 Now, this handout is supposed to be recorded. This way. So here's the first page. Here's the second. And here are references. Okay. Yeah, I think it's good. Jill, you have 30 to 35 minutes. And there is also a handout in the chat. So I will be posting it from time to time. If new persons will be joining, you can follow the handout. So Jill, please begin. Great. Thanks so much. And thank you for having me. Also, I don't know if it's 8am for anybody else, but if it is, thank you for coming. So there is a handout, which has been posted in the chat. So I hope you can all see that. Great. So I'm going to be talking about the idea of perspicuous representations and symmetries. So in general, there are many different ways of representing something. Different mathematical devices can be used to represent a given physical or mathematical object as when we use different coordinate systems for the Euclidean plane. Different languages can be used to express the same content as the proposition that snow is white can be expressed in French or English. Different languages, excuse me, different formalisms can be used to state a given physical theory as quantum mechanics can be formulated using either wave or matrix mechanics. How we represent something moreover seems to be entirely under our stipulative control, essentially a matter of arbitrary decision or convention. We may use any coordinate system we like to represent the plane. We may choose to speak any language. We may equally use wave or matrix mechanics. The choice seems to be governed entirely by pragmatic considerations as to which representation will be most useful or expedient for us given the aims we happen to have. So choice of representation, in other words, doesn't seem to be a matter of getting things right, but is something that's completely up to us, just like a country's decision of which side of the road to drive on. That said, some choices of representation seem to be clearly better than others. And a recent buzzword in philosophy of physics in this context is perspicuous. Philosophers of physics generally agree that some representations can be better or more perspicuous than others. In particular, they agree that some representations of physical systems or theories can be better or more perspicuous than others. A further question that people disagree on is what makes these representations perspicuous. So does the goodness of a representation, when one is particularly good, does it flow entirely from the aims of creatures like us in the context at hand, or is there a more objective non-pragmatic sense in which representations can be perspicuous, something having to do with the intrinsic natures of the representational vehicle and the represented item or the target? Or maybe the answer lies somewhere in between. Many philosophers have thought that the answer lies significantly, if not entirely, in pragmatic or other subjective factors so that perspicuous representations are better for us, not better in any objective, intrinsic sense. I agree that we do often choose to use certain representational devices for pragmatic reasons as choosing to speak French is better for getting around Paris than speaking English or using one coordinate system rather than another might make certain calculations easier for us to perform. But I also think some representations are more perspicuous than others in an objective, intrinsic sense so that some choices of representation are objectively better than others. That's what I want to argue for here. One consequence, which I'll turn to at the end, is that there's an interesting non-pragmatic sense in which representations that differ only in their level of perspicuity are not equivalent. As a quick note, my focus here is going to be on representation in physics and mathematics. I aim to remain neutral on the nature of representation in general. Okay, so now we're on section 2 of the handout. Excuse me. So you might think our representational choices are just a matter of convention or stipulation at the very least unconstrained by non-pragmatic factors. This permissive view of representation has been defended or simply assumed by many philosophers. So in this view, the representation relation between a chosen representational device and a target is achieved by mere stipulative fiat in the phrase of Craig Callender and Jonathan Cohen. So one representation might be better or more perspicuous than other, but that's in time. That's entirely a matter of whether it's more useful for us given our aims and not because of any essential features of the vehicle or the target. So good representations aren't objectively or intrinsically good. As a result, we're free to use all sorts of things to be our representational devices, even ones that don't resemble the target in any clear way. As in Callender and Cohen's examples of using a salt shaker to represent Madagascar or your left hand to represent the platonic form of beauty, we can imagine entire communities who make very different representational choices from our own, such as a community which stipulates that writing a sentence in a particular font signifies negation or one which stipulates that only positive numbers represent time-like vectors and two examples from Trevor Tytel. These representational choices might strike us as odd, but they're perfectly coherent. There's nothing objectively wrong with them. No one's making a mistake. More generally, it seems we can use any device we like to represent anything at all. No device in itself has representational content. We have to stipulate that it does. And it seems that anything can be stipulated to serve as a representational vehicle for anything else. The Putnam has called this idea trivial semantic conventionality. And in Tytel's words, the idea is the familiar platitude that any representational vehicle can in principle be used to represent the world as being just about any way whatsoever. Now I think this idea can't be quite right for the reason that some representational stipulations can't be made. So no stipulation on our part will allow us to use the integers as our mathematical device for our physics of Newtonian systems, for instance. And the reason isn't pragmatics. It's not that it would be more difficult for us to use the integers or if we had different aims and interests, we could get by with that mathematical representation. The reason is the integers just don't have the right nature or structure. There aren't enough integers available to represent all the different possible physical states of the world that the culprit isn't us and our interests, but the intrinsic natures of the mathematical device and the physics in question. So there are limits to the idea of trivial semantic conventionality. Some choices of representation can't be made irrespective of us and our interests. And so there are some non-pragmatic constraints on our representational choices. Still, there seems to be a lot of leeway and arbitrariness to our choices of representation. And you might think the only non-pragmatic constraint is the cardinality constraint, which says essentially that a device must have the resources available to represent distinct aspects of the target by means of distinct aspects of the device, or in other words, that the mapping from the thing we're representing to the device we're going to use to represent it isn't many to represent the case of Newtonian physics and the integers. So you might think there's no objective sense in which some representations are better or more perspicuous than others beyond the basic point that a representation which lacks the requisite cardinality is objectively less perspicuous in the sense that we can't even use it. Okay. So now section three of the handout. We're going to talk about our representational choices in physics and mathematics, and a reason is just that we generally want more than the capacity by brute stipulation to represent things, which is a capacity that any candidate device will have so long as it satisfies the cardinality constraint. We wouldn't be satisfied with expanding our mathematical representational device in Newtonian physics beyond the integers to include the real numerical representations, even though that kind of thing will be able in a sense to represent the physics of Newtonian systems. So we want more than just a crude or forced assignment of distinct representational entities to distinct aspects of the target because we generally want our chosen device to represent more than just the cardinality of the thing we're representing, but various of its other features too. So this now raises our earlier question. In what sense do we want to represent things well? A permutation of our usual numerical assignments wouldn't be calculationally convenient. It will make it very hard for us to come up with the correct inferences about these systems. So is representing things well just a matter of pragmatic considerations like these? The permissive view of representation says that it is. So consider that there being a resemblance or an isomorphism of some kind between a chosen representational vehicle and a target is unnecessary for a representational relation to hold, as in the case of using a salt shaker to represent Madagascar. Calendar and Cohen conclude from this that resemblance might make one choice of representation in their words more convenient than another given the scientific purposes at hand, but that there's no further reason for choosing a device that resembles the target. So I agree there's a lot of leeway in which representational devices we can use so that our choices of representation are to a large extent up to us, but at the same time when it comes to physics and mathematics, the choice isn't entirely up to us and not just because of the cardinality constraint. So think of the Euclidean plane and different mathematical devices that can be used to represent it. We can represent the plane by means of real numbers. This satisfies the cardinality constraint. There are enough real numbers available to design a distinct real number to each point in the plane. However, this is clearly a worse representational tool than ordered pairs of real numbers. The one one mappings from points in the plane to individual real numbers will be highly convoluted and unnatural. That representation is going to obscure just about every feature of the plane other than the cardinality of its points. With the right stipulations, this can represent the plane, but I think we can all agree that it would be a bad choice of representation. Now I am assuming here that a space like the plane has a structure as opposed to philosophers like Poincare and Reichenbach who say there's no fact of the matter about a space's structure independent of various conventional stipulations that we make. But as long as you don't have that kind of view, you should agree that there are facts about the match in structure between a representation and a space. I'm just going to set aside here the view that a given space has no intrinsic topology or other structure. There is still the question whether a mismatch in structure makes for an objectively worse representation. And you might think that the badness of the single real number representation of the Euclidean plane is fully pragmatic. The calculations required to recover features of the plane will be extremely complicated. And that's true, but the badness doesn't have to do only with us and our aims and the context. It's hard to imagine a context we could be in or aims we could have for which this would be a better representational tool. This suggests that it's a worse representational device regardless of what we're planning to do with it. This suggests that it's objectively better to use ordered pairs and that someone who didn't use ordered pairs of real numbers would be making the wrong choice. And the reason doesn't seem to be the reason for the badness isn't just us and our aims, but the plane's nature and the fact that some devices are inherently better suited to that nature. This is a bit different from the case of the integers and Newtonian physics where the cardinality constraint rules this out as a choice of representation. In this case, it's possible for this device to represent the plane, but it's still worse to use this device. And I think in just about as objective and non-pragmatic sense as the ban on the integers for Newtonian physics, the stipulations required to enable this device to do that representational job will be so unnatural and contrived that even the gods or a Laplacian demon could reasonably be faulted for choosing to represent the plane this way. Now it's true that a Laplacian demon presumably can grasp all the features of the plane on the basis of this representation even without performing any complicated calculations. So that might make it seem like an equally good representational choice for a creature like that. But there's also an objective sense in which it's worse just as a salt shaker is worse than a geographical map for the purposes of getting around Madagascar, but is also worse as a representational tool in an objective way. We could reasonably be faulted if we were to present to you the salt shaker as our best representation of Madagascar, even if you were never planning to visit the place. With the right stipulations, we could represent the physics of Newtonian systems using a permutation of our usual numerical representations, but this too would be a worse representational choice, again, in an intuitively objective and non-pragmatic sense, one that flows from the intrinsic natures of the device in question and the physics we're trying to represent. In particular, it will fail to represent the relationships among different physical states or quantity values at all naturally. So all of this suggests that there are constraints on our representational choices in physics and mathematics that go beyond the cardinality constraint and are not pragmatic or subjective. Some representations are intrinsically or objectively better than others, intrinsically or objectively more perspicuous. So some representational choices are objectively better than others. So this idea of an objectively perspicuous representation has been called by Thomas Muller Nielsen a metaphysically perspicuous characterization. So it's one that in his words corresponds to or limb's reality structure in some suitably faithful way. However, he also says that a perspicuous representation faithfully represents the fundamental physical ontology, as for example, a formulation of classical electromagnetism in terms of the Faraday tensor rather than the potentials more directly reflects a physical ontology of fields. And I agree, but I want to add that a representation, one representation can be more perspicuous than another in an objective sense, even if there's no difference in ontology as in the case of ordered pairs of real numbers versus single real numbers for the plane. We might go on to spell out this idea further by saying that a perspicuous representation more directly matches the nature of the target by means of some kind of resemblance. Maybe that a representation is perspicuous to the extent that there's the right kind of morphism between the structure of the device and the represented item, as in the case of the match in topology between ordered pairs of real numbers in the plane or the match in structure between our usual real number representations and Newtonian physical quantities. I do think something like this is probably right, but I'm not going to say anything more detailed about it here since I am trying to remain as neutral as possible on an account of representation. But I will add that I'm not suggesting that a morphism or resemblance of the right kind is necessary for representation, I don't think it is, but I am suggesting that something like this is necessary for representing well. Okay. Let me say something about the relationship to symmetry. So one way for a representation to be objectively more perspicuous than another is for it to more directly reflect the symmetries of what's being represented. And this is because one way of characterizing something's nature or structure is in terms of its symmetries. So on a Kleinian conception, this will be a way of defining a given mathematical structure. The representation of the plane in terms of real numbers is less perspicuous because the set of real numbers under its usual ordering isn't topologically equivalent to the plane. The real numbers do not possess the topological symmetries of the plane. Or a permuted assignment of real numbers won't respect the symmetries of the values of physical quantities like mass. The original but not the permuted assignment will have an ordering that's preserved under translation, for instance. As a result, the alternative representations can capture the symmetries of the represented thing only very indirectly by means of highly unnatural or contrived stipulations. Now, Josh Hunt has argued that some representations can be better than others, not just for pragmatic reasons, but neither for fully objective or at least fully intrinsic ones. Instead, he says that some representations can be better than others to the extent that they impart greater intellectual understanding to us by making certain features manifest in particular contexts. So this is a kind of middle ground between a pragmatic and an objective conception of perspicuous representation. To me, this undersells the sense in which something like ordered pairs of real numbers are better suited to the plane. It's true that this representation makes manifest to us particular features of the plane, but there's an underlying reason for this which doesn't have to do with us or the context, which is that these features are naturally or directly characterized by this device. This vehicle is inherently well suited to the nature of the plane, and that's why it makes the plane's features manifest. Or consider using polar versus Cartesian coordinates for the plane, for the Euclidean plane. Hunt says that neither of these results can be more objectively in an objectively better representation than the other. Each just makes different features of the plane manifest, which can be epistemically significant to us in different contexts. And that is intuitive. We can represent the plane using either coordinate system, but it doesn't follow that we can do so equally well. And there is a sense, an objective sense, in which Cartesian coordinates do so better more perspicuously. In the same way that ordered pairs of real numbers are better suited to the topology of the plane, Cartesian coordinates are better suited to the plane's affine structure, for instance. And polar coordinates circles on the plane can appear straight according to the coordinate function describing them. Polar coordinates can be better suited to other kinds of spaces or to certain figures drawn on the plane. They can make certain calculations easier. But the plane itself, the intrinsic nature or structure is objectively more aligned with Cartesian coordinates. In the same way that cylindrical coordinates, in another example, are more objectively aligned with the nature of the cylinder. Hunt says of cylindrical coordinates that they merely make manifest to us features like the cylinder's length. And again, notice the relevance of symmetries. So when the thing we're representing possesses certain symmetries, some representations can be objectively more perspicuous in better matching those symmetries, as in the case of cylindrical coordinates and the cylinder. Let me quickly address a few concerns you might have here before moving on to the next section. Sorry, I'm just checking the time. Okay. You have ten minutes. Ten minutes. Okay, so I'm a little bit running out. Okay, let me quickly do a couple of concerns and then I'll move on. So you might think, first of all, that the kind of perspicuity I've been discussing is just a pragmatic matter. So ordered pairs of real numbers better serve our aim of discovering the nature of the plane or our usual representation of Newtonian systems better serves our aim of making predictions about these systems straightforwardly. The cardinality constraint allows us to do physics. If we didn't care about discovering the nature of things or doing physics in any way, then some other representation would be just as good. Now here, I suppose you could call the desire to get at the nature of things or the desire to do physics at all pragmatic. So part of the scientific purposes that Callender and Cohen mentioned. But at this point, I'm starting to lose a grasp on the distinction between the pragmatic and the non-pragmatic. Once we're in the business of aiming for objective truth and understanding and having a physics in any form, we seem to have left the realm of the pragmatic unless we want to go for a more thoroughgoing pragmatism, for instance, with respect to truth itself. The cardinality constraint, for instance, doesn't just let creatures like us do physics, but any inquirer, even a Laplacian demon couldn't get by with the integers for Newtonian physics. So to say that we have a pragmatic reason to reject the integer representation seems to unduly stretch the meaning of the term. Relatedly, you might wonder about the idea that some representations are objectively worse than others. So even granting that some representational vehicles by their nature are so unmatched to the things we're trying to represent that it's only by means of some weird contrived conventional choices that the representation can even be implemented. We might still question whether that makes for an objectively worse representation. The gods, presumably, could get by with it just fine. Now at this point, we might just be starting to wrangle over the meaning of objectivity and also getting into thorny questions concerning representation. So suppose we say instead that objectivity and objective goodness is a matter of degree. Then I think we can all agree that the two-dimensional representation of the plane is objectively good or perspicuous to a greater degree than the one-dimensional representation. And at the end, I'm going to suggest that this difference in degree is significant, that it matters for science. Okay, I'll skip one thing. You might finally worry that the idea of a representation sort of being well adapted to something's nature is too imprecise. There are different versions of this concern in some work by Thomas Barrett and also Casper Jacobs. Here, I'm just going to say that even if it is imprecise in some cases, it seems clear-cut in others in the way, for instance, that ordered pairs of real numbers are objectively well suited to the nature of the plane. Okay, so I'm going on to Section 4 and apologies, just one last time check. So I have about 10 minutes. If I go to 35, sorry, I was hoping to keep it short. Okay, okay, quick, quick. Okay, Section 4. So Goodman said... Goodman said, the conventional is the artificial, the invented, the optional as against the natural, the fundamental, the mandatory. So I think our representational choices in physics and mathematics are conventional in that there's a lot of leeway in which devices we can use and what to use them for. So the choice of representation will be a matter of decision or stipulation or convention, optional, in other words, not mandatory. The choice is up to us in that any of the options is legitimate. At the same time, the representational choices we make aren't completely up to us because they're constrained by the nature of things. Some representational tools are objectively better suited to that nature than others. And in this sense, the choice of representation isn't completely up to us. Not any of the options is equally good. Again, for the plane, which device we choose will be a matter of conventional decision in that neither of them is mandated by the nature of the plane, whereby contrast the nature of Newtonian physics on its own mandates that we can't use the integers. Even so, the representational choice we make isn't a matter of arbitrary decision because it's constrained by the nature of the plane. The choice of these devices fails to align with that nature. As a result, there is a wrong or inherently misleading choice to make. We could fault someone for making one of these choices of representation. So nature of the plane is such that the two devices both can represent it, but not that they can both represent it equally well. So all of this is to say that the choice of representation in physics and mathematics is conventional, not arbitrary, which is something that Poincare once said of the laws. An arbitrary choice is made capriciously not for any reason, especially not for the reason of the nature of things, and choosing to represent the plane by means of ordered pairs of real numbers isn't arbitrary in this sense. So when a representational stipulation isn't possible, as in the case of the integers in Newtonian physics, the fact that we make a different choice of representation won't be conventional. Not everything about our representational practice is a matter of convention. But once we're in within the realm of possible representational choices, the choice we make will be conventional in that it's made from a variety of legitimate ones. By assumption, any of the options represents the thing in question. They all contain the same information about it. The choice is up to us in that we could choose differently without loss of content. But even so, the choice isn't arbitrary because it's not made from among equally good options. Some are inherently better suited than others. The choice isn't entirely up to us in that we couldn't choose differently without losing something else of non-pragmatic value. Okay. So now we're in the last section. So this presents a puzzle. By assumption, the different representations we can use all represent the same thing. They all have the same content. That makes it very hard to see how there can be any objective non-pragmatic differences among them. If there were any, that would seem to make the choice of representation a substantive choice. But a disagreement over how to represent something is the paradigm of a mere notational or verbal disagreement. A disagreement or a difference in how to describe the facts, not the facts themselves. There's nothing of significance as at stake. Or another way to put it, if there is an objective sense in which some representations are more perspicuous than others, then there would seem to be an objective sense in which representations that differ only in how perspicuous they are are not equivalent. And that just sounds wrong, right? And mathematics were very familiar with their being different representations of a given object or structure, which are then regarded as fully equivalent. The same for all intents and purposes. In physics, we regard coordinate changes as passive transformations, just changes in our descriptions of things. Two people who disagree, who assign different coordinates to a given location, they're not really disagreeing. They're just labeling things differently. Similarly for different formulations of a physical theory, which Feynman once described by saying, there is always another way to say the same thing that doesn't look at all like the way you said it before. So a difference or a disagreement over how to formulate a theory would seem to be a verbal dispute. So Chalmers puts it, intuitively a dispute between two parties is verbal when they agree on the relevant facts about a domain of concern and just disagree about the language used to describe that domain. In such a case, one has the sense that the two parties are not really disagreeing. So this seems to be what's going on in the case of different representational choices in physics and mathematics. Now, in a way, this is true, but there can still be good theoretical reason and not just pragmatic reason for choosing from among all the different representations we can use, certain ones over others. And in cases like this, there is a kind of substantive choice to be made, even if it might not be evident which choice we should make. There is something of significance at stake. There's a sense in which they are inequivalent. So I think examples from the history of physics reveal that a theory's mathematical representation can have consequences for the nature of physical reality and vice versa. There are examples both in which a change in representation or formulation followed from clarification or alteration in our conception of physical reality and examples in which a theory's formal features were shown to have consequences for the nature of physical reality. And these examples demonstrate that how we represent things matters to theoretical progress in physics. So take Heaviside's reformulation of Maxwell's equations. Maxwell initially formulated the theory of classical electromagnetism in terms of the vector and scalar potentials and the result was a messy 20 or so equations. Heaviside came up with the improved version we know today. And he did this by aiming at a perspicuous formulation, one that directly reflects a physical ontology of electric and magnetic fields. So by assumption, Heaviside's is a reformulation of Maxwell's. It has the same content, but it is more perspicuous and this resulted in theoretical progress. So for example, it made evident an asymmetry in the equations when applied to moving bodies that's not present in the phenomena which led Einstein to the special theory of relativity. It also made clear just one other example of mathematical symmetry between gravitational and electromagnetic phenomena which led Heaviside to note in 1893 the possibility of gravitational waves. So are the two formulations of classical electromagnetism equivalent? So in a way, yes, by assumption they depict the same physical reality, but in another way, no, one of them led to important theoretical advances. And so particular scientific discoveries were enabled by a perspicuous formulation. In a similar vein, a theory's mathematical representation can guide us to discovering new things about the nature of physical reality. We didn't previously know. As Bell noticed that as a mathematical consequence of the formalism, we can't have a local theory that reproduces the standard predictions of quantum mechanics. Or as Arnav and Bohm derived their eponymous effect from the quantum formalism, which led them to conclude that in the quantum domain we must treat the electromagnetic potentials as physically real. So in cases like these, formal features were shown to have profound physical implications, which were later on experimentally verified. So when developing a physical theory, it's not as if we start out knowing everything about the nature of physical reality and formulating a theory that directly mirrors it. We start out knowing some things, and we formulate a theory that reflects the things we know pretty well. And at this point, we might revise the mathematical formulation in light of our conception of physical reality, or we might discover new things about physical reality on the basis of the mathematical representation. So we go back and forth between the representation and our conception of physical reality, adjusting one in light of the other. And different representations can affect this process differently. So in this sense, they can be epistemically, theoretically inequivalent. There is a substantive choice to be made, something of significance as at stake. So you might still regard this inequivalence as fully subjective or epistemic as hunt things, just a matter of making certain features manifest to us in certain contexts, or that the representations are inequivalent only as Feynman put it for psychological reasons. Again, a god or a Laplacian demon has no need to be guided to the nature of reality like we do. At this point, let's fall back on the idea that objectivity comes in degrees. Then we don't have to settle the question whether Heaviside's formulation is objectively better full stop, given that Maxwell's will work just as well as far as the gods are concerned. Instead, we can say that Heaviside's formulation is perspicuous and therefore better in this way to a greater degree. And what we're seeing is that this difference in degree matters to physics. Okay, last half minute. So in general, different representations can be equivalent or inequivalent to one another in different respects. And one of these is the degree to which they're perspicuous in a relatively objective sense. So representations can differ in this way, even if they otherwise say the same things. And this difference can have important consequences for science. So the choice of representation in other words, which is often treated as a fully pragmatic, up to us arbitrary, mere notational choice can in fact be more than that. So this isn't to say that every difference in representation is like this as using one or another set of Cartesian coordinates for the plane or speaking English versus French is not. But it is to say that taking all representational choices to be on a par with the choice between French and English is to lump too much together. The difference might be a matter of degree, but the degree matters. Some representational choices are better for physics. Okay, that's it. Almost on your time. Thank you very much. Yes, you are very good with respect to time. Okay, so if someone still needs the handout that we know, I will post it again in the chat. And thank you for telling us about symmetries. So broadly, I talk with anyway in the workshop because it's also about equivalence and inequivalence. So now we will have your response. Sebastian from the University of Amsterdam. Sebastian, you have up to 10 minutes. So by 57, we should finish. Then you will reply and then we will have up to European time to discuss in this slot. Okay. Thank you. I'm going to share my slides. Yeah. Can you see them? Yeah. You can begin, please. Thank you. Yeah. So thanks first of all for the invitation to reply to Jill. By the way, if you hear any construction noises in the background, that's because there's construction in the neighborhood. So yeah, it is an honor to respond to Jill's interesting and thought-provoking talk. And I'm going to focus on some of the aspects of the paper that she sent to me in preparation for the start that she wrote for this occasion. And I wanted to begin by just reminding you of some of the recent discussions about theoretical equivalence in the literature. So two theories are theoretically equivalent if they say the same thing. So various authors in the recent literature, but of course older literature as well, have taken theoretical equivalence to be a conjunction of two conditions. On the one hand, the formal criterion of equivalence. On the other hand, the interpretative criterion and different authors have filled in these criteria in different ways. Now in her book from 2021, Jill filled in the interpretative criterion of equivalence as metaphysical equivalence. And so there she emphasized that these two conditions come apart. So you can have formal equivalence and interpretative or in her case metaphysical in equivalence. And to make that clear, she introduced the notion of direct representation. So one representation is more perspicuous than another if it represents the nature of the represented item, the target more directly. As an example, she gave the example that she also just discussed the potentials versus field formulations of the Maxwell theory. The fields, the electrical magnetic fields more directly represent what is physical, what is physically real. And so that is a more direct more perspicuous formulation. So against this background in the paper that she has read today, there is another thesis of which I would like to mention one aspect which is that we can have theoretically or metaphysically equivalent theories that are representationally in an objective non-paragmatic sense. So that is how I read the paper. And here are two quotes from the paper to substantiate that one representation can be more perspicuous than another even if there is no difference in ontology. So we could have no difference in ontology to theories could be metaphysically equivalent and yet one could be more perspicuous than the other. And also where she notes that same things in different ways, in other words theoretical equivalence is too crude a measure of their equivalence. So I have some questions about this novel thesis. Of course there's a lot from Jill's work to learn from and to agree with but I have two questions about this particular this novel thesis. So what is the sense of representational in equivalence that is being suggested and in particular why can these representational differences not count as for example metaphysical differences. So normally when we talk about scientific theories and Jill has mentioned this we distinguish epistemic, semantic and metaphysical aspects. So could these why could these representational differences at least some of them not be classified as either epistemic semantic metaphysical or a combination of those. And in particular are there any clear examples in physics that of theories that are theoretically equivalent but representational in equivalent. So with regards to symmetries that is one way that Jill argues that a theory can be more perspicuous than another if it more directly reflects the symmetries. And again the example of Maxwell theory comes in here where the vector potential formulation the old or the original Maxwell formulation with the 20 equations is less perspicuous and it also represents the symmetries less well than the heavy side version and that also led to theoretical progress as Jill has emphasized. Now I have a slightly different reading of this example. So I think there might be two mundane reasons why the heavy side formulation the one with the four equations that we are familiar with why it was adopted rather than the old one with 20 equations. One reason is that Maxwell formulation contains additional quantities and assumptions that are not general such as the velocity of a point particle and ohm's law. And so it seems less general than the heavy side theory. And those are besides mathematically simpler but that simplicity doesn't seem to be only related to the representational capacities or the representational but rather it also has to do just with epistemic factor such as simplicity. So Maxwell's 20 equations mixed potentials, electromotive and displacement fields, current density and total current so it is not a familiar gauge field formulation that we think of when we think about gauge potensions. So even if some textbooks may say that they are equivalent on the phase of it it seems that they are not and it would be interesting to see whether they are really equivalent in a formal sense or in some other sense. With respect to symmetries, is it the case that the heavy side version better reflects the symmetries in particular, the Lorentz symmetry that let Einstein in 1905 do his formulation special relativity. Well also that is not clear to me because the four vector potential reflects the Lorentz symmetry I would say better than the electromagnetic fields. The electric and magnetic fields are covarian they are not scalars or tensors and the Lorentz transformations to make them covarian you have to define the Faraday tensor but the Faraday tensor is an antisymmetric two form that contains many redundant components. So it is not clear that the electric and magnetic fields by themselves even though they represent the fields what is physical about the fields very well they don't seem to represent the symmetries as perspicuously as say the four vector potential or the Faraday tensor itself. So those are at least some skeptical points that one could have with respect to this example also if we look at extensions of the Maxwell theory if we couple it for example to a source a particle point particle source or a spinner or some other kind of source then it is then we normally use the gauge field for that because that allows us to have a Lagrangian or a to write down a Lagrangian or a formulation sorry or a Hamiltonian also the formulation of classical non-abelian gauge theories writing down smooth monopole solutions or instant solutions those are given usually in terms of the gauge potential. Now to speak about quantum extensions which actually also mentions the no Harunov bomb effect of course we often say take the Maxwell theory in vacuum as is and forget about extensions since we are in the end interested in the possible world described by this particular theory so we don't want to extend it to other theories however what I am going to argue is that there is a tension between the idea of directness and the scope of the theory so that we cannot ignore the extensions because those extensions might overrule the directness principle so in my last slide I would like to contrast two principles which are really not in contradiction but they emphasize various different aspects so a direct realist principle would say that we prefer the formulation that represents our target most directly on the other hand there is an alternative thesis where we generally agree with this direct realism but when faced with inter theoretic relations of for example equivalence or correspondence we should take those inter theoretic relations into account when we determine the ontology of a theory so those inter theoretic relations help us individuate our theory they might also imply a logical semantic relations between the theories so a more cautious realist principle would be that even though formulations with additional variables or which gets the symmetries maybe might be more indirect in some respects they can give better descriptions of the structure of the actual world as opposed to a single possible world if we want to find the structure of the actual world we should not isolate our theories only in the possible world described by the isolated theory after all scientific theories are general they describe a wide scope of targets so we should also ask how does our system of interest interact with other systems and in that interaction it also reveals part of its nature so I would argue then that the formulation using the gates field is better than most extensions even if it has some redundancy and so it has a wider scope in this sense since the successor theories use it it gives us a better representation of how our system interacts with other systems and in this sense one might take it to be more perspective thank you thank you very much do you probably want to report something on this sure although just briefly because I need more time to think about it all thank you so much that was super interesting and this is the first time I'm seeing the comments I don't have anything prepared but I'll say a couple things and also that's not to complain I only sent him the paper a week or two ago I just so super interesting and I need to think more about your claims that sometimes the less direct formulation is better for various reasons I was just going to say a couple of quick points on your earlier questions or comments that's a very interesting question in what sense is there an inequivalence among representations in terms of what I'm saying this difference in level of perspicuity and you're wondering can't we essentially fold that kind of inequivalence into some pre-existing physical inequivalence semantic epistemic I can't remember if you those are the three I jotted down yeah that's a really interesting question I want to think about it more my initial thought is well I don't I'm arguing that there is a kind of sui generous kind of inequivalence here when it comes to representations that differ only in how perspicuous they are by assumption they're metaphysically equivalent they present the same physical reality they're semantically equivalent in the sense that they have the same content epistemically equivalent that's something we can wrangle over maybe that comes in degrees but I do think there's intuitively a kind of sui generous type of inequivalence here that I'm trying to explore and you also wonder whether there are examples the relationship between theoretical equivalence as a whole and this idea of representational inequivalence so can there be theories that are theoretically equivalent and nevertheless representationally inequivalent here I'll also say I'm not completely sure but I'm not sure that it matters it's more like in one sense let's call them theoretically inequivalent or another sense let's not what I am trying to suggest is that there is this sui generous kind of inequivalence and that it matters that it's significant um the well maybe I'll stop there there's so much more I really appreciate your comments thank you very much I have a lot more to say but maybe I should pause just in case other questions and then I can talk to Sebastian on my own yeah okay I hope Sebastian is at least a bit satisfied with his questions so everybody else if you have questions this is a great time to ask them so you can raise your hand or write in the chat if you want and if not I'm happy to go back to your comments yeah I also have questions but John was the first question John you have time for a question uh thanks yeah thank you for the talk this is um very interesting to me I have a question about I mean I guess the short version of my question is how perspicuity is related to correctness so when it comes to the sort of stuff that you're putting from calendar and co-in one of the things that they're up to in that paper right is like trying to pry apart representations and look at her from like correctly representing or good representing or something like this and so like when it comes to the case of Newtonian physics representing that using integer is it seems like what they should say is in fact you can represent Newtonian physics as integers you'll just be in representing the world incorrectly or the world that is Newtonian as having a discrete space or something um now you're wrong to do that because the world does not have a discrete space in Newtonian physics but you know you can represent you can misrepresent things that's like a thing about representation um and so I think this is the kind of thing that they sort of want to pull back on and it sounds to me like what they ought to say in response to your criticisms is that um when you say that perspicuity is directly reflecting the symmetries of the target that is an important thing it's one of the sort of making features alongside truth now I don't know I mean you raise the question whether truth should be pragmatic as well and I think that's an interesting question I don't know what they should say there but I mean even if we do privilege truth it seems they it seems like goodness is going to have to be something much broader than this including these pragmatic factors and so you know would you I mean um would it be a kind of well I don't know this seems like what they ought to reply to you does this seem like something that makes contact with your worries or is it just missing the point um good so I heard a couple things in your question and actually I was thinking about my response to the first one second part so I might come back to you no no no that's fine um but yeah so that's a very interesting question good point there's we can misrepresent things when is it a case of misrepresentation and when is it a case of something that I'm calling uh essentially an impossible representation um uh this is getting into the stuff about representation I'm trying to set aside but I will say um I do think you're right someone could say what we can use the integers is just misrepresent Newtonian physics um that feels so I have to think more about it that just feels wrong to me it feels more like uh someone representing they're not representing Newtonian physics they're doing a different they're providing a representation of a different theory in that sense they're not misrepresenting Newtonian physics they're just not giving a possible representation of Newtonian physics um that's how it feels to me but I want to think more about that um and then I'm sorry you had a second thing about what yeah the second thing was just my attempt at taking the first question which you answered fine so okay okay great thanks thank you okay good uh anyone else has a question while I was like uh yeah you got you got a little pinta oh hello uh thank you for for both talks I um though I guess my question is uh is to Jill but regarding the response so as I happen to have uh a red part of your work on structuralism and um what I was wondering is whether you take uh just in the maybe in the in the sense that Poincare originally also took into consideration you know uh epistemic structuralism and the history of physics it seems like the response that Sebastian gave you was focusing uh particularly on the aspects of representation that make those theories amenable to you know generalizations in the course of history uh and that you know that that seems also very much um you know in line with structuralist uh approaches to physics so would you say that his responses are in that sense uh very much compatible with uh with your view of physics in general or or am I confounding everything no thanks that's a good question um I so I'm not sure um I do uh I do think that it's important to pay attention to the structure of our theories when it comes to thinking about what these theories say about the world and whether different theories say the same things about the world um but I'm not sure that I'm on board with a more full-throated structuralism of the kind that I think you're suggesting um if only because uh I think there's a lot more to scientific theories than just their structure I think that there's an important sort of metaphysics you know this bad word in this context uh to different physical theories um and so I don't yeah so I don't think I would adopt the sort of full-throated structuralism you're suggesting um that said I and I didn't respond to Sebastian's comments on the fly but something I do want to think about is this question um he is suggesting that certain formulations as you put it are more amenable to further generalizations and that's an important a significant feature of a formulation and is maybe comes apart from its directness in the sense that I have in mind and um I'll just say that that's a really interesting question and I don't have an answer off hand I agree that there's a question there or a concern um uh yeah I'll leave it at that thanks okay uh yeah we have the minutes left uh uh hi Joe thank you that was really interesting talk um I wanted to just ask you to say a little bit more about your argument regarding coordinates on the Cartesian plane and and maybe just some of the details went by too quickly for me uh but if we sort of set aside the singularity uh at the origin and polar coordinates um I take it the claim was sort of that that Cartesian coordinates more perspicuously represented the affine structure then uh then polar coordinates um and I think there's certainly a sense in which like that that's right uh in so far as like the the affine structure is going to be given by linear equations uh in the Cartesian coordinate system uh and not in the polar coordinate system but I can still sort of see somebody saying well look uh you know uh a preference for linear equations is is pragmatic it's a feature of our own psychology it's not in some genuine sense objective uh and so I using to sort of want to say no no there's really an objective sense uh in which the Cartesian coordinates are better representing the affine structure of the Euclidean plane and I just didn't see how you break that tie or how you break that impasse in in your argument so I just this is just sort of a question asking you to expand a little bit on sort of how you think that that impasse should be broken or or what your reasons are for breaking it one way rather than the other yeah good thank you um so boy I don't know if I have more to say than what I said in the talk and I'll think about it more um but I see so you're suggesting that um even if in a certain sense the Cartesian coordinates better reflect you know the straight lines in the plane as it were um you're saying the sense of betterness doesn't seem to be an intrinsic or objective one or one independent of us and which sorts of things we take to be nicer or simpler uh in other words representing a strict I take it your main question is this is it true that representing a straight line using linear equations is objectively better than representing a line in terms of non-linear equations or or vice versa um rather than just pragmatically better is that right yeah I think that's roughly right that like whether or not I mean I agree that there's like you know there's no question that like one would opt to represent a straight line in terms of linear equations right like uh but but then there's sort of this lingering doubt like well is that is that because there is like an objective fact about which representation is better or is that a feature about how my processes work and what is easiest for me to handle um and so I don't even want to say that like there isn't an objective sense I'm just not entirely clear sort of how we break that how we break that sort of impasse yeah that's a I see and when you say break the tie how do we decide whether it is just a matter of our psychology and our calculational abilities versus some you know we can't get outside our own head that's right that's right yeah um so that's a really good question I don't know um I have the intuition that their linear equations are you know as I would put it inherently better suited to representing straight lines um independent of us and our calculational capacities that you know a Laplacian demon who did otherwise would still be intuitively missing something about the nature of a straight line by representing it in this other way um but if you come back to me and say no no no that's really just a matter of our psychology our calculational capacities it's fully pragmatic um I have to think more about where we go from there yeah thank you that's really really interesting thanks okay what is this I think we should finish the discussion of the first talk