Synthetic a priori knowledge in geometry

0:00 Thank you. Hello, I am Marcus Jacuinto from the University College of London. In the department of philosophy, Marcus will talk about synthetic apiary knowledge in geometry. Thank you. I apologize. I have to speak in English because my French is too poor. So I'm sorry about that. I'd like to say thank you very much. Every single one of you here. I mean, it's not a big number, but any number at all on a day like this is great flattery. So thank you. Okay, I'm going to, let me just say by way of background, I don't know how in Paris, you know, how this idea of synthetic a priori knowledge, a priori knowledge is regarded. But when I was a student in Britain, the very idea was regarded as bizarre, a little bit silly, is that what you say? Foolish, a bit foolish. If you took it seriously. But now I'm arguing against my teachers. I'm saying, yeah, we should take it seriously. Now I want to justify my attitude. So, now, in the first part, I'm going to, this is just the plan of the talk, I'm just going to argue that there can be synthetic a priori judgments. And then, in the main body of the talk, I'm going to address the question, can such judgments be knowledge? And I think there are two big problems when we're talking about geometric knowledge, there

2:30 are two big problems with the idea that there can be synthetic a priori judgments which are knowledge. And this is the space problem is the first one, the geometrical truths are spatial truths, but how can we know truths about space a priori? The second problem is the generality problem. When you're reasoning synthetically, you tend to focus on the number of cases. So the problem is to explain how we can reliably reach a general conclusion by reasoning about specific cases. So those are the two big problems. and the bulk of it all is going to be just a generality problem. But now I'm going to try and support my claim that there can be synthetic a priori judgments. And I'm going to try and do it with an example. And the example is a way in which somebody might come to believe the triangle inequality. You might come to believe this proposition in the following way. You say, well, look, suppose it's not true that the line segments in a triangle have this property, that the length of one is going to always be strictly less than the lengths of the other two put together. Suppose it's not true, okay? And we've got two cases to consider. The first case is where you've got one line segment, which is strictly longer than the other two put together. Now, if you imagine the longest side represented by the horizontal baseline here, and the other two sides as fixed at the endpoints and you rotate them, you can see that there's going to be nowhere that they'll meet the other two sides, and so you won't get a triangle. So in the first case, where the longest side is strictly longer than the other two put together, you don't get a triangle.

5:00 Now that leaves just one more case to consider, which is that the longest side is equal to the other two put together. Okay, let's consider that. okay now what happens when you consider this case you imagine rotating the shorter sides around the around the common endpoints you can see that or you can tell you can come to believe that they're only going to meet when they coincide with the baseline and so again you don't get a triangle so you so we can come to believe in this way that whenever it's not the case that when you have three line segments one is shorter than the other two put together it's you don't get a triangle all right so now contrapositively you know that whenever you've got a triangle it's got to be the case any line is going to be shorter than the others we put together I mean this is a way you might sorry, you might try and convince somebody if you didn't have a lot of time and you just wanted to get them to believe it if they ask you to prove it this is a different question but if that wasn't the problem you just wanted to get them to believe it you could get them to believe it that way some people I mean everybody here will have already believed it but still it's possible to see that this is a way in which somebody could come to believe the triangle inequality and if it is a way I claim it's going to be your judgment your belief is going to be synthetic apri it's going to be synthetic Why? Well, first of all, if you think of the claim that in any triangle, any line is going to be shorter than the other two put together, if you think of that claim, it's not analytic in the way

7:30 that Kant thought of it where an analytic claim one which you can get by unpacking the subject concept, the triangle and then getting a creative concept out of it because the concept of a triangle, as Kant would say has nothing of quantity in it it doesn't tell you anything about the length or the relative length So it doesn't look like an analytic proposition. But moreover, if we're talking about this way of coming to the belief, this is clearly not a standard way of unpacking concepts or looking at definitions, unrolling the definitions and seeing the proposition fall out, what you're doing is you're looking at the... You're putting together, you're synthesizing some visuospatial bits and pieces. You make a construction. So in that sense, it's synthetic. So this looks like a clear claim of a synthetic judgment, not an analytic judgment. Now, what about the claim that it would be a priori? If you came to believe this way, I argue, your belief would be an a priori, sorry, I must say, a priori belief. The reason is this. If it's going to be a posteriori judgment, the experience has got to be being used as evidence I'm taking an a priori judgment to be one in which no experience is used as evidence for the proposition or for anything from which you infer the proposition then it's a priori that's the notion that I'm using I think it's true to I think this is the idea This is Kant's idea when he explains this notion right at the beginning of the critique. Now, in this case, one could come to believe the triangle inequality and be very confident about it,

10:00 even though you know you've only looked at most two instances. This is a general claim. It's a claim about any triangle whatsoever. Now, we're looking at a sample of at most two instances, right? It doesn't worry us that we've got a particular ratio of lengths, the length of B to the length of C. We don't think, oh, well, maybe I should try a few more cases and have the length of B very small in relation to the length of C, for example. but if you were collecting evidence if this was the point of a diagram like this if this was how you were doing it then this kind of consideration would matter to you but now when you reflect on the fact you've got at most two instances to illustrate the contrapositive of the triangle inequality it doesn't affect our confidence in the belief And here is another factor, too, which is that one can see that, in fact, one hasn't even got a single instance of anything geometrical here in front of one, because these aren't proper circles, and it's all very rough. If you look closely, the lines are fuzzy, they're not straight, none of this matters to us, but we're supposed to be drawing a conclusion about perfect circles and perfectly straight lines. Okay, this is about geometry, but real geometry is not about approximate geometry. But we've only got approximately good circles and lines in front of us. But now when we reflect upon this fact, it doesn't destroy our confidence, or it needn't destroy the confidence of a rational person who comes to believe the triangle inequality in this way. And what does all that mean? that they were not using the visual experience they have when they look at

12:30 these and manipulate these diagrams before them. They were not using it to provide evidence of instances. So this means that the nature of their path of judgment was a priori okay so this if you come to the belief this way I claim it's a synthetic a priori judgment let's forget about Plato okay so now I'm going to now I'm going to address the real question here suppose you we do have we do make we do get synthetic a priori judgments in geometry suppose it's possible, as I claim. The question is, could that kind of judgment, could a judgment in geometry arrived at in that way, be knowledge? And the first problem is the space problem, geometrical truths, the spatial facts, how can we have a priori knowledge of spatial facts my story is this I agree with those people who say that we can't have knowledge about actual space a priori I concede this if you want knowledge about space, it's an empirical matter, it's physics, you have to get empirical evidence to find that geometry of actual space. But I think I think there's another way, well I'm sure there's another, a more appropriate way of thinking about geometrical sentences that shouldn't be too unnatural to somebody today who knows that there is investigation into different kinds of geometry different geometrical theories in mathematics departments we're doing pure geometry you could be studying hyperbolic geometry or you could be studying Rumanian geometry or some other kind of geometry or Euclidean geometry. And you could be studying

15:00 three-dimensional geometry or n-dimensional geometry in height. What are you doing in these cases? Why are people doing all of this kind of thing? Are they really concerned to be telling you the truth about actual space? No, they're investigating kinds of space. Ways that space could be. And I think that now if we go back to Kant, now he identified actual space with the mind's representation space. I'm not asking you to do that. I think that's wrong. But if you focus on the mind's representation space and you accept Kant's claim that the mind naturally and pre-theoretically represents space in a certain way, then you can ask yourself about that kind of space, the kind of space that conforms to the way that we naturally represent space. I mean Kant's idea I think is that even before we start making any geometrical judgments even before we start before we acquire any geometrical concepts the mind represents space in a certain way as infinite he says as singular there's just one thing space, there's lots of spaces within this one thing and I would say a lot more than that I would say also the mind represents space as without gaps we find it difficult to understand what somebody means if they say well space has gaps in it I mean of course we can understand this mathematically now because we can understand what a mathematical model of the Euclidean geometry is but if we go back to if you forget all your sophisticated and you just think about how you might have felt about space before you had all this apparatus to help you along. You should remember, you should be able to recall, that the idea of space with the gaps in is a bit puzzling. Now, what's going to go in the gaps?

17:30 Isn't there going to be more space? I claim that our natural pre-theoretical representation of space of gaps, and I claim also that it's space of zero curvature. You should still, despite mathematical sophistication, you should still be able to capture the puzzlement or recall the puzzlement, or at least experience it when you try and explain curved space to somebody else now how can space itself because I think can be curved in space okay maybe you know maybe a library could be can be curved in space but space itself it's puzzling right it's initially puzzling and why is it initially puzzling that is because our natural represent the mind's natural representation of space is the representation of the space and zero curvature so now I'm agreeing with Kant I would say a bit more than Kant along this line the mind has a natural representation of space I think we acquire this in infancy I mean of course exactly the nature of this representation is a matter for cognitive science it's not a matter for philosophers But my hypothesis is that this representation space is a representation of Euclidean space. It's a Euclidean representation. So if you ask me, what is the kind of space that the mind represents, I would say it's Euclidean. It's that kind of space. Now the question about, we can return to the question about your knowledge. The question is this. But could we have a priori knowledge about that kind of space? Space as it would be if it were as the mind represents it. Now we're talking about the kind of space. I could say, look, maybe this kind of space is not physically realized. That's a separate question whether this kind of space is physically realized. But now I can still, by synthetic a priori methods, investigate this kind of space.

20:00 Just as in the mathematics department, you might be interested in investigating hyperbolic geometry. You might be investigating it, aware that actual space may not conform to hyperbolic geometry. so the question is now is it possible that we have synthetic a priori knowledge of a certain kind of space namely space as the mind represents it and my claim is my suggestion is yes that's possible that's a serious possibility it's not knocked out this possibility is not knocked out by the realization that you couldn't have reorient on a Jabbat about physical space okay so now I'd like just to just to kind of Can I rub this stuff off? Can I erase this? Thank you.

22:30 Thank you. Thank you. Okay, now, here is an argument, uh, it's, I mean, it's, uh, not, um, it's a bit, it's too simple to be accurate to Kant, but, you know, if you allow me a little, I mean, Kant's argument is more complicated than this. I know. Right? But it could be rough. Can you see this? Good. Okay. But there's an argument roughly like this in a particular period. We have a priori knowledge of geometry. The geometry is about space, therefore we've got a priori knowledge about space. But, and I can't argue seriously to this, if we have a priori knowledge about space, space can't be independent of mind. So space is not independent of mind. So this is an argument for transcendental idealism.

25:00 Now, a lot of people say, oh well, this conclusion can't be right. So, something's wrong with the argument, let's spell it, this premise all seems okay, it's obvious that geometry is about space, right? therefore we don't have a upper urinary knowledge of geometry okay I think this is that the move that most people make especially you know in the Anglo-Saxon philosophical world when I was a student you know we've got an argument we've got an argument against the claim that we have we could have proper knowledge of geometry even euclidean geometry okay because you say well this is obviously wrong so we've got a full point of the premises but this practice is okay this premise is okay so it's this first premise that is the mistake all right well i'm saying what i'm saying to you is no the problem comes with uh premise two premise two is too simple you have to distinguish between about space, meaning actual space, and the claim that geometry is about a kind of space. When you've got lots of different geometries, Riemannian geometry, reading geometry, and so on, it makes sense to think of geometry as being about, here I'm talking about pure geometry, the kind of thing that we study in mathematics, as opposed to in cosmology, right? It's about a kind of space. Not about space. So I'm saying, look, instead of saying no to this, we should say no to that. Even though it might look a bit odd before you start thinking about it. This is where the argument goes wrong. Not here. I claim that we can have some upward knowledge of geometry. This does not commit me to saying space is ideal because I can reject this premise. That's what's going on here. So that's just to link my spiel to the map.

27:30 Okay, so that's the first problem. much too long there. I think if, you know, if we make this distinction, we think of geometry as making claims about, well, including geometry, making claims about the space as it would be if the mind represents it, then this problem, the space problem, can be overcome. yeah sure I should have had that that on the board before I lost my notes I had speakers notes I'm good in front yeah okay so this is my answer we cannot know a priori the geometrical facts about actual space but we can know a priori the geometrical facts about the kind of space that is represented by the mark Let's go on to the big problem, the main problem. How can we reliably reach a general conclusion by reasoning about specific cases? Okay, I'm going to talk about this example because Kant talks about it. So if you're fed up with hearing, hearing talks about it, I apologize. But he looks at the angle-sum theorem and he follows pretty much Euclid, but not exactly. draw a triangle, okay, extend one side, then we are instructed to draw a parallel at the vertex of that vertex of the triangle that meets the extended side. Those are your parallels. And then we apply the theorem, which has already been proved in the course of Euclid's development, that alternate angles are equal and we apply the theorem already being proved that the angles

30:00 on a straight line on one side of a straight line sum to two right angles okay this is how we get it synthetically according to Kant let's just be clear about what the problem is forgetting the fact that we haven't even got a single properly geometrical figure in front of us because the lines are fuzzy the lines are squiggly I mean if I get close up here there's no single point that's the vertex and so on so we haven't got geometrically perfect figures here, forget that still we've got a very big problem which is that there's only one roughly one shape of triangle here it would look very very different if we alpha much closer to 180 than it is at the moment. Okay, so we made it a great big fat obtuse angle. All right? Would the argument still go through? I mean, maybe it would. Maybe it wouldn't. All right? But the fact is, if you come to it this way, you don't draw lots of figures, or at least if you do it the Euclid way. He just says draw a triangle, right? He doesn't say draw lots of triangles. do it one when you've got three acute angles and one where you've got one right angle at the base and one where you've got an acute angle at the base it doesn't say that, just one triangle right this is a generality problem how can we get this way to a conclusion about all triangles, all Euclidean triangles whatsoever and Kant has an answer to this go through what he says. First of all, he says the representation must be constructed. This is clearly very important to him. And this is what he says. This is the English translation of this. Thus I construct a triangle by exhibiting an object corresponding to this concept, neither through mere imagination and pure intuition or on paper and imperial intuition,

32:30 but in both cases completely a priori without having to borrow the pattern from any experience. I think what he means just by this is simply that it's not a matter of recalling a figure that you've already seen. It's a matter of you producing in the imagination or on paper a perceptible figure. And then he says this. The individual drawn figure is empirical, and yet serves to express the concept without damage to its universality. Without damage to its universality. For in the case of this empirical intuition, we have taken account only of the action of constructing the concept, to which many determinations, e.g. those of the magnitude of the size and the angles, are entirely indifferent, and so we have abstracted from those differences. Okay, now, I think this is insufficient as an answer to the generality problem. He says, look, we don't pay attention to the specific angles. Okay, that's true. But he says, what we do pay attention to is the action of constructing the concept. Now, which action are we talking about? We're talking about the action that resulted in this particular figure. I take it it's that particular action that particular action itself had a lot of component parts first of all you've got to draw the triangle produce it in your let's say you're drawing it you draw one line another line, a second one by the time you've done that you've already made three specific movements so your action has components to it and then of course you extend one side Your action has specific components. And that's why you get a specific figure at the end of it, one particular figure at the end of it, because your action is specific. Your action is a sequence of actions, of sub-actions, if you like. Total action of construction is sequence. And these component actions are specific.

35:00 So you can't get generality out of considering only the action. So this is insufficient. say more about it. He also, you know, he shifts attention away from the action that you've performed in constructing the figure to the general conditions of the construction. And here is this very famous and much repeated quotation. Mathematics cannot do anything with the mere concepts, but hurries immediately to intuition, in which it considers the concept in concreta, although not empirically, but rather solely as one which it has exhibited a priori, i.e. constructed. And this is an important point, which what follows from the general conditions of the construction must also hold generally of the object of the constructed concept. So here he's saying, look, when we're doing the reasoning that leads us from, after we've got our construction, we do some reasoning to get us to the conclusion, we're reasoning only from the general conditions of the construction and that seems much more promising to solve the generality problem because if we're talking about general conditions they must be the same conditions for all triangles for all triangle construction we're talking about general conditions, right? and so it looks as though we can get universality out of the general conditions Okay, so it looks as though Kant is heading towards a solution here, but he doesn't say more than this, and if he doesn't say more than this, it's a big problem for him, and a problem which shows that what he says is at least incomplete, if not wrong. The problem is, well, there are two problems. problem is, if you pay attention only to the general conditions of construction, you're not paying attention to the figure that you've constructed, right? You've done this, produced this figure, you've got a certain kind of experience, an intuition, as Kant would say, but now it seems to be playing no role, right? Because you're only paying attention to the

37:30 general conditions of construction. And that seems to me wrong to the phenomenology and to the, not merely the phenomenology, but the any proper analysis of what's going on in these cases. The experience of the images does play non-superfluous part, but here he's wiped that out. The second problem is, if Kant were right that we're only reasoning from the general conditions of triangle construction, you have to ask, well where do we get these general conditions from? The general conditions are independent of our acts of construction and our experiences they're general right we get them it's very difficult to see how we could get them from anything else but the concept of the triangle so now if we if we're reasoning from these general conditions and these general conditions are provided by the concept it looks as though really what we've got although not obvious is knowledge from the concept and that looks more like analytic knowledge than synthetic knowledge. So here I think we've got to say Kant's attempt to give a solution to the general problem. It doesn't seem to work, or at least it's insufficient. So now I'm going to rush through my alternative response to the generality problem, this problem about how we can go from particular to general. But before I focus on the generality problem, I need to say more about the roles of visual experience. Okay, now here, let's just go to Plato's Fink's example in the main note of coming to the theorem that the square on the diagram, the given square, has twice the area. we have to we have to come to believe various things one of these is that the small triangles in this figure

40:00 that we end up with are congruent another is that the square on the diagonal consists of twice as many of the small triangles as the original square If we think about B, my guess is that if we follow the course of thinking that Socrates takes us through here, we do it by counting. We can see that there are four in the big, two in the middle. This seems to me as though we're using the experience as evidence. of course it's a little bit more complicated it's complicated because we're using we're using these these areas that we see in front of us to stand for perfect perfect triangles and we know that these are not perfect triangles okay but nonetheless there's an empirical element I claim in the way we reach we reach B so here is a case where you can be using the experience evidentially and therefore in a non-a priori manner but what about the claim A that the small triangles are congruent this is much more complicated why do we believe this? One of the things you've got to do, you've got to believe that the triangle's either side of the diagonal of a square or a concrete, or a rectangle, maybe a rectangle, but let's say a square, okay, we're dealing with squares here. Why do you believe that? What's happening here? It's difficult to explain that. and my story is that these beliefs these congruent beliefs as well as a load of others they result from the interaction of the actual visual experience with concepts that we have I claim that possessing a concept

42:30 in possessing geometrical concepts in the way that we do provides us with belief-forming dispositions. It causes us to have these belief-forming dispositions. You couldn't have these dispositions without... You couldn't have the concepts if you didn't have the beliefs. I know that people find this puzzling, but look, let me just illustrate. Let me give you a non-geometrical example, you've got the concept that's smaller than, right? Is relation smaller than? Now, if you have this concept and you come to believe that A is smaller than B, for any objects A and B, and that B is smaller than C, if you have the concept, that will give you the disposition having formed those beliefs that A is smaller than C. if you didn't have the disposition to form that belief that A is smaller than C I'd claim that you wouldn't have had the concept merely having the concept smaller than gives you the disposition to form the belief that A is smaller than C when you get input beliefs A is smaller than B and B is smaller than C so this is a way in which having a concept that gives you automatically a belief-forming disposition. Now, I claim we've got a lot of belief-forming dispositions as a result of having simultaneously in play various geometrical concepts, in this case a concept of a square, a concept of a diagonal, a concept of sameness of shape and size, and so on. If you have all those, you're going to have an interesting array of dispositions, and they can help give you they can give you a belief-forming disposition but in this case what I'm claiming okay, now I will put this

45:00 I've lost the array, what did I do? Anybody see it? Thank you. Thank you. Yeah, I know some people hate these box-arrow diagrams, and I agree that they can be misleading, but I am sure that we have, besides our geometrical concepts, our geometrical shape concepts, some of which can be defined in terms of others, we also have perceptual shape concepts. So I can say that that tabletop is rectangular, meaning rectangular, not geometrically rectangular,

47:30 but it's rectangular in the perceptual sense. We can say this of a region of land, that it's rectangular, or a square, and be speaking literally correctly, providing we're using the perceptual shape concept here in the center of the circle. And I think that these perceptual shape concepts are activated when we have retinal stimulation which which results in our seeing something in our environment as shaped in a particular way. So I see the rim of this cup here as a circle. I'm not really employing a geometrical concept here. I'm saying it really is a circle, but it's a perceptual circle. I'm employing a different concept, but it's a concept which is... departing from Kant is related to the geometrical concept in a specific way. So I don't want to go into that. But experiencing a figure as a figure of a triangle, as a figure of a circle, results not merely from the retinal stimulation but the activation, the laser activation, of the perceptual shape concept. So now you've got the perceptual experience of some array of lines, maybe, as representing a geometrical figure, So, providing the figures are not too wired, our geometrical shape concepts can be activated by having this experience. Yes, do you want to say? As well, thank you.

50:00 But when you say that what you call perceptual shape would also correspond to some geometrical concepts, say, because something like smooth, so-called, not really topologically, because they would distinguish where they are. Right. But you might find some, you know, plausible geometrical concepts that make the same difference. I'm sure that's true, and important, and relevant here. Because of the connections, because there are correspondences between geometrical concepts and perceptual concepts, that is the reason why the experience you get from the perceptual activates the geometrical ones, because of the correspondence. My point is that if that's true, that we wouldn't distinguish as, say, perceptual and geometrical, but rather as geometrical and physical sensing, natural science, or something. the perceptual concepts, because they, how do I put it, the geometrical concepts are derived from the perceptual concepts, although I mean you can get looser concepts than geomagical ones, pathological ones. You can define them precisely. For example, there's a precise definition of symmetry which allows for degrees of symmetry. okay so there there is a kind of rough mathematical mathematical concept of rough symmetry but that's a very sophisticated concept which you build up out of definitions and it's not it's not the same as the perceptual concept so I mean this is a bit of a side issue that let's not stop but this look this So this is what I think is going on. There are two lots of concepts operating here.

52:30 And what happens when we draw figures, it's only going to work for us if our drawings are sufficiently good. they've got to instantiate perceptual shape concepts pretty well. If they don't, if they're too wild, then the kind of thinking goes on when you're following the Euclidean argument won't happen. Right? So, and this is why, you know, Mick, Mick Deppler, says, look, why can't they let that dot triangle okay now I've got something there visual in front of you it represents a triangle I mean it's my intention that decides that I represents a triangle right now he can say say that that's true and now this means that we have to explain why it is that a dot won't work why can't you do the triangles the Angle-Sund theorem just using a dot let this represent a a triangle with one side extended and a parallel line drawn in the appropriate place, it's not going to do any work for you, right? You need to, the drawing needs to instantiate perceptual concepts for it to do any work for you. So this is a response to that kind of problem. So now, Okay, I'm sorry, I've gone too long about that, but look, what I'm, let's go to the next slide. Yeah. All that was to kind of put some, to try and make it less incredible that having geometrical concepts provides us with, having certain concepts provides us with dispositions,

55:00 which are dispositions to believe things as a result of having a visual experience. The visual experience is what activates the disposition. The input would be visual experience of a certain kind and the output would be a belief. Now, if you go along with this, I mean, I know I haven't justified this claim, that's much too big, And I've been trying to make you understand what the claim is. We've got these roles, these possible roles for visual experience to provide evidence to activate belief-forming dispositions, which is what I've been talking about just now. But sometimes, also, we already have the belief, right? Let's say the triangle inequality, okay? but you can still go through this reasoning or this thinking. You can go through it or the Engels-Sund theorem and you can... the prior belief can be brought to mind as will your belief that the triangles either side of a diagonal and a square are congruent. It will be brought to mind. didn't bring it didn't cause you to acquire the belief but it will have brought it to the front of your mind okay so these are three three roles that visual experience can have okay now I would just want to illustrate another another role for visual experience if you remember how we go through the angle sum theorem we build up we build up a picture okay we need to retain that we build up okay this is another another role for the visual experience which is to acute accumulate the information okay with it so that here I'm saying the visual experience serves to hold in mind successively added information so the total info information put together leads you to the theorem so here I'm not experience, but what I'm saying is that there are at least these roles, right? To provide

57:30 evidence, to activate belief-forming dispositions, to bring to mind prior beliefs, and to hold in mind accumulated information. Now, suppose you reach a belief which uses experience only in the last three ways, or maybe in small ways, but not to provide evidence, not in its evidence providing a role. Now, in that case, the way in which you acquire the belief can be a priori, even though visual experience is playing an important role. So this is how it's possible that visual experience does play an important role in the Engel-Sund One that you can't ignore. But now this leads us back to the generality problem. Given that the visual experience is important here, and it's not just the general conditions, as Kant says, we've got this problem back again. Let me be a bit more specific about the problem. The problem is that the path by which you reach the theorem, and a feature of the constructed figure that's not shared by all figures. Okay, that's stated in general terms. Okay, here is a typical example. The area of a triangle with a horizontal base equals half its height times its base. Okay, this is true, all right, but here is a fallacious way of reaching this conclusion. You draw your figure, you drop a perpendicular from, sorry, you place it in a rectangle with the same base and the same height, same vertical height, you drop it perpendicular from the upper vertex, and then you can see that the triangle is divided into two parts, each of which is a half of the rectangles composing the whole rectangle. So the area of the triangle is half the area of the whole rectangle, rectangle, and the whole rectangle is the height times the base. So you can reach it that way. But this is fallacious. Why? Because you can't carry out this operation for all

1:00:00 triangles. You can't carry it out, for example, if there's an obtuse base angle. So this path is fallacious because we're relying on a feature of the constructed figure, which is not shared by all triangles okay we're generalizing to all triangles with a horizontal base but we're relying on a feature that is not common to all triangles with a horizontal base namely that the base angles are a cube okay so this is this is the main big problem the main problem the big problem for reliable generalizing when you're using and using a figure there are conditions for reliable generalizing. To explain these conditions, you need to distinguish between the initial figure, which in this case would be a triangle, it's the angle-sum theorem. And then there's the elaboration of the figure. You extend one side and then you draw a parallel in the appropriate vertex. That's the elaboration of the initial figure. So here are my conditions. The type T of construction that elaborates the specific initial figure of kind K has got to be applicable to all members of kind K. Think of kind K as being triangles. We're reaching a theorem about all triangles. But for a theorem about all triangles, the type T of construction that elaborates the specific initial triangle must be applicable to all triangles. And then the reasoning that you carry out once you've got the fully constructed figure has got to be applicable to all figures of kind K, all triangles, with that elaboration, with an elaboration of type T. Those are conditions for reliable generalizing. But I claim there's a further condition which refers back to those two, which is that when you make your generalizing step at the stage where you say yes it's true for all triangles meaning more Euclidean triangles that the internal angles sum to two right angles that step of generalizing has got to depend on

1:02:30 the holding of conditions 1 and 2 and what I mean by that is that if conditions 1 and 2 were not to hold in similar visual reasoning you'd detect the failure and you'd hold back from generalizing or you'd be sensitive to the failure you've got to be sensitive to the possibility that conditions 1 and 2 fail you may not articulate these conditions 1 and 2 but it's got to be the case that something would stop you two didn't hold why because why have you got to have this sensitivity because if you if you weren't that sensitive then if conditions one and two do hold it's it's just luck that they hold for you okay it's just an accident so your way of getting to the belief isn't really reliable right it's not genuine knowledge if there's if luck has played a role in your hitting the truth yeah so that's important okay so here is my answer to the generality problem how can we reliably reach a general conclusion by reasoning about specific cases answer by fulfilling these conditions one two and three yeah no no sorry yeah that's misleading, yeah, I meant specific, yeah, thank you, yeah, specific construction. Okay, so now I say some things which go way beyond what I'm entitled to say, but let me say, I think that these claims I'm making are plausible. I haven't established these claims, but I think it's plausible that the three conditions can be met if we reach the angle-sum theorem in the way outlined by the way that Euclid takes us through. I think that that way results in a synthetic a priori judgment.

1:05:00 Given that the theorem is providing that you take the theorem as a proposition about triangles and spaces the mind represents it, as opposed to about physical space, this way, I claim, I think it's plausible that it's a reliable way, right? And also, I think you can reach the judgment this way without any other violation of epistemic rationality. knowledge. So here is my conclusion at last. Kant claimed that all geometrical knowledge is synthetic apriori. That's too much for me. I quarrel with that. But I agree with him that some geometrical knowledge can be synthetic apriori. And I think that this is made more plausible when you pay very careful attention to the actual ways in which we get geometrical beliefs. Okay, thank you very much for your patience. That's it. Thank you. You're saying that space has equal to the mind that presents it as video-filiers. I must be curious about your responses. You asked me to remember, or to forget what I know, to remember that space. And I think I remember quite well that the idea that space is very hard to solve. And if one looks at the representation of the universe, you can find all the knowledge is an extension, on which we can take evidence that other people will not think about it. And the second, let's go back to the previous slide, how to grow about, that the young child, the intuition,

1:07:30 when we think about points, they are all expensive. Okay. Let's go to the first one about the infinity of space. It's a very controversial claim, and actually that claim isn't essential to my story. and ultimately I think that it's really up to the scientists so it may be wrong but yeah the infinity of space is a it is a difficult one we have a lot of quite specific intuitions relating to this for example people don't generally have a problem with Wallace's idea that for any line segment, sorry, for any triangle, for any triangle, any given line triangle, and any line segment, however big, there is a triangle on that line segment similar to the given triangle. now if we thought of space as finite we might have a problem with that I mean this isn't conclusive, I agree it looks as though we've got it may be too crude, given what you said it may be just too crude to say the mind represents that space more fine-grained than this to be said. Yeah, I'd go wrong with that. So that's the first point. The second point, yeah, I remember reading this from Klein, but you see I'm not claiming that that the geometry of phenomenal experience is Euclidean. In fact, I'm sure that this is not true. And I don't think that we get the geometry out of phenomenal experience.

1:10:00 I think when we're talking about the geometry of space as a mind represents it, we're talking something more abstract than that. It's something, in particular, it's not visual. We're talking about something of a blind manner, a representation of space. It's something I believe that we get very early on in infancy as a result of our innate dispositions our innate faculties and our very early experience tracking objects and navigating our way around the world so I'm really talking about a very abstract kind of representation which is not tied to any particular sentimentality so that's but it's still it's controversial I have a question what is perception shape why do you define it as a concept because there are some kind of characteristics that we have a tendency to add Well, I realized that we couldn't be satisfied with geometrical concepts when describing our experience as a result of reading cognitive scientists who work in visual perception. in particular Stephen Cosselin who also works in imagery we know from experiments that very early on infants already make visual categorizations then when they acquire language they can start to talk about They can start to think about circles and squares and so on. Now, what do they mean? They don't mean the locus of points equidistant from one single given point.

1:12:30 Now, this is a very sophisticated idea. You get taught that later in school. They mean something else. But they already have got a concept of the circle. So what is the nature of that concept? I claim that it's a perceptual concept, and I think it's very related to the perceptual categories that they have in the visual system that they've developed as a result of their experience. So that's my story. Thank you. Actually. We have a related question. Those concepts, perceptual shape concepts and geometrical concepts, play such an important role in these forms. Doesn't it threaten the authority? don't you think there's an information analytic 2.5 again I think that it's analytic if you're if you're analyzing a concept and you're this is a very this kind of thing that you can do if you've done some logic you know, you've got a definition of Z in terms of Y and a definition of Y in terms of X, then you can see by unpacking some proposition involving the concept Z, you can get some proposition explicitly about X using the concept X and so on. Now, the kind of thing that we do. now that's using a concept that's analyzing a concept or unpacking as Christian as Moses Mendelsohn said unpacking a concept and when that kind of thinking is happening that's that makes it analytic but since I think any way of reaching a judgment does involve concepts I think

1:15:00 that not, unless we think that all belief formation is analytic, we're going to have to allow that the involvement of concepts allows for synthetic judgments as well. So it's true that the concepts play a very important, indeed essential role here. But if you think to reach something analytically. I mean, forget Kant who had a very restricted notion of what analytic was. Let's widen it and talk about Frege analysis. the kind of story that I, the kind of path that I illustrated for the triangle inequality at the beginning, that wasn't doing the kind of thing that Frege would have counted as demonstrating the analyticity of the theorem. There what you've got to do is you've got to have premises which are either logical laws or definitions and then you've just got to do some logical deduction and reach the theorem. Now that would show that that would be analytic thinking. This wasn't that. So that's my story. Of course I think it's absolutely justified to distinguish, say, perceptual concept, But what I actually doubted is this platonic story about perceptual being somehow imprecise and say geometrical, pure, and mathematically being more precise, and I even wonder how this element actually enters, say, Kantian philosophy. I cannot see how, say, that it enters in a really natural way. And it makes me doubt it might be just kind of historical, I don't know, habit to bring this whole idea in some new context. Because I think if we even trying to analyze it, say, in canter terms, let's bring one, it's rather that we might have, say, interpret differently, the same picture is more as like this double face or something.

1:17:30 So we just, there is one thing which we can interpret as a sort of, and as, you know, more complicated thing. So, so we have kind of oscillation between concepts and say that, that make, yeah, that's related to perception, of course, that possibility of the oscillation, but in this, in the sense it's just, say, in arithmetic, it really takes this kind of idea seriously, that just thinking number we might have kind of, you know, this construction for a number, and then it, it It does go through so naturally that we might have three dots, it's perfectly three, even if we can also imagine a case where we can't really say three or four, but in some cases at least perception doesn't make any harm, I don't know, it's perfectly, so we have this kind of coincidence between perception and concept formation. And sometimes we don't have it, but I'm just trying to understand, perhaps it just kind effect of a particular separation, or it still has some general sort of stuff? Well, I don't want to generalize what I've been saying beyond geometry. I think that there are many more problems for Kant's story of arithmetic. So I'm not generalizing, but But just going back to your original point here about Plato and Kant, I mean, it seems to me that one has to distinguish between imperfect triangles and geometrical triangles. I mean, sometimes we can illustrate a theory or we can get somebody to think about circles which is visibly non-circular. It's visibly imperfect, visibly imperfect, but still it can do some work, right? And you can immediately see it as a circle. So we definitely need some distinction now. Kant, agreeing with Plato,

1:20:00 think that our geometrical concepts are prior to the perceptual concepts. somehow the geometrical concepts are the ones that I mean in Plato of course they're kind of literally prior in the sense that we acquired them in some former existence possibly and we're just recalling them Kant's not telling that kind of story we construct them we do define them our geometrical concepts And then our perceptual concepts are just imperfections of the geometrical ones, right? But I think it's the other way around. I think that we start at least with basic concepts. We start with basic perceptual concepts, which allow for imperfection and approximation. And then we get geometrical concepts as a result of thinking of improvements. It's not abstract topology or something, but something like law theory, and some simple law theory, not generalization or something. And its defendable position, I would rather claim that, that here you have a really perfect, let's say, the peak between, even empirically, your perception, visual and other. Yeah, of course, of course. I mean, you could actually have a physical knot. I agree, I agree, look. But then you make theory about that. Yeah, yeah. That's true, but I'm only talking about geometry. I'm not, you know, I depart from other philosophers of maths who want to make grand claims about math, mathematics as a whole. No, but that's geometry. I'm just talking about geometry. But not theory, geometry, smart. Oh, look, this is verbal. Okay, yeah, I'm just talking about geometry as Kant thought of it, okay, Euclidean geometry. So I accept, okay, you're making a point, you're making a good point, Andre, which is that the kind of things I'm saying about Euclidean geometry cannot automatically be generalized. Yeah, I accept that. It's restricted. It's of restricted importance. But look, if I convince

1:22:30 people that we can have synthetic a priori knowledge of anything, that would be a big victory for me. John? So, you said that you weren't trying to argue for the mildly have the retinal stimulation you just kind of that's what it is and then you also said that it's a terrible matter about what you represent that space is this also a terrible matter your claim the support that you want to hear for or do you want to complement it a terrible matter absolutely yeah I mean I I came to this kind of a story as a result of reading a lot of stuff scientists. So it's all vulnerable to, I'm making empirical suggestions, and it is all vulnerable to new findings in empirical science. Okay. It's just that I was thinking also in the first question about the kind of evidence that at least it would be the job of the cognitive science to define, to support or not support what the mind represents. You naturally have that chronological background, right? Yeah, yes, okay. Yeah, yeah, so... Yeah, this doesn't tell the whole story. I agree. I mean, this box-arrow thing doesn't tell you how the way the mind represents space in general, the role it plays. but I think it plays an important part in which perceptual shape concepts we get okay so this is a question about how mind represents the better things you see the thing look I think we've got to go empirical I mean otherwise what are we doing when we say you can't get knowledge this way us philosophers or you can get it this way we're just supposing that the mind has a certain set of cognitive facilities, and no more, all right? Now, we've got to stop doing this, especially, it's especially

1:25:00 stupid when cognitive science is making such big advances. We've got to look at what the cognitive scientists are telling us about what cognitive faculties we have, all right? And then we've got to draw our conclusions about the possible ways of acquiring knowledge. So this is my attitude to it. We've got no option but to pay attention to empirical work. If you don't, then you're going to stay in the Dark Ages. I mean, this is the advantage that we have over Plato and Kant and people who wrote before us, that there is all this good work in cognitive science. Of course, a lot remains to be done. The big questions still remain to be answered. I'm not a believer in saying let's abandon the philosophy for cognitive science. I'm just saying let's pay attention to it. Thank you very much. Thank you. Thank you very, very much. Thank you for your patience. Thank you. Thank you. It's... I don't give you a chance here.

1:27:30 Are you? I'm sorry, Marcus, I had to... No, no, never mind. No, thank you for enduring it a second time. No, no, no, it's really, I would love to do this novel. It's just all this, my case should have a price, I know. Yes, I know. Thank you.