Walter Knorr, they have become active research areas, again, often conducted at the mathematical engineering departments rather than the physics departments. As you may know, Truesdell has wonderfully rude remarks about books that you thought were good, like Goldstein's classical mechanics, many physicists introduced classical mechanics, which he disparages in his unpublished review, because vetoed by the journal editor is too rude, he disparages it as, you know, just treating classical mechanics as a mere prelude to quantization, and as an exercise in teaching students how to solve ordinary differential equations. Because for the finite dimensional case, You can make classical mechanics look like that. So the fact is, I think, that analytic metaphysicians tend now not to know that these problems look.

22:30 And I think that partly for several reasons. One is the split in natural philosophy around 1700, so that philosophers, typically, they know their Leibniz, Locke, and so on, and Hobbes and Descartes. they're Euler or Lagrange. Once you get into the 17th century, 18th century, you're only reading Hume, Barclay, and Kant. You're no longer reading natural philosophers in the sense of the 17th century. But it's also true that analytic metaphysics, of course, was forged in this revolution, logic and conceptual analysis are siren calls, luring the next position away from the deliverances of science. And it must now turn into attack philosophers of science, a different community. They have been lured from these wonders and problems by quantum and relativity revolutions themselves. of course one politically said historians of science as well, tended after Kuhn to treat the period 1700 to 1900 as just a single paradigm, but an awful travesty of the complexity and subtlety of the issues as well. And of course, as Truesdell's disparaging review of Goldstein brings out, physicists also forget the demands of physics pedagogy, getting people to learn quantum and relativity means that the physics community tends not to look in such detail at classical continuum mechanics, or fluid mechanics, and in some ways. Okay, so the next point is to urge that quantumism is quite widespread, and the first example here is the great and greatest family-led physician of our age, David Lewis, who, as many of you will know, has a doctrine that he articulated in mid-career as an overarching theme of many of his specific claims and arguments, called human supervenience in honor of human, and formulated here as a matter of, in a possible

25:00 world like ours, the fundamental relations that describe the world and the fundamental properties are as follows. There are these basic temporal relations, and then their relations, They're not, in that sense, intrinsic to any of their relatifas, their relations, but in addition to these spatio-temporal relations, there are what he calls local qualities, by which he means intrinsic properties of points or point-sized bits of matter or other kinds of occupant of point, and which are perfectly natural, which actually is a technical term in his metaphysics. But for the moment, we just need only worry about the intrinsic. Everything else about the history of the world supervenes is determined by this space of chemical arrangement. Now, two comments about this. One is that there's an elementary paradox that is often discussed in philosophy class. does the length of the line get determined by the lengths of its parts this is a xenolite paradox if the parts if you're allowed to decompose it into non-overlapping parts by taking points, the length of a point is certainly zero and now admittedly there are an uncountable number of points but presumably the length of a line segment should be the sum of the lengths of a non-overlapping decomposition their lengths. And an uncountable sum of zeros should surely be zero. So surely the length of the line is zero. This paradox is, well, you can answer in the discussion period, but there are various things to say about it. And as a concession to that, Lewis and others admit, in effect, that you need to postulate spatial or spatio-temporal relations, geometry or chrono-geometry, as not given simply by the point-by-point intrinsic properties. And the question then is, is it only for geometry and chrono-geometry that you need these non-supervegan properties of

27:30 And the metaphysicians discuss, to some extent, how vectorial properties, like electric field or the instantaneous velocity of a point particle, seem compulsory for signs, but being represented by a vector, they're apparently extrinsic to a point. And quantumists, in fact, respond by typically, by defining the property in a heterodox way. want to make the point here that at least in some metaphysics literature you will find metaphysicians feeling the attraction of quantumism so strongly that they will work hard to redefine velocity or vectorial property or electric field in a way that does that's not according to the textbook treatment in a careful physics or mathematics book. And examples here are Phil Brecker and Michael Tooley and Piccolo and Parjica, and also recently David Alvarez on the idea of instantaneous states. Okay, so that's, let me now try to introduce a bit more this perjurance and endurance thing, and I think what I'll do combine the discussion to about three more, four more slides, then we've had it some time to talk. So, as I admitted at the beginning, classical mechanics that it's normally written down certainly looks as if with that QI of T, that it is assuming the unproblematic, fully-fledged identity of particles over time, the label I. and in any case most people in science when they hear philosophers to make this distinction their eyes will glaze over they will say this sounds like a verbal difference talking about the part or stage as the thing that really exists at a single time, and in fact, I can make that equivalence, also, that suspicion, solid by rendering it as an equivalence, and I've done it, though, in a very simple way on this slide.

30:00 You could say, just as the worldwide of an enduring particle, here it is, defines stages simply by restricting the function, I'm taking closed variables only as the restrictions, so a meshing collection of world lines of stages, here they are, defines a unique world line. You see, there's an equivalent in the sense that each object mathematically defines the other, and those who speak one way can easily, by mathematical definition, construct the objects of the other way. So a meshing collection of world lines and stages just means it's a collection of functions outlabeled them by their domain, a closed interval. And they need to mesh just in the sense that when you have overlaps of domains, they need to agree on the value, as in the picture. So those red lines are meant to be in the same place They're not meant to be just beside one another. And that's for a single-point particle. But what this point, this remark, as you can see, generalizes to any number of point particles labeled by lambda, and this can be done in two ways, only the simplest way is on the slide. If you imagine that you have a collection Q lambda, well, this lambda is like I in my out loud discussion earlier. you imagine you have these Q lambda defining various stages, then obviously you can introduce a double indexed family which meshes for each lambda and just say the very same remark again, this is trivial. Actually, you can also write down, it's not on a slide, a singly indexed family of Q's, call them the Q mu's, and you can lay down conditions on a collection of Q mu, all of them functions from some or other closed interval of reals into R3. You can lay down conditions on that very easily, you know, freshman mathematics, which make it provable that their index mu could be decomposed into lambda, comma, A, B. So that the the apparent invocation of underlying endurance by having a double-indexed family can be avoided,

32:30 though it's not on the slide. But let me stress that when you do that, this little equivalence depends upon the Q mu's having a domain which is not just a single point. The closed interval cannot be generated. It must have a little bit of temporal width. talk about the red lines meeting, matching, whenever the domains overlap. But if you're just given the Q-mues with all the domains of all of the Q-mues being a single point in time, then you can't talk about meshing, because all you're doing is saying, well, there's certainly at each time a whole bunch of field points, and at another time a whole bunch of field points and no threading going off. Okay, so that actually is of course the germ of what metaphysicians might know already as the so-called rotating disks argument against perjurance. But that's coming off the next slide. Your land idea is ranging over a set of any finality. Yes, that land could be... That could be cut. Yes, it could be cut. Okay, so is it a distinction without a difference? Well, as I said at the beginning in the preamble, I do think actually much as I would like to resolve philosophical disputes by proving the utter equivalence of the two sides so that it can condemn the two disputes to talking past one another, I always find when I try to do this that there's a philosophy still to be had. And I think, therefore, that there are these philosophical differences that remain. And there's a common question that both sides face in the so-called discussion of criteria of identity over time. It's a discussion which predates the endurance versus perjurance fight, which has happened perhaps predominantly these last 50 years. but the discussion of the criteria of identity over time, in a sense, goes way back. And there are various aspects to this and various replies to a given precise version of this question.

35:00 But the two things I want to say here, one small, one large, are just to specialize down to classical mechanics again and to emphasize first of all that the notion of instantaneous velocity does presuppose, well it doesn't presuppose endurance, but it certainly presupposes the notion of persistence. Sorry, I shouldn't emphasize. Persistence, throughout my discussion, is the neutral word. For somehow or other, in some way or other, metaphysically not yet decided how to think about it, an object exists at two times. That's persistence. Endurance and perjurance are the two rival ways to think about it. And instantaneous philosophy certainly presupposes the notion of persistence. After all, you can't give me an instantaneous velocity that's superluminal by saying, well, you can't give me an average velocity that's superluminal by taking as the distance traversed divided by the time of the traversal, by taking as the distance Butterfield at noon and some piece of the sun at 12.03. It's got to be Butterfield at the later time, whose position you consider. So in that sense, the notion of persistence is involved in the notion of instantaneous velocity. However, for point particles, there's a sense in which you can answer this question without invoking instantaneous velocity by, in effect, saying track through space-time the points of occupation, track through space-time where matter is. Since we're, I'm imagining a point of articles without collisions, so the idea will be that looking at a time slice, there are certain points of space that are occupied, And assuming trajectories are continuous, if you like, you can simply talk about tracking occupation by matter. If the particles are not distinguished by mass or by some other intrinsic property like their color,

37:30 then you, in a way, are condemned to talking about just tracking occupation. but of course if they are distinguished by their mass or another intrinsic property like color then you can talk about reconstructing persisting objects by saying well the color doesn't change so of the five that exist at noon there's only one that's yellow and just identifies the persisting yellow thing all the instantaneous stages each of which are yellow So what about continuum? Well, the rotating disk argument actually has its precursors in remarks of Leibniz, but it was first stated really by Broad and independently rediscovered by Pripyat and Armstrong in metaphysics 20 years ago. And the idea is that it's a compulsory distinction in our metaphysics of matter to distinguish a disk which is not rotating and made of homogeneous matter, a continuum, and a disk that is rotating. and indeed amounts of rotation should no doubt be distinguished one from another. Now this argument is not necessarily cast in terms of disks. It could be cast in terms of spheres or rivers and it isn't necessary to just make the distinction of compulsory stationarity versus rotation. It could be amounts of rotation. But I'm going to choose disks and I'm going to choose rotate versus non-rotate. is that if you are a perjurantist and you are temporally quantumist, so that you wish to construct the persistence or reconstruct the persistence of objects over time by looking only at instantaneous intrinsic properties, you will have trouble with a homogeneous disk because you won't know whether the, you don't have the resources to recover whether or not particles are like spaghetti when it first comes out of the packet, all vertical and

40:00 parallel, all helically twisting, as in rotation. Well, that's the idea of the argument. It's an argument against perjurans. It's saying, perjurgists, you don't fly. You're not going to work because there's a compulsory distinction concerning persistence of the spatial parts of homogeneous matter. And this compulsory distinction is one that you do not have the resources to recover in your metaphysics. Now, there is various things you can say about this. And one of them is said by a good friend and colleague of many of us, Craig Callender, who pointed out, in effect, that the literature about this, first of all, didn't take sufficient of the literature about the philosophy of space and time, Newton's market and what not. And furthermore, the dynamical effects of rotation were not sufficiently recognized. Well, I am going to play the metaphysician's game to this extent. I'm going to say, after all, rotation is definable without dynamics. You can talk rigorously about the rotation of a time-like conference or a time-like vector field without having dynamics. You need spatiotentrical, metrical structure, but we already gauged of that on a previous slide called space, right? So stress or strain, I'm imagining to be suppressed, in effect, in the rotating disk. Just imagine that it's not there. And there are various replies that analytic metaphysicians have given to this, but the—well, let me skip over this. We can talk about what David Lewis himself said and how he got zapped by Dean Zimmerman, a U.S. analytic metaphysician, and we can talk about what's happened since Zimmerman was at Lewis, Ted Seider, this is, I don't know, to my very partial knowledge, perhaps the latest move in this, Ted Seider

42:30 in a book that came out two years ago, tries to revive a Lewisian proposal for dealing with the rotating disks at the very end of this book, and this is cast in terms of, now if you, my ultimate decision, you can just forget all this, this is cast in terms of the fully fledged Lewisian ideas of a Mil-Ramsay Lewis account of laws of nature, together with the clever Lewis simultaneous functional definition of problematic theoretical terms. So the side that one should simultaneously define both dimension of persistence of what he calls gen identity and the laws of dynamics as the best system in total simplicity and strength in recovering facts about gen identity over time and causal laws of how things move. Okay, so never mind what all that is. If you don't know about it, The point is it's extremely programmatic. It's cast in the language of modern analytic metaphysics. And I think one can do better with an analogue which is less metaphysically committed and it's closer to the detail of empirical inquiry. And this is what this slide is about. And I'll say this and then I'll stop. we can talk more about the anti-quantilism across space of classical mechanics in the discussion period. It seems to me that, as so often in philosophy, a certain theoretical framework or program can lead philosophers to, so to speak, not say the obvious thing or to tie their hands behind their back intellectually so that they have to struggle to either assert and proposition they want to reserve or understand the notion that they'd like to understand. Because to some extent in philosophy, that is the game. I mean, the whole idea is that a competent philosopher knows where not to understand a notion. Because having said, my God, I don't understand that notion of substance. You mustn't then say, oh yeah, I understand alpha, when your opponent has said, well, what about the concept of alpha? And then they'll leave you from alpha, which you admit to, back to your problematic notion. So trained philosophers are savvy about when not to understand.

45:00 And that's because they're suspicious that it will lead them to understand things that they've officially declared. But, although that is part of the art, it's a very odd subject. I think there are some obvious things to say that it would be nice to be able to make this distinction between the rotating and non-rotating this from a perspective of perjurantism. And, of course, the fundamental intuition is to allow you to have a little bit of instantaneous velocity or full momentum or temporal extent in the notion of state that you use. And, well, this could be made formal in many different ways. And in particular, when we actually look at dynamics, we know that in Lagrangian Q at T-naught and Q at T-one, for sufficiently nearby T-naught and T-one, so there aren't these problems of the focal points of what I want, there's a unique determination of the world-wise in simple cases. And in Hamiltonian mechanics, with Q at P, P, well, they're put on a par with Q, which does interpretatively have the meaning of instantaneous momentum, momentum, and therefore, like philosophy, it presupposes the notion of persistence. So with a little bit of mild and uniform acceptance that you are going to presuppose the notion of persistence, you can, in my opinion, make some sense of the rotation-non-rotation distinction from a plagiarist perspective. And so the way I think this happens is that in the case of white particles, all kinds of criteria of identity in fact converge, following the unique color that I mentioned, following the track of matter occupation, and therefore in the actual practice of elementary classical mechanics in the schoolroom or the research lab, we're constantly not worrying about identity over time because we have many convergent everyday criteria which we then find when applied gives us a successful dynamics of the type laid down by the founding fathers

47:30 and using those criteria these dynamics thereby earn their status as something like laws of nature or law life. And at that point, the Quinean web of belief takes over that, in effect, the success of these dynamical schemes begins to give them the right to dictate, to contribute to the determination of when things at one time are the same object as things at another time. You solve one dynamical problem and therefore know at time final what Q and P Well, Q and P at the time final of that dynamical problem are, thank you very much, Q and P initial for another dynamical problem. Similarly, as I said about the Lagrangian, the determination of world lines by the dynamical scheme can act as one of the several criteria of identity to which you can appeal. Well, then, in that case of particles, all done, with that case all done, one extends both the doctrines of persistence, formal and informal, and dynamics, continua, on analogy with one and two, in an inhomogeneous fluid, of course. some quality like mass density or color imagine mixing paints as a child so it varies across the fluid and you can appeal then to unique color on the other hand dynamics is set up by various kinds of limiting processes or analogies with particles and using these two efforts of these two, dynamics can earn its status as law, and the web of belief begins to apply again. Okay, so in that sense, I hope to have sketched how to be a quadriarchist, if you allow a little bit of temporal width in your materials with which to begin. I'm sorry, I don't mind, I'd state it. That's fine, thank you, John. I'm going to let Peter leave in the first question because he's got a zip file. It's not a question, it's just a remark. One metaphysician, I don't know whether you call him an analytic or not,

50:00 but probably would, who escapes your critique of metaphysicians in the 20th century is Whitehead. He was very anti-politicalist and I think would be very much in agreement with an awful lot of what you said. Yes, thanks, Peter. I actually completely agree, and I think that, I mean, of course, I shouldn't say it's a forgotten tradition, because the literature, yourself and others, talks about it, and indeed this very man, Dean Zimmerman, talks about it. So it is actually part of the prehistory of the modern dispercussion, endurance, and perjurance, that relativity theory, and its apparent suggestion of an event ontology, extended space-time regions and their properties were the constituents, rather than some kind of enduring Aristotelian substance. That led into the advocacy of perjurance in modern analytic metaphysics, and it is generally true that, I suppose, that metaphysicians who have an eye on the deliverances of science are more persuaded by perjurance in the light of relativity theory and kind of philosophical construal of it that occurred in the 1920s and 30s, but especially in connection with method of extensive abstraction. I'm going to allow more time for the discussion because it's not Jeremy's fault that we started late, and I don't think he should be penalized for the start of 15 minutes late, and we'll just shift the program 15 minutes, we can be relaxed about that, so David. The way modern physics introduces continually as some sort of approximation to some sort of form, probably. And the way, say, I tend to motivate the Commonwealth's and Curves definition of notation is the Commonwealth describes the notion of the very large part of the approximate of the continuum. But doesn't that suggest that we could give a naturalistic response, such as related to the canonarchist, that says something like, well, if you did it well, it would genuinely the world would be like ours

52:30 but it wouldn't be physically possible to have a different discipline we just know they're like ours it's not in the sense that they're not like ours well this is a very good comment and as I had on a slide that didn't show it I was, for the sake of exposition, just assuming a classical world, but I actually, of course, in the written version, say several times that classical physics, classical technique, is a house built on sand, and that when you look very carefully, you're, of course, led to the quantum underpinning. But I did want to imagine that we were not taking the normal textbook gloss on things like the current vector as being an average over a microscopically large number of quantum goings-on, and on a microscopically large volume, but so small microscopically that one can model the system by attributing it to a point. I wanted to play the game of what is the metaphysical picture of classical mechanics itself. However, as regards the consequences of your perspective, which of course I think is a perfectly valid one, for the discs, it's amusing to report that what you say is actually what some of the literature, In fact, Lewis himself, in his first discussion of the disk, more or less says, we hereby learn that truly homogeneous matter isn't conformable with our conceptual scheme. It's remarkable what you can learn from a bit of conceptual analysis. And that, in a sense, is a bit like you, except that you said you put a different twist on it that we can say they're not physically possible. But your general idea that one could run it that way is very well.

55:00 Well, my question was sort of related, but I didn't want to go into quantum mechanics. I'm just thinking that, apart from the rise of relativity in quantum theory, there may be a third component which led to the sort of neglect of classical continuum mechanics, namely the atomic hypothesis, which convinced some people that there aren't really continuous, probably speaking, you can easily classify the mechanics and basically everything would be composed of particles and I understood, part way to your talk you said that particle classical mechanics could be proteins, so it's only continuum just mechanics that is not proteins. Is this correct or is the time analysis going to change that? Thanks. I should have said not just quantum but the triumph of atomism, never mind quantum nature. I mean first of all it's a grand theme throughout my two centuries I'm gesturing at, of course, that indeed, perhaps, the large federated bodies are swarms of particles. But as regards the second half, roughly speaking, quantilism breaks down into a spatial and a temporal aspect. And I'm saying, this is the paradox of the length of a line, that everyone has to agree that you need some spatial width, some spatial extrinsicality as regards geometry. That applies in the point particle case. Everybody, I think, has to agree that the notion of instantaneous velocity, even for a point particle, presupposes the notion of persistence. My silly example of me at noon and a piece of the sun at 12.03. That doesn't give me a superluminal costume. So those are the two compulsory, uncontroversial ways that quantilism has to be modified, even for point-particle mechanics, or for any physics at all. However, I wanted to say that apart

57:30 from that, well, within the various theories of classical mechanics, it's point-particle mechanics that quantumism fits best because of two things. As regards to the spatial extent, you can work in a kind of action at a distance for us. Whereas when you go to a continuum mechanics, the laws of Euler, at least are on a slide that I didn't show it, those laws of Euler of continuum mechanics involve universally quantifying over arbitrary sub-regions of the body you're concerned to describe the motion of. overlap, so you don't decompose it. And as regards to temporally, I want to say point particle, pointillism for point particles, temporal pointillism for point particles, only needs to make this mild and uniform admission that of course the notion of instantaneous velocity needs the notion of persistence but apart from that things look pretty quantity you can track from instant to instant where is the occupied spatial point in that sense you're rather pointless but when you go to continue up the rotating disks argument hits you and in a sense I mean, it looks like I am replying to the rotating disk argument, but it's a bit subtler than that, because the rotating disk argument is grist to my mill. It is the argument that you can't be temporarily quantitist about truly continuous classical matter. Sorry, I may have slightly disprezented that, but is that fair? Well, I'd like to query that because I'm aware that I may be very amateurish in what I say that I don't understand. And I don't understand the link that you've made between instantaneous velocity or velocity and persistence, which you take as obvious and uncontroversial. I think that's because you're insisting on velocity being a derived quantity but the message I learned from when I was learning fluid mechanics and Harrison with classical, with point, with particle mechanics is that indeed in particle mechanics you can take positions being primary and that's all you need to define what's happening temporally and then velocity is indeed a derived quantity.

1:00:00 You can't do that, the rotating disk line is telling you that you can't do that with a continuum. But an alternative to your insisting on persistence on covering each point of the disk is to say, you did briefly say something like this, but it's too quick for you to catch, is to say that the velocity field is fundamental. And that as well as giving, I mean when you're doing the mechanics, as well as giving the density field, you have to give the velocity field as an independent, fundamental description of the situation. Now, what's wrong with that? Why shouldn't a situation in any continuum is that there is an occupancy of points of space and there is a velocity field, so this is an extra property at each point of space, And then that the systems, the material lines of motion, are the derived ones, which is derived from the velocity field. And why isn't this a possible way of being quantilised about ? You're right on the topic and not at all amateurish, in my opinion. So I think, in a sense, I want to endorse just that, that vision. Well, there's quite a lot of things that, there's a lot here that I don't really understand, and maybe it's a matter of a private discussion, but So, when I was talking rather fancifully about the trouble with philosophers as they tie their hands behind their back, I was saying, rather like you, grab the textbook vector that determines the world lines and make it part of your metaphysics of continuum and don't deny yourself it. On the other hand, there is a sense in which the velocity is, after all, conceptually different from the position. I mean, you can write it as Q and V and so on, but there's nothing particular singular here. You could go through a Lagrange description where you probably get the Q dots.

1:02:30 And in that sense... Not for fluid. Sorry, what do you mean? Not for fluid. that you can't, can you? Can you write a description in which the philosophy field is? Configuration is basically . I mean, I think it gets complicated, but there's a sense in which you can introduce this Lagrangian description. In fact, this is something that David Wallace put me onto, that there are these Lagrangian descriptions. Well, Hamiltonian and Lagrangian descriptions, both. outrun what you normally see in a fluids book, of course, but which I think, yeah, I think we should probably call a halt there and thank Jeremy for a really interesting talk. Okay, our next speaker is Roman Free, recently appointed to the LSE, The title of his talk is, take a deep breath, in what sense is the Kolmogorov-Sinai entropy a meant for chaotic behavior, bridging the gap between dynamical systems theory and communication theory? That's it, your time is now up, so we'll have to start. All right, so the starting point of my talk is the observation that the historical matter of fact the chaos theory has been with us for quite some time and the methods it provides by the use in theoretical and applied physics. So what is chaos after all? Two answers have been given to this question. First has it that we have

1:05:00 chaos with some kind of random behaviour. Can you see that? If you point at the board rather than... No, no, no. No, no, no. That's very good. I think that if you can stand over and point by the board and point at the wall I'll try that and turn it back to the audience, I don't know, so I'll give you that. Okay, the second family of replies tries to cash out chaos and terms of exponential divergence and discuss the up-and-up exponents and it's kind of six feet off here. My talk today is concerned with the first family of behavior, although one might argue that the second or is down to the first, anyway, but that's a different matter. The problem of the having in doing so is that characterized behavior as random is not very telling to say it needs to cause the notion of randomness as much in negative analysis as the notion of chaos itself. There is one suggestion how to do that, however, namely that an ergonic theory provides us with the notion of help here, namely the Kolmogorov Sinai entropy. is often claimed, as it's known as Terpichlund said in the physics book, but it has also been introduced in philosophical literature by Newton-Ehrman and Gordon-Bellett, that the transition from zero to non-zero Kolmogorov-Sein-Arient could be marked as a transition from regular to chaotic behavior. The problem with that is that when you look at the definition of Kolmogorov-Sein-Arient, The only thing you find is partitions of the phase space into their measures. So it's a topological notion, basically, and it's just not clear, to say the least, how this is supposed to be a measure for random behavior, and to explain how actually that is possible. That is the purpose of my talk there. But in order to get to that, some preliminaries are needed.

1:07:30 Okay, so elements of dynamical system. So I take a rather simple definition of dynamical system, thanks, because nothing of the details matters. I have some measurable space, a phase space, have a measurement on it, and you have an automorphism phi mapping the space onto and this is used to induce a time development. So you take time to be discrete and you apply phi to the phase space at every time and this induces the dynamics on the space. I make two additional assumptions. The first, M is mobilized, so the measure of the whole phase space is one. That's rather unproblematic because we deal with bounded motion normally. The second is less trivial. I assume that dynamics would be measured preserving. So I restrict it to something like Hamiltonian dynamics. As an example for a dynamical system in this sense, you can simply take shift map, you map x onto x plus c modulo 1. So you start here, you add c, and you check it back onto the unit. That's a dynamical system in this sense. Okay, we also will need partitions, that's very important what follows. Okay, a partition basically, if you take a partition, you cut out the face space, in finitely many sets that are pairwise disjoint, so they have the overlap is set. And they have to cover the whole phase space together up to measure zero. That's quite intuitive if you see here examples. Moreover, we need the sum of two partitions. The sum is simply defined as the intersection of the cells of the partitions. space, and you have two partitions on it. So the sum of the two partitions is this partition here. So the sum of the partition on this yet developed partition. Introduce all that

1:10:00 because these are the notions that we need to define the power of sine i n to b, which is this object here. So you take the sum of all the time developments of the partition. h is the standard function we know from discussions about entropy all over is the x times log x function basically, then we let time go towards infinity and we take the supremum over all possible partitions of the phase space. If this discussion, if this definition is not telling to you, So we're welcome to the club because the directly important line of this little introduction was that this notion is really not very useful, we can't read off anything from that and we need further discussion to understand what's going on here. The problem we face, to start with, is that the homo-borough side-range of the us defines a purely topological notion, and no reference to any concept related to randomness is made, and more generally, it's rather opaque, and as I just mentioned, we can't reveal of anything useful from this formula. Physicians' reaction to this are twofold, standard reactions just don't bother, but quite often in physics books we also see people using information talk explaining what this amounts to and my suggestion is to take this news talk serious and to see where it actually takes us so in order to do that I need to say something about information and information theory that's my next section. So the setup is the following. We have a source that at every discrete time step selects one of the messages S1 to SN and sends it to a receiver R. And this is a probabilistic source. These are the probabilities P of S1 and so on that one of the messages will be sent. And further on, the receiver receives the messages and superscript, just for the sake of order, and takes it down on a paper. So the result is a string of that sort.

1:12:30 Here just the message isn't when they have come in. Given this, this is the standard definition that Shannon gave of the entropy, and they call this just entropy of one step, because it's the entropy that that one step, and the step is the process of the source sending out the messages of this process. So, and this can be interpreted as a measure of uncertainty about what symbol will crop up at the next step. So far, that's a summary of what Shannon has done, so that's nothing new, I just mentioned that for the sake of completeness. And the problem we face is that this doesn't bear any resemblance to what we call Mogorosh-Sami-Entropy as I presented both. First, we keep a totally different contextual framework. So we have a continuous phase, space versus discrete messages, we have a deterministic dynamics in the case of a dynamical system we're dealing with, and the probabilistic evolution in the information case, and we have an automorphism versus a source. Moreover, we can't resort the formal analogy, we can't say, well, it's a bit like probability theory, for instance, where we have a mathematical machinery and you can interpret the terms and bonds as actual frequencies, and this is a philosophical matter, but the theory is the same. The theory just isn't the same, so this move-between interpretation doesn't really work. Hence, we have a real problem to deal with. And what the solution to this problem, I suggest, is the following. Namely, first, we should generalize Shannon's notion of entropy in various directions, or especially we have to conditionalize on the entire past history and not necessarily on the step, at least one step back. We should consider more than Markov processes in other words. And second, we need to bridge the gap. That's a reference to my impossible title. We have the proven equivalence theorem between this generalized version of the Shannon entropy and the Kolmogorov-Sci entropy as defined at the beginning. This is what I'm doing in the rest of my paper.

1:15:00 And from what I know in these, please tell me if I'm ignorant about something, both these steps involve new results. The gains we get from this is not really a technical exercise, is we should get a deeper characterization of randomness in chaotic systems. And finally, this has a consequence for a philosophical understanding of randomness as well, especially in connection with the so-called process of product randomness types. It has a counterintuitive result that we'll come to at the end. So let's first generalize Shannon's entropy. The first generalization we have to make, as I mentioned that briefly already, that we have to conditionalize the probabilities of the entire past history. So we don't only have to give you the probability of a message cropping up at the next step. We have to consider the probability of this message cropping up at time tk plus 1 given that that's t1, this message came in, that t2 is 1 and so on. whole past history of the system. When this is merely a formula move, then you can rewrite Shannon's entropy and what we get is this. Then we have to generalize our question slide. This was merely a formula move, you might say, nothing has been added to the substance, but now we generalize the question, instead of asking what is the uncertainty about the symbol at time k plus 1, given that a certain message, namely this one, has been produced, we answer, what is the uncertainty about a certain symbol coming up at time k plus 1, whatever that message has been produced so far? And we can answer this question, I think, by taking these terms here, and averaging them out, and taking a weighted average to justice that not for every past history is equally likely. So we weight the sum, we weight the terms in the sum, the probabilities of the histories that occur in them. And this then

1:17:30 is a measure for the uncertainty of what the next symbol of the source will be, whatever the past history of the system. The fourth, then the third step in our generalization is we have to give up the argue focus on one step. We want to have the whole process. This step is again straightforward, we just average all the entities, all the entropies from the beginning to where we are now, so we just take the normal arithmetic average, that's straightforward. And finally we can say that the average of the source as a whole, or the entropy of the source as a whole, is defined when we take this and let time go towards infinity. So this is the last move you have to make. Gathering all the pieces together we have this term here which I call the communication theoretic entropy and which is Shannon's entropy generalized in the ways that I have just explained. The problem with this is that we even see a few further away from the Komodoro Sinai entity than we have been before, because still this whole story doesn't bear much resemblance either to the formal or the philosophical aspects of the dynamical systems set up. So we still have to do something, and that something we need is the equivalence theory. That's what I'm getting at now. This needs some preparation once more, so as I mentioned the concept of frameworks are different so we have to make them compatible in a way and the first step in doing so is to remove the mismatch between discrete messages and the continuous phase space There seems to be an actual answer to that because the Commonwealth of Sinai entropy is defined in terms of partitions of the space anyway So it seems natural to use these partitions to do that. So consider this is the phase space and we have this partition here. What we do then is simply we take the trajectory of the system and we choose discrete time steps.

1:20:00 So exactly the times when the information source pulls out the messages and just check in what cell of the partition the system is. And then we take that out of a string of paper. At T1, the system was in cell alpha A, at T2 in alpha 1, at T3 in alpha 2, and so on. And we see there that the string generalizes, structurally isomorphic, to the string, the information source produces. And from this point of view, we can say the atoms of the conditions, the alpha i's, correspond to the symbols of the information source. The next step we have to make is we have to get probability in the game. But by assumption we have a normalised measure on our dynamical system. And we can interpret this as a probability. I should mention that there is nothing compulsory about doing that, especially in physics contexts. not every measure we account for is a probability, but I'm concerned with the mathematical achievement so far and when you have a measure space with a normalized measure, then we can interpret it as a probability that's perfectly possible and that's all I need for the time being. So I can say that the measure mu, alpha i, for that to correspond to the probability Psi, by obtaining a simple SI but there is an epicycle here because as we have seen the formula before it's essentially that we have not only the probability of individual messages but of entire strings and that's rather non-trivial to get in the case of dynamical systems In the information case, we have it by definition, that's where we start off with all the computational probabilities. But here we have to rework together, and that's an important point that you have to realize that the probability of being on set A, of basements at the time TI, and set B at the late time TJ is given by this formula, which basically means, take A, then take the time development of the system, see what's

1:22:30 the image of set A under the time development from TI to TJ is, and then intersect the two, And then the measure of the intersection is the measure of the probability of this string. I think this picture should illustrate this nicely. And this is clear that this generalizes in a straightforward manner to more than one set. But what you get generally is the probability of the string alpha, L1, T1, and so on, is the intersection of the time developments of these. So, well, that's really a technical issue, but it's important to say that this is the main idea. Then the third point is we have to say that the optomorphism corresponds to the source S since they both did the job of generating the string that's relevant. So we have set up the analogy, the cells correspond to the message, the probabilities correspond to the measure of the cells, and the dynamics correspond to the information of the source itself. given this we can carry over the communication theoretic end to the dynamic consistent space and we get this formula here which we basically just get by replacing the probabilities of the messages by the measures of the of the partitions. There is neither an important disanalogy between the two cases that I should briefly mention, while in the information case the messages are an integral part of the definition of the source, while in the case of dynamic resistance partitions seem to be just an auxiliary there is nothing intrinsic in that, then I'm a consistent don't oblige, you can take this or that partition, so we should get rid of the dependence of the partition somehow because when you choose a sufficiently convenient or sufficiently stupid partition, you always end up with no uncertain deal at all, so you always get to be down to zero

1:25:00 if you just consider particular partitions, it doesn't really get you very far So what we should look at, rather, is the Supremo over all possible partitions. That's a way of getting rid of his dependency on the partitions. This was a move suggested by Kolmogorov, I have a slide justifying that. But look at the watch, I shouldn't get into it, please press me on it later on if you want. Okay, all these things in place, I get to the core, or at least the technical core of the argument, namely the equivalence theorem. On the assumptions that I have outlined so far, one can prove this theorem, namely that the Kolmogorov-Sinai entropy here in the first line, as introduced at the beginning, is equivalent to the communication-theoretic entropy as carried over from information theory to dynamic dynamical system theory by associating cells of partitions with messages and so on. This is a, and this is a strict mathematical result. There is no philosophy of the depth as it were. I just want to briefly sketch what the proofing was. First, you introduce a technical device, the so-called conditional entropy. Then you prove five layer mass. Then you can prove theorem one and theorem two. Again, they don't have a straight forward physical interpretation. They are not interesting as theorems themselves, so don't bother with formulation and then we'll put it out. If you have a question of how the proof runs, I'm using these two theorems then you can prove the equivalent theorem that I have shown you just before. So, I have set all these lemmas and theorems in between there of technical interest and that's why they dwell on them. What I want to do now is get them away from the technicality, let's explain what we gain from that. What is this equivalent theorem doing for us?

1:27:30 And I think what we get is a rather detailed characterization of randomness and dynamic persistence. In many respects this characterization is not new, but I think it's just now that it's been set on the first theoretical basis because we have this equivalent theorem. So first, a system with the positive thing is one, that on average at every step there is some uncertainty but what certain of the particular system will be in next. Then second, the qualification on average indicates that this does not mean that there are no instances of time. In such a system we can have certain instances of time at which there is no uncertainty. And so, if knowledge does not improve the situation, you can know as much about the past of the system as you want. It will never get you in the situation you can actually predict what happens with certainty. And this will not cease to be the case. There is no cutoff time, as it were, after which we have gathered enough knowledge of the past to actually do that. When you can go on as long as you want, you never reach a time where you have these predictive abilities. Furthermore, the dynamics of A is the so-called 0-1 law of probability. Even if you have complete knowledge of the processes behavior in the past, the only event you can predict uncertainty at the next step are those which have probability 0 or 1, independent of the past history. But finally, one can say that non-refining entropy is a quantitative measure of the magnitude of unpredictability. So the value of the entropy actually gives you a measure of how big unpredictability really is. Summing up, we could say that a system which is called a difficult model of cyanide, entropy is one, in which the past history never conveys certainty upon what can happen at the next step. On the other side, we have the case that the entropy is zero, we can say on average there is no uncertainty if you can predict what the future will look like. But there's because the qualification on average means that it is not precluded that there are some

1:30:00 instances of time in which the process is indeterministic. So that for these two points, it is true that for a deterministic process, the entity is indeed zero, but it is not the It converts false. It is not true that any process from which it is, the entropy is not true that when the entropy is zero, the process must be determined. That's false. Finally, my last section is a remark about philosophical discussions of randomness. I think, well I know it from John Ehrman, probably seems to have been made earlier, but it's things between process and product randomness. You have process randomness, when, loosely speaking, when we're faced with a process that involves probability or works without the harm caused principle. and we have product randomness if the product of some process is somehow out of shape and these two notions do not generally coincide for instance a sequence that is a product of a random number but generated looks random but the generator itself is normally a computer which is not a genuinely random machine and on the other hand side process like flipping coins for instance, it does not necessarily mean that the product looks random, it can toss it 100 times in 500 heads, that's very unlikely to happen but it's not impossible. So this led some to claim that chaotic systems, they exhibit product and if what I've been saying so far is right, I think we have to cast some doubt on that claim and the argument is the following, I want to use algorithmic complexity for that very roughly speaking the algorithmic complexity of the sequence is the length of the short computer programming that we haven't had in order to get the universal Turing machine to calculate the sequence

1:32:30 and we then define sequence to be random and show the program has roughly the length of the sequence itself. So that's very rough, that's the main idea. And then, underwritten complexity is obviously an notion of product management because it is concerned with the of a given series and the computation of a power unit for it. Next, the present discussion of the communication theorem shows that this is clearly a notion of process because you are concerned with the way in which signals are produced and how they are sent out and so on. Next, we have a mathematical theorem called Kolmogorov-Sinai entropy is equivalent to the algorithmic complexity of the trajectory of the system, that is the string of its positions. And then the next theorem that I have thrown here states that Kolmogorov-Sinai entropy is equivalent to the communication theorem. Therefore the algorithmic complexity is is equivalent to the communication theoretical entity and what we have now is we have an equivalent between two notions that are one is a product and one is a process randomness notion. Well obviously as Jeremy said before mathematical equivalence doesn't imply philosophical equivalence. One could now go on and say well the mathematically prevalent about the concept, as I'm going to say in any way. They certainly might be said about it, that's why I say you just should cast some doubt on the distinction. But this result of these suggests that a cutting of theopsis, more specifically Bowes, process...