Visualisation in Mathematics — Roundtable (contd.)

0:00 Finally, from a line of Spears papers, Spears said that immersion in practice is what makes the actions obvious and the actions that are not present foundational. To take out the word actions, obvious, and foundational, and put it in the proper way, we need less training and less immersion when we're dealing with two-dimensional geometric equations. And that all the other differences ultimately play a play in that this immersion that's responsible for our relegating algebraic fluidity to the distribution. I mean, I'm not sure there's a lot to be said. One thing I'm, I'm, I'm, I'm, I'm also, I'm, I'm, I'm, I'm, I'm, I'm, I'm, I'm. I think we have, even if what you're saying is right, we still have to be able to say that part of what's involved in this is the terms in which we found an initial problem are in fact not the most profound way to represent the problem, but it's an original way of saying it.

2:30 It's easier to introduce students to the explanation that I didn't show you, which involves the questions. Five questions and three minutes. The author is Jeremy Gray, an advocate of Janusz Jastrzański, Michael Jastrzański, and you have time to pick your questions. So, okay, I'll try and be brief. Picture proof. We just have rational fields that they find that is a picture of a particular mathematical concept. In analysis we would call that something kind of a theory for real. I can give you other fields that are a bit of good pictures, but they're false again. I want to know what's wrong, I'm being able to think what's wrong with this argument in my case. Supposing you want this argument to be honest. You want the intermediate value theorem to be true. So you formalize the argument. You tell it it's got to be true. Now you've formalized your argument, and you come up now with proof in analysis. Well, you're going to have, whatever arguments it is, to deal with the question of what is that picture. To be able to say that it is a picture of the theorem of truth, and not one of the delusive pictures. That's one of the things I'm getting at. This picture presented in your talk is an unambiguous picture. We all recognize this is a theory. We don't know how it's done. So I'm saying, let's take a picture, do some formalization. Now we have to check that we have correctly formalized what we are interested in and not something else. So we have to say, does the formalization give us the results we want? If it doesn't give us the results we want, We suspect that we have not formalized correctness, so there's a parallel planet to ours, I suggest, where people have a geometric intuition and they only accept real analysis because it gives them the geometric theories they want instead of visualizing the geometries that are in control. What's wrong with that? I'm just lost, but okay. I mean, I don't want the discussion. Why don't we get the rest of the discussion?

5:00 I'm not committed to the view that there's only one way that our view about space could go, but I'm not sure that that's true. It could go in the direction that it did go, classical analysis, but then why can't we have other kinds of analysis? Maybe that's, in some sense, much more faithful to at least certain of our intuitions, so that we can go different ways and maybe say, well, look, some other kind of analysis could be another subject we could get a picture of. I don't know. But I was discussing the screen analysis as it actually was. In your talk, you rejected the identification of pencil contiguous with epsilon-delta, but I don't think you have a reason why to reject that, but I can just offer a quick one. It's intuitively clear that any pencil contiguous curve has a finite length, because if you draw with a pencil, you have no groups of that. But there are other continuous functions that have infinite lengths, so the two presumably have the same, you know, have the exact same size. But God tends to use these ones. I don't think that's true. Of course, these examples are very informative, but unfortunately, I was forced to elementary stuff, because the rest of it, so if you ask you proof theory, why don't we develop a suitable proof theory of these issues, is that you can encode geometry's originality, which is true since it was proved, but that encoding is conservative with respect to consistency, that's why the consistency issue is very important, because it carries on consistently to maximize the geometry.

7:30 But in internal proof theory, along the lines, for example, the very few results we have, we cannot cut in three an angle, that is, with pressure, but are very few, compared to the depth of an angle of induction, like per-bar induction, which are beautiful and very deep. It's kind of similar. I've heard once, but in geometry, Pierre Chastier mumbling about the use of chirality in quantum physics. You know, chirality is the fact that you have... I mean, I know nothing, but clearly we missed something there, because this claim was that this could not be dealt in a formal way, because you need this erection space to specify the reality, and at the same time, the reality of the erection space and the direction of space. I mean, if this is just improved... I don't know where exactly these key terms are, but they are very important. In the light of that, we give us what we believe is an essential use of location, space, time, and instruction, the choice of reference space of a metric, where encoding for it cannot be done, or is missing. It is missing the crucial information we are carrying out. As long as we stay to very elementary constructions, And, of course, there is always a possible complete formal encoding of the image that, I would say, distains the level of heuristics. Only concrete completeness examples, I know, really require that, and, of course, a reasonable proof theory of geometry, yet no other way to provide reference to explanation which grounds on quantum imaging. Thank you. I just wanted to make, it's very late, I feel guilty even to speak a little, but I wanted to make some very simple points, simple points about vision, because your discussion is about the possible role of visualization in mathematics or in proof, and we have to know to which concept of vision we refer, because if we take vision to be the receiving of a pixel table,

10:00 A purely passive, nothing happens, no thought. And then only after we have the whole process of thought, which is beginning with categories and ideas and so on, and shapes and forms, then in that case it's very clear that vision cannot play any role in mathematics. But this, if we see vision in that way, there is not enough to explain human vision. Human vision has never been that. Human vision is much more. There is... In any kind of empirical human vision, even if I take the point of view of cognitive science, there is much more. There is generality, as Federico was arguing, but there is also the figure of ground opposition and a lot of other things. So if we take seriously vision as something that we don't completely understand, then the question of the function of vision with mathematics, accompanying mathematics, As a kind of ground of a certain level of formalization, it becomes completely different. I wanted simply to make this point. I'd like to make it that we are talking about... I'm taking a vision in general, for every one of you. For example when, I forget your name, when you think that probably the vision of the trajectories on the torus are in a more intimate relation with the mathematical theory of the torus, I think he may very likely make a plea for that, if you consider that this vision is not stupid. The experienced vision of the trajectory in the torus, there is much more in this that we are able to say. Well, certainly I think you're going to consider a picture of something. There's some kind of cognitive component there, and I'm suggesting that I talk about it by using vaguely articulated words like include, reveal, name, and so forth, just to suggest that yes, there is that contextual component. So I'd better be talking about an adaptation of things which have a spontaneity involved.

12:30 I have a question about whether the discussion about visualization is really couched at the appropriate level of generality. You seem to want to identify the question of whether visual information is as physiological or as value or something like that. Is the question enough? I'd like you to ask people to play along with me in the following, because I think there's a tendency for it to be a sabbatical in every game. Now you've got a guy who can give you an analytic proof, a perfect analytic proof of the intermediate values. If you say, oh, is this what you're saying, and you draw the picture, and you just don't, I have no idea what you're talking about. I'm going to say that guy's got a real cognitive condition. He's got an epistemic condition. It may not be something that I can describe by saying, such and such could play this justificatory role and that justificatory role. Maybe a better model is something like this, that knowledge consists of significant parts, in a way, in drawing relations between different representations. And if you leave any of a family of representations out, there's something deficient in their knowledge. And I'd like to say, and I mean, I like your writings on this, Jeremy, and Jane, I haven't seen too many of yours, but I've read a lot of Mark's. It just doesn't seem to me to take on that kind of question. There seems to me to be something definitely cognitively efficient about a guy who just doesn't get it, and he's getting efficient. You want to say epistemically efficient? Yeah, epistemically. Yeah. I'm going to say he does not understand. I'm going to have to work more closely with this student, because this student does not know how to do that. She should have learned.

15:00 There's more than one epistemic. There are other epistemic visualizations, I think, that may explain what is called a legitimate application, and the case that you mentioned does suggest to me that there could be a problem with the theorem. It couldn't be either way. I would like to explain why it's true, and to prove that it's true, and to justify it, or believe in the theory. Well, maybe he had some other information, but he couldn't get this one. Like, well, he couldn't tell, he couldn't try and say what's going on. He couldn't, he might have difficulty in seeing why he could have died of fear in certain areas of history. And so, I would agree with you that he survived. I'm interested in the efficiency of the degree, and this efficiency does extend to an academic degree. I guess why not call it a degree of efficiency? I believe that there is a special double-edged war behind certain subjects represented by certain sorts of understanding. What if the student came back to you and said, no, no, no, that doesn't tell me anything because here I have described two curves like, oh, you know, either one of those could be this picture, right? Would you still say he's cognitively deficient or just that he happens to be? I want to make sure I understand your case. Yeah, the case is this is the person who just sort of took to analysis like a fish to water, excuse me.

17:30 And then when he tried and figured out these intermediate values there completely abstractly, and he comes to show him this picture and says, this is what it looks like. And he says, what are you talking about? That could be either of the following two curves. Describes the curves of Marcus as given and I say, so that picture tells us a completely different case because he does recognize that it's describing Marcus's function, okay? He does. That's not the only thing he realizes. That's not the only picture he equates with or identifies with, okay? But he does identify at least that one. But okay, I'm talking about the one, the case where the guy makes no connection at all. I show them the diagram. They go, is this your argument? It's all the blame. Don't, don't. And it just gives me the analytic argument again. I guess I want to say, this is some kind of idiot's fault or something. I guess he doesn't understand what he's done. But I never, well, I was never satisfied with my own children in their understanding of mathematics until then. Could be things like associating pictures with it. And I think that this is a standard part of pedagogy, right? When we ask young students to diagram functions, I think that's part of what we're wanting to do. We're wanting to say, oh, yes, now you should have a picture in your mind of what this thing looks like, too, otherwise you don't understand it. Thank you very much. I don't make that kind of distinction between the data that I have explained or by your understanding. I think maybe that's it. I don't question it. I want to say, this person has a definition of understanding, okay, and I guess what I'd call what adds to understanding, what we call justification, what he tracks from, I'm going to call the opposite of that, okay, and I don't know if that's a good definition. There are some cases where I think we can get knowledge by means which don't actually sound true to discover it, and I haven't got time to testify to this, having not come up with an ancient or profound knowledge.

20:00 Maybe I won't try to run out of time. Well, fair enough. I'm not asking you to run out of time. Okay, I'll go on with your experience. Go ahead, if you want to. So, go ahead. In this case, we would want to say that something's happening here. And that means that we need to arrive, I believe, in a way that, you know, But he's also known to be actually thinking, you know, I still need to prove this. I need to check it. I need to provide an explanation. I've discovered it. I haven't tested it. I think I've discovered it, but I haven't checked it. He allows, he allows these two to happen, but he's already made a distinction which is basically one thing, which is a different kind of sense of the word. I mean, we've got two visualizations. One would give you the discovery that there's a lot of knowledge, even if your belief is justified in some sense, it's wonderful, but there would still be the extra kind of justification that you don't have, but once you make that distinction, then you think you should at least be open to a possibility that... There are loads of other academic values, but rather than making justification and explanation, it's always going to turn to science or physics. They're left with a lot of that. They're detractors.

22:30 Wow, that's very short. Very short. The remark was that the example with remote surfaces and elliptic integrals is a bit dangerous, because there are two things that appear together. One thing is the fact that you can have a visualization of computations on elliptic integrals, which is one thing. And the other point, which is the main point for the history of mathematics, that using these geometrical tricks, using integrals that were simple integrals... ... have a new meaning. They appear as invariants. And so the transformation from numbers or integrals to invariants is something that changed radically the nature, the very nature of these integrals. And it's a bit dangerous to mix the two things and to say that visualization was really transforming the point of view in integrals. No, no, no, I'm not saying that. I'm saying that this representation... In terms of Riemann surfaces, in fact, quite the opposite there. I want to emphasize part of my example that Riemann surfaces are completely valuable, completely independently of any connection to visualization. However, it is also true, interestingly enough, that part of the form, and more generally for Abelian integrals, generally, being able to form certain kinds of pictures. Seems to be part of why this explanation has such a psychological force for me, and my question is, is this an accident or is there some deep connection between what we're representing in the future and the underlying, and the conceptual story that gives it its mathematical fruitfulness, or are they just accidentally connected in a way that... That was simply my point. I agree completely that there are two things going on and they have to be kept separate. Thank you very much. I must close the discussion.