The unseen world

0:00 Okay, testing 1, 2, 3, testing, testing 1, 2, 3, testing 1, 2, 3. Testing 1, 2, 3, testing 1, 2, 3. Thank you. And therefore, if you look at the biology, you're looking at the genes and so on, you can explain the characteristics underlying the organisms. But there are also more abstract entities, such as energy and entropy, which are not part of our immediate sensory experience. And these, again, those who have gone to school fast, who introduced ideas of potential energy and kinetic energy, you certainly can't seem to play a big role in the scientific story. Entropy, quite a little rather than that. You can't actually see it or touch it in a kind of indirect quantity. Well, I come on at a moment or two to discuss how science doesn't get used in touch with these things. Well, now, apart from these things, which are not part of our mutually sensitive experience,

2:30 entities which seem to figure in science, things like numbers and mathematical points. Not just in the physical space, but in still more abstract spaces like the Hilbert space in form of mechanics. There are lots of abstract mathematics going on in science. How does that figure, how does the mathematics get involved in science? How do you get to know about these things? You can't touch the numbers, you can't hold them in your hand, I guess somehow this would have shaped quite a big role in science. So what I'm going to call now is what I'm going to call the unseen world. This is the world we can't directly see. I'm going to use the word unseen, I guess this type of today. I'm going to say that. and I don't think it's actually a black circle. I don't think, while it's actually a constitution, it's what you might call the most important role of it. Now, this has been his mission. It's a time of action. A page where he was giving his tower lectures, came in and he talked about the two tables in it. He talked about the table in front of him, with a solid heart, he could rest his elbows on it. that was the table of everyday experience and then there was another table a kind of ghostly table which was mainly empty space with a few atoms lying about with large distances into the atom and this empty space was permeated by coarse fields between the atoms and molecules so there seemed a very different type of entity in front of him the ordinary solid table and this mysterious ghostly table with all these unseen entities constituted. And of course, Eddington famously asked the question, which is the real table? Which is the true story? The scientific story or the everyday story? Now, I'm going to sort of pick up more or less from Eddington's questions and answer it in my own way and see what was anything in that lecture in the 1920s and see what's happened to that sort of question in more recent times. So in the lecture what I want to do is to explore the cognitive credentials of the unseen world. To what extent do we believe that it's true? Is it a reasonable thing to entertain that there is this unseen world, the world inside the table of life?

5:00 And I'm going to look at it from a historical and also modern perspective. Because people's attitude to these unseen voltages in science has changed. Sometimes people don't like them, try to get rid of them, sometimes people do like them. There's a quite long sort of history of different attitudes to the other side of the world. Now the philosopher David Hume in the 18th century famously warned that the ultimate springs and principles are totally shut up from human curiosity and inquire. A human didn't think that human beings were capable of getting to the truth behind the world of appearance and the ultimate springs and principles are totally shut up, as he puts it, from human curiosity and inquire. But science seems to have not to heed issues morally. Let me begin by reminding you of the famous medieval woodcut. Now, can everybody see that? Because we don't put the lights of it. Well, I suppose you've got to put up with the camera. But can everybody see it? I'll sort of use my stick now instead of using it to walk me about. is that here's a little curious chap who's poking his head through the vault of the heavens. Here's all the seen world, the sun and the star and so forth, but he's a bit inquisitive, this chap, he's poking his head through and looking at all this secret mechanism beyond the vault of the heavens. Now that sort of illustrates this idea of the unseen world. Behind the visible universe is this unseen world of apparatus and contraptions of various sorts So even the medieval universe, people are beginning to wonder whether you could poke your face through, as it were, behind this visible world and have a look at what's really going on. My first question is, what can we directly observe with the unaided senses? What is it that we can actually see? What do we mean by seeing things? Well, I suppose the first shot of answering that question would be macroscopic objects in an unneeded vicinity, perhaps such as the table in front of me, or the chair next to But is it the table we see, or the light reflected off the table, or is it the electrical stimulation

7:30 to the retina caused by the light, or in the optic nerve, or what exactly is it that we Now this is something that has perplexed philosophers. They've argued a great deal about what it is that we actually see. And there's a kind of physiological story which says, well, in the past we do actually get some direct experience inside our brains. This comes to us somehow down the optic nerve. We don't see the table. The table doesn't get inside our brains. There's some sort of electrical stimulation that gets inside our brains. And you can sort of naively, and this is a bit naïve thought, but I suppose what the man on the catamomaly bus might think, because we have a representative of the catamomaly bus. He told me he was going to represent the man on the catamomaly bus in the discussions, so maybe I'm speaking for his views. But we can think of a sort of homunculus inside our brains, our conscious selves that read out and interpret these input signals, There's these sensitive signals coming down to the retina of the optic nerve. But if our brains and minds, if we do have minds, and that's a kind of vexatious philosophical question, whether we have minds in addition to brains, but anywhere they seem to be just part of nature, then the whole idea of the homunculus, or the ghost in the machine, as the philosopher Bill de Guile called it, seems patently absurd, it just seems ridiculous. there's just really a little man inside my head who's sort of weeding out the little television screen interpreting what he sees in terms of the external world. Now that's a kind of ridiculous view, really. Although I suspect that most of you draw on philosophers, perhaps, well, maybe everybody secretly does think there's a little man inside their head, or if a woman inside their head, reading up what goes on in this individual experience. Well Gilbert Ryle called this the ghost in the machine. He ridiculed it in his book on the concept of mind and it does seem really patently absurd. Well that of course is the problem of consciousness. What it is to have conscious experience. If it's in the little man then what is it? We have great experts in the audience on the theory of consciousness. I'm not going to hazard any attempt of even my own views about it. It's not the problem I'm going to consider today interesting and important that it is. I'm not going to be concerned in trying to analyze consciousness. I'm going to start with an assumption

10:00 that we do in some sense see tables and chairs I mean it seems a very natural thing you're always suspicious of philosophers that come along and say well let's start somewhere, let's start with an assumption let's assume that we do have to see tables and chairs, immediately everybody goes up in philosophy what do you mean by seeing the table and then it all starts in arguments and discussions but anyway, my own way of doing philosophy is always to start what's going on, a simple scenario, then when we see what's wrong with that we can try and polish it up and improve it. We've got to start somewhere. So I'm going to start with the assumption that in some sense we do see tables and chairs in a good light, possessing normal eyesight. The question of whether I'm allowed to wear spectacles, it's much been debated by philosophers because if you're short-sighted, then fully on spectacles seems all right, maybe. But then, it's a short-sized person, actually, he has a kind of, I'm a very short-sized person, so I'm not the spectacles, I'm a very fuzzy viewer viewer. But that's my view of the world, so if I'm looking with my uneven senses, that's all I get. So am I allowed to put the spectacles on? I'll come back to that vexatious question in a moment. Well now, even if we don't actually see them, if we don't actually see the table, which I don't turn my back, for example, so I don't actually see it. Nevertheless, they're observable, observable, in the sense that it's possible to see them. If I turn around and look again, then I see the table. So I'm now getting a distinction between things you actually see and things that you could see. Now, some philosophers of science, and indeed historically many scientists, have thought that scientists can serve in discovering wicked energies in the behaviour of observable entities. They'd be very suspicious about this unseen world that I've begun to talk about, and they've thought that science should eschew the unseen world, and only talk about the things that we can actually directly perceive. And these people are generally called positivists, and I'll explain why in a moment. Scientific knowledge can be checked out in a positive fashion by direct observation. Now, I must warn that labels such as positivists, and more particularly is cognate empiricists, have many shades of meaning in philosophy. And I don't want to get into all the different disputes about whether this man is an empiricist or a positivist and so forth. So I'm just going to use these terms with a broad brush just to give you the general idea. So the positivist thinks science is only concerned with things you can directly see.

12:30 At first blush, the positivist position sounds rather attractive. Isn't it the case that the scientific attitude has progressed by getting rid, first of all, of supernatural spirits of gods controlling the world? If you think of old Greek mythology, get rid of that, that's childish stuff. Well then, get rid of the theoretical or metaphysical concepts, of the famous dormitory virtues and other substantial forms beloved of the Aristotelians. Remember in Moliere's play in the Moliere, the doctor is off to explain why opium puts you to sleep, and he says that's in virtue of a dormit in virtue, and this doesn't seem to get you much ahead, just giving a name to it. And so this kind of thing was very much the type of explanation given by Aristotelians before the scientific revolution. And so all that got swept aside, We don't want normative virtues, we can't see them. So let's finally arrive at the culmination of what the 19th century philosopher August called positive, that is to say non-speculative knowledge. I definitely can see something in front of me, and that's something that we can all agree about. Now the question is, has science really followed the positive of his programme? There are all kinds of difficulties. If we are restricted to direct observation, then what is the point of scientific instruments? like telescopes and microscopes. Surely these are supposed to enable us to see things which we can't directly see. But if science is only concerned about things which we can directly see, then what's the point of telescopes and microscopes? They just seem to have their own role in science, and that seems a very odd conclusion to draw when you're trying to discuss the nature of science. We have something counterintuitive, I'm saying, there's no role for microscopes and telescopes. Now, there's a significant difference here between the telescope and the microscope. telescope enables us to see things that we could see directly if we were differently located. Move closer to the distant tower or close up to the moons of Jupiter or whatever. So if we were right up against the moons then we could see them although with the naked eye we can't see the moons of Jupiter. But for the microscope it's not a matter of relocating ourselves. For the virus or the cell to become directly visible you would have to change our normal sensory apparatus. You'd have to have some much more acute sense than we actually possess we might have to adopt the perspective of what I call the incredible shrinking

15:00 man. Now he, the chap, would shrink himself down to the size of a virus so that he could directly inspect the virus. So to count the virus or the cell as observable in the same sense of the telescope needs rather more science fiction than does the case of the telescope. You need to have this idea of the incredible shrinking man. Now historically the first practical version of the telescope were employed by Galileo at the beginning of the 17th century. The telescope revealed all sorts of oddities in the heavens. For example, there were mountains on the moon, and the satellites of Jupiter I've talked about. And he has a famous picture taken from the Starry Messenger, Galileo's famous book of 1610, in which he announced to an astonished world that the moon would be pure, perfect, with no imperfections at all, mountains and depressions and had that sort of spotty kind of appearance. So Galileo astounded the scientific world by showing this picture of the moon in 1610. But what was the reaction of Galileo's Jesuit opponents who didn't like all this new science of astronomy spoiling the perfection of the heavens which God had created in a kind of perfect way? Well there were two kind of reactions. Some refused to look through the telescope. And the reason they gave was that if God had intended us to inspect the heavens so closely, he would have equipped us with telescopic eye. So it was pure presumption to go looking through the telescope. But there was another line of argument that others claimed that Galileo's observations were simply artifacts of the instrument. They were nothing to do with the moon. They were somehow created inside the telescope. Now let me turn to the microscope a bit of the unseen world. And there's a famous picture I want to show you, dated 1625, Francesco Stolucci looking through an early microscope with rather low magnification, but this is Stolucci's picture, 1625 picture of a bee. Now that's not exactly the unseen world, we can all see bees to some extent, but look at the amazing detailed structure that Stolucci finds and the texture of the wings and those nice little legs and so forth. You see how the microscope is beginning to penetrate into this unseen world. Well, of course, sometimes people saw what they

17:30 wanted or expected to see. There's a famous discussion in the history of science about the science of embryology in the late 17th century. There were people called preformationists who thought that actually inside the spermatozoa they could detect a little preformed homunculus in complete detail. Well, now you say that's ridiculous, but this was supposed to be based on observation. So I'm now going to show you a picture by Nicholas Hart's subcrap, which is dated 1694. He just took the microscope, this is nearly a hundred years later than it's the end of the 17th century, and he drew what he saw. Now I think you can all agree that there definitely is a case of preformation. You can see the little man hooked up in the head of the planet as well. So it's a little bit, one has to be a bit careful about observing things. What you see is sometimes what you want to see. and what we see is largely determined by the overall theoretical background of our thinking. The slogan here is the fairly lateness of observation. There's no such thing as pure observation. It's always involved in some degree of interpretation. And that picture of Hartzucker is a good example of that. Now, we've already an occasion to question whether the table or chair is directly observable, but isn't observation always a case of probing or interacting with the physical world? Isn't that what we always do? observe what we do affecting light or logic then observing it with the eye or taking the light through a microscope whatever it is don't we always observe things by the effects they produce ultimately in our conscious mind isn't that really what observation is all about in science now we often talk loosely of observing fundamental particle reactions and I want now picture of a bubble chamber. This is one of the little bits of hardware that people doing particle physics use. It's not a very good picture. This is the Brookhaven bubble chamber 1964 which was used to discover the famous omega minus particle. You can't actually see the bubble chamber. I apologize for that. You can think that that's the bubble chamber. That has the magnet round the bubble chamber.

20:00 The bubble chamber is all in that long. This up here has all the refrigeration It's very dangerous these bubble checkers, you know, the early ones used to blow up regularly, terrible explosions. This is a two meter bubble check. You can see the scale of the thing. Here's a little man adjusting the magnet. Here's a little man up here doing something, sort of refrigeration and so forth. So that's kind of sort of scale of the thing. So that's the kind of bit of apparatus that's used for seeing particles but now what you actually see when you get out from the bubble chamber well what you actually see are a photographic plane recording the tracks, the little chains of bubbles that are released when a charged particle goes through the superheated liquid hydrant On the left hand side is one of the most famous pictures in physics the discovery of the over the minus particle You can see it's just a maze of these little tracks in the photographic plate, it's a photograph of what was going on in that big bubble chamber I showed you a moment ago. The only good minus is that little tiny kink, do you see, I'm trying to get the shadow to it, no I can't quite do it. But that little kink, can you point to the kink, sorry? They're up, up a bit, they're down a bit, left, that's it. They're a little sharp bit going upwards. That's it, that's the king. That is the particle which won the Nobel Prize for the people who predicted it. This was one of the famous predictions of particle physics to be such a particle. Now, of course, that looks all kind of difficult to interpret if we're just showing that. You wouldn't think this would be the Nobel Prize for it. But if you look over here, this is the interpretation of what you see in the photograph, what's going on in the bubble chamber, which is telling you what the parametric particles are doing. You see how this long chain of communication between what's actually going on in the physical world and what you read about in the textbooks. So here we have a K-minus particle which is decaying into the omega-minus. You see that's your tiny short space, and the omega-minus almost immediately decays like that. So it's all for pion, so that little bit there corresponds to that little kink. And if that kink hadn't been there, the whole of particle physics of the 1960s would have taken a very little bit of blue. So it's amazing what you would mean to a picture. I mean, you would probably not look at some of that, but as I said, it's one of the most famous photographs.

22:30 So, from this perspective, electrons, quarks, genes, viruses, all these things are, after all, observable in a very sensible sense and a more observable. So, what does all this talk of the positivists say? We don't want to look at unobservable things, but, after all, we can observe these things. And so, do they really belong, the genes, the viruses and so forth, to the unseen world? And on that account, should they be instituted by the scientists? Well, now, this debate was carried on particularly vigorously at the end of the 19th century in respect of the reality of atoms. For example, there was a group of scientists at the end of the 19th century, people like Ernst Bach and Ostwald and others, who regarded the atoms of the physicists and the chemists. They were just fictional entities, introduced speculative mechanisms for explaining empirical regularities combination of the properties of gases. They were not to be thought of as real in any robust philosophical sense. And this may rather amaze you, but if you read physics textbooks at the end of the 19th century, you find a whole school of people writing on the nature of physical sciences telling you just that, that the atoms don't exist, there's no such thing with atoms and all this talk about them in the box is just kind of fictional stuff and all there is are just the observed directly observable properties of gases. Now to the modern scientists it's usually assumed that these debates have long been settled in favour of a realist conception of these so-called theoretical entities rather than their positivist dismissal. And surely all that 19th century stuff about atoms not being real isn't that just history and nobody But again, things are less simple than they seem. I'm sorry, that's a familiar refrain in philosophy. You just get what seems a simple story at the sounds of an ongoing point. I hope you're all persuaded at this point that there really were things like but I now want to, I haven't got you in the frame of mind for thinking that's alright, I now want to whisk you round again and show you that these things are actually things are not quite as simple as they seem because if we look at the history of science we can see it as a series of U-turns about this explanatory theoretical structure that lies behind or beneath the world of macroscopic experience.

25:00 There were entities like phlogiston, the nivoliferous ether, caloric used it. Heat was thought of as a kind of liquid, a thing that could rub off when things were hot, the particles of caloric could rub off on you. And this is what heat was all about. But these entities, like the logistin and the luminiferous ether who explain the provocation of light, or caloric, have simply disappeared from the scientific vocabulary, and the nature of atoms and molecules is quite different from the modern perspective of quantum mechanics, for example, from the billionth-alt inception of the 19th century. So there's a lot of change about people's understanding about this nature of these fundamental, explanatory, theoretical entities. Well, now this is again some philosophers to what is called a pessimistic induction. If we've been so often wrong in the past, is it not pure hubris to believe that our present scientific theories won't look equally ridiculous a hundred years from now? So, you see, the point of view being put forward here is that we look with a kind of superior a view of these old 19th century textbooks, talking about the uniferous ethers and so forth, and I think how could they be so stupid to believe all that. But now imagine yourself a hundred years from now, looking at some book on particle physics and seeing talks about electrons and quarks and so forth, how are we to say that a hundred years from now all that will look equally ridiculous? But if it's going to be kind of turned out to be just ridiculous, then why do we believe it today? This is the so-called pessimistic induction. It would have been so wrong in the past that isn't it reasonable to think that our modern theories are going to be wrong and they're going to be replaced by a totally different theory. This idea of fortification, of course, was famously espoused here on the LSE by Karl Popper. Now, to diffuse this pessimistic induction, some philosophers have tried to of progressive fashion. And here I'm going to refer to the views of John Wall, who will be kindly sharing the session, so I'm going to be deliberately provocative and I hope it will become the discussion. John will defend his views, so I'm going to have one or two things critical of him. It has been argued by John, for example, that although the ontology of physical theory changes abruptly, nevertheless there might be what we could call a structural continuity in the

27:30 sense that in many cases the mathematical equations survive, if you look at the kind of theory, the equations seem to survive, only the interpretation of the quantities entering into the equations is what changes. So the actual nature of the entities is being revised all the time, but there is a kind of continuity at the level of the mathematical structure. Now there are two versions of the structure's philosophy. In fact we have pushed John in discussion to say which version he actually believes in. Now in the extreme, and I think I've been provocative and say bizarre version, an ontological version, then it's only a structure which really exists. There aren't any objects on which the structure is imposed is just abstract structure, everything else is just imaginative fiction. Now, I'm not really happy about having structures but don't even care in objects. This is really Aristotle versus Plato, for those who are familiar with his arguments that go back, like so much in philosophy, go back to the Greeks. Aristotle didn't like the idea of the forms being independent of material or physical objects, where Plato was quite happy So I'm following Aristotle, essentially. So as a more prosaic, epistemic version, structure is all that we can reliably claim to know. Things have natures, but we can never actually probe what their true natures are, we can just know these mathematical structures. We don't deny that atoms or quarks exist, just that we never know what their true natures are. Only the mathematical description of how one is structured related to another behave in various experimental contexts, and so on. argument here is that the continuity of mathematical structure defeats the argument of the pessimistic induction. Now, there are various comments I'd like to make about this. First of all, does it make sense to talk about things we can never come to know? I mean, I've suggested that we can never know the true nature of the electron. We can only get to know the equations governing its behavior for it truly. But if you can never come to know something, Does it make sense to even, is it even meaningful to talk about things that we can never come to know? Well, of course, that line of thought would drive us towards ontological structurism. It's really meaningless to talk about the objects because we never can come to know them. This would drive us towards the ontological structurism.

30:00 I've already dismissed that as bizarre, so I don't want to get driven in that direction. You see, in philosophy, you always get driven to things you don't want, and then you have to wiggle out somehow. Well, this line of thought, of course, is linked to the verificationist theory of meaning espoused by the illogical positivists. Statements that can't be verified are simply meaningless. Of course, any strict interpretation of such a principle would arguably render every statement in science just as much as, for example, in theology as meaningless, and this was the view of the illogical positivists in Vienna in the 1930s, that all these things were totally meaningless. Well, I think that is almost a reductio and absurd proposition. I don't think there are any positivists that really strike around these days. We never know anything for certain except perhaps in logical mathematics, and I say perhaps because even these things are not entirely clear about another lecture. So if there are so many things I'm not certain about are the same things that I personally am quite happy to accept that there are things kind of an inactivity about. I don't know whether I'd sway to you on that, but that's the personal point. But is it true that mathematical structure really is intact? Because this was the point about John, that would be the pessimistic induction. Now in the most revolutionary episodes of modern physics, like relativity theory and quantum mechanics, that actually is just not right, I'll put that politely. The new mathematics involves parameters like the velocity of light, which will C, as it's often written in the case of relativity, or the famous Planck's constant, little h, in the case of quantum mechanics. It's only by letting little c go to infinity, like moving infinitely fast, or little h going to zero, that we recover something like the old mathematics of classical physics. But these limits are in general highly singular. Now I want to try, then of course the flavour was meant by a limit being singular and in the sense in which new structure is really observed, is being developed in these new theories. A world in which H, for example, the quantum mechanical constant, is actually zero, is qualitatively quite different for a world in which H is different from zero, however small in magnitude it might be. So with different values of H coming down and down, closer and closer to zero, and they all look roughly the same, except some say they're slightly different, they've got different values of H, then put H equal to zero, you can't go back to the classical world, and the thing dramatically changes in a totally singular and discontinuous way.

32:30 To illustrate what I have in mind, consider squeezing a circle so as the trun turns it into a line. This is one like making it smaller and smaller, but you can go on squeezing it and squeezing it so you can get more and more elongated circles, but the final point of the line is absolutely not an elongated circle. For example, it has no inside, and whether a curve is open or closed is an all or nothing matter. bit of it inside or partly got it inside, either got it inside or it hasn't. And so the line is quite different from this successive squeezing of the circle. Of course from the practical point of view, a very elongated circle for many purposes looks a bit like a line, but my point is that in no sense is it a line. Now that's the sense of singularity and what mathematicians mean when they talk about singularities in structure. Now, as another example which is relevant to quantum mechanics, I want to consider, and this takes you back to your school physics, of course you're all deep physics in schools, I'm quite happy to remind you of your school physics, about waves travelling backwards and forwards on an elastic string, and see what happens if the velocity of the waves tends to infinity. So the next transparency is going to show you this, so here we've got a string, and this bus takes you back to school I can't believe any of you didn't have to have a string at some point we have the string fixed ends and I've care of a finite see the string vibrates up and down that's what's called the antenna point of maximum displacement and the space was a function of time is that rapidly oscillating curve on the other hand if you look at the equation where the velocity is actually equal to infinity if you try and take that limit to see that the solution is just zero. There's no displacement at all. Well, now, that oscillating curve is not really like that curve. On the other hand, there's a sense in which there is a sort of connection between the two. If you average those oscillations over some resolution time, and if the oscillations are going fast enough, the velocities are higher now,

35:00 So the average oscillation is going to go down roughly to zero, or ten to zero as the velocity becomes larger and larger. On the other hand, it's also true that for any velocity, however large that you calculate, I can always choose a resolution time sufficiently small that in fact the average of that displacement won't be equal to zero. I've just got to go in there and take the average over that first bit of an oscillation and then obviously not zero. And so what happens is that the average displacement in some sense tends to zero as the velocity tends to infinity but the limit is not what mathematicians call a uniform limit. It has this singular nature and you can see it that you can't just turn an oscillating curve into a flat curve. It's rather like my other example of the circle of the line. So in structural terms, relatively important mechanics genuinely involve new structure. That's the oscillation, if you like, rather than just the flat curve, not just the preservation of old structure. So isn't that just another example of a U-turn by the abandonment of calorical resistance? Have we really I'm not going to have a head at all in this moral program, what I should call it. I'm not sure whether he's shaking his head or nodding his head. We'll discover that major discussion. Well, the best I can do here is to say that the way mathematical structures develop in physical theory maybe has a certain natural, although not, of course, an inevitable aspect to it, natural, that is to say, to a mathematician. So maybe if I can see a sort of element of continuity in these mathematical structures, same structure, there's a sort of natural growth of structure. Whenever a philosopher says something natural, you should always be on your guard, because it means what was natural to one man is totally unnatural to the other person. Very awful. Now there's of course a long tradition in natural philosophy that the physical world is constructed according to mathematical principles. For Plato, for example, in the Timaeus, everything is constructed are two sorts of triangle. It's an odd kind of mathematical atomism. And Galileo famously remarked that the book of nature is written in the language of mathematics. So the cosmologist James Jeans, he described God as a mathematician, but he is actually designing the world according to mathematical principles. So in this vein, in discovering the new mathematical structures

37:30 in quantum mechanics or relativity. Are we learning to read the mind of God, as Stephen Hawking famously claimed in his bestseller, Ephesians of Time. You remember the very last sentence in the book, those of you who have read Stephen Hawking's book. Most people claim only to have got to page 20 and never got to the last. But on the last page, he claims that maybe we're reading the mind of God. So that's the kind of view which does have this role of mathematics. But I want now to pursue this question of the role of mathematics for a moment in a slightly more prosaic way, not with this kind of mystical appeal. Now there are two quite distinct cases to consider. In the first case, mathematics provides a language to represent physical reality. So if we just look at this next stop, apparently This is how mathematics gets into physics. Here you've got a bit of physics over here. You've got weights and beings being balanced on balance and that kind of thing. And then over here you've got mathematics, numbers, that kind of thing. And so the way you do theoretical physics, I must admit this is not the only way of thinking about it but this is a rather simple way which I think I'll introduce you to instead. You start with the physical world and then you go into this world with the numbers, you do those little calculations with the numbers and then you translate back again, this is a kind of translation, back into the physical world. So if you add two weights, you want to see what that balance is against. You do that with a mathematical calculation, just adding numbers together. So you get this kind of idea of mathematics representing the physical world. But this is not really typical of the situation that actually obtains in theoretical physics. It's very much more like this. You've got the physics, and you've mapped it onto some sort of mathematical structure, but that mathematical structure itself is embedded inside a bigger structure, This is what I sometimes call surplus structure, these extra bits. Now, of course, there are many examples of surplus structure. It comes in all sorts of different kinds. You've all met with the famous family with two and a half children, the average family that loved statisticians. Now, the two and a half child family belongs out here. There aren't any families with two and a half children, but it's perfectly possible to do a mathematical calculation and get two and a half of the answers.

40:00 So these are elements out here, what I call surface structure. In the other case, bits of surface structure that seem completely nonsensical, like the famous negative energy solution, the Dirac equation, turn out to have very clear physical interpretation. And that was the discovery of the positron and the antipotron, all that kind of thing. But the thing I want to talk about is something which is going on very much in modern theoretical physics, and that is that the surplus structure controls the bit of mathematics that can use to represent the physical world. So out here I can actually control that mathematics by doing things out here and then that control can feed back by that translation scheme into a sense of explaining or controlling what goes on in the physical world by something operating right out here. Now, I want to try and make this clear to you, what I mean by one bit of mathematics controlling another bit. So I'm going to once again take you back to the school, because you all did some geometry. And they're very nice theorems about triangles. One of the most famous theorems about triangles is Desaugs' theorem, which was actually not a theorem of nucleus at all. It was proved in 1639 and then forgotten and rediscovered in the 19th century. says, and I'm not going to prove it for you, you may be relieved to hear, that if you've got two triangles which are in perspective, he was an architect by the way, he was producing a theory of perspective essentially in the 17th century. If two triangles are in perspective with a spectral point, that implies they have another, what I call a dual property, that in perspective with a spectral point, then also in perspective with a spectral line. I'm not going to explain what respect to a line means or anything like that, but that's what the theorem is about. Now the question is, how do you prove that theorem? Many of you may have met the theorem at school. Certainly when I was in school, we used to do projective geometry. Maybe they don't do it anymore. It shows how long ago it was since I was here in school. But the way we use the situation is to say, there's the plane of the two triangles. Now the first step was, take a point outside the plane. Now that seems a very curious thing to say, and I remember when I first met the Zag's theorem, I thought that's very odd. Why don't we take a point outside the plane, because this is a theorem of plane geometry. We seem to be going out here into, in fact, into a three-dimensional geometry.

42:30 Now it turns out with a very simple axiom, like the fact that two points determine a line and two lines determine a point, you can actually prove the Zarg's theorem in a few lines, which have a little construction, and there you have the theorem. But the whole thing hinges on moving out into the third dimension. And it turns out that using these simple axioms of instance, you cannot prove the Zarg's theorem in play. is it does a very pure just a little bit of that matter. Now if you want to prove the Zarek's theorem in a plane what you have to do is to have some more powerful principle operating in the plane geometry something like the famous axiom or theorem of Pappus which can serve as properties of hexagons in a plane and using Pappus's theorem you can deduce the Zarek's theorem but you can't do it just on its own except by going to this wider structure. So this third dimension, three-dimension, is controlling what goes on in the two-dimensional plane and gives a kind of nice explanation of why did Zarg's term be actually true in the plane. Well now I've got one other example that I'm not going to give you because time is getting on a little bit, and it's a slightly technical answer. I'm going to move on straight to the remarks that all this is rather familiar to pure mathematics. Well, the surprising thing is that this sort of thing is also going on involved in theoretical physics. I don't think philosophers have sufficiently won't have been up to this great revolution which has happened in the last 20 years in theoretical physics. But if you don't believe me, I'm going to now read some quotations of books in theoretical physics that prove my point through life that this is actually what's going on in modern physics. In particular, there's a famous theory of elementary particle interactions which are called gauge theories. I'm not even going to define a gauge theory for you, but I just want to explain that the explanatory principles of these theories all operate in this realm of superstructure. It's like going out into the third dimension to prove design theory. Now, let me just quote you for a very well-known monograph of these theories by Heidelberg. Well, this is a direct quotation. So this is what's actually going on. I'm now taking the legal theoretical physics for you. Now, this is not all for done in public, by the way, because physicists are feel ashamed about this, and they don't know what to make of it. So I've got to expose them to what they're actually doing. And this is how they say it.

45:00 Physical theories of fundamental significance tend to be gauge theories, which I may not have so much. Now these, they say, are theories in which the physical system being dealt with is described by more variables that are physically independent degrees of freedom. So you have the number of physical degrees of freedom, and then you increase them. It's like going from two dimensions to three dimensions in this other scale. You're actually increasing the number of degrees of freedom. But of course these extra degrees of freedom aren't physical degrees of freedom because, as we've already said, we're augmenting the physical degrees of freedom. what we do is that the physical significant degrees of freedom re-emerge as being very much invariant under a transformation which gets the valuables. So in other words we can start with this big augmented set of valuables and then we as it were project down to the physical degrees of freedom by imposing invariance under this so-called gauge transformation. So to put the thing just crisply, this is how and a part of one expressive by introducing extra valuables to make the description more transparent it's just like this arc they never mentioned it out because the professional theoretical physicists going to go back to the school or to tell you how things work but I just point this analogy to you and then they bring in the same time a symmetry to extract the physically relevant content so you go out into the structure and then you come back again world. And they go on to say it's a remarkable occurrence that the road to progress has invariably been towards enlarging the number of variables and introducing a more powerful symmetry rather than conversely aiming at reducing the number of variables and eliminating the symmetry. You see, what you might expect is, if you go back to the desired example, the way to do particle physics would be to just operate with a physical quantity, physically significant quantity. But like this papacy's axis, some powerful principle, and nobody has ever discovered how to do modern particle physics, except by doing this excursion into this realm of surplus structure. Now these gauge theories are complicated by what are called ghost particles. Now physicists quite seriously talk about ghosts, you may think that surely they don't have to think about Goethe. Well, to prove that, here is Volume 2 of a great monograph published in

47:30 1996 by Stephen Weinberg, the Nobel Prize winner on the quantum theory of fields, and I've taken page 25 of Volume 2, which is headed section 15.6, Ghosts. I mean, you may think that's very curious. What's Ghosts doing in a book of theoretical physics? Well, there's all kinds of that, and so forth. But let me just explain how Weinberg explains the world those particles and I links up with what I was saying about going into surplus structure the ghost field he says represents something like a negative degree of freedom whatever that could be and these negative degrees of freedom are necessary because they're really over counting the physical degrees of freedom of the components of the gauge field best the parameters needed to describe a gauge transformation to these ghosts are all operating in the realm of up a structure. But it's impossible to do these modern gauge theories without introducing these ghosts and of course if you have a particle you must have an antiparticle so as well as the ghosts of course there are friendly anti-ghosts around as well as the area. And so these ghosts and anti-ghosts play an absolutely vital role and nobody knows, if you ask any of the great professors of theoretical physics at Cambridge or Oxford or Pell College, none of them know how to do these theories without introducing this palaphanalia of surplus structures I call it these extra degrees of freedom these ghosts and of course these ghosts are not intended to have real physical significance remember that they're out in the world of surplus structure and they belong to the unseen world with a more extreme sense than electrons or photons I can't that be reminded of the famous Tibetan ghost trap now let me just show you this is But if you live in Tibet, you know, you've always got lots of ghosts around, and so what you do, you hang up the ghost trap. I got this from the Horniman Museum, by the way, in South London, this photograph. You hang that up, and you say a little spell, and repeat a little mantra, and so forth, and the ghosts all climb inside the ghost trap. And then you don't say any more, they're caught there, so you have them hanging up. If you don't like your friends, you take the ghost trap, you carry it round to your neighbour's house, you say another little spell, and that lets the ghosts out, and they go all over his house. So it's actually a rather convenient little bit of metaphysics, this ghost trap. But let me just remind you very quickly of the bubble chamber again.

50:00 That is the ghost trap of modern theoretical physics, where all these elementary political interactions are supposed to be going on. And if we go back again to the Tibetan ghost trap, I think you can see what Thickenstein might have called a family resemblance of the Tibetan ghost trap and its modern particle bubble chamber. Now, what sort of world is the unseen world? This is my last bit I want to talk about. There's an ongoing theme in writing about science that behind and beyond the complex, of allegated diverse world of sensory experience that lies a simple, unified, integrated world that science is gradually revealing, that the unseen world knits together the patchwork structure of the world of appearances and provides a true account of the reality referred to in Plato's famous simile of the cave. Remember the prisoners locked up in the cave to see the shadows on the wall. Of course, I must add, when I talked about this, John Lucas immediately said, think the way to get a reality in the methods of science, which is actually correct. Nevertheless, science is supposed to investigate this reality. T.H. Huxley put it like this, the aim of science is to reduce the fundamental incomprehensibilities to the smallest possible number. Well, this theme of unification has generally been expressed by a scheme of reduction, which the sciences are raised in a hierarchy, in a sociology and psychology, some of the top of the load-out biology and then chemistry. The whole time rests on the bedrock of physics, and physics itself is reduced to a uniquely theorem of everything, a capital T, capital O, capital E, theorem of everything. Now, such is the rhetoric, particularly espoused by Nobel Prize winners in physics, when they apply for huge government grants to work on problems in fundamental physics. I may say, coming back to Weinberg again, that he wrote me a letter when was applying for a large grant to build it while the new accelerator in Texas say could I give him all the references on reduction in philosophy of science he wanted to get this straight for making this application in fact because the thing was being built in Texas it failed the other states were jealous so there's a lot of other story of that well anyway you might be forgiven for believing that the ultimate aim of science to achieve a sort of one-off of called Humperdinck's Law, the final law, Humperdinck's Law, which everything else would be accounted for in its play. If only I could write Humperdinck's Law for you, then I would have the Nobel Prize, and then everything would be explained from Humperdinck's Law.

52:30 But a strong reaction against this sort of wild talk has set in recently, a philosophy of science. The penitent has swung strongly in the opposite direction, promoting the disunity of science and the virtues of the dappled world. The title of Nancy Kopp writes, a most recent book here at the London School of Economics in the United States, Nancy Kopp writes in the audience. She can make some comments on this. Well now the arguments here in Nancy's book to double well, look at detailed case studies of what science is really like. And not just in moments of wishful thinking how we would like it to be. The description of real science provided by this work is much closer to the experience of the research work of the cutting-edge of the sciences and the sanitiser count, given in much of popular science literature. For sure, warnings about the tendency of human beings to jump the conclusion about unification go back at least to the 17th century, when Bacon wrote, the human understanding is of its own nature prone to suppose the existence of more order of regularity in the world than it finds, and though there may be many things in nature which are singular and unmatched, yet devises form of parallels and conjugates of relations which do not exist. ask my question, has the pendulum swung too far? Now I would like to explain my own point of view about the dappled world. The idea of unification, to me, is essentially a regulative idea. One might even want to define a concept of scientific rationality as one which invokes the simplest, most unified theory to explain empirical phenomena. On this account, creationism, for example, this famous rival to evolutionary science, would be rejected not because science chose it to be false, but because its acceptance would violate the canons of scientific rationality. After all, if you define scientific rationality in terms of reducing these unified simple types of explanation, well then you can use that to argue against creationism with all this special production of the fossil record and so forth. But this argument in defense of the scientific account is obviously viciously circular. If I define science in terms of reviling unified explanations, I can't use that same argument to justify the fact that science should be preferred over a nasty, dappled, undriified account by the creationist one produced in evolutionary theory.

55:00 The justification of the scientific account can, however, in my view, be provided in terms of the past record of scientific theories based on these pragmatic explanatory virtues of simplicity and unification in producing successful novel predictions like that omega-minus particle we saw before, which is the usual gold standard of scientific progress. So is it not rational to expect the same criteria to produce more successful sounds in the future? These meta-inductions are always liable to fallibility. Perhaps at some deep level of explanation, physics will just get more complicated, rather than increasingly simple. But that's why I talk of a regulative ideal. It doesn't have to be indefinitely achievable, but its past successes provide justification for pursuing the ideal as a leading principle of scientific investigation. The difference between myself and Nancy is essentially that she likes the double world in the style of Gerard Manley Hopkins, whereas I want to get out my needle and thread and try to stitch the whole thing again. So let me try to summarise in one paragraph the status of the unseen world. In philosophy, there have always been two attitudes to the senses. The first is that the senses are linked not to reality, but to their appearances. In the words of Armendie's poem, they access the way of seeming, not the way of truth. are in effect a barrier between us and the world. Here's reality and here's a slightly lugubrious philosopher cut off from reality by the barrier of the senses. Now if he's going to get through that barrier, that reality can only be known, if at all, by reason or rational insight, the sort of thing that Plato thought that we could get to know about reality. and now look at the happy empiricism you see the smile on his face is the other view it's a liberal and relaxed empiricism the sense is linkous notice the link in an admittedly tenuous and fallible way with reality and that science in pursuing that link has an elevating path revealed to us the unseen world that lies the world of every day experience.

57:30 So, as I said, there'll be a few, just a couple of weeks right now. Anyone who needs to leave? Okay, well thanks very much Michael, I'm going to decline your kind invitation to abuse the hour of the chair and respond to what he said about me. At 10 or 8, until afterwards, dark alley in summer. But, because I know what I think and I'd rather hear what other people think, Hassett Chang was the first person to catch my eye. We have a Robin Michael Well, I'm terribly careful, Shab, actually, like Michael, in the simple advice, I'm always very careful with these things. First of all, it is epistemic, not much of what you call it, it can't make sense, just like Michael come up with a relation that doesn't relate something. And secondly, I keep saying that, well, it's all in Poincaré and I think we like it. And secondly, as Poincaré himself says, it would all be very simple if one example we operate on then with a switch from Fresnel and the elastic solid ether to Maxwell and the electromagnetic field were typical. Because that's the one case that anybody can ever completely intact. You get exactly for announced equations, for example, for the relative intensities

1:00:00 of the reflected, refracted, polarized in various directions when you shine for being applied on an interface between, say, air and glass. You get exactly the same equations, no correspondence, for example, nothing. You just get the same equations in Maxwell as you did already for announced. But of course, in other cases, clearly in the Einstein-Newton of the quantum mechanics classical mechanics case, you're going to have to talk about continuity of mathematical structure modular of the correspondence principle, and allow, for example, the sort of things that Michael, much more cleverly than I could have done, mentions here, some sort of continuity of mathematical structure, which will be a bit of a tricky notion, once you allow for continuity not just to be completely taking over the same mathematical structure about modifying this in this way so i want we're not really against each other at all i think he's elaborating on uh the program that quite very started and admitted very clearly that it was only based on one example obviously one example right now of many cases Thank you very much for the chance. Nancy. Well, you said I was going to startle John by defending structural realism and it couldn't be the second time in 24 and a half hours that I've done so. So could I just add, I think there's a little bit of a history of philosophy of science that helps make John's little more comprehensible. You'd like to start, you start with one of Michael's simple stories. We'd like to start with the idea that there's a lot of continuity of our theoretical knowledge because, for instance, we say that in the limit as the velocity of light changes, Newton's laws are approximately true. And in the limit as h-bar changes, the laws of classical physics are approximately true from the point of view of quantum mechanics. So that we have, lots of people said for years and years that theoretical knowledge was cumulative because not that we had it right, but that we can see that within the range of some parameter, which

1:02:30 at that time we didn't change very much, we had it approximately right. And that was what we all wanted to say for a long time. And then along came this big revolution in the 60s, thinking of anti-positiveist revolution, Kuhn, Feyerabend, and people like that who pointed out, particularly Kuhn, that it was very hard to say that because, after all, Newton's theory and Einstein's theory conceptualized the world very differently and it had very different kinds of things in it so the equations might look kind of the same in the limit, but Newtonian mass when you write an M, Newtonian mass just ain't anything like relativistic mass that the whole meaning of the terms had changed what was actually being referred to changed then along comes John and Elie Zobar who say, well, that isn't so distressing because there's still something to be salvaged. We didn't have it right about the quantities that we were studying, but we did have it approximately right in the limit about the relations among the quantities. So I see one of the advantages to what John and Zahar and structural realists have done is to bring us back to being able to say that knowledge accumulates in the way we wanted to be able to say before we started realizing that the actual subject matter had changed dramatically. I'm sorry, just to respond to that, besides I really agree with you Nancy that quantum mechanics and classical mechanics in terms of the equations and what they say about the world are approximately the same. And this was the point of that example of the wave equation, you remember that kind of thing and how that got, in some sense, 10 to a line. You know, for that, it's all the way it's meant to infinity and it's caused 1 to 2H to equal 0 in quantum mechanics. There's no sense in which an oscillating curve ever is, in a mathematical sense, to just a straight line. There's a well-defined sense in which certain aspects of oscillating curve like the ambient value of some resolution time are similar to what you get, the close approximation of what you get.

1:05:00 I was going to say, the point is that some equations are possibly equal to each other under some conditions, and that's what we can do, and that's what we can do, and that's what we can do. Well, I think you see there's been all this very interesting work by Michael Berry, with Oswald Society Research, Professor of Wisdom, Mr. Schutau, working on this classical limit of quantum mechanics and this is really what Meri's work has been all about, showing that there's no easy limit in which you can say a classical physics can be recovered. There's a kind of interface between the two theories which has got all kinds of delicate phenomena. I mean, not that it corresponds to the fact that if you have a wave being defected through a slit then there's some fluctuation of appropriate scale whatever the weight length is unless you actually let the thing go to zero so there is a quantitative a quantitative change that we watch you get from the old theory and the new theory and I think that's something that's only really emerged from the work, very careful work of fetal isomite and so forth rather exposed this and it's quite true that simple-minded books on quantum mechanics, you say, oh, well, classical physics is somehow approximately extracted in a certain unit from classical physics and quantum physics, but it's really not like that. And the thing, I think, well, I think it's not more correctly expressed in terms of what I call the structural discontinuous. I don't think that's structure, but I think somebody else. I didn't want to ask the question, I just wanted to say that perhaps one way of looking at what Michael is saying is what Kuhn was saying, but about the structure. So there's a sort of a paradigm shift that becomes evident when you look at the structure and the chaos sort of way that Barry does. So, whereas, so in the 60s, as Professor Carver said, there was this sort of realization that there was a paradigm shift and we had to look at the meanings of things like N for mass more carefully. and then the structural radius came along and we could see that perhaps there's

1:07:30 something that can be salvaged there but perhaps now with people like Barry coming down we can see that actually maybe even that isn't salvageable because there's a paradigm shift in structure perhaps there's an analogy Thank you for coming to my rescues. What always relies on one research students on these occasions. Thank you for the wonderful lecture. All I'd like to ask is, in terms of the creativity about quantum mechanics, I think of the 1920s, I'm not an expert, I mean I studied here but not science, international relations, but there was a book by Baris Hoffman, a strange story with content, which talks all about their tremendous intellectual creativity in the 1920s. And I think at Göttingen, partially there at the University of Germany, I feel like Andrew Tello still were involved. Could you say anything about that that's relevant to your talk? Yes, well of course the 1920s were a wonderful time in theoretical theory, where everything was happening over a very short space of time, but gradually the people's understanding of quantum mechanics, initially there was a formalism, a mathematical formalism that was developed rather different ways by Werner Heisenberg and by Hermann Schrodinger, but the question of what it meant, how you look at the world, what it means to the world, was largely expressed in the 1920s by Niels Bohr, who became the great guru of the whole Bohr tried to introduce what he called the principle of complementarity, which was a kind of perspectival view of reality, that there are lots of different ways of looking at reality, that these ways are mutually exclusive, so we can't simultaneously entertain these perspectives, but on the other hand they're complementary in the sense that unless we take account of all these different ways of viewing reality, then we can't get a total grip on reality. It had sort of two aspects to it. The fact that we couldn't put these perspectives together in a single perspective, that was the kind of idea, and the idea that these different perspectives complemented each other, so you had to take both into account and so forth. And this view was regarded by Paul as a profound philosophical discovery.

1:10:00 And he thought that the same things ought to apply for the case, for example, of life. He thought that there were two aspects to life, the kind of life as we experience it, the subjective view of life, and also this view that we get by doing biochemistry and so forth. and that these were essentially complementary and also exclusive. That if we poke it in to the human brain and really try and find out what was going on in terms of biochemistry, that we would actually kill the poor patient and say we're no longer have a subjective experience. And Paul would have a lot of essays trying to extend the principle of complementarity, complement mechanics into things like biology and so forth. Now, the thing is, it's always dangerous to make great metaphysical discoveries from bits of theoretical physics, because, in fact, later on, somebody who's working here in London, in Bowen, discovered a new way of understanding quantum mechanics, which didn't involve this quantum mentality idea at all, it was a kind of guiding field, it was moved to what it was about, it had certain non-local aspects to it, but it was certainly a much more traditional, kind of classical, the way I'm describing the world. And so along comes Bohr saying, look, I've discovered a great new metaphysical principle of physical complicity that somehow rules out the possibility of any way, other way of understanding what we can. But then in the 1950s along comes Bohr and actually produces a version of the interpretation, which is totally a valiant So if Bohr is not quite about the interpretation of physics, then what to all this extension of the great metaphysical discovery to biology and so forth as Bohr, taught a lot of, and many of these more public and popular lectures are devoted to extensions of this philosophical principle of cosmitality. So it's an opening lesson to be very careful not to draw too hard and fast conclusions and then extend them towards other branches of cells and body, because you may find yourself underpired by some technical developments in the original interpretation. That's mostly what happened with David Bowen. So does it all share with us, you know? Is John Lucas next? Well, very first, rather briefly, after the rescue of both Professor Ray and Professor Cartwright,

1:12:30 Where the issue is the same and the answer is it's a place and I think the best example I can bring is special volatility rather than on to the campus. And on the one hand, I'm not going to say that the return of mechanics is just a special case that if you can see it as utility, we then get the borax transformations turning to the anterior components. But, on the other hand, as a good head side, there is a very big difference between any change of space in time and the structure from cost-based time with four dimensions. and it may have an effect on which sector of your country is most important, but there will be implications. In the case, it's perfectly fair to say, A, that the new business is a requirement rather than a reputation of the airport, and B, what do I try to expect? It may well be a reputation of the airport, and those two positions can live together And one more, this is, again, Aristotle's book, as it's always a commentary by the use of identity. What Aristotle was concerned with was one form of explanation, a form of explanation, which is what he turns his diagnosis. And diagnosis in medicine still is the main thing, but right to the scientific world, we are often concerned, A point that came out later is a kind of perception where we perceived by being able to fit our sensory stimuli into some literacy pattern. And the lesson I've given to this is that biologists have sold them in these photographs, and they always use diagrams to tell you what you will see in the pictures of a microscope, and you have to know what you're looking for in order to see it. And that then leads to a more point on the other sea level, that if, as Michael wanted to prove on paper, that the access to the other sea level is a state of the world, and there are many important explanations, we are quite likely to get to a whole series of rules, getting more and more and better explanations of what the other two had not been expected.

1:15:00 Thank you very much indeed, John, because I absolutely agree with your first point. I think I go back to my example of squeezing the circle. That may be nice to illustrate it. For lots of purposes, a very elongated squeeze circle is just like a line. If you look at it for a long way, it looks much the same. But if you look at what I would call the deep structure of the thing, the topological structure, then a closed curve is quite different to a line. In the case of relativity, I take relativity to be the theory that the geometry of space-time is a metrical geometry. And the Newtonian space-time only has a singular metric. It doesn't have a non-singular metric. And so that would be an all-or-nothing matter. That's where the discontinuity comes. There are other aspects of things that you can measure, of course, that do have the continuity, the approximation, So that's a little bit of what you mean by the deep structures you like. Just about Dissarg's theorem and stepping outside the plane. In a sort of simpler way, that sort of thing is familiar even within plane geometry, isn't it? When you prove a theorem about some given figure by adding auxiliary constructions. I just wondered how that sort of consideration fitted into the general theme of the unseen. Couldn't once if we say, oh, that's just sort of mathematics, these auxiliary constructions are not real. Well, yes, one could make the point with many other examples. and the reason why I chose this large theorem is a rather nice example where three-dimensional predictive geometry is not a conservative extension of two-dimensional predictive geometry with respect to the axioms of incidence, and there are really many good examples of non-conservative

1:17:30 extensions of that sort. But the point about these extra constructions or the extra variables or the extra dimensions being just mathematics, well that was the point about the gauge theories That is exactly what's going on in modern theoretical physics. The people are producing axioms through the principles that operate out in that realm of what I call surplus structures, a purely mathematical structure which control what's going on in the physical world. Another really good example of that is the famous S-Maprich program in the 1960s, in which people extended the scattering aptitudes and the same force of will-variables into complex variables and looked at actions and coding the behaviour of these experiences as functions of will-variables. It sounds like the singularity structure I can be in this context plain. he said it's a principle of maximum smoothness now of course maximum smoothness must be taken as maximally smooth consistent with the things which prevent it being completely smooth if something's completely smooth it's just a constantly complex variable So it's not really a principle of maximal smoothness, it's a principle of maximal smoothness, as good as you can get, given the fact that the thing can't be completely smooth. But nevertheless, that was very much a theory that operated in some of the structure. It got through the table of like age theories, why works allow, and so forth. But my point is, the example is, in fact, that tactic, which was rather laughed at with the aspect of the theory at one time, ridiculous having axioms of that very mathematical nature has actually been totally resuscitated by the modern creative programs. This is something I think philosophers might want to take account of. Michael, I'd like to ask you to say something about what you think specifically is the relationship between theories of the one hand and the physical systems. You alluded to the translation. translating from the physical objects to the mathematical structure and then back. Translation also applies to languages.

1:20:00 So in a sense, it doesn't seem that that is the precise notion that you may want to use in that case. It's not that we have a set of statements on the one hand that we can explain to a set of statements on the other. So there must be some more specific relationship. And philosophers, of course, have put a number of proposals for what their relationship can be. I don't particularly find any of these proposals very appropriate. This range from resemblance and similarity of the physical object and the structure, to isomorphism between the two structures, and so on. But it seems quite crucial to get at the precise nature of this representational relationship. Yes, well it's a really good point. To begin with, of course, when you're talking about the relation of mathematics to physics, you must have some sort of philosophy of mathematics in mind. and I was being outrageously platonistic in thinking there were actually abstract objects to sit out there in the mathematical world. The relationship that I call the translation was essentially one of isomorphism. But isomorphism not with the real world, but with some scaled-down, rigid, emasculated version of the real world, which you find in the textbooks of physics, the British kids play, and the simple and modern opinion, and that kind of thing. So that's what you sort of model, as they say, onto the mathematics. We think about what mathematics is doing in physics and the structural philosophy of mathematics, for example, which sees mathematics as just being a theory about structures, including physical structures as part of that theory. There's all kinds of other ways of understanding how mathematics gets into physics, but the sort of simple idea of surplus structures to separate the mathematics sharply from the physics, I introduced this idea of what's called representation, representational approach to the word of mathematics in physics. And of course it's very difficult to make that sharp distinction between what actually is the mathematics and what is the physics. And that's where I held up the example of a family with two and a half children. And this obviously is something that's just a mathematical calculation. You take the average of the family and try and become the average family who's got two and a half children. but that doesn't correspond to any actual kind of example where just bits of mathematics can't relate to the world but if you take things like energy for example it's a very nice example when

1:22:30 energy was first introduced in the 18th century it was just a mathematical device that was the first integral of newton's equations of motion for a particle moving under conservative forces and then in the 19th century with the rise of thermodynamics and the law of conservation of energy, energy got focused on the real ontological significance. It wasn't just the mathematical construction or calculation of p square plus q square kind of thing. It was actually itself the primary ontology in the 19th century was very much that energy was considered more fundamental than force and so on. So the kind of shifting attitudes to bits of mathematics and whether you're going to think of them realistically or not, and it's fascinating to look through the history, particularly the energies of the gradual change in the attitude to the concept of energies through the 19th century. Just a question I wanted to raise. You mentioned very nice, very intelligent systems that don't really exist. I wonder if you comment on systems such as inertial record systems which don't exist, on which a very complex and beautiful theory is built, not only atonium, but also special relativity and the hot thing, so to speak, of inertial reference systems of thermodynamics, the karnel engine, which also doesn't exist, on which the science of thermodynamics is built. The second question concerns translation, that's not exactly the first one. What is your point of view on Taurus-Pongebob principles being translations between possible worlds, so to speak, or different languages, C going to infinity, moving back from special relativity to Newtonian physics, which is perfectly continuous, and h bar going to zero, and moving from quantum physics to the Newtonian science. And I think you'll agree, one of four is great insights in complementarity, which I decided others just considered to be a theory of Mr. Fielgit, was that h-plot going to zero decouples the wave particle duality. And that's also a perfectly reasonable continuous limit. Well, I'll take the second part first, back on the question of continuous limits again. I want to be slightly technical through that, but if you take my simple example about the vibrating string,

1:25:00 I mean, if you write down the equation for a operating stream of a finite velocity, what mathematicians call a hyperbolic equation, if you just put the velocity equal to infinity and scrub that term, it turns into what mathematicians call a parabolic equation. And that's the striking structural difference between what I call the limit equation and the equation of finite but really R of C. It's structurally a totally different hyperbolic equation, a totally different property. I mean, for example, they can't absolutely support parabolic equations, whereas hyperbolic equations can. So they're totally different types of equations. I think this is something that I was trying to emphasize, how these mathematical possibilities do actually reflect in the physics. Well, that was corresponding to the H characters, yeah, I don't know. It's getting higher than you. It's getting higher than you. No, that's right. It was only a kind of analogy. If you get the vibrations of the waves getting faster and faster, that you can, in some sense, model the fact that you haven't got oscillations at all by taking averages. David Lewis, and I think it's perfectly possible to consider idealized worlds with frictionless planes and inertial reference frames and so forth. Now, these are not actually our worlds. It's perfectly possible to speculate what the physics is in these ideal worlds. And, of course, if you go back again to Galileo, remember that was Galileo's great contribution to doing science. He said, don't tackle the world in its full complexity straight off. start with assuming there isn't friction, that kind of thing, and get some simple first approximation, then feed back what you call the accidents as a correction term. I mean, it's interesting, I think, to speculate what would have happened to physics if we started with nuclear physics.

1:27:30 It just happens that Newton's example and Galileo's examples, there was a kind of major effect, if you like, which could only have the accidents feed back as approximation. But that's the trouble with something like nuclear physics. There is no kind of simple basic approximation. You've got to tackle the thing head on. This is the quite sort of messy part of physics. And that's why, of course, I do very much agree with Nancy about the DAPL world. I think nuclear physics is full of examples of DAPL and it's a kind of messy subject. That's why I started doing research in nuclear physics but rapidly gave it up and went to the quantum electronics It just seems a lot quite serious. Okay, well, I think we should all thank Michael for a very non-messy talk. Thank you. It's very nice to reflect on the circle, so I took your... I thought that was like your... Yes, you know, I actually have to do it. Trailer, I'm quite concerned with this, but you look more distinguished from there when you're golden. You look younger and less distinguished from there when you're golden. You look younger and less distinguished from there when you're golden as well.

1:30:00 It's the same phenomenon for very good. Let's see if we get this real load up there. What you're measuring is match the resistance to change the vote. Thank you. There's no... There's no... There's no... There's no... There's no... There's no... Why did they do it? They took his back on. They took Carol Daz He walked back Thank you. I'm going somewhere now. I said, you can't be going now, but now you're here. You can't be going now. You're going there in a few moments.

1:32:30 And if you're a moment, we'll get there. No, no, I'm going to be on a beach. Literally, you can't be able to go somewhere now. Well, he is afraid of all the time. I'm reading. Thank you. Oh, yes. I'm in politics. And therefore, you know, he's rather speaking dismissal of the orthodoxy. It's from the orthodoxy. You know, I know. I think so. Where is something like that? Yeah, Michael would sit when I sit on the phone. I'm sorry. I was going to... I was going to... I was going to... I was going to... I was going to... Yeah, but then some of the drivers on the south west is already down, and I thought, oh, there's a hell of a lot of them here, and I had to go and pick up some books because I've got to go away tomorrow, it's going to be my last chance. I don't know where he's going to be. He's probably... Well, do you want more? If you were coming up, if you were meeting or not? No, I don't agree. Oh, right. Oh, right. Oh, well, maybe you can go into bed. Well, do you think I've almost had a while on the scene? I don't know. It's not. I've got a special show on Friday, and I'm going to be away from the end of the month.

1:35:00 And there's a very interesting... You know, that discussion is simple, it needs to be complete. Well, all of us will talk about the next week about... Well, an act that's been shared is going to be John Perrin. I'm sorry, it's the week after. I'm sorry, I'm so sorry. You're quite right. I didn't want to get back to him, but he's so far away. Absolutely right. John Barrett's talking about the television station. It's the week after that, but I think it's... I think it's a 15th anyway, I've got it. I'm not quite sure about that. Hang on, I'm not looking at the different things there. No, no, no, that is a 15th. Hello. When are you doing that last year? No, you're right, it is going to be gone. No, I'll tell you what I'm getting mixed up, because he's talking at Bristol. He's talking at Bristol, that's right. That's why I was getting mixed up. He's talking at the Bristol group on the 15th and 15th and 16th. Well, I was going to ask if you could ask me I'm working with Chris or somebody to make a tape of it. I'd really like to be in trouble about that. Yeah, I'm really interested in stuff. And as I said, it's one that I will look at. Yeah, no one I'm sure. Okay, well, what I could do, I don't know if it is, is to send, I could either send the tape to, I could either send it to Chris or I could send it to, What I'll probably do is give you a ring sometime tomorrow, I'm actually going off on Friday, just to make sure I know who I should do. And I really wouldn't be interested in it, it's very interesting. It's about, yeah, you've heard, and I like that stuff, because it really can't change the whole terms of the absolute relative debate. I just want to get back and look at the impact of the framework. I'd really like to learn. So I'd appreciate it.

1:37:30 Okay, that's it. Well, let's go on a drink somewhere. This is really stupid, we've got to keep going. Are you serious? I saw you coming in on the back. I let the rest find me down, not knowing where the over-go-inders are. Hi, Leslie. Hi. How are you? How's it been? I've been ill, and... I'm very well in here. And you, I'm sorry if I feel late. Well, only five minutes after you've scarred a little bit. You're coming down, actually. The general luggage. Yes, yes, yes. Michael rang me up yesterday, so he said... The World Railway. Yeah, which has been pulled off, but only at the last minute. Well, I hope that some of us want from the south side to meet each other as well. Oh, that is your life. Let's go and have a refresher. Thank you. That's right. But now there's no moment from boat. A lot of people come in the morning line. No, that really puts me up. When you have enough to sleep in, it's embarrassing to me. I can't use that screen. I'm waiting for 10 years, sir. To be correct. I'm waiting for 10 minutes. You're welcome. There is something to be said about gaps, I think ten years probably, five or something. My mother is one of three. Thank you.

1:40:00 Thank you. I suspect it's just some sort of advertising.

1:42:30 Is that where your base is? No, it doesn't. In the philosophy department? No. Right. We must have had these games later. I'm locked in. What is he? He's a great one. He's a great one. He's a great one. He wasn't there himself to hear it, was he? He was. Oh, what? I was hoping to get him in a ring before going over there. We'll be right back. I'm assuming that people don't know which way.

1:45:00 There's somebody else we were worried about. It's so rather sad it isn't anymore. It could be, that isn't it. Well it was, for years and years. But it isn't fair. I don't know if it really is the one on which... I really don't know. I think it's been very substantial if you can stop the actual, you know, the actual thing. There's been a building on that side. Oh, yeah, yeah. I think it's been very substantial if you can stop it. You know the drummer? I am so sorry, we haven't met you, it's Michael. Oh, how nice to meet you. I'm very good to meet you. I'm Michael's other brother. I haven't met an idea, it's a great brother. Well, Michael Wright, I would like to think that I'm a lovely old friend of mine, but we go back the best part of 20 years being the Chelsea day. Oh, Chelsea, yeah. Yes, and even after that in Cambridge. Right. No, no, no, no, he's a very own friend, isn't he? We've known, uh, since, uh... How pretty good to see you again. You're looking very well.

1:47:30 Thank you. I don't know. Mm-hmm