Space-like Categories & the Continuum / Dissenting Views

30:00 If I had taken directed graphs that would be called irreflexible, that is, not necessarily reflexible, with no dependence, it is definitely not true. In that case, a picture of two arrows on their own. The reason why you can't have Schroeder-Perservers or Schroeder-Perservers products is because another way of talking about speed in delta-1 is it's just

32:30 Reflexive co-equalizer data. That's the same thing as the graph of redundancies. And so gamma-shriek is actually the co-equalizer of reflexive data, and therefore preserves product in general. In fact, in many, many kinds of categories. And there is a kind of intuitive explanation for it. When you take co-equalizers, of course you have to build up an equivalence relation. And this equivalence relation, you know, is built up in steps, so each instance of the equivalence relation has a proof of so many steps, starting from the basic data, and if you have then two spaces, equalize the data on each, a pair of pairs of points, this equivalence may have a proof of length m, this one a proof of length m, how can you combine that into a single proof of whatever length that the pairs of equivalence. Well, in general mechanics, if you have reflexivity, or equivalently degeneracy, you can just put in trivial steps. Reflexivity means that you have given yourself a proof that x equals x. It sounds stupid, but it's all important. You can bring up the length of the proofs until they match just by filling in these trivial steps. If they have the same length, then of course the pair of proofs is a proof of the pairs. So that's one way of looking at it. It also shows that it must have something somewhere to do with some kind of finiteness. It's certainly not true that Gamma-Street is sort of an infinite process. Now, actually in this case, in this example, and again the E could be, the S could be getting closer and much more generally, there's even a further adjoining called Gamma-Upper-Street

35:00 A co-district graph is one such that between any two nodes, there exists a unique arrow in that direction and, of course, a unique one in that direction, so it is here. But it's like a complete graph except that we also have the identity. Usually when you are speaking to a complete graph, they explicitly omit the reflexivity. So this is kind of a reflexive version of a complete graph. In other words, most of the things that are known about a complete graph should be applied, but you have to be very careful. Now, this upper shriek, as you know, both need notation, and it is this symbol which I translated as the word amazing, because it's nice to have a word and not just a symbol, as the disastrous experience of the comma categories proves, because I had no word. It's still called comma. So, in order to read upper sheets and lower sheets and all that sort of intractable stuff with some kind of meaning, you could sometimes call this the amazing metronome. That was the idea. These are sort of extra things that you don't have in all cases, so remarkable exist. So, this leads us then to consider such situations before. Cohesion and non-cohesion. A very nice instance of what Hegel called the unity and identity of opposites that I started with, and look only at the other three, we have two categories, we have two inclusions, two quite different inclusions, mainly gamma upper star and gamma upper street, so any one determines all the others, and I think this

37:30 Well, we can notice that even if we go down to the one point category, sometimes the category has an initial object, a terminal object, and these are left and right adjoints, and this also applies at the level of E there, so that, you see, how does it go? So Engel says, well, we first should look at meaning, recognize it exists, and then start analyzing it from the bottom. So, the terminal object, in some senses, is the pure being, the being itself is this incredibly complicated thing E, but the purity of it is just its terminal object. Notice that we often name a category to its terminal object or vice versa, if that's not accidental. But then, of course, if we have pure being, we have also non-being, which is opposite to being. But it's the same, also, because although these conclusions are almost always totally different, the actual category is just one, so we have total unity and total oppositeness. And this is achieved not by some strange sort of logic, but by simply recognizing the distinction between an object and a sub-object. The empty space and the one-part space are both the same, namely because of the one-part category, but as conclusions into false spaces, they are quite opposite because they are the opposite of what you would imagine. Now, the first stage which includes both of these is Kepps. In other words, For example, if this were a topos, the smallest sub-topos which contains the empty object as a sheaf, because the sheafs are on this side, and these other things on the other side are opposites. Companions of sheafs. Sheafs are always in the left, right. But actually the empty set is a sheaf.

40:00 I think this, in a way, may be part of what Hegel was getting at with his elephant movement. Subsuiting them under the right-hand point at the next level. And if you look at, say if you look inside of simplicial sets, you will find that the next step after, in other words, the one-dimensional, yes, a discrete, that means coming from the left, simplicial set, is in particular also under. So the left and right, which were opposed at the lower level, have both been subsumed under the... Now, there is the question of computing in the specific case of some special sets. What is the alphabetical function? In other words, if you look at n truncated There should be an M, bigger than M, such that both the skeleton and the co-skeleton of dimension M are always actually co-skeletons of dimension M, and as long as such M, according to a student of Ross Street whose work has never been published, is 2n-1 if you go beyond 3, 0, 1, minus infinity, 0, 1, 2, just adding 1 as you like. But then beyond that, it becomes 2n-1 rather than plus 1. As I say, this unfortunately has been published, and I don't understand the calculation. It's something very intricate about implicit sets. But there is an even better, even more basic, in some way, example.

42:30 Finite, non-empty sets, then the appreciates on that are actually very important, topos. This topos contains, as a full subcategory, the classical categories of superficial themes, such as complexes and monomorphisms. Anyway, these are just those factors which are sort of living on their one-dimensional part. Remember that's going to change. On the other hand, the category of groupoids is included in this. Because those categories are included into the question sets, well, really, groupoids, in some sense, more naturally fit into this case, because of the fact that you've got maps that aren't order-reserving in correspondence. This has a left adjoint, and this composite is essentially the... So just to give you some of the aspects in which this topos exists, but of course we also should look, at least in this case, at what kind of classifying topos is given to topos. This turns out to be the classifier Boolean Algebra. The functor 2 to the blank, blank ranges over finite non-EV sets, this is of course a Boolean Algebra object. But it's the universal Boolean algebra of any Boolean algebra in it, and there will be any geometric morphism from that topos to this one, so that the inverse of this particular object is your given Boolean algebra.

45:00 Something like a singular complex realization. If we realize that the realization of this human algebra has to be an infinite dimensional sphere, not an integral, which we're forced to see, and this calculation is also in both of these pursuing stacks or part of it, but really the infinite dimensional sphere is a kind of quantum mechanical integral. In the sense that you not only can go from zero to one, you can go back and forth many, many times, sort of a typical problem test, it's just an analogy, a model of quantum mechanics. The idea that this infinite dimensional object is actually, in another way, really just a complicated version of the integral. If the apologists would have accepted that, then they wouldn't have had to come up with simplicial sets because the good old simplicial schemes, the problem with simplicial schemes always was that if you think of realizing the base of one as an integral, then the realization factor does not preserve products, you see, so it's a bad, it's a sort of bad way to go about computing because you have to constantly replace one product by the other and so forth, so. But if you accept this infinitely fat version of the interval, then all sorts of more complicated spaces can be built out of that, then the realization is this, the fact that this anti-conventional sphere has already been forced by the relation between the energy distributed lattice and the movement algebra that it generates.

47:30 And so forth. You are successively building up disks and spheres, disks and bounding spheres, and then, and so on. You get very concretely out of that construction of the Boolean algebra from distributive lattice, you get the classifier for distributive lattices. There's just a few shoes on the final process, and these are, of course, related to that year of what we're going to be passing through the natural sphere from what you thought was an interval. Anyway, so this is just to sort of promote the study of this particular topos. Topos theorists haven't studied the particular topos as a lot, but they should. And this is one of the first, I think, that they should study more of the sets. But anyway, I've got to introduce this to say in particular, there is this out-of-the-womb question about this topology, because again, if you just take, well, the endomorphisms of a two-element set and the endomorphisms of a three-element set and so forth, these all fit inside by means of all these punctures. And at the very lowest level, you can easily compute that the lowest level which contains both the skeletal and co-skeletal aspect of a given one, as contains as co-skeletal, is just the next one. And then one would expect that just as in the simplicial case, if you do a few steps more, that it will no longer be just the next one, but some particular number theoretic function. Describing the level to which you must go in order to achieve this, bringing the opposites together on one side and producing a new opposite on the other. The alphabetical impression, in particular, for this fluid algebra classifier.

50:00 In a way, every object in this topos is really a formula in some infinitary intuitionistic theory of ordinary fluid algorithms. So you can think in those kind of terms by trying the congenial way to approach the problem. One of the parts also has the property that the components of it deserve to find the problem. So to go back now to the general theory a bit, typically between the various essential subtoposites, the various subtoposites that have both skeletal and askeletal. They have shelves including as well as the usual social systems. You have just these three functions. But it's sort of only in general, it's only from sort of the top level to the lowest level that you have this fourth function. It's very much like a target system here. It does not exist between the stages. So, now, we have to sort of keep in mind these two possibilities. Cantorian view. That we have the points bumper and that even in many cases we have this extra co-skeletal or co-discrete extension in the Kantorian notion in a way that's usually implicit in examples like this.

52:30 I'd like to say that it sort of explains why the contradiction, the sharp contradiction, which is contained in Kantor's notion of an abstract step is not a logical contradiction. Namely, the idea of an abstract set is that the elements have no properties whatsoever, on the one hand, and yet they are perfectly distinguished from one another. Offhand, it sounds like it couldn't be so, but with this unity and identity of opposites, you see that the typical space has got its points, it's got its co-district inversion, Emphasizes the unity, whereas the discrete one emphasizes the distinction, and you have this, in substance, very non-trivial map from one to the other. Why non-trivial? Because there are almost no maps from the back. Typically, the only maps from the back are constant, and maybe have a finite number of values or something, depending on the setting. But the contradiction in the idea sort of spread into two very clear parts, and now you know that. We have a graph that has just dots, so we can have the complete graph. These are actually the same in a way, because the category of discrete ones and the category of coded ones are both isomorphic to the category of sets. So they're totally united as things in themselves, but totally opposite as they are plugged into their world. So it helps to listen to it. So, explicitly recognizing the philosophical notion of unity and identity of opposites and still further modeling that in this vaguely way, here's a very sharp way of resolving some of those things that philosophers might argue about if they didn't have mathematics. Philosophers and mathematics might argue about if they didn't have mathematics. That's part of the abstract theory here, has this general notion of limit-potent, monadic, preserving no sums and products on a lex-nensive basis, which is basically the kind of theory I'm going to consider, so imagine the two categories of all those things, okay?

55:00 Now, I have another two categories. The central item book. Remember that the center of a category means, well, there's the identity functor. And like any functor, it can have natural endomorphisms, and those are called central because if you look at it, you immediately see that by naturality, this means that to every object, you've got an endomorphism assigned, which is totally natural. That is, every map in the category is a homomorphism with respect to it. So multiplying by a constant in the category of vector spaces has that property, and indeed the center of the category of vector spaces is the ground field. But now I'm assuming not only is this essential, but it's significant, so if I compose it with itself as a natural transformation, it's the same thing. Now, I claim this is actually a special case of the above. In fact, it's a special case to put in the intermediate case where Cantoria's case, so to speak, where gamma lower shape also exists. So in other words, if I assume this, if I assume that gamma should also exist, then this component is included in that one, and I claim that this one is also included in this one. Why? Because I can split that input. And if I split the input, if I split the input,

57:30 And so on and so on and so on and so on and so on and so on and so on and so on and so on and so on and so on and so on It's quite unlike Cantor's idea in which S is just a fixed category in many things, right? S is the algebra for this moment. It's a little bit harder to suggest how we can fix. So the subcategory E, where theta is the event, where theta is the event, has both left and right adjoins. But moreover, these are equal. This very special, this very special situation. I decided, at least tentatively, to identify with a category of qualities that would be some kind of very extreme, like the category of spaces that the student was talking about at all, until you come up with instructions that one normally makes in analyzing spaces. Because of this setup, you can see that I imagined that, well, at least this includes

1:00:00 Often have both left and right adjoints. That is to say that if you're given an extensive category, then you have to take it from the one end to extract part of it, and on the other end, sort of forge it into something that's lopping on the edge. And I claim that at least to a good first approximation, Those are two of the most common movements that we make in analyzing category spaces. Well, or at least, yes. I have two constructions, I have two constructions, which I don't know for sure that these are the adjuncts, but they are certainly in the flagrant. And one is the Eurewicz construction. So in this case, we assume that he is also Cartesian closed, as well as... Category and functor, in particular, lead to S, and S is really just thought of as the subcategory there, which is fixed under that and so on, or all equivalently fixed under the other one, as a category. So, when we speak about, when we write E of x, y, x and y objects would be the Ham set, Well, instead of the homsets, let's interpret it as an object of S, so I'll define this to mean gamma lower star of Y to the X. So this means that any time you have a, well, if you have a map of closed categories, that is a function that actually preserves products in this case, well, certainly in particular, the first one becomes enriched over the second one. So here's the formula for the, for the enrichment.

1:02:30 Sort of automatic. But Horowitz did the opposite thing. He said, let's define instead a square bracket column, which is the pi zero, or in other words, the gamma lower series of what is the base of E, and then we call it pi zero, and take this discrete object, whatever discrete means, and make the only partial of it, and take the discrete object and maximize it. This gives a different enrichment of the NMSW category. It does give an indifference because if we want to compose these things, how can we map that into e of x and z? Well, we just use the fact that, as I said, this is gamma-shreded to y to the x times gamma-shreded to z to the y. And now here's where the assumption that gamma-3 preserves product becomes crucial. This is gamma-3 times y to the x times z to the y. Now, many times you can go to the category where there's a canonical map which takes this kind of a product into z to the x. So applying the functor of gamma-3, you're going to... The term itself determines my position here because of the product-preserving nature of the components' puncture. Oh yes, to point out that this product-preserving feature of the components' puncture is almost never true for sheets on topological space. The actual meaning of that is that every topic can be covered by a locality, one in a touching way, with the inverse image-preserved logic and other stuff. But just in a straightforward sense, it cannot be so that as such a simple property a product reservation, which is typical of locality spaces, is not true for a locality.

1:05:00 But if you have a locally connected topological space, it means that you have a kind of, if it preserves products, then it's also less exact, and therefore it is a point, it is a privileged point, so that the space is actually irreducible, it's only irreducible space, in fact, that's the problem, irreducible space, one which is locally connected and for which the components are not very useful. These are products, hence it's being looked at exactly. And that is quite even palatable with another condition, which I'll discuss in a moment. Now, this Heravism instruction will give a category of qualities. There is, of course, a functor into this. A functor of S categories into this very rich construction. So it's basically an idea of a momentary type. This functor preserves exactly its sums, products, and even exponentiation. So this is again a Cartesian closed category, but not locally Cartesian. It has sums and products. It's both Cartesian closed and extensive. But it can't be extensive because it doesn't even have a whole lot. It has many, many more properties, and of course it is still an S category. Now, my condition is that is homotopy actually an extensive quality? Extensive could mean that there's an extensive intensive here in a way that's not clearly connected with the previous use, just the last time, about modes of variation of quantity. Perhaps they are more related than I can see at the moment, but at the present, I'm just taking this from Grassmann, I haven't been able to find any Grassmann scholar who understood, who understands what Grassmann meant, but he said that extensivity has to do with inequality, intensitivity, intensiveness has to do with equality.

1:07:30 What you have left over is things that are really very very different. Things that are irretrievably different. They can often be formed. They might be different, but they're not really really really different. What's left over after you have the homotopy type is things that are really different. So, in that way, not only should homotopy type be the quality... But maybe you can deserve to be up to that extensive quality, at least that helps to distinguish it from intensive quality, which I'll mention in a moment. Now, the Hayover-Weibach instruction does not always satisfy this condition of being determined by a sample of inner programs. Reflexive graphs, or indeed any category with finite onsets, finite topos, and in any such case the two, the green one or the two yellow ones, correct. Or, or, through either finite common, really finite common theory, or else continuous, you take the ordinary homotopic category of space and it's thought of as being based on continuous paths. Equation, because essentially the equivalence equation that's involved in forming the half-components is already represented by the integral. If you glue two integrals together, that's actually embeddable in the integral, and it's a topological. And so there is no true infinite process of building up the equivalence equation. It's already sort of self-transcending. But the pro-equalizing there, it is reflexive, but it actually preserves products. Infinite products. It's preserving infinite, okay. It's preserving infinite products in here, because I can say that this new pair of guidelines, okay, so you have Gavalor, Shriek, and Wei is...

1:10:00 The maps in the array which category from 1 to y. Because if you put x equal 1, the maps in the array which category from 1 into y is just the components of y. So you do have a components factor, namely, that becomes representative by 1. Totally not representative over there. And you have a left adjoint, of course, to the inclusion. Because of the street spaces, you know, have their own rules, right? Now, these, and of course, the points, the points functor has been completely wiped out, so there's no points functor as such. But there might be another functor, which is right adjoint in this new situation, to the discrete inclusion. And indeed, it even might be the same as Gamma-Street. And as I said, it will be so in either the finite, really finite, or the continuous case. But Witten noticed that in dealing with combinatorial approximations to the continuous, there's some kind of gap there, which meant that, in this case, one has to take a further category of fractions. In order to force this property to be true, if you start, let's say, with simplicial sets, this is explained to some extent in the book by Gabriel and Wiesmann. But I wanted to say, on the necessary and sufficient conditions, that the gamma lower street, which exists, and the gamma upper star, which exists, are adjoint the other way around. There are in which category the sense of quality means that there really will be a category of quality in the sense that I defined. This will be true if and only if the components culture preserves infinite powers in the sense that for any discrete space,

1:12:30 So, in addition to preserving finite products of two objects, the infinite discrete product, so to speak, really just means the exponential of the universal property, really, because anyway, that should be, and as I say, this will automatically be true if S is the category of finite sets, because eventually they're finite. To force that equation on the realist category in general by fractions or something like this, I don't understand exactly how that can be done, so I'm going to lay out some open problems to clarify that. Now, clearly the intensive quality then should sort of be on the other side. We will take those spaces for which? For which gamma-strict of B is isomorphic to... I'm sorry, I put it the other way. In other words, for other points, we'll map by adjoining these two components. Every point is then a definite component. So now, take those objects for which this is even an isomorphic. Every component contains a unique point, so to speak. Again, be careful about thinking that you really are adjoining the points, But clearly this category, at least the premise of this equation, is perhaps not clear that it has all the properties. Is it still a topos, for example, and is it still extensive and so on and so forth?

1:15:00 So I have some nice conditioning for which I can do a very good thing with this. I'm assuming he is a topos just in case he makes some construction and requires that. But then the condition I'm going to propose is what I call the Nullstellensatz. Now, the Nullstellensatz, it has two clauses. One is that General Oerstar, he maps epimorphically. This is certainly to favor the Nullstellensatz. There are plenty of points, you see. And not too much in the dynamic category. This is just at the level of S. This is the math in the lower category. And the other one is that, yes, I'm assuming that all four functors exist as well, and that, so we have the map from Jammu Upper Star, now this is in the category, not just in S, but the things that are in the special phases, and the speed one and the code speed one, so that's where you have them working, so I think these two clauses mean those seven signs. Question. It might well be that condition 1 implies condition 2. I'm not quite sure. But I will need both of these to make it work for a moment. And, oh, I should like to say, why do I... We should also have some picture of how this code of streetness works. It's like a more general example. So if I have a category C in S, and I look at the internal pre-sheets, Then I will always have the three functors, like this, but when do I have the amazing one, which is the algebra.

1:17:30 And this exists only if C has a terminal on it, of this linear importance. In which case the formula for it is, given that the speed space is S, then you have to appreciate the value of C is S to the power of the mass from 1 to C. That's the formula for the extra right angle in this case and so you can see that of course gamma upper star of s of c is just s all the time and so this map is a this is really just a sort of diagonal map where s is the rest of the circuit power for each c but if this is to be a monomer for each c You have to have that there exists a map from 1 to C. If there's not, you have a zero exponent, and so the exponential is 1, and this is not 1. So this is mono for all C's, if and only if for all C's there exists a map from 1 to C, which is certainly a ball and face type of Nostradamus. So, this deserves to be thought of as most of them, but whether the one implies the other, I don't know. What about the other position, one, the discreteness? Let's see, is it humanistic? Oh yeah, well this depends on the presentable, if this is some kind of quotient. No, it's still not. I mean, it depends on the topic you're discussing. I think I heard what you were going to say. You've had the same conditions.

1:20:00 Oh, really? Ah, good, good. That's good. Anyway, so I want to say something else about the important concept of the renunciation of thoughts before we come to the intensive quality. There's a place where this map is a biface, an isomorphic one, and so the idea is it's going to be easier to handle that subcategory if you're going to do the whole category just like this. Well, in fact, I'll just tell you in advance what happens. In this case, there is an essential connected surjection from E into E sub. This is an inclusion. In fact, it's a very reflective subcategory. With this, you end up with this number one. You show that these objects are closed under sub-objects, closed under products, and closed under products and sub-objects. We have an essential surjection, which assigns to each object its intensive quality. Each is sub-dotted by construction, a capital of quality, but there's some intensive quality of a particular space, because, well, we imagine this, these spaces, you see, are

1:22:30 All of these terms are of the nature that every component contains just one point, so they sort of look like this. These little blocks don't necessarily reduce the point, but there's only one point in them. Now, a typical example of this is an algebraic geometry. Interdecimal neighborhoods of points which are not necessarily rational, they don't necessarily come from one, but they're represented by field extensions of the base unit. So essentially the discrete things, discrete things, yes, so now the great geometry over field K, then what the S is, is essentially the atomic sheaves, really discrete value sheaves, that's not the S itself. On the category of finite field extension. So that, I think that deserves to be called the Galois topos, it's a Boolean topos, and so forth. And it does have a good property, the Hamas shriek does preserve tolerance, whereas going to sets it would not. But we're taking out only the global cohesion and thinking about it, so it's essentially the same thing as the Tom Mathur singularity of the space, you know, the space is defined by some equation and some information, throwing away the global cohesion to remember the catastrophe... Now the other remarks I'd like to make about an important document that the notes comes up is that one and two together imply that

1:25:00 It implies that every space is included into a connected space. Take an arbitrary space, look at its points. We take the points of the space, and we take the screen space for responding to that, but we also take the co-screen space for responding to that. So, since we've assumed that points are included in two, then we get a monomorphism here. On the other hand, this is the discrete core of x itself, the thing which Cantor claims exists, but no minimal is denied, right? So, this is that. So, if we take a push-out here, probably less depth of x, this is automatically an inclusion because push-out is an inclusion. To this diagram, we will get a push-out diagram. It's a left-hand joint, so it preserves push-outs, and of course the components of this space is just a star x. Now this space, I'm going to put one here. I'll explain that in a moment. That's the pi zero gamma-shriek of x maps into gamma-shriek of f of x.

1:27:30 This is assumed to be an epimorphism, but in all Stalin's eyes, this is still a push-off, so this is also an epimorphism. So, x is included in f of x, and yet the components of f of x is the image of the components of a protospeed space. It ought to be connected if it's not empty. Now more precisely, I'm just assuming that, I'm assuming that, but that's, I think that's an interesting question that I don't quite understand. You see, we have these four functors. Any one of them determines the whole thing. And the adjacent composites are the identity. The composite going around like this, it's a functor from S to S, which is product-preserving. See, the intermission in the simple case is that this ought to assign to every set simply the truth value, whether it's empty or not. It's an image. It's image is one. It's support. It should be a huge collapse, except for the empty case, when you get one. And therefore, when you get the specific one. But I don't really know whether perhaps much more general endorphin progress might occur by composing the outer two So this condition that a space should be embeddable into a connected space, by the way, in the topos, this immediately implies it's embeddable into a contractible space, not just a connected one. So in a way, this argument is a special case of that one, because we're saying that the cone of the street is connected to the two points of the street, and so in particular, every power set will be really just using that one little fragment here.

1:30:00 Of course, the points are discreet because when they come back down, they remain just two properties together, and the cone is going to preserve finite products, and every object can be embedded in a connected object. are incompatible for the locality of topology and geometry. So it shows that really we need to seriously develop the theory of categories of spaces when we need to consider other kinds of topology. I'd like to comment on my observation. The assertion that the code of trig space is a method for the case where E is a machine is actually unimapable within the field of that. It's the opposite of the field of that. It says that some object is unimapable under it. So that's the second clause, I think. Certainly the first clause is both of them. Well, I quite appreciate that. Both of them are equivalent. Okay, well, so we need non-Greece. Yes, yes, I mean, the example that I had in mind. So again, that was another Grecian.

1:32:30 Was it the outlook of the composite mind? No. They should be able to find a new harm, which is the components of the goal of the experiment. Just a small correction. Yes. When you consider the chemistry of machines of planned non-engineered satellite tests, it is classified as topos. Non-engineered. There are a number of different types of subjects, but they are all different from one another, and each has its own speciality. Thank you very much. At 11? It's 11 o'clock. I don't want to move the clock, that's fine. The speaker is John Bell, and the title of this talk is Descending Voices, the Virtual Conceptions of the Continuum, Minding, and the Early 20th Century in the 9th century. On this talk, although it's a topic I've actually long been interested in, it was felt that this meeting would be an appropriate place to present some account of the different views of the interview. I won't be able to, it's too long, I think, to do all of it, but I will select what I think they could be.

1:35:00 The opposition between continuity and discreteness is, as we know, animating the development of mathematics as activity, a tradition, in fact, defying mathematics as the science of discrete and continuous magnitude, when we find that there are definitions of that sort in literature. A striking example of this opposition, amounting, one might say, to a collision, is the Pythagorean discovery of incommensurable magnitudes. Here the realm of continuous geometric magnitudes resisted the Pythagorean attempt to reduce it to the discrete form of pure numbers. The theory of proportions later invented by Eudoxus to resolve the problem of commensurability was in essence an extension of the idea of number, that is, of the discrete, adequate to the task of expressing relations between continuous magnitudes. With the emergence of the differential and integral calculus. Here the controversy centered on the concept of infinitesimal. According to one school of thought, the infinitesimal was to be regarded as an actual, infinitely small, indivisible element of a continuum, similar to the atoms of Democritus, except that now their number was considered to be infinite. Calculation of areas and volumes, that is integration, was thought of as a summation of the infinite number of these infinitesimal elements. An area, for example, was taken to be the sum of the lines of which it is formed, which is what's called indivisible. So, we may say the continuous was, in some sense, again reduced to the discrete, but now with the intrusion of the concept of the infinite in a subtler and more complex way than the earlier attempts. Now, the conception of infinitesimals as actual entities was gradually displaced by the idea originally suggested, certainly by Newton. The infinitesimal is a continuous variable which becomes arbitrarily small. By the start of the 19th century, when the rigorous theory of lineage was for the purposes of being created, this new conception of infinitesimal gained general acceptance.

1:37:30 In general, a continuum such as the line was now understood not to consist of points or individuals, but to be the domain of values of a continuous variable. So we may say, I guess, at this stage, to be discreet and give it away to be continuous. But of course, the development of mathematical analysis in the latter half of the 19th century led mathematicians to demand greater precision in the theory of continuous variables, and above all in fixing the concept of real number as the value of an arbitrary set of variables. As we know, the 1870s saw the emergence of the modern arithmetico-theoretical conception of number, of real number, largely of course in the hands of dedicated categorists. The newly fashioned ordered field of real numbers became known as the arithmetical continuum because it was held that this number system is entirely adequate for the analytical representation of all types of continuous volume. In particular, a line, or the domain of values of a continuous variable, is represented as a set of distinct real numbers, identified as points. In this scheme of things, there was no place for the concept of integer testing, which accordingly departed the scene, well at least for a time. So we could say the continuous was once again reduced to an assemblage of set of discrete points. A reduction again of the continuance of the discrete. This last reduction, underpinned by the development of set theory, as we all know, becomes a reigning orthodoxy among mathematicians. Even so, the doctrine that the continuous is fully explicable in terms of the discrete has never remained unchallenged. Typically, the doctrines of moments accept that a continuum is an inexhaustible source of points, but deny that it can be reconstituted, so to speak. And here are a few examples. Aristotle himself says in Physics, No continuum can be made up of indivisibles as, for instance, a line out of points, granting that the line is continuous and the point indivisible. One of Leibniz's doctrines, of course, was that nature was continuous, that Turing on fire sold us, and Leibniz actually says that a point may not be a constitutive part of a line. Cobb, of course, also says in the critique that space and time upon their continuum, points against in severe positions, and out of mere positions viewed as constituents capable of being given prior to space and time, neither space nor time can be constructed.

1:40:00 Now, among philosophers of the late 19th and early 20th centuries, we can mention two, Bergson and Whitehead, both of whom emphasized the primordial nature of the phenomenon of continuity. It played a central role, of course, in Bergson's philosophy. However, there are six figures of this period. Dubon, Raymond, Veronese, Brechtano, Peirce, Wilde, and Brouwer. Who seemed to stand out as champions of the irreducibility of the continuum concept to discrepancy. Now, I'm going to talk a bit about all of them, to bring down the curse of vile, but who made significant contributions. Now, Dumont-Raymond embraced the idea of the actual infinite, but he rejected it. He was committed to the reduction of the continuum. There is no doubt that Pesce was the founder of the notion of a sharply defined uniform line from a series of points, however dense. Four, after all, points are devoid of size, and hence no matter how dense a series of points may be, it can never become infinite, which must always be regarded as the sum of intervals between them.

1:42:30 He was the one who famously said there are ignorant innocents, things we can never know. And of course it was Hilbert who later challenged this view. These solutions would somehow completely elude the understanding of mathematicians, thus seem to run in the family. I mean, Paul Dubois-Ramon also says the continuum has its sorrows. Nevertheless, this did not prevent Dubois-Ramon from developing his own theory of the mathematical continuum, his so-called calculus of infinities, during the 1870s and 80s. As well as offering an account of mathematical magnitude in his scheme for comparing and distinguishing actual infinite quantities, It also incorporates a theory of actual infinitesimals, something that a notion of infinitesimal that Dubois-Raymond actually had long championed. I think I'll skip the quotation there. Now, one of Dubois-Raymond's severest critics, as I'm sure some of you know, was Cantor. Cantor fought an unceasing battle against the country, and more generally, Against the idea that the continuous was in some essential way irreducible to the continuum, while Dubois-Reymond did not hesitate to employ geometric and visual intuition wherever he felt it necessary. Kandor, of course, by origin was not a theorist, was naturally inclined to the discrete, as his work from Analysis of Discontinuities to Set Theory shows. Kandor's fixtures against the work of the next member of our section, Giuseppe Veronese, were, if anything, even more virulent. I'm sure we need an audience of a number of Italian mathematicians who are amazing. Anyway, let me say, he was an outstanding member of the Italian school of geometry in the last quarter of the 19th century.

1:45:00 In 1891, he published his exhaustive work on the foundations of geometry, whose title in the proximate English translation reads, In this work, Veronese develops n-dimensional projective geometry, including non-Euclidean geometries, from first principles in a synthetic and unified way. Controversially, he also introduces non-Archimedean geometries, containing both infinitesimal and infinitely large segments. On publication, this work attracted the scathing criticism not only of Camdor, but also of Piano and Killing. Yet Hilbert later called it profound. And he incorporated some of Veronese's ideas into his own way of unlocking their geometries. As a geometer, Veronese naturally took an essentially geometric view of the continuum. He begins his foundations with a complaint about the use of real numbers as the basis of geometry. Spatial intuition, he says, is what furnishes us with the basal geometric objects and their inherent properties. And he goes on to say, this synthetic approach always treats figures as figures, works directly with the elements of the figures, and separates and unites them so that each truth and each step of a proof is accompanied as far as possible by intuition. In answer to the question of what is the continuum, Veronese writes, this is a word whose meaning we understand without any mathematical definition. Since we intuit the continuum in its simplest form as the common characteristic of many concrete things, such as, for example, to give some of the simplest, the time and the place occupied by the external neighborhood of the object sketched here, or by a plumb line if one takes no account of its physical properties and distinctions in the interior of the sense, for Veronese, points are nothing more than signs indicating positions of the uniting of two parts in the rectilinear continuum. They are, as for Aristotle, a product of mental abstraction, not parts of a rectilinear object. Veronese in contrasts his own account of the continuum with that of Cantor and Dedican in the following words, and I think it's worth quoting this.

1:47:30 He says, Cantor and Dedican assert in their valuable works that the one-one relation between the points of the line and the points forming the real continuum is arbitrary. They certainly obtain the continuum by means of a sequence of abstract definitions of symbols which, although possible, are arbitrary. According to Dedican, the numerical continuum is necessary in order to clarify the idea of the continuum of space. According to us, however, it is the intuitive rectilinear continuum, which by means of a point without parts, that serves to give us abstract definitions with respect to the continuum itself, of which the numerical continuum is only a special case. In this way, the definitions appear not as the force which keeps our mind in check, but finds its complete justification in the perceptual representation of the continuum. One must take some account of this representation in the discussion of basic concepts, but without leaving the field of pure mathematics. Moreover, it would be truly marvelous if an abstract form as complicated as the numerical continuum obtained not only without being guided, Without being guided by the intuitive, but, as is done nowadays by some editors, from near definitions of symbols, the fact can then find itself in agreement with a representation of simple and primitive is not a correct way to continue. Well, of course, the emphasis of all of these on intuition is something to make this. And also, of course, there was the corresponding work also of Frege, who at about the same time, Now, I do want to devote a pretty well, quite fully worked out theory. You may say that Brentano, the philosopher most concerned, a major portion of his, he was very much a critic at the beginning of the 20th century writings.

1:50:00 Without Brentano's philosophy, it's worth noting parenthetically that Brentano was the teacher of both Rousseau and Meinheim, and Brentano also introduces the idea of intentionality. But Grandona's philosophy has its starting point in the continuum doctrine, and its conception of the continuum constitutes no exception. Aristotle's theory of the continuum, as you may recall, rests upon the assumption that all change is continuous, and that continuous variation of quality, quantity, and position are inherent features of perception and intuition. Aristotle considered itself evident that the continuum can't consist of any pair of unextended points he observes are such that they either touch or are totally separated. In the first case, they yield just a singular, unextended point. In the second, there is a definite gap between them. Aristotle held that any continuum, a continuous path, say, or temporal duration, or motion, may be divided ad infinitum into other continuums, but not into what might be called discrepant parts that cannot themselves be further subdivided. Accordingly, paths may be divided into shorter paths, but not into un-extended points, durations into greater durations, but not into un-extended instants, motions into smaller motions, but not into un-extended pulsations. Nevertheless, this doesn't prevent a continuous line from being divided at a point, constituting the common border of the line segments.

1:52:30 But such points are, according to Aristotle, just boundaries, and not to be regarded as actual parts of the continuum from which they spring. If two continuae have a common boundary, that common border unites them into a single continuum. Such boundaries exist only potentially. Deciding potential exists as potentiality is an important component of Aristotle's philosophy. Boundaries exist only potentially since they come into being when they are, so to speak, marked out as connecting parts of a continuum, and the parts of their turn are similarly dependent as parts upon the existence. Now in its fundamentals, Bertano's theory of the continuum, his scenicology, is akin to Aristotle's. Bertano regards continuity as an essentially perceptual phenomenon rather than as a mathematical construction. Indeed, Brentano took a somewhat dim view of the efforts of mathematicians to arithmetize the continuum, that is, to construct it from point A. His attitude seems to have varied from rejecting these, the mathematicians' constructions of the essence of the continuum, as absurd to interpret. Well, this isn't surprising given Brentano's Aristotelian inclination for physical theories to be genuine descriptions rather than idealizations. Certainly if such theories were to be taken as literal descriptions of experience, they would amount to nothing more than What he called misrepresentation. Indeed, Bentano actually wrote, we must ask those who say that the continuum ultimately consists of points what they mean by a point. Many reply that a point is a cut which divides the continuum into two parts. The answer to this is that a cut cannot be called a thing and therefore cannot be a presentation of the script.

1:55:00 There are only presentations of contiguous parts. The spatial point cannot exist or be conceived of in isolation. It is just as necessary for it to belong to a spatial continuum as for the moment of time to belong to a temporal continuum. Bertano actually held that the idea of a continuous is derived from primitive sensible intuition, in fact he called it a kind of unitary intuition. He says, the concept of a continuous is acquired not through combinations of marks taken from different intuitions and experiences, but through abstraction from unitary intuitions. Every single one of our intuitions, both those of outer perception as also their accompaniments of inner perception, and therefore also those of memory, bring to appearance what is the continuous. He thought it was a primary form of intuition. He's done a highly elaborated theory of the continuous. He suggests that the continuous is brought to appearance by sensible intuition in three phases, he said. From such objects, the concept of boundary is abstracted in turn, and then one grasps that these objects actually contain coincident boundaries. Finally, one sees that this is all that is required in order to grasp the concept of continuum. Continuity, of course, is manifested in sensation in a variety of ways. If visual sensation was presented with extension, something possessing length and breadth between any two of its parts, provided these be separated, Every sensation possesses a certain qualitative continuity in that the object presented in the sensation could have a given manifested quality color, for example, in a greater or less degree, and between any two degrees of quality lies still another degree of that quality. Finally, each sensation manifests a temporal continuity. This is most evident when we perceive something as moving. For a total regularity, which caused them to possess multiplicity, a continuum may manifest in several ways simultaneously.

1:57:30 This actually led him to classify a continuum into primary and secondary. A secondary continuum would be summarized as a continuum. It would be represented rather nicely as a domain, and a secondary continuum as a co-domain. Continuous objects and correlations between continua and arrow. Then, given any arrow, let's say, and its co-domain B is a secondary continuum. In Brentano's examples, which I haven't actually read out here because I really don't have time to read all the quotations, The primary continuum in that case is the given spatial surface. The secondary continuum, B, is the color spectrum, and the correlation after signs to each place in A is color as a position in B. In the case of a corporeal point moving in space, which is another of its examples, the primary continuum A is the interval of time. The secondary continuum B is the region of space, and the correlation F assigns to each instance in A the position in B occupied by the corporeal point. Finally, in the case of the varying direction of a curve, which is not alone inconsiderate, the primary continuum A is the curve itself, the secondary continuum is the continuum of measures and angles, and the correlation F assigns to each point on the curve the slope of the tangent there. Thus, F is nothing other than the first derivative of the function associated with the curve. Well, for Brentano, the essential feature is its inherent capacity to engender boundaries. The fact that such boundaries can be grasped is coincidental. Boundaries themselves possess a quality which Brentano calls pleurosis.

2:00:00 Pleurosis is the measure of the number of directions in which the given boundary actually bounds. Thus, for example, within a temporal continuum, the end point of a past episode or the starting point of a future one bounds in the same direction, while the point marking the end of one episode and the beginning of another may be set to bound up. In the case of a spatial continuum, there are numerous additional possibilities, of course. Here, a boundary may bound in all the directions in which it is capable of bounding, or it may bound in only some of those directions. The boundary is said to exist in full pleurosis, is the terminology he uses, in the latter in what he calls partial pleurosis. Grantano actually believed that the concept of pleurosis enabled sense to be made in the idea that the boundary possesses parts, even when the boundary lacks dimensions altogether. Thus, while the present or now is, according to Brentano, temporarily unextended and exists only as a boundary between past and future, it still possesses two parts, so to speak, or aspects. It is both the end of the past and the beginning of the future. It's worth mentioning that for Brentano, it was not just the now that existed only as a boundary. Since he held that existence in a strict sense means existence now, this was also an important constituent of his general philosophy. So it necessarily holds that existing things exist only as pathways of what has existed, or of what will exist, or both. We continue to center it on its phenomenological and qualitative aspects, which are by their very nature incapable of deduction. Brentano's rejection of the mathematician's attempt to arithmeticize the continuum, that is, to represent it in discrete terms, thus highly surprised him. He might, on the other hand, have been more sympathetic to the accounts of the continuum put forward by the other members of our sextet, all of whom, as I have observed, took the continuous as something necessarily transcendent or discreet. Well, now we come to Peirce. Peirce's account of the continuum is published in sort of fragmentary. He wrote a number of fragments on the issue.

2:02:30 But his accounts of the continuum, as I've been through the test pools, his sort of musings on these matters are really led to scatter. Anyway, was in some sense intermediate between that of Brentano and the arithmetic. It's sort of in between in some sense. I mean, like Brentano, he held that the cohesiveness of the continuum rules out the possibility of it being a mere collection. On the other hand, he also held that the continuum harbors an unboundedly large collection of points. In his colourful terminology, a super-multitudinous such collection, we probably call it a proper class now or something, but anyway, yes, a problem. Purcell, that if enough points were to be crowded together by a carrying insertion of this ultimate limit, they would, through a kind of logical transformation of quantity into quality, lose their individual identity and become fused into a true continuum. Here are his observations on the matter. Well, one of them, he says, well, I don't mean to quote that. Yes, all right, I'll just read the end of this. Supposing a line could be a super-multitudinous collection of points. To sever a line in the middle is to disrupt the logical identity of the point there, and to make it two points. It is impossible to sever a continuum by separating the connections of the points, for the points only exist by virtue. Anyway, to sever a continuum is to burst it, to convert what was one into two. Now, I just remarked, parenthetically, there's some resemblance, and John Landry also made this point to me, resemblance actually between Gers's conception of a continuous line and Conway's surreal numbers, when John Conway developed this sort of universal theory of numbers, which happened to be called surreal numbers. Conway's system may be characterized as being A certain type of field, an e to alpha ordinal field, for every ordinal alpha, that is, a real closed ordinal field, S, which satisfies the condition that for any pair of subsets, x and y, for which every member of x is less than every member of y, there is something strictly in between.

2:05:00 And it's not hard to show that in that case, between any pair of members of S, there's a proper class of members of S. And Peirce's terminology is super-multitudinous collection. Nevertheless, S is still discrete. Its elements, by super-multitudinous, remain distinct and unfused, but of course, if it weren't for that fact, Conway would hardly be justified in calling the members of S numbers. On the face of it, the discreteness of S would seem to imply that the presence of super-abundant quantity in Peirce's sense, which is what Peirce actually maintained, is not enough to ensure common unity. He never really finally makes the connection between this super-multitudinous collection of points and the actual continuum from which these points are extracted. There have been various attempts, as you may know, to find ways of modeling or discussing Peirce's ideas in modern terms. I think that Hilary Putnam and, I don't remember who the other person was, Kettner, and Putnam characterized Peirce's conception of the continuum. There's a possibility of repeated division which could never be exhausted in any possible world and then of course they they need to go on to try to compare this with non-standard analysis but I don't I don't really think non-standard analysis is the right way of thinking about a person. But anyway, and it should also be said that Grant Conner himself, although might have been a bit more sympathetic to him, I don't know actually if he knew any of those words, I haven't looked into that, but anyway he certainly would have said it was his idea of constructing a world. But interestingly, Peirce's conception of the number continuum is also notable, with the presence of an abundance of infinitesimals. He was a great champion, of course, of the idea of infinitesimals. A feature it actually shares, both with the Jouvardian laws and Veronese, is the non-Archimedean system. Again, I don't know whether Peirce was aware of this. Peirce championed the retention of the infinitesimals in the foundations of calculus, both because, as the efficiency of infinitesimals, he regarded infinitesimals as furnishing the glue.

2:07:30 The very word continuity implies that the instants of time, or the points of a line, are everywhere welded together. In defending it through the decimals, he said, It is singular that nobody objects to the square root of minus one as involving any contradiction, nor since cantor infinitely great quantities much objected to, but still the antique prejudice against infinitely small quantities remains. Kearse actually held the view that the conception of an infinitesimal is suggested by introspection. I mean, he also here and there made this connection, you know, between the preceding, the law of time, a kind of carnary, subjective, if you like, linear intuition. And the idea of infinitesimals as something emergent, this, in fact, re-identifies the present, the now, as an infinitesimal. I think an apropos of this occasion is a quote from a letter addressed by Peirce in 1900 to the editor of Science, Journal of the Day, in which he defends his views on infinitesimals against the strictures of Josiah Royce, who is, of course, one of the major art luminaries of the... He says, Professor Royce remarks that my opinion that differentials may quite logically be considered as true infinitesimals, if we like, is shared by no mathematician outside of Italy. As a logician, I am more confident by corroboration in the clear mental atmosphere of Italy than I could be by any second inkling from a tobacco clouded and bemused land, if any such there be, where no philosophical eccentricity misses its champion, where sane logic has not found faith. I must read these carefully. This is actually material, some material that I have written before about Weyl.

2:10:00 So I'll just extract what I think is really appropriate. Of course, we know Weyl's philosophical outlook was certainly influenced by Brentano. He mentions Brentano here and there in his writing. But of course, the primary philosophical influence was on Weyl. Of course, it was Husserl. Weyl adopted the principal tenet of Husserlian phenomenology that the only things which are directly given to us, that we can know completely, are objects of consciousness. This is a view he held. I don't know whether he, he certainly deviated from phenomenology later on in his life and I don't think there has been really of the development of his own philosophical thinking, but it certainly was not any more. He did a great deal of creation free of the taint of the actual infinite. There were several things that he was trying to do. One of them, kind of a, he accepted the potential infinite, but really not the actual infinite. The Cantorian reduction. In 1910, he's quite serious about Cantorian combinations of the continuum, and then at some point after that he decides, and in fact it's really not acceptable. Anyway, as he saw it in 1918, there's an unoriginal gap between the intuitively given continuum, on the one hand, and the discrete exact concepts of mathematics. e.g. you've got a real number on the other. The presence of this split meant that the construction of the mathematical continuum could not simply be read off in intuition.

2:12:30 At least that appears to be his... The mathematical continuum must be treated as if it were an element of the transcendent realm. I mean, that is something that's not directly presented to intuition. I mean, this is the way you follow physics too. I mean, physics also... He also thought that was the reason why mathematics really had to transcend the intuitive together, really because he thought that was necessary for physics. Well, so in other words, the mathematical theory in the end had to be justified in something like the same way as the physical theory. In Weyl's view, it was not enough that the mathematical theory be consistent, it must also be reasonable. It does continue, which of course finally was translated into English. I'm sure it was translated into Italian long before it was translated into English. It was translated into English about ten years ago. It embodies Weyl's attempt to formulate the theory of the continuum which satisfies the first, that is, consistency, also as far as possible the second of the requirements. In the following passage, he acknowledges that the conceptual world of mathematics is so foreign to what the intuitive continuum presents to us, that the demand for coincidence between the two must be dismissed as absurd. The continuity given to us immediately by intuition in the flow of time and motion has yet to be grasped mathematically at the totality of discrete stages in accordance with that part of its content, which can be conceptualized in an exact way. He also says, exact time or space points are not the underlying atomic elements of the duration or extension given to us in experience. On the contrary, only reason which thoroughly penetrates what is experientially given is able to grasp it. And only in the arithmetical analytic concept of the real number, belonging to the two formal spheres, can we find the real number.

2:15:00 So, in Das Continuum, he was trying to give a kind of secure foundation for the idea of a real number. On the other hand, he did not claim that the arithmetic continuum is a construct of unbridgeable chasm between them. Anyway, however much he named it so, in Das Continuum, Weil did not aim, as I said, to provide a mathematical formulation of the continuum as it is presented to intuition. Rather, his goal was first to achieve consistency by putting the arithmetical notion of real number on a firm wall, and then show that the resulting theory is reasonable by employing it in the foundation for a plausible account of continuous process in the objective physical world. Actually, it's worth mentioning here that, of course, at the same time he was writing this, he was also writing space-time matter. And you can see that there are really quite subtle affinities between the two works. Anyway, if I were to come to believe that the official mathematical analysis at the start of the century couldn't stand up to the mythological scrutiny, then he says that every cell of this mighty organism is permeated by contradiction. So in Das Continuum he tries to overcome this by providing analysis with a predictive formulation. Of course, he was one of the earliest... Parker too, of course, was concerned with that kind of problem of providing a... he saw that the set theoretic encounter would really involve what he felt regarded as logical circles, possibly ones that would lead to contradiction, as, of course, also did Russell. He seems to regard this as too complicated. He says, rather, he defines the comprehension principle to formulas where bound variables range over just the given initial entity, so he tries to ground the whole thing in numbers, in natural numbers, which he takes as being in some way intuitively given.

2:17:30 On the relationship between the two. And he asks himself the question, does the mathematical framework that he's actually provided already in Das Continuum, does it really provide an adequate representation of physical or temporal continuity as it is actually experienced? He considers this to be an important problem, or at least he sees it as being within his own briefer, his own amulet. Now, I don't have time here to give you all what he says, but he gives various examples to show that somehow the account that he's given of the Archimedean continuum really can't do justice to the phenomenological continuum, if you like to put it that way, the phenomenal continuum, because he actually focuses attention, directly, on what he calls the continuum of immediately given phenomenal time. That is, as he characterizes it, to that constant form of my experiences and consciousness by virtue of which they appear to me to flow by successively. By experiences I mean what I experience exactly as I experience it. I do not mean real psychic or even physical processes, which occur in a definite psychic somatic individual, belong to a real world, and perhaps correspond with the direct experience. So he goes on and says, That really, having given him a definition, so to speak, or some kind of characterization of what he means by intuitively, medially experienced time, he then goes on to show, essentially, that they count no account of the type of keys-giving of the continuum is going to do justice to this notion. Well, I mean, since I don't have a lot of time, I'll just say it. Vital actually comes to embrace, as I'm sure most of you know, Brouwer's views on the intuitive continuum. He was very impressed with Brouwer's account of the continuum and with the whole intuitionistic program. And as you may also know, he did make attempts to try in some way of, you know, of overcoming the opposition between formalism and intuitionism.

2:20:00 The fact that Weyl and Brouwer had become quite close in that respect, really along Hilbert, as you may know, since he thought Weyl was being infected with this Brouwerian lunacity, and It wasn't until later that Wilde thought a lot about how to reconcile the two, or at least to show that there were two aspects, if you like, of the whole problem of the foundations of mathematics, and he wanted to effect some kind of synthesis of the two. And again, there is devoted to this problem about, or this discussion, if you like, idea of, of course he actually rejects it, he sees, or he sees Brouwer's account of the continuum as going much further down the road to providing what is a kind of, if you like, a rigorous or mathematical account of the continuum, which is actually closer to intuition. And this was a very important consideration to him. Now, of course, on this side, he'd come to repudiate animistic theories of the continuum, including the one he'd actually given in his own work. He'd actually sort of moved away from the theory that he presented in Dawson's continuum. So, of course, he didn't actually welcome Brauer's construction of the continuum. We don't have time for that. All right, well, in that case, let me just turn to that final student, someone you could call one of the most dissonant voices. A lot to say about Brauer, but since it's a natural... Of course, in Brauer's philosophy, it's the temporal continuum which plays awareness of the temporal continuum in the subject, or, if you like, the primordial intuition that engenders the fundamental concepts and methods of mathematics. In mathematics, science, and language, for example, in 1929...

2:22:30 He describes how the notion of number, how the discrete emerges from the awareness of the continuous. He thought the continuous was primary. Discrete is some kind of abstraction or at least something extracted that leads to the continuous. He says, mathematical attention is an act of the will that serves the instinct for self-preservation of individual man. It comes into being in two phases, time awareness and causal attention. The first phase is nothing but the fundamental intellectual phenomena of the falling apart of a moment in life into two qualitatively different things of which one is experienced as giving away to the other and yet is retained by the act of memory. At the same time, this split moment of life is separated from the ego and moved into a world of its own, the world of perception. Temporal tuity, born from this time awareness, or the two-membered sequence of time phenomena, can itself again be taken as one of the elements of the new tuity, so creating temporal free thought. In this way, by means of the self-unfolding of the fundamental phenomena of the intellect, a time sequence of phenomena is created by arbitrary multiplicity. On the other hand, in his doctoral dissertation in 1907, he regards continuity and discreteness as complementary notions. Neither of which is reducible to the other. There does seem to have been a shift in his account, as far as one can tell. He says, we shall go further into the basic intuition of mathematics in that every intellectual activity has a substrate of domestic or quality of any perception or of change, a unity and continuity of discreteness, a possibility of thinking together several entities connected by and between, which is never exhausted by the insertion of new entities.

2:25:00 Since continuity and discreteness occur as inseparable complements, both having equal rights and being equally clear, it is impossible to avoid regarding each one of them as a primitive entity. Having recognized the intuition of continuity, of fluidity, is as primitive as that of several things conceived as forming together a unit. The latter being at the basis of every mathematical construction. We are able to state properties of the continuum as a matrix of points to be thought of as a whole, so he's trying to see some kind of synthesis, really, of these two opposed notions. In that work, though, Brouwer does state quite categorically that the continuum is not constructible through discrete points, at least in the usual sense. He says, the continuum as a whole is given to us by intuition. A construction for it, an action which would create from mathematical intuition all its points as individuals, is inconceivable and impossible. Now later, again, as I said, Barb, well, he actually modified this doctrine, and you can say, really, in his mature thought, he radically transformed the notion of a point. The endowed points with a delightful sufficient fluidity, this fluidity was achieved by fitting its points not only fully defined discrete numbers, such as the square root of 2, which have, so to speak, already achieved B, but also numbers which are in a perpetual state of becoming. In that the entries of their decimal and dyadic expansions are the results of three acts of choice by a subject operating throughout an indefinitely extended time. The resulting choice sequence can't be conceived as a finished, completed object at any moment only in an initial cycle is known. In this way, Bauer retained the mathematical continuum in a way compatible with his belief in the primordial intuition of time. That is, as an unfinished, indeed, unfinishable entity in a perpetual state of growth. I don't doubt that Brentano would have found Brouwer's account of the continuum considerably more congenial than that of the Aristoteles.

2:27:30 My question also sort of builds this morning. The question is, at which extent the one-dimensional curve with points is a default model for thinking about at the end, historically? Actually, I can think about it as one exception. Aristotle also mentions it, you know, just as a parallel. You mentioned, it's an idea that comes down to an extra chorus about Gourmet Merits, sort of myriology, sort of pointless continuum, and that was also discussed then by Similicius and, like, in this Nevsky myriology, which people separate, it was also... They have a really strong example of case-territorial procedure of reduction of quantita that doesn't work. Yes, well, it's true. If I were to... This was supposed to be... Yes, I think... Also, of course, the later, certainly Blastewski, and in other words, it really is a kind of this dissenting tradition in Mariology, which is still here. I mean, Mariology in Italy...

2:30:00 They may not be primarily concerned, if you like, with the continuum, mathematical continuum as such, but clearly it seems to be sort of implicit in genealogy that it would resist the idea of reduction. Parts of the continuum are still... Take the example that Tarski used. He took the regular open sets as an animal's fluid algebra and thought of it as some kind of model of the physical energy of the equation. I don't think there's really such a good model. I mean, a more interesting one might be measurable sets, modulo sets, and measure zero, because there is a sense of emotion there of equality of size, unless you could say two measurable sets have the same measure, so maybe an axiom of equality of size might be interesting. So it's sort of a question of... Composing the arithmetic structure from above. Now, such a completely different point that I learned from Daniel Martin's beautiful book, Science Awakening, is that in Greek times, the success of Marx's theory of proportions was not due to dedication. But proportions allow you to compare ratios of magnitudes of different kinds. So types are different from areas, but you can still say that the ratio between And so, I mean, I think that the idea that you can compare things of different dimensions It's still a good point that we should keep in mind.

2:32:30 Oh, absolutely. I mean, in the Doxon case, of course, also the ratios are some kind of universal, you know, they're just presented, the theory that was presented, and it's simply laid down, right, that any two magnet strategies are the same kind. These are universal. It's a very striking thing. I mean, it's true, but usually, often in the accounts, that point is insufficiently emphasized. I agree. I wanted to ask you about what the system-in-dust continuum looks like. It takes a kind of classical view of the natural analysis. It's only the power set of the natural. Look at that from a modern point of view. He does take them as given. I mean, he doesn't, he doesn't, as far as I remember, he doesn't, he doesn't dwell on the source of these. I mean, I think you're right in a way, that is, for all intents and purposes, he's just got some kind of, he's given as much as he can for us. But I, on the other hand, I think probably, if pressed, he probably would have... Felt that as for Poincare, the natural numbers were a sort of primitive starting point. Certainly Poincare, and we know Lyell, certainly, I'm not sure if that point is made with any very much emphasis in Delft's continuum, but it's sort of implicit in his whole outlook towards open foundations of mathematics. In other words, this is why he would have been opposed, I think quite opposed, to example, to the Freudian program. And he didn't see the international numbers, I think, as sort of logical entities, as something that comes from some kind of a more linear issue in some way.

2:35:00 All of this does enter into Weill's thinking, but I think it does continue that he's trying to do, as I say, quite a few things at once. But what I mean is no account of that sort could possibly do justice to the intuitive continuum. I mean, at least that seems to be good. And that's why when Bauer comes along, he does something really startling in the original. Bauer's theory, he recommended De Waal, a much more faithful, if you like, of the continuum, certainly at the intuitive level, which was something very concerning. Bauer sort of took the natural numbers in granted. Yes. Well, they come out of the period of time, you know, the pituitary. It's the repetition. He gets the numbers from the continuous, you know, from continuous flow. Well, essentially from boundaries again, you know, like putting boundaries into it. One can distinguish between various kinds of continuity. For example, one in the power set or the collections of all sequences of zero and one, and the other is divine. Of course, there's a relation between the two, but these are two kinds of continuity. But it seems to me that all the problems that arise when one wants to study one of them, arise for the other two.

2:37:30 Somehow, although they are quite different, it seems to me that the fundamental problems that one faces in analyzing one are also there for the other. What is your view of this? Are they essentially different? Or do you consider that the continuum is totally disconnected? Yes, that's it. So, possibly one should not consider that a continuum since it is totally disconnected. However, it is a concept, too. So the continuum seems to be, I mean, is it really necessary that the continuum should be connected? Well, that's a good question. I mean, I didn't come to that issue because there was a bit of a notice that I left out. While thought the continuum was not only connected, it was actually in decomposal. I mean, there's a famous quote, he says, like he quotes Anaxanderus, he says, he says, let me see if I can find the quotation here, he says, he says, oh yes, he says, yes, yeah, okay, he says, he says, a genuine, a genuine continuum declares vital. Cannot be divided into separate fragments. In later publications, he expresses this more colorfully by quoting Haneke Sangeres for the effect that a continuum defies the chopping off of its parts as with a hatchet. Of course, because, you know, I mean, you don't, this is a problem of what happens at the bottom. Aristotle maintained something like the same thing. We all know Bauer, of course, did too. I mean, Bauer was looking for, in his model, continuums.

2:40:00 That it actually had this extreme cohesiveness, that it's actually a decomposable. So I don't know the view of, certainly of the, of Vial Engelhardt was that it's actually very cohesive. I mean that you can't, that you can't chop it up into pieces at all and reassemble those pieces, so to speak, to produce the original continuum. Look, I mean, yes, in that respect, I guess, maybe your first comment, but yeah, I mean, these various representations of the continuum sort of all suffered. Right, for the fact, not only, you know, the actions of the disconnected, the distributed, the very, very far away, string codes, and, and, and, I don't necessarily, you've already, or, or, or a person necessarily, I've had you, but certainly by the practice of, of, of problems, of, of, of problems, of problems, of problems, of problems, of problems, of problems, of problems, of problems, of problems, of problems, I mean, now, as you know, of course, if you do this, make some of these constructions of the right kind of topics, you can bring these sort of elemental constructions closer in the way to intuition. I mean, for example, if you take a model of synthetic differential geometry. You know, smooth travels. And you, of course, I mean, of course, the thing is designed with the usual, the real line there.