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

0:00 Oh, there we are now, this is recording, yes, sorry, now this is recording, yes, now this is recording perfectly, it was, yes, no wonder, yes, testing, testing, 1-2-3, 1-2-3, just seeing if this is recording, and it is, yes, testing, testing, testing, which has the property now that it's classifying topos, that for any Boolean topos, the geometric number is not exactly the same. The models of the first-order theory were, for example, sets. So we show them that there are enough set-value points for that positive or coherent topos, and that implies, in particular, that the first-order theory, which I used to derive the presentation, has enough models and sets, or indeed any boolean topos. We actually lived in three different places. We lived in a place called Colette and Bertholdt, which is between here and Perugia. We had a place there for only one month. We didn't like it, I don't know why, it was too far away or something. So then we moved to Uteladarno, which is between Perugia and Assisi, about halfway. For about a year we stayed in Perugia itself.

2:30 You see, I had just got from the ETH, this pre-election of SGA IV, so there was a delineous theorem, and I studied that, and I thought, oh, I can do this, and I derived the hurdles of completeness theorem from delineous, so I told everybody about that, so then it occurred, of course, in Gonzalo Reyes' book with Makai without any... I just want to thank you for reminding me of this, that I found this in Colleen Barrett's book, and it's in Ray's book, of course. I mean, this is the equivalent of journals here. It implies, it's more general, it implies journals. You might think Earl's theorem is more general, because the first-order theories seem to be more general, but in a way, I think the positive logic is the basic one. We talk about alternating quantifiers arbitrarily often, and the axioms that you actually assume for a theory will be a certain number of positive formings when you transfer them, and that's enough to generate the classifier. In a way, if you consider different presentations of the same first-order theory, these might be giving you different topos, so there's a kind of refinement process going on there where you can distinguish between different positive equivalents, quote-unquote, of the equivalent modulo moving topos as values of the same first-order theory. In fact, this is systematically used by logicians, without giving it such a formal status. The positive logic, for example, Morley's theory about rank and all this stuff, they always use the positive formulas in effect. You see, you can even take all the formulas in effect, let's say that's an example, all the alternations you want, a full first letter theory,

5:00 and then look at the positive theory that's generated, you know, relatively freely generated by that. So this is a typical move in model theory and in logic, but they don't actually say so. So it's four refines, isn't it? Not with the same primitives. If you start with thinking of a fixed number of primitives, well then, okay, let's first look at it more in general. The process you were describing is really one of systematically introducing new primitives, the ones that you actually need. You say, well, I want to make a rule of negation of that statement, so I have to introduce that new primitive. So, if you consider all the positive theories, it's very easy. So, it's a generator, it's more, it's a refinement. So it deals with the issue for the theory of description traditionally had a lot of problems generated by the fact that we have to deal with unlimited sequence of quantifiers in the graphics of the description. The basic form of a description, of a definite description, is a term, is a term, a closed term, because it's a free variable, but anyway, which says that there is an x such that for all y, such and such, but the basic form, so you have already an alternation of quantifiers in the basic form. But, with this approach, you can eliminate that, for the uniqueness is obtained in sequence form, says that if, say, x satisfies phi and y satisfies phi, then x equals y.

7:30 And the other view says that T, true, implies there is an X such that phi, and we put together this, so this... A political theory is what you would need in order to deal with the description terms that are used in general. You use entailment, not... Entailment, right, entailment, yes, entailment, entailment. In other words, there's no need to introduce a third statement, A by B, if in the end you want to say it's true. This implication is only interesting when it's not always true, when it goes around, you see. If you want to assert that it's true, use a sequence, or a quote. I mean, this habit that clinicians have of putting a for all in front and everything to make it into a for all and then an implied, make everything into a sentence, you know, true entails a sentence, and it's just... It's one of those things, it's an alleged simplification, but it's not at all, it's actually a massive complication, a massive complication, yes, yes, yes, yes, yes, this is why the free, and the good news is my receipt, the good news is my receipt for this approach from my logician, from the colleagues of mine who were logicians, and who expected that I dealt with the provenance of an incredibly complex, say, sigma n, that means it doesn't, that's... Description. Wow, you are dealing just with the segment 02, the description in first order language. Because they still look at it from the traditional point of view, you know, that I have to deal with every time I add it and quantify it. This way, it's just what we're told by astronomers to say, you know, to confront it with competitors, of course. So it's either really bad things or not great things. You see, not that he stole my idea, that's fine. But that he propagated this, you see, so that all the topos theorists, with almost no exception, do not follow my, well, what you're saying. You know, I always say, whenever I get the opportunity, I say, positive logic is the fundamental one. Don't write for all in front. For all doesn't even belong to the language. You have this implication, and so on. But nonetheless, because there is this book, you see, which did that, and then other books which copied that particular thing,

10:00 You know, this is, they don't understand, most people don't even know topos here, so they don't really understand this point. And this, so this is the real crime of Reyes and, you know, writing it without consulting me, is that he's obligated this, you know, This from where we start sticking the universal quantifier in front of everything, which of course immediately puts us in the wrong place. We'll only consider restricted sentences in first order logic. We'll only consider those in the form, for all x, a implies b, implies, where a and b are positive. We'll only consider sentences of that form as axioms, but within the first order logic. Because what they expected when I dealt with the descriptions, they expected that I provide characterization theorems for the different classes of complexity of the description. But the only thing I could say there was, look, all these entailments and several entailments are geometric, so they are preserved by the inverse part of geometric models. So I have a complete control on what, on changing, say, the model of my theories. Provided the models are related through geometric models. Exactly. The whole point is a completely different way of thinking about what a model is. You can't think of a second model. We are sure, yes, that there's... We want to put things around to quantitize in order to say what a model is. Instead of saying... Following the traditional model-theoretic approach, which says, oh, this is a pi-0-2 sentence, so it's presented by the unions of chains of models. Well, if it's instead of pi-3, which chains, which models... Each time you add up on pi, you have another constraint on the preservation of the truth of the formula. What I call Tornay model-theoretics. For the models can be related to any kind of math. Which is completely uncategorized. I mean, if you have to consider a category, where maps, if these categories are models, they're related to functions. These functions are such and such.

12:30 Your very simple definition is the total essence of the problem, of definition. Of course it can be applied to all those things they care about too, but only as a special case. You don't see that... part of it, you see, is this view that you have to deal with... And you make the point at the end of that paper, very beautifully, which I think the people who have this traditional... This is the key to thinking about the structure in the model, which later, the thinking of the structure in modern terms, in terms of the way the syntactic machinery and the semantic machinery for which you've got to have a kind of set of theoretical semantics because that gives you... I've actually tried to present the ideas in that paper. I, of course, helplessly copped at it in a raving way, to try and isolate this crucial point. What's absolutely fundamental is structure that's preserved and pulled back in geometric morphisms, but this is probably a failing me. They don't get it, and then when they read that last paragraph... Even in mathematical practice, that's supposed to be considered calculus, the definition of women, so this is almost... Basic examples by logicians might claim that alternation of quantifiers is really a crucial thing, because it says, for every epsilon, there exists a delta, so that for all, you know, for all... Remember the time I had to waste going through quantifier elimination and alternation? But now you see, this is not what, and there is a claim, but this is the only way to define the effect. In fact, we have discovered something better. I mean, I think that Toklas goes even beyond that, of course. We don't have to stick with that. But if we want to define, in 20th century mathematics, to define a limit, we introduce topology, i.e. we introduce the notion of neighborhood and the notion of final part of the sequence or something like this.

15:00 So that each of these entities that you introduce is... so it's not just the formula that you introduce, it's also there's an entity. Each of these entities is defined by a relatively simple, so that it's like any universal property, once it exists, you know, maybe it's defined by maps going into it, but now there are also maps going out of it, because it's treated as an entity, so the neighborhoods and the… You know, the eventual subsets of directed sets and all these are introduced, you see, and then this drastically reduces the number of quantifiers in the actual statement then, and it's also more conceptual. I think in every case where we have to deal with something that's complicated... Logicians might put it as some alternation of quantifiers, but what mathematicians do is they introduce appropriate entities, not that they, you know, they know about Occam's Razor, it's not that they introduce entities all over the place, but judiciously chosen things that are really coming up in practice by the neighborhoods, and that's what they do. So this should be, actually, according to me, a very calculus definition of women. If you want to talk about women, somebody should actually admit that there are neighborhoods and stuff. Rather than rebuilding all the Cauchy-Gillis-Strauss-Tuckin-Sci set theory. Well, no, you could even build this, you could talk about the infrared, you could think of them as sets of topologists. It's not even an issue. It's just that you... Well, I guess you're right, anyway, that in sets, they need to be tempted to define things by rule and combinations and so forth. Yeah, that's true. Judiciously, but crucial, things are comprehensible.

17:30 Well, remember, he said, don't multiply them beyond necessity or beyond need. Well, the reduction, well, what struck me when I graduated in a monetary field at the time, Products for the solution of any problem concerning elimination of quantifier, very simple, very simple solution to this problem, which signs that something has been wrong with arriving at this point. For the data language, you consider all the possible granite forms of any sentence you have. And then, to each of these formulas, you assign a new formula, a new symbol, a new predicate symbol, and you get, for any theory you express in this language, elimination of quantifiers, which quantifies the sphere. So, by enriching the language in the way that which is used in the proof of the compulsive theorems for you only, you're not only... Yes, yes, of course, the proof of the completeness. It also involves introducing a larger presentation and a simpler doctrine of theory. That's right. It always did, even before the beginning. So something, the problem for me at the time was where... Just as that presentation is concealed, I'll leave all of this machinery pre-next normal form. Because in the competence theory you have new constants for terms, but you can apply the tool for predicates, for formulas, and use new predicates for each formula you like, or whatever complexity it is. So, but to say that every theory is decidable in this way is very, what does, it doesn't say anything, there's no problem, it's sort of taken out of your, of your, of your, of your, of your, of your, of your, of your, of your, of your, of your,

20:00 You have to meet some students. Yes, I have to meet some students. How long does this go on? Say, half an hour. No, it will take as much time as that. Do you have more appointments in the afternoon then? No, no, no. Okay, so then, so we can come back in, what, an hour? An hour, yes, that's okay. If you, well, if you go downstairs and make a copy of this, or otherwise I'll make it after when we meet and go downstairs, okay? Well, I won't take long today. Yeah, yeah, yeah. Can you tell me to take a minute to do it?

47:30 Because I won't go down for this otherwise. Initial. I know there are three different things, I realise that, it's just that I want to... That was just musings, yeah, this was the pot of stuff, but all the same, I just want to remember what was... Because of the interesting point about, yeah, that's the point about quantum groups. And then lastly, let me jot this down, a bit about quantum groups.

50:00 I don't think, I have to say, that is taking... Well, I think that the whole talk about quantum groups is taking the algebra of quantum... In other words, because these quantities are, this is an objective idealization of the measuring force. Or else, of course, it comes out as some deeper underlying metaphysics of process, which the bone annihilates. See that space itself has just emerged from this self-organizing process that's going on here, which is the hollow movement. And you get projection, you get the standard ket, just a projection out from the lowest level from this Kauffman algebra, which has got this interpretation in terms of the hot. Algebra structure and the Braille groups and it's all a clue to what's going on in this. See, points just are little things that everything's intrinsically non-local. But I just wanted to sort of understand what's so confused about their way of thinking. It's not getting attention, as you say, to the spectrum, I think. It's coming from this huge thing and then spotting down to get the tensor product and then treating the...

52:30 And that's what gives these weird things which are kind of, they claim are groups but you're really not groups.

55:00 So, Mr. Gidney there, you are very good, thank you for taking me so long to get this out. You're welcome. What? What the fuck are you doing to me like that? I've never seen anything that can't be explained like this. And now I really don't understand what's going on. See, the answer should be back at the same time. I see it now. Instead of set value, it's truth value. Yeah, yeah. And therefore, we're on a regular set. The regular sets are kind of enriched in... Truth values. Well, rather than seeing it all coming down to truth values, there's something in there as well, in the way that they do. Oh, that's why I guess I wondered what was going on there. Yeah, I've never seen it in that way before.

57:30 Of course it is. What's the connection with Cayley? How did I tell them that? In the abstract, maybe you wrote it back on yourself, couldn't you? Yeah, then the connection, I mean, the reason that you want to look at things at the same I think we may be able to wait until the very small round of lunch time, I think possibly before, but I think I've got most of this now, just let me, I just want to take down the dedicated cut, I know we've written it once, but this one I'm not quite sure, but you're absolutely right, this does really need to be written down, because they're not written down, we're going to have others for the show, I've never understood it, well thinking of it, it connects up with the whole way of thinking.

1:00:00 The sub-bubble will be the setting of the Higgs space. Do you have anything from the Cartesian? Yes, yes, yes. Once again, where's your authority on the Higgs matrix? This one will not be there. This is the cover of the Higgs matrix.

1:02:30 That's a, this is the sheet of the Higgs matrix. And this is the corner of the Higgs matrix. I'm taking off of the whole thing. Ah, I see. Whereas the other one, in the first part of the Higgs matrix, I'm taking it straight, without the Higgs matrix. I have to stop. The Higgs machine is insane. You can see I think why I wanted to bear it for a little frame, like he didn't even get this subject down, but I said I would get it down, and he should kind of get this subject up somehow, but there's none of our first time, he's not set for mathematics, and should be, but I have to check again. See this is so basic to me, I'm just assuming that's the one direction that he would be in.

1:05:00 Oh, hi! I just don't know where to start. We don't know where to start. We'll just be trying to find as far as we can. There is no one at home who would like to say something about the thank you thing. It's not okay. Oh, okay. Have a safe day. Ciao!