FW Lawvere / Discussions, incl. M Wright & A Peruzzi

0:00 1, 2, 3, 1, 2, 3, testing, testing. We're in Como on the morning of July 1990.

2:30 Let's go here, yeah. Simple construction is then possible, justified by the problem, because that's what this relationship says to us, not being the smallest, but the kind, intuitively, that goes back to the boundary of B, the intersection of B and self, and the continuity of all that.

5:00 Now, actually, this paradox is exactly correct. Classical is a classical example of a topological space. And in that case, this is at the boundary of E, so that's this part here and this part here. So some of you may know this . It seems as though topology is pathological, but this is very simple. Maybe it's true for this picture.

7:30 The first looks too simple. Some of these are irregular before, rather than very controlled, and then put down. An observation of Walter Morrow, I think, is that he made a remark in the context of attempting to define, in a simple, perceptible way, what are the sub-bodies of a human body.

10:00 But elementally, one could also find a unifying category of switches, and it should be pictured as an example, if you look at it somewhat like this, an object B might have a substantial part, and then some here, just of all that, but then they ought not be just a substantial part. So if we're looking at pictures like that, another... Which is that any, any algebras is a deep and very complicated interaction between these terms as all these are so close that some objects can never be understood. Some objects are made out of the magic of all things.

12:30 I think some people, philosophers, the discussion actually started across the state today, In a way, the most basic lemma on the track about topoglyphism, which here may not be true, is even true in greater generality than topoglyphism, perhaps. But the basic point is that the Heidegger structure for omega is not a code, it's a theorem. So the idea that preserved by going back, preserved in a simple sense, implies the hiding area of some object is collapsed. So, therefore, it is already known, no place for topos in topology, no hiding area for your topos in quantum science.

15:00 The story about a white cat that might be flying and stuff like that, it's possible we're thinking about it, but in modern terminology, what's happening is that we have a lack of that in the spirit. About necessity, let me just say that this is not considered as a necessity. Necessity means something which is not a veterinary object. It's a Western identity. It's a true-to-true. There are natural necessities out there. Omni, on the other hand, there are natural ones that exist.

17:30 So, on the other, General Margaret's direction, which has already been considered by John Boehner, and mostly many other people, complicated in which context, very often this laxity is not out of control. In the sense that the operators, if they only lagged in the control for three cents, there's some hope of controlling them as far as we're concerned. So, a very important example of a co-height problem is that the lattice of a co-height problem is an edge, the boundary, across the model. I can just illustrate, in a very, almost degenerate case, how to make it into a finite category.

20:00 The argument that all subcomposites are essential, in fact, all subcomposites are in this form, so this algebra of subcomposites, which in principle, in this particular case, is also what it consists of, but it's indexed by this, which would mean A is finite number, but it's not true that we can see any finite number, so that particular case of a subcomposite does not consist of all A times. Which is better than not, which is not interesting, not just because of the level of time and parameters that are placed there.

22:30 So, really what I'd like to do would be to calculate the nature of this more continuous going case. I have to learn to hang on to the prejudice of considering hiding algorithms which are so special that you don't want to look at them. In particular, mind that it is a subaltern with an object and a category. Subtoposism is a three-sheet category of numbers, S and ZI, and so you take XZ, the object of the subtopos, and you put a three-sheet in front of it, and that's how you come to C. But I'm saying that the subobjects of course I can't compute it pointwise, therefore I need sets. Or if you want to perform any of this, this is all the earthwork. So, this is a mantle of naturalities, or what lacks naturalities, substituting them together.

25:00 If we sum all these up, let me say, first, looking back, precisely, substituting terms into disjunctive forms, that doesn't reduce the pathway. These are not equal, if that's wrong or not, you can figure it out in a few minutes. There's always been a problem under it, and a problem. In fact, this is all very concrete. So let me just say what not-a is, what not-a is. So, not-a, on the one hand, it's union with they has to be that. On the other hand, it has to be the sub-particle. So you just take the sub-particle under it.

27:30 In other words, this is the function's value of speed, the sum of all x, the sum of that of the small u, and there exists an x-bar, which is an x-d. The x-bar is simply not a sensor. And of course, this is actually slightly more general than the projective topos, and the timing continues to be exciting to start with on the subalphas, on the other hand, for most topos. So again, the co-line instruction exists mostly for each max. On the other hand, we've got an account front, an operator on omega, not a signal map of omega, The naturality of the special case, as I said before, is more abstract, but since I can just verify, it does not mean that there can be a theory, a natural law, etc., which is the Cartesian product of the base and the height.

30:00 It's clear, geometrically, that the surface, the tip required, is the boundary of the Cartesian product. Now, the fact that there are two ends is subsumed under the fact that, in this picture, the first boundary of H is just a discrete point space. So, there is this formula, which is sometimes true and sometimes not true, on topology, on spaces.

32:30 But in difference to what I wrote down before, this one involves Cartesian product, not perception. And it's always true, if you're writing algebra, in particular it's always true for the sub-objects of an object in an appreciative category. But this one, as it turns out, is sometimes true but not always true in an appreciative context. So, really, the function product is a special case intersection. Here, of course, we're imagining that v is inside some larger space x, say, the plane, and h is inside some larger space y, say, this example, the line. So, it's really an intersection. v cross intersection h cross x, or the intersection is taken. This formula shouldn't follow from the formula for intersection, and a similar remark about it itself is preserved by substitutional logic projection. This underlies something that I've preached for years and long back preached for years and so on, namely thinking now about formal logic.

35:00 A formula is not really meaningful unless you specify what the three variables are. You change the three variables, it may very well change. This may change the meaning of the formula a lot, especially with regard to various logical operators that you might have thought, such as modal operators, negate. So, how come it's not on the other hand? On the projections, you want to join the moral variables. The idea that that ought to preserve various operators is more correct than the idea that everything should. Here's the theorem. The theorem is that I don't have an answer. It has the problem of a split logarithm. The problem is with x.

37:30 This is much, much more general than that seen with the group work. You know that non-digital group work, non-projection, has the general. This is the proof that this is just an easy calculation using the formula for the O-hat integration that we get. But let's see, what does it mean for any category which has finite products? This is the sort of formula that would want to be true in the syllabus of all spaces. The picture is suggested by that small category, the Penrose, Witten, Connes, and so on. But if you have finite products, the idea is that they could be equal to, for example, an i equal to the graph. If all concepts are non-linear.

40:00 The concept itself is self-dual, even though you could have a small category of products that don't have much to do with the co-products and appreciating them, but that doesn't matter. This property is true, so finite co-products and non-empty homosets in C also. To make best use of C, I'll satisfy this formula and more generally the substitutivity along projections of this type of negation and so on. It satisfies this property, because it's not a good boy, but it does not have, namely, gold, metal, any order-preserving map, finite, linearly compacted, split model, the third, usually larger, in the middle, because that's obvious to all of us. So, in other words, official sets, we'll be glad to know, because it satisfies.

42:30 Alberto, sorry to interrupt, but did you actually manage to get a fairly good set of notes, because unfortunately I couldn't see the board very clearly from here. Did you manage to get a fairly good set of notes, because I didn't see the board very clearly. Oh, Darcy, yeah, I heard about it. You've got so many connections there, aren't there? I think more or less. You know, what I was saying to you the other night, I mean, look at this closed, this re-opened, second-stand substitution group. It's not false and accurate. Because clearly this does affect the callback long reaction as well. And also I would like to understand more deeply this condition on the function of the Boolean in the case where with the extensionality well pointed. Perhaps a new way, a very geometrical way of thinking of decidability and so on. I don't know, lots of ideas. I want to go away and spend the next two years or so trying to think of that.

45:00 Also from your point of view. Absolutely. Well, and yours too. It's a very interesting area for science to look at from that point of view. I think I'm asking you to leave. Thank you for watching.

47:30 And of course it does, it is interesting. I think also there's just a little bit of work on capturing the Derrida-Bitko modality for the German operator by using quantum mechanics. It's an interesting area because it does obviously capture an area where substitution is not necessarily observed. And the fate of some of these podcasts connect, or, partly, the fate of some students will be preserved long-haul, but I think they connect with the intentional extension of the convention, and I think it's going well. It would be very interesting to look again at this work of Nadal Kiara and the group on the modal operators in quantum logic. What I'd modalize the point of view of this insight about the state of substitution. I'd have to know far more about the... But certainly the preservation of substitutional projections does seem to connect in a way with that way of thinking of separators of variables, separators of arrows, variables in terms of the behavior of separators of arrows. The object, the base on which there is variation is meant to be a bit of an anthropological structure and a more anthropological, I don't know, self-anthropological structure on which there is variation. Yes, I agree. This is a problem because one has, well, it really has no interesting topological structure in the classical case. It's really quite trivial to have identity-symptom separation. This, again, connects up with the inside, but the way that one thinks of identity theory in the classical context in terms of domain-based semantics is one where there's really no development going on. In this sense, the whole question, I guess, becomes trivial as it's an absolute necessity in the domain of mathematics. I think the tools are safe at the beginning of this course.

50:00 I think I might actually have a coffee. A coffee, please? Thank you very much. Yes, that is, I'm sure, right, the needs of the planet. Yes, tell me more about this. You don't want another one, then? We were just talking a little about Bill's paper. The one of today. To say a little bit more about Gennady, about the neck operator. The major change it is in the usual model for nodal logic. ESL has shifted from the situation in which you have just an accessibility relation between two possibilities and instead you substitute theirs as a class of errors. So you have the determinants of the neural operators which are dependent on the collection of errors. All the errors are named by hand-to-word words. My coordinates as the French state. If you put some nice public behavior on a morning of the 8th of September, you can elaborate in some manner. First order.

52:30 Thank you for your attention. These results were presented at the meeting of the Italian U.S. Society for Logic this winter in their region, and they will be published in the same series that I published mine. Well, I very much want to get that. I very much want to get theirs. And, of course, the paper you were telling me about this as well. What I would say is that in that case of heaven, you maintain a notion of identity, which even if relativized to any given object, any case is that, for example, if you remember the situation with the theory of counterpart, the theory of identity of Lewis. I remember very well, it was on this side, it might be my MPhil thesis when I was, when I, yes, I was actually doing, well, this was a long time ago, this was 15 years ago, but it was actually on quantified modal logic, but I originally wrote my first, my MPhil thesis was on quantified modal logic. No, I'll have it. Go on, go on. The problem was that you didn't have any information. ...on which could be the counterpart of an individual in another world. It was just a formal notion. So, there's your counterpart in another possible, my counterpart. Why? It's just there, that, and not another one. If you substitute the accessibility relation with a collection of errors... In order to have your code in that world, you just look at the arrows through which you are sent there. So it is something in which you can operate, you can manage.

55:00 You look at the image under these arrows of a given constant as defined on a preceding world And you see what the extension of that constant, where this predicate constant is in that world, it can be in terms of, but obviously you don't have to assume a kind of totality of worlds to have, okay, with fixed extensions in each world for all of the predicates. So in other words, you do, yes, I mean, the variation is built in at the level of the way that the arrows capture the variability extension of the predicate constants rather than having to assume. So we're totality predicates for each world without gaping out to the other side. Yes, it's a very good idea. It certainly makes the whole apparatus of modal logic, semantics of modal logic, it strips away a lot of this extremely inflated, corrupt metaphysics. It allows a very flexible treatment of identity, because you treat identity everywhere through composition. So, for example, you can make simultaneous treatment of all fusion of individuals, which is the easiest situation, but even of divergence of the same individual through two different arrows in the same world, in another world. You can treat simultaneously all this way of divergence, for example, of the same individual who did for an hour in the same world, and you look at which is the structure of the resulting collection of possible countries. Image there. Image is there. And the theorems you obtain are nice. And suddenly the guiding idea is a very appealing one to me, that you do treat identity in this context, as I say, through. Looking at the arrows and the way that, as I say, you can think of the case where, as you say, in the case of extensionality, strict extensionality, the identity is just, you assume that there is an identity over the whole domain, which is trivial and separated for any paradigm. It explains why the strict extensionists like Quine have always been so uncomfortable with modalities. Because, I mean, if you have this conviction that identity is an absolute relation. And there is no question of variation at the level of...

57:30 You know, there is this talk of contrast between constants and variable structures is misconceived, because in the Platonistic universe there really is only one domain, which is an absolute idea, but then, I mean, that's not why any semantics, which gives you this kind of flexibility in the identity of a language, the strict extension is what it is about, this kind of, in fact, of course, it's the old kind. There is no objection to modal logic from way, way back. But there is no. There is a very clear notion of, a very clear criterion of identity for the objects, for the possible objects. You know, is that possible thin man in the doorway the same or different from the possible fat man in the doorway over there? You know, the famous... But it now becomes, it's what traces back this antipathy to the modal way of thinking, to the intentional context. The way the extensionist has of thinking of identity, or the way he has to think of his domain, in order to have partition theorems, in order to have the cost at the bottom of everything. At the bottom of this extension is this desire to hold on to absolute identity. The fact that the identity is trivially separated of any two arrows in an extension setting is the gap. Whereas here in this setting you can actually look at the separators of arrows and see how the separators of arrows in different worlds happen. You might even have, as you say, it's composition of arrows that's important, but maybe you have to ask what additional... Assumptions do you have to build in in order for the composition structure to exist in the category that you're in, in order to give you an accessibility relation across worlds? Maybe there's some connection there with... Where it does seem that you need something like a direction of time or, at any rate, a notion of identity control in order for the composition structure to exist, in order for functoriality to evolve.

1:00:00 I think this is a very helpful, to me, a very helpful guiding idea, that one looks at composition of arrows and whether one, yes, how... How one actually thinks of arrows as separators, or how one thinks of the separators of arrows, in order to get at the notion, the required notion of the variable, and it may be that under substitution, you know, you don't necessarily have operators preserved under substitutional pullback along projections, this is the case where your notion of variable is not that of, that one has every single sorted in there. In a certain sense, I would say that the main shift going into the case of the philosophy grounded on the existence of the semantics for modal logic as designed by Christie and Raddatz and to the situation in which we have the categorical treatment of Raddatz. As a shift which is able to answer Quine's doubts lies in the fact that in the case of Christian semantics, the focus of attention is on the notion of plurality of situations or plurality of words. Why, in the case of a categorical question, the question is on the way through which you get a narrow world from a given one. It is this shift which can offer you the opportunity to answer one's objection.

1:02:30 Because if you just multiply the ontological, your presumably unique ontological domain... If there are problems in only one domain, why shouldn't they be multiplied in a... Where you simply enlarge the domain in the sense of increasing the cardinality of the world. Yes, because the very notion of cardinality itself is assuming, you know, one is assuming after all, all of that structure of the behavior of the existence of the single set, the unit object, and hence one gets all those confused. Ideas about, very platonistic ideas that the numbers are abstract objects, these retain their identity of course across possible worlds, mathematics is the science which is true in all possible worlds, and restriction one thinks of the kind of domain of physical possibility as restriction on that of mathematical possibility, I mean it seems to me that this is a completely confused way of approaching the subject. In fact, they'd be the combinatorial theory of possibility which Armstrong developed, although I think it's self-wrought because it's resting essentially on a nominalistic metaphysics of objects participating in situations which are, where they're completely specified by fixed predicates and really old tracterium thought. Nonetheless, I think that particular combinatorial practice is at any rate a much more promising point of departure, although I think nothing like as rich as the categorical language. It can't address the questions, the combinatorial question, as I say, doesn't allow you to address the questions of the possibility structure of the domain, you know, the fact that, you know, the non-Boolean possibility structure of the case of author frames or other relations. You assume a Boolean possibility structure for situations that are built in from the beginning, but at least you do have this way of kind of building up from an inner sphere of possibility defined in terms of first order combinations over an actual ground domain and then further outer sphere of possibility which kind of supervenes on that, which at any rate is metaphysically much less... Costly, it doesn't give you the very inflated ontology of the modal realism.

1:05:00 No, as I say, the thing which I find very helpful is the connection between this and the idea that one looks at separators, the way that separators of arrows behave over a domain, and the capture that needed to capture the variable. But bringing it all back to the quantum situation, we still do not have a satisfactory way of thinking of the variables in this domain The structure of the truth value object is not such that you can take sub-objects and form properties and products in the way that you can in the classical case. Have you got the source for these papers of Gennady? Have you got the references for these papers of Gennady? Sorry, what's Gennady? Gilardi. Gilardi, I'm sorry, Gilardi. I'd very much like to look at them. Yeah, I would very much like to look at them. I didn't know his work at all. Actually, now I'm mentioning it, I think that Gonzalo gives several references to his work in that paper. I wrote this paper with interest, but I hadn't seen the connection between this, to me, very important point in your own work about the way that one thinks of the separator of arrows in the category, and in the case of extensionality, this gives you this deep connection both to the way that variation is arrested, in the case of the actual choice, the way that... The way the completeness is built in at the level of the type theory and the way in fact the behavior of identity as the trivial separator of any pair of arrows reflects the metaphysical commitment to a, as I say, to a notion of objects there, the same or different absolutely to be the values of the variables taken over a single sorted domain rather than a... and the refusal of any relativized notion of identity. All of these connections suddenly appear more clearly when one... He starts looking at things in terms of the behavior of separators of pairs of arrows. And indeed, the quantum concept is the kind of failure of the existence of equalizers and coequalizers of pairs of maps in certain categories that might be candidates for quantum theory, would also become explicable in terms of the behavior of the separators of arrows.

1:07:30 I mean, Holdsworth talked a little bit about this in the context of his account of... What he called quantum temperature is an account of the constitutional relation of complexes in the world. Things which don't retain their identity and interaction in the way that classical objects do retain their identity and interaction. I think there's a connection there as well. I think that the step I'm working on, in order to bring the stuff in my paper on, is to find a treatment of the things in which I am simultaneously faced with the notion of... The difference between local and global is the local weight of the final separation of arrows and the notion of... In the case of purpose theory, and you have this from the point of view of this situation, generally they are not externally extensional in the sense that most of the purposes are not but internally all because what is internalized in the purpose is just the local point of view. But in general, this is one kind of thing that is the everywhere situation. So I think that we should... I think it's also connected to the funtorial of mathematics, the fact that H1 is an equivalence of categories, the relationship between the internal and external form in the category which... This is what I've been thinking. I think it could be interesting for a kind of those two quantum contexts. Because there you have not these localizing of entities. So, you could have a universe in which the ways of internalizing semantics cannot have the local character they have in purpose theory.

1:10:00 So, we have to manage these different possibilities, not only in the case of Eisenhower's theory of internal-local, this merging of local and internal. And the situation which, for example, one could think of, in quantum theory, of an internalization which has to do with global aspects, but not localizable. But even the other two possibilities, only external semantics, which has to do with local aspects and internal, these four combinations, and to study simultaneously those. I think that is absolutely on the right line. I think this is on the right line. This is also the way that I have been thinking, but you have expressed it much more clearly than I have. This is very much so now. Thank you. That was why I was so interested in the various constructions of the projectile, the size of it, the cost, the home function, the internal-external home function, why it breaks down because of the lack of functoriality in the first place. And so you understand the non-commutativity at a level of kind of deep dimension on the side, which is to do with separability, as perhaps the durability of the errors in the domain. And in the same way as the perfect separability in the case where identity is strictly separated anywhere else, it covers the intentionality. Okay, very interesting.