Indistinguishables, a framework / discussion

0:00 That there is a sort of solid scientific consensus. This is not Dr. Kenovich. But the book, actually, is it a state of fail? It's sort of presented everything as a gigantic scientific consideration. You get funds for research. That seems to have done a lot of damage in the United States. are you honest tim yeah i actually get to be honest i i get virtually no my own funding actually is very little bit if any is for climate change uh most of the actual funding i get is for doing much short-range what we call seasonal climate forecasts so i i you know people say oh well you're just you You just get money from wherever. Actually, that's not true of me. You know. Well, as you say, you can't even get money out of Brussels. Well, that's right. I mean, something as big nonsense that... Who was the guy who read that book? Well, it was Michael Crichton. Michael Crichton. Michael Crichton. It's like State of Fear. That's reinforced the position of the know-nothings in... I mean, it was so ridiculous reading it. I don't think it's had that much impact, even in the States, to be honest. No, I don't think so. I think most people treat it as a bit of a joke. I don't think so. Actually, even in the... Actually, even, you know, I was up at the Royal Society a couple weeks ago, and we had a deputation from ExxonMobil. Now, in the past, they have been, you know, the black, you know, the evil empire in this area. But even they now have said, we accept the science of climate change, bad boys in not accepting that this is a problem, and they came over as contrite and, you know, wishing to do penance and all that sort of stuff. I mean, I'm sure there was, you know, some kind of bullshit in this, but this is a radical change of colours, and so I do see, you know, I'm sure there's commercial reasons for doing this, but they realise if you really kind of maintain this totally sceptical approach to climate change, you're really out on a limb these days. There are very few credible scientists. I think there's one general problem here, that there is less interaction between politicians, there are more interaction between politicians and economists than between politicians and scientists. Yes, yes. I mean, unfortunately, as I say, the scientists tend to, sorry, the politicians

2:30 tend to view that the sort of science is being done now, and the important question now is technology, whereas actually some of these particularly, I don't think the temperature issues are so much, but the rainfall, the precipitation issues are a big one. You know, in the UK, we've had this drought for two winters, and the question about whether to invest in a national water grid is quite an important one. It's a billion pound investment. And at the moment, our best forecasts are that the winters will get wetter, and so this water grid is not needed. but I'm consciously aware of the shortcomings in some of the models particularly these precipitation forecasts and I'm slightly uneasy actually about this advice we're giving government and I feel that you know to have because the next phase of decisions about infrastructure investment, how to adapt climate change is inevitable now to some extent, so the real question about infrastructure investment to adapt to climate change and this is where I actually I'm not so happy about the level the accuracy of the computer models because of this computing power issue. Well, adaptation to a seven-meter prize in C-Level is going to involve programming for trillions. I mean, I would have thought seven times present over the world GDP. So measuring... Just to say Venice alone, they're not going to be a national prize. Yeah, that's right. So trying to really understand how the temperature and effects and the disintegration of the ice sheet is a critical one. There. Okay, thank you. Okay, now. Quick question, sir, then, George. I'll just say something about what I've been doing in life, right? What I'm going to do. Well, I was pretty sure that I was going into science in some way or another. and at the time at that time and as well as now it was important to start with mathematics that's what I did and I ended up I think to the very nice offered a very nice job and being an undergraduate undergraduate in Gothenburg and and also having some teaching which give me a lot of money compared with it so I stayed

5:00 mathematics for some time and mainly in analytical complex analysis this was a fascinating for me a very fascinating subject and also in an infinite series on an individual I am by Knopf so my main education mathematics was a really in functional analysis. Not so much an algebra, unfortunately. It should be more much insight. It should be more algebra. And then I choose physics and first I choose experimental physics. But somehow I was not the guy who should work with the experimental physics. I'm not practical enough. My daughter has as much even in simple experiments. So I went into theoretical physics, and there, when I do my PhD studies, I became fascinated by collective systems, collective systems, and particularly phase transition models. And from there, phase transition is, of go into irreversibility, and where does it come from? That was my main question, my kind of philosophical question, where does irreversibility come from? Well, I couldn't avoid but say it must come from the collapse of the wave function in quantum mechanics. I can't see otherwise, if I couldn't at that time see otherwise, so I was led into then into foundation of quantum mechanics. So I studied and went over to study Foundation of Quantum Mechanics. And during this time, of course, I met, I read Foundation of Physics, where Bassett and David Brown published in around 1975 or something. A New Order in Quantum Mechanics, what was the title of it? Indicating a New Order of Physics or something? No, it was 171 in the... 171, sorry, sorry. International Junior Service of Physics. No, Foundation of Physics, Foundations of Physics. Is it? Okay.

7:30 I'm not sure. Anyway, so what I did are, I went, once I was in London, I went to Weber College and just knocked on board, as Professor Bowman said. And he was extremely gentleman and he took time, ideas, I think, in set theory about orders and ordering. Looking at set theory a little bit different and to introduce order in a very... And of course, I just asked a few questions and then he talked all the time, more or less than two hours. I just asked a question here and there and he just continued. So, and since then I think I have been following, really following what he's been writing in his thought also. Also in a more general philosophical point of view. So I think I, my profession is on the other hand, now I'm working in herbal medicine, and this is just a coincidence, it was a hobby that by coincidence developed into a profession. So this is what I've been professionally doing for more than 25 years. It means that I'm not working at the institute anymore for a long time ago. So now and then I work a little bit or think a little bit or whatever, when I have time. Mostly in philosophy and science, I would say. And in particular with David Bowes thinking. So this is a little bit of background. Now, my contention is that mathematics is the most fundamentalism in the science for understanding and description of physical gravity. I think very few really say something against that. And I think that any limitation in the scope or in subtlety of mathematics will entail a limitation or aspect of how to understand reality, physical reality. This is an act of belief, right? This is what I believe. And in the nature of mathematics, I think Bessel says something about that.

10:00 mathematics, if you really consider what it is, it must in some way reflect basic rules of thought. And David, in a conversation I had just quite, I think, one or two years before he died, about mathematics, he gave, orally, verbally, I don't know if he overheard it, Bessie, but a very good definition of mathematics, as rules of thought, given the of coherent or consistent necessity. I think it's a very nice definition. I'm sorry David. But take that together with von Neumann's definition. The stereotypical antique is, quite a lot is there. If there's something like definition of mathematics, if there's anything like it. Now, David took the order as most perhaps universal or more than most of our other basic conceptual categories. order is primary and common to all we conceive and perceive, that was earlier. And this means relating both to abstract thought and external reality. Now, my interest is this, I think, to get some, to understand the illusion of orders, and David said that David quite penetrated analysis of order in various contexts, and that order is based, ultimately, on differences and similarities. Where difference is the most, is the primary category. Similarity is a derived, as it were, on difference, in the sense, because of when two things are considered as different, as similar, you must disregard some difference between them, otherwise they can be seen as similar. Similarity is

12:30 is derived, as it were, of difference. And thinking about that, that difference is really such a primary notion, sort of primitive notion, difference. I was thinking about if there is only one thing of difference, and just by accident, I came across a book by and a book of the theory of interchangeables, the theory, sorry not that, the theory. So I immediately wrote a book, I wrote a book, sorry, and read it. But it was extremely hard to read because this guy was he had a complete own terminology he was very very particular it was syncretic and also that there were a lot of errors unfortunately in the books and mistakes but I was trying anyway to to to grasp what was written there, really, to see what could be done. You see, if difference is such a basic notion, primitive, one can think of then, just in set theory, when two elements I think we'll take a little bit of this away and put this on. When two things here, down here, I just skipped it to the time being. When two objects are to regard distinct, there is an implicit, tacit assumption that, when we look at it, distinction really comprises two aspects of difference, distinction in set theory, with two elements of distinct. This is one aspect as it were a collective derived difference of two objects because there are different parts of a collective

15:00 That is one kind, aspect of difference, because they contribute differently to the collective. And you can say this is then reflected in coronarity, this difference. I also think about an individual particular difference, because there are a direct relation between two individuals, and this is always used when some of the two objects are named or labeled with respect to another, or in any way labeled or indexed. So, from this one I can say that distinction really based on two aspects, as it were, are different. So, if this reflects cardinality, this one reflects ordinality. So, the idea is to take away this to see what's left. Perhaps I could use this then, yes, good. So this is where the notion, in my mind, where the notion of indistinguishables comes in. You can take away ordinality and you keep cardinality. They collect the difference, not the individual difference. So, there are two, one can say verbally, they are collected plurally but severally identical. means von selber, which is cut or take apart. Another way of saying it, verbally said, can be told together, but not told apart. So it means, essentially, what I'm saying, that they lack ordinality, but not cardinality. And this difference should be lying, as it were, between identity and distinction. there. Now, I will jump now to some main feature and concepts. Is the element lack ordinality?

17:30 I think it's called for a new novel parity relation between elements. It's called twinship in Popper's book, so I keep to this notation mainly. When two elements are indistinguishables, twins is a shorter word, A, distinguishable from B. Then we have really three parity relations. It is identity or equality, Equality, indistinguishable, indistinguishability, didn't it? Oh, I can't even pronounce the word anymore. Indistinguishability. Indistinguishability. Indistinguishability. I should have kept away with this. Indistinguishability. So, this means that having three, and that they have the negations of those, right? and you can take negations in various ways. But there must, first I say, there must be some relationship between those parity, there should be, also a relation between the parity relations by them, by themselves. One such, and also some more property of this, for instance, this new one, twinship, When you look at it and think about it, these indistinguishables should be semi-transitive. It comes a little bit later. And the negation of one parity, that is, for instance, it is not distinct, the negation union of equality and twinship. And there is a certain slant to this, that this is a union, because to jump a little bit ahead, this parental relation induces a fundamental ambiguity in in the whole theory. So when I say that not distinction is a union of those two, it also

20:00 incorporates cases. It's either quality or twinship or both. It's the same type. So there There is a parity relation which says that A, this is it, is indistinct, an element is indistinct, indistinct, the negation, right, for distance, indistinct from B. could mean that one can't say, one can't decide whether A is equal to B or A is twin to B. It's a fundamental ambiguity. That's why I say the negation of one is a universal. and this ambiguity really defines a new plural quantifier, which is also in front of Parker Rhodes' notation called Blur. One can look at the operator, come to the data perhaps, and so, and so, and so. Now, the third is of course, one of the main features is those ideas cannot be individual indexed or labeled, but only given some collective index, and that collective index must be, If it's been to a Gbg, it must be coronality. It will be inconsistent. Now, if you look now on this that couldn't be ordered, and so on. Look at a very simple definition of order, that a pair, an ordered pair, how normal is defined. And the order A, B is defined like that in set theory, right? Now we have really three possibilities here. We have that A, B could be equal to B, A, or A, B, the pair of A, B could be twinning with the pair of B, A.

22:30 Or they could be distinct. This is, of course, could not, we couldn't use that if we are talking about indistinguishables. It is, of course, they can't be, if you're ordered, they cannot be distinct. If they're distinct, of course, then they could index them, etc., etc., right? So, there's no really two distinct ordered pairs. That is impossible. As long as A and B are twins. If we choose, no, sorry, distinct, sorry, sorry, sorry, I was just misreading my own thing. Distinct is not, it's not written here. This is indistinct here. The loser, sorry, sorry. It could be equal, there could be twins, and they could be indistinct. It means that be either equal or twins, but total ambiguity, you can't tell, whatever. If you take this one, a starting to define order, as it were, order pair, you take this, then your theory will be very, very... it will really be a poorer structure. so I'm not using this but now in line 2 when they are twins sorry? in line 2 when they are twins there, yeah so I have three possibilities, right? yes, you do label them by A and B but you said this is exactly what's not possible if they are twins that you cannot label them individually no, I'm not doing that either sorry, here they can't be labelled, right? At all. Here they cannot be labelled, because you can't tell whether equal or they are twin. In this third, and this second, it seems as they could be labelled, but could only be labelled by an ID, right? They could not

25:00 be labelled by a member of a set, that could only be labelled by an indistinguishable, two indistinguishables. Because, you see, this one is labelled by this one, George. In what sense would they then be labelled at all? I've missed the point. I don't see how, as you put it, labelled by indistinguishables, that they're labelled at all. The whole point No, we're saying here there is some kind of difference between them, you see. In this case, there's a difference between the pairs. But the difference... Which is what? I mean, not their order type, because we can't say which is one or two. They're just barely... No, we're just saying that we have the three definite parity relationships. Yeah. I haven't quite understood which is which, which parity relationship is which. I understand that the top one is just... Oh, I see. This is twinship, and this is the union of those two. Oh, I see. The top one is union. This is the union of those two. The bottom one is union. Which means, I'm sorry, the union is the case where you've got... I'm sorry, I don't understand this parity. But you cannot tell which of those you've got. You can't tell. You can't tell whether you've got a pair of individuals or a pair of indistinguishables. A pair of equals or a pair of indistinguishables. That's one or the other that you can't tell. As I said, the first, the choice, oh, sorry, it should be, this should be, if we have chosen that they are different, the pairs are different, but they are still indistinguishables, but they are different. Take the pair of the pairs then, right? And look at that. Then we have a second choice. Should we at this level have the equality, the twinship, or the indistinction?

27:30 like the three choices we have, right? Three choices. Now, I'm looking at these three choices. If we have chosen two, for instance, then we can ask, then what about forming a pair of those two? Should this be equality, or anew, or twinship again, or indistinction? First form this, and say we keep to this first, we select this. If we select this, forming a pair of pairs, you ask the same question about pairs of pairs. What is the relationship between them? Well, again, we have a choice of three different, right? So, if we again would choose this twinship, we can go on, pairs of pairs of pairs of pairs of pairs, but at some, if we never stop, right, if we never stop, if we choose the same all the time, a person, the same twinship as a parity relation between two, actually we end up I think with a theory where there is no really distinction between twinship and common distinctions. So somewhere we need to have a choice, we need to make a choice. So in these three cases you assume that A and B are twins? Sorry? You assume from the beginning A and B are twins? the confusion. I got one of these cases. They were always twins. These are all twins. So I understand. Oh yes, yes. That's where I was getting confused. So far, I must underline that. So far, if you're not saying something else, they're all twins, right? They're all the cases. I'm sorry, I understood, because I was thinking about this. I don't even know. I see. I'm completely understanding. Now, I choose this one, to be the most interesting. And, of course, in a common context, this would imply that A is equal to B.

30:00 In fact, that's not a choice. You have to do that, otherwise you will distinguish them. At some point or another, yes. this is this was implied that equal to right this must not be the case that could not be the case in order to proceed this is one way of seeing I can say there are different ways of going into this but this simple way a more direct intuitive so to that A and B are not identical here in this equality event. And from that, one can say that A really does not refer to that A, it only is a notation for being one of A or B. And then you can be consistent about it. And it means also that the pair could be written differently, if I want to. It could be written as, symbolically, this blur. Total ambiguity, whether it's B or A. And this, we can say this, oh, it's writing AB, of course, not CB. This is called a blur of AB. And it could be as a purely quantified, but not one particular. something about parity relations could somebody give me a time when I have used half an hour I that I already did, I guess. Anyway. 26 minutes. Oh, okay. So those are the parity, and those are negations. The transitivity, or semi-transitivity, we've got this parity-twinship relationship.

32:30 I guess one way to say it, I can say it in this way, that given AB, A is twin with B, B is twin with C. It can never be the case that A is distinct from this or C. So, if it were known that all are different, then of course we have total transitivity. and but it has to be this is this is also complement so to say if we have but if that doesn't happen then you have equality sorry if that doesn't happen then you have equality right which is the only other possibility yes yes yes exactly yes yes exactly yes Explicitly also, if B is distinct from C, it can never be the case that, if A and B are R is related to B through one quality indistinction or twinship, and you have B distinct from C, It can never be the case that R is equal, indistinct, or twin with C. But what is annoying, or what I don't understand, if they're equal, you don't call them twins? Oh, then identity. That's not that identity. There's only one, in fact. Does that actually mean there is only one? They are the same thing. Because then you always have that. thing wouldn't happen, you know. Yes, that's right. There's a sort of ghost, ghost image. I'm not going to agree to the details here. I just say that there is a difference of distinction between twinship and ordinary distinction, and somehow that should be reflected in the theory. One way of doing this is that one can order the parity relations according to difference, increasing difference, identity or equality, indistinction, twinship, and distinction.

35:00 Also, there's also ambiguities coming into the picture here. Also, if I can say that there is an order of ambiguity in the way that this is more ambiguous than, of course, the definite parity. So somehow this should be reflected into the theory. I only briefly take one of the postulates. Whereas this is a, oh I must remind you, this is not, I say, it's not yet a theory also, this is just as a discussion for a possible approach. If you have two expressions, object expressions, containing X related to Y, and one of the this Ari denotes one parity relation, equality, twinship, et cetera, right? Yes, it is. Oh yes, it should go up here, sorry. the Y should be up here. And they have the X and Y are the same, have the same parity relation in the two expressions. And if you change now the parity relation here, in a Anyway, let's see here, something, something is missing here, no, no, that's right, that's right, yes, I was reading, misreading my own thing. So if you have a parity relationship between the two expressions, which are objects, right,

37:30 objects in one way and other, object expressions, and of course belonging, not of course, belonging to the same set of parity relations, if you change the parity here, exchange it, one for You can never create a situation where the two object expressions are distinct. See, if you work within this class, as it were, of parity relation, and you have two expressions with the same parity relation between the two elements, So, and if you exchange the parity relations, both, you can never produce distinction. What it does, it guarantees that ordinality cannot be created for non-ordinality. I think there are many, many ways to do this. This is just one example of how it could be done. And then there are some other postulates, with only taking the verbatim here. Yeah. I said there was an order of distinction between the parity relationship. This order here, going from parity to distinction, to twinship to distinction. This is also reflected in similar between two expressions. You can say that, let's see here, do it, didn't I? Ah, I thought I was happy with writing a little bit more. Something is falling away here, I think.

40:00 I don't know what to do with this one. but these expressions that you're using are they are they using only those relations or can i also use other things in these expressions yes you can use other yes other statements that not depend on the twinning yes yes so that's a very strict condition then it's a very the very very strong condition that you cannot go through this thing because i could make elaborate expressions using using other relations that are not depending on the industry miscibility but what i the change i do i only change yeah but the ions the absence of the e's or what are they the big e's themselves objects is it only objects compound compound object at this well you see at this stage I'm reframing from going into higher orders. I keep at the object and function level so far. I think I skipped that because I didn't find it easily. I thought I was going to... and there's another tempted postulate also that will somehow take care of ambiguity in order to read it but just that I was looking for this part of the other one destruction of ambiguity cannot create ambiguity cannot create ambiguity and creation of ambiguity cannot destroy ambiguity This is when we shift parity relation between elements in an object expression, a compound object, if you so. one. So, this is about the relationship between the parity relationship. It's not that important,

42:30 That's something more... Now, as they cannot be labeled or explicitly ordered, they can only be considered as different when they are collected together. so that must be the theory must contain a theory of what is collection and what is context being in the same context this is I think crucial to whatever you do about it it must contain that aspect of context dependence that can only be directly two elements can only be directly related when they are in the same collection or in the same context For instance, if you look at two, say this is our two collections of indistinguishables in the pan pieces here, and you look at these two collections together here, look at them together, those are in different collections. You cannot relate A to D, for instance, saying whether A is equal or identity to D or it's twin with D, because they are in two different collections. And they only make sense when they are in the same. They can only distinguish when they are the same connections to say. But what is a collection? It's not a set. It's a collection of indistinguishables. I'm coming to that. You have to define it. Or at this point we're going to say, just put a parenthesis around that collection. I don't buy that. No, no, of course not. No, it was a joke. So, to establish a parity relation between two ideas, A and B, essentially three different ways of doing it.

45:00 If that could be placed in the same collection or context, you could relate it directly. And for that purpose, indistinguishably, there's something I call a constitutive level. It's a level where you can make formal manipulations. I come to that later. then the second is that so what's SC stand for sorry SC oh so called oh so called and they can also be related there are sets of functions and operators and this in a way that for instance if if you have a operate an a and b or a function of a and b then then that a and b are in the context of f as a function operator then they are the same context right as a way also defining context when they are in the same operator function and the functions are defined on the coefficients sorry where are the functions defined from Not to find here, it's later, but I'll find that, sorry. So, yeah, where is this? In fact, this is a different... I introduced three kinds of operators and relatives. One is that, let's see, now I wait for this. Transparent operator means that if you have a function or relate operator on x and y, f, x and y, If you have an X outside here, it could relate directly to, that is, it could be said that this X is identical to this X here, when F is a so-called transparent operator.

47:30 So I'm introducing two classes of operating functions. One is a so-called transparent, where it could make a direct relationship, saying whether different to Y, and X is identical. The first X is identical to the second one. The other one is the non-transparent operators. When they are non-transparent, you cannot, as a repair, you can't say that this X is identical to this X. You can only say that this X is one of X and Y. We are completely in different contexts here, this one here. Here you can say there is, you can go across as it were. I did that because it, it leads to a more interesting theory later, later on functional operators. Now, I just list some of the difference of a set theory. And this must be axiomated. If C is a collection, then you create a collection of a collection, you get the collection right. I think this is necessary to avoid mishmash with set theory, in fact. And also there is a different meaning of singleton here. If you have one element, and the collection of one element could be X itself. collection i'm not saying yeah i'm waiting for the collection you might get me there i don't know that's that's also i also on the other hand i think it's also will be really problems if there are two different kinds of empty soul. So the empty soul is identical to empty set. Yeah, and this doesn't set theories. Yeah. The separation axiom is not really,

50:00 doesn't work here. not the full extent in a way but that can be some kind of what I call diluted form of separation axiom and and the main difference is that that the property P defining if I define the subsector does not necessarily define unique collection I also not another another I'm using really you see I have two words here, two terms. Collective is a non-neutral form for saying that there are many of something, right? Or sort of ideas, many ideas, collective. And collection has a specific meaning, right? So sort is general, sort is just collective. Collective notion of, yeah. And this is then the some, this is the ambiguity notion of A and B, it's either A and B or, but indecisively which one. It's of course different from set theory completely. But you can use it if there's a notion in set theory. then there's a this is just this is this is not ready yet I would say but how to define this little bit more formally well how to define that well is this is really not definition it's more like I don't know what to call it but it doesn't work somehow anyway set is a ghost it's not really an element it's a ghost element right so don't take set as literally as an element here just a set is something right set is something which is equal to this have any properties that it is not a collection of what is inside here. It doesn't matter. You can say, sit here, if you want to. In this parenthesis,

52:30 it's all normal parenthesis. If you want to, say so. But if you insert this, is, it's quite clear that it can neither be X nor Y. So if you ask for the current net, X, Y, and Z, it must be 2. Of course, it's neither of those two. They cannot be equal. This is not equal to Z or Y. So actually, this is a ghost, as it were a ghost on the Trying to generalize a little bit of a border, you can use predication, saying something like that. that, this is the blur, set, and for almost any p, set, it implies px, py. On that one, a hand, for any, it also implies that what every p, p prime x and p prime y, must imply p trim set. I don't know, I haven't found a direct, I say, inconsistency here, but I'm not happy with it either, so. you can say something with every predicate which is true for x and y but nothing else you'd certainly buggered out Russell's theory of definite descriptions because you know these things are not these things by definition are not names of they don't have barriers that's the whole point they can't be picked out name so it's obviously you can't get back from definite descriptions to names in this presence of this framework. Then there is one thing I'd like to show so that there is a blur could be you can transform as it were blur into object you can objectify

55:00 what it means really you add new elements you make just an operation operate operate a B on this blur oh sorry is something missing here oh this is just this H here does just means a collection or a sub sort a collective of of n various ideas. This means just n various, this is a sub-sort of collection. You just made an identification and there's of course you have to say was this generated novel element something about that, right? And that is just The cardinality must be one, for instance. It's one element. Things like that. It's not very... And also that for every P has a predicate, not inconsistent with the definition here, given here. It must hold that it should hold this one. This also should hold it. So if we have a sort, a collective of instinctuals, and say that we have the cardinality, then we have of course this amount of, as I said to you, sub-sorts, sub-sorts, if you take away the elements. It means that we have, it means also we have that many blurs, possibly blurs. It's really sub-sorts. You can think about sub-sorts and taking away them, the sorts and the elements by themselves. So they have that many blurs that could generate, generated into elements, so we can generate that money more of new elements, which in fact then could be used as defining boundary. Can we talk a bit later?

57:30 Ah, let's see if I have a little... I'm looking for a little piece of... Ah, there it is. I'll make a jump here, just to say something visually. Say that we somehow define mappings on individuals, and you have a range, could be, if the range is a sort over here, well, this doesn't have to be in fact, this could be a set as well, but the main thing here, you have a sort here, as a range, as a sort. Then, as we have this blur, not yet elements, you can still talk about mapping into an element here could be mapped into a blur, which is a shadow really. It's not a real element here, it's a shadow, I call it shadow element here. But then, you see, if we want to expand it, to expand the sort and making loose blur into distinct, to elements, to blurify, to objectify them, right, to objectify this, then you have but the boundary here becomes really a set of elements.

1:00:00 I just want to show that before you go. So you can define boundaries, in fact, in three ways. The first I've just shown, where the boundary is not really an element. It's a kind of shadow element. A boundary also could be, if this, to a sub-sort. it's a sub-sort you can in the sub-sort you can form all kinds of ABC you can form all sub sub-sorts of this collectives it means that you can have you can when you objectify it you can have a whole you can have a number, of course, of new elements. And the other thing is that you can also define a boundary just taking one. If you have a sub-sort, for instance, here, you can take the union. You can take every sub-sort of this sub-sort, you can take all the units of every, oh, this should be, you objectify, you form all sub-sorts and then you objectify all sub-sorts and then you have a very large boundary, in fact. is larger because the boundary is the cardinality of 2 to the n minus n minus 1, the new elements which are added to this sub-sort. Then we can have closed sub-sort and open sub-sort.

1:02:30 whether you regard this belonging to the boundary, whether you objectify it or not, you end up either with a closed sub-sort where the boundary belongs to the sub-sort, or you end up, when you objectify it, the boundary does not belong to the sub-sort. Because there are extra elements here. Here by definition, this belongs to age, because it must yield, this is, A and B belongs to age, and it means that this must mean that one of them believes it belongs to age so you have a membership here a kind of membership it's a shadow of a membership I don't know what to call it for the logic you can only have this the logic of this but if you objectify it because the cardinality increases by one for each objectified element, uh, blur. So you, you have, you have a choice of two different boundaries. Uh, sub-soft, close. Let me see. No, I, I don't know. It's an extension there. Yeah, there is, of course, there are differences with set theory and logic. It's possible to select elements, which is true, by some procedure. But in this case, we can't really choose particular elements. so this makes only mean to talk about the coronality of a sub-sort for which this is true you can't go further it's impossible you can have conditioned quantification because that defines

1:05:00 because that defines the condition defines a sub-sort And then there's something I consider level of sort, that is the idea here is that some level can guarantee the construction of compound, new compound elements, which is the context for former manipulations, which you can create structures which will not be otherwise definable, For instance, in power, you can define a power sort of a sort, and you can define a power of power sort. For instance, this is an example of two elements in the P2S, power sort of power sort of S. And those, you see, this is just a, this is a commutation of this one, or a commutation of each other. And at this level you could regard them at different entities, but when you work with and when you map them in any other context, they reduce to one element, whatever you do about it. As soon as you lift them out from this level, they are identical. I will, just something here, that at this level you can form, this is a sword, the collective called a sword, and this is a power sword, and this is a parallel power sword, etc. Could be defined in this level, but any two elements belonging to one power sword, two elements, compound element that is, they are said to be equivalent if they could be derived from the same sub-sort. That means the constitutive elements of S1 and S1 are from the same sub-sort.

1:07:30 And given also that they are not distinct. Of course, compound elements could be distinct. You see, compound elements, of course, if you have a compound element of two here and three there, they are distinct, because they are different, and they are different indexed. So if they are not distinct, they are said to be equivalent, as long as they are derived from the same sub-sort. Yeah. So, in principle, two equivalent elements, when they are, when they are taking out from this level, constitutive level, they, whatever, by a function operator, a function operator, they must be seen as identical. So, we have, now looking into mapping functions, they are usually synonymous, but here they are not really, because if we look at, we have the, just for four different cases, set to sort, set to, sort to set, and set to sort, and sort to sort. and here two sorts elements of two different sorts could be distinct. I'm leaving that open. I mean, this is what I'm supposed to assume. I don't think that is... I think this also gives much more of structure to the whole theory. But the functions are not defined if you don't know the elements of the sort. No, we have to define it. We have to define it, yes. But mapping is a function defined in two ways. As a subset of ordered pairs, right? Order pair must be selected from ordered pair, right?

1:10:00 it is then the the problem is how to select how to make that selection how do you find how do you find that i you use the word subset in the sword sorry you use the word subset the appropriate subset all right that's every misprinting sorry sorry it's a misprint No, perhaps I was thinking about the usual first. But later you have to know. You have to say some sort. And then the functions are not well defined. Yeah. But then you also operate the notion of all the functions. They exchange, you exchange some element for another. And normally those two approaches are equivalent, as far as I know, but they will not be equivalent to it. You see that? Yeah. First something about... The question is just here. How to select pairs? X is the main range from the bullet. The next is... Okay, sorry. How do I say values, assign values to a function of f of x, right? I don't know what is happening here. I blame my secretary. She, of course, has no idea what she's writing so that I don't really, not doing your labor. Yeah, and this is only saying that, this is x, this x belongs to the domain, and this f of x, which goes to the big y, to the range. Anyway, if we have now, let's say we have a mapping from set to sort, and we try to define it by a selection of pairs from this product.

1:12:30 And, say we have one, one is giving, we have a pair here, right, X. But this product is a set, this M times S. And it contains that sword. But it contains the sword. Yes, yes, that's right. Is that good? I don't know why otherwise it was juicy. Yeah. Well, okay, after your talk I will say something like this. So, I try to, first I take x1, x1, that is no problem, that is a set, right? And I show some other maintenance, that is some here. Same way I can do it, I can do it by just taking anything, pops up, right? As long as that's pre-selected, there is no problem. Now, take X2 in the domain. What's happening then? Well, I've chosen X1, and then X2 is somewhere here, but X2 in this, You have, yeah, as the first coordinates, let's say here, there are many X2, following by elements from the salt, and all those are twins. if you have the same, if you have the same, if you have pairs, which, these two, those two pairs, for instance, those two pairs here, those are two pairs, because they have the identical set of elements, right? But when you try to choose the next one, you're in for problems, because you can't tell whether the second component here is an alpha or something else. So apparently this is not the way to define mappings in this way, right?

1:15:00 It doesn't work. Let's have to put an end for a sequential. You cannot have a sequential assignment to value for the function. And if you have to turn the other way around, mapping, look at that pairs defined by the product of s times m, where s is sort and set, of course the As we also knew it was a common number, somehow we need, we need mappings, we need of course mapping the sort of set. I cannot find a way around it. You're welcome to think something out. So somehow we need some kind of collective, I call it collective mapping, where not a particular argument is taken out at a time, but some kind of simultaneous mapping, say from sort of set. We will, therefore, later we postulate something that takes the hold of a sort into a set. Here is some ingredients, I would say. This is very tentative, you see. It's almost like we need something, collective mappings, which are defined from top to bottom, so to say. We start with mapping whole, the whole domain. saying we can define a partitioning, that's fine, we can do that in this theory. That is not a problem. So, first of all, we say the whole of us maps into the ring. And then we take, we make a partitioning of the sort into sub-sort and let the mapping act on those

1:17:30 simultaneously. Then you have images as it were. This is the this is a partitioning of yes the partitioning just is it belongs to the most power sort somewhere something is written down. So this is a partition of the sort partitioning in sub-sort, where each sub-sort belongs to the power of sort, right? This is defined, right? And the first is that, what is this, something, yeah, this just says, take away this, forget about that. Just say that the mapping of is just the whole of the range. That's all. And for each H here belonging to this, this partitioning, you will have a value in the set. You will have a set. And next, you will say that if FA is equal to FB, there is always a sub-sort in this partition, such as A belongs to H and B belongs to H. And secondly, this, if those are different, it means that A is So, this, you define a collective, this actually defines a collection, this is the way of defining a collection, by postulating this kind of collective mapping. This is what I'm about anyway. It's slow forward somehow. I'm sure that's a much better way to do it, but just as a... Yeah, I would suggest a categorical way. I would suggest a categorical way again to make it consistent. For example, take two rings. And now I go to my cupboard and I take

1:20:00 head of a group theorist, and I look at them as objects of additive groups, and suppose they are isomorphic there. Good. Now I take my other head, semi-group theorist, or I look at the semi-group for the monoid of the multiplication. Suppose they are isomorphic there. Now I have no more heads. So these things are the same in my vision, because I see in what I would One called a collection is a kind of putting them in categories, as objects, not as elements. You're not allowed to use elements because they're not sets. So then you look at these objects in different categories and that's a collection. And the number of categories, that's the number of hats that you have, allow you to distinguish or not distinguish the objects. So then these two rings, for you only having two hats, are indistinguishable. They are the same in the group category, they are the same in the semigroup category. But now I have a third hat that you don't have, I put on my ring theory hat, and I see that they are not isomorphic. That doesn't follow from the two isomorphism, they can be very stupid groups or semigroups, they can be the same without the rings being isomorphic even. So that, I think if there's an application of this, this would be a kind of structure theory where you look at... Some kind of... information about the structure of certain things so you view them in a collection of categories that you can do and you can call that a collection maybe. So this is a number of properties of your objects and for example they can be identified in these after these observations and therefore you cannot distinguish them but they are different if I find another hat, if I find the right hat then I could distinguish them. So in this sense this puts of course a limitation on the level of industry we should build yeah that was the word that is very different but but it is a it is i think that's a useful notion of correction you don't have to use any set theory anymore and everything is is expressible in corrections or family of categories and your your things become objects they're not elements and then you can freeze and these categories may be categories based on sets that's not important anymore i mean there are there are They don't care what your collection of your objects are, they don't want to say set, they don't want to say class, they just say collection or something like that.

1:22:30 So you have to describe objects. I suppose if you want a theory that has a meaning, you have to be able to describe something about the indistinguishables or the twins. and in this way, how can you describe them if you're bound to set theory which you cannot apply really so the only way I see is to use families of categories as collations and look at their properties there which is essentially what you do anyway when you're looking at the peculiarity of the category of sets or specifically the topos of sets vis-a-vis the structure of the topos, because we were saying the other day, in connection with this work of George's, the thing which picks sets is that they're the decidables of the topos, and you've got an injunction from category of spaces with some topological structure into the topos, which precisely ensures that all coverings localize, so that you've got decidables with points. That's the thing which gives you the category set. Here, obviously, you haven't got that, but you might be able to express what you're doing here in terms of other weaker conditions on the topology that restricted the degree of localization of the covers. That's one way I thought I might So the spinning comes from an incomplete knowledge. You don't know enough to distinguish them. You've got equivalence relations. You're saying that you don't know enough to distinguish them, but in principle they are noble. You're saying that they could be known, no? It might not. It's irrelevant whether they could be known or not. You've got an equivalence relation and the talk of partitioning is what's leading you astray because the very language of partitioning assumes this kind of set theoretic picture underneath that in the last resort you've got an absolute identity relation on the domain relative to which you can order any equivalence relation with respect to its weakness or strength vis-a-vis the absolute identity relation. And therefore, you know, under that relation, things are just the same or different absolutely. Here you haven't got that. So you can't think of your equivalence relations as inducing partitionings because you can't think of the domain in the first place as consisting of objects there to be the same or different as values of the variables. You've got to think of it in a different way. I mean, what that way is, I think you've got to think of it in terms of the way that topological structure or other kinds of structure on it restrict or localize. You've got to have another way of thinking of it. The definition here, the mapping that, the collection that is defined by the mapping you see,

1:25:00 is by no way unique now, right? Because, as I said, those are considered as, if you take another, for instance, if you exchange, you can take other elements here, other subsorts here, as long as they are equivalent, they give the same mappings. Yeah, as you said, the mappings are only defined after equivalence. Except that still there is no definition of correction. Exactly, because the equivalence is, you're not thinking these equivalences is ordered with respect to some underlying absolute identity relation. There isn't one. Okay. Yeah, let me just see. Let's see what I've got here. This was my first attempt to define coronality, you see. that identify to every sort as a set, there's a new map, and that mapping must not be perhaps of its collective character. It might be defined differently, it might have it. And for every set with colonnities, lesser than n, there's at least one onto mapping. And then Esther I don't know if we could see sort-set mapping, that was the referring to earlier definition somewhere. But for N-set, where this is a bit larger than N, there is no on-set or on-to-map thing. This is exhaust, exhaust-based often, you define the color analogy. Larger and lower sets, until you get the set large enough, there is no on-to-map. Do you envisage any applications to statistical mechanics? Ah, no. No, not yet, no. It's too much to... But I assume that one major motivation for this was indistinguishable particles, was thinking about both... Yes, that's right. I mean, of course, that's... I assume, so... So, you are thinking of particle statistics as a motivating idea. this kind of mapping I show to you first, right? Which is an onto mapping. And then I put the set to zero set afterwards, the L to set.

1:27:30 That is a kind of operation, you can say. This defines the collection, this operation. So I'm lifting it out here, the whole collection from this constitutive level, lifting it out. Oh, yes, that's right. Let me see if there's something that is worth mentioning. This is more like... No, this is what I said. But I'd like to show this also briefly, but... If you defined, for instance, just an example here, if you define functions as operators, no, this is not what I want to say, but I think it's better... I don't have to go to the camera. I guess something was left out here. Just one second. define, what I should do is not use SS, a function defined on operator. And if you look at this as exchanging one element from another, say this is a function, say the sort, sort, this is defined on sort from sort to sort, it's going to be an endomorphism.

1:30:00 Define operator, you can say that, you can look at this as exchange of one element for another. And if you, now if you exchange that amount, you have two possibilities. Either you don't make anything, you don't exchange, no exchange, that is on one operation, or an exchange. exchange. If there is no exchange, you can say, this is well defined, right? And if there, if you exchange, you say that you don't, if you, it means you take something from this or even you take, you can take any element which has, it's very democratic because you cannot select a particular one, right? And it means that it's you can choose that one, but you cannot read it. This one could only signify any here. So it's It's the only value you can give to it. There's no other way to give a value to an operator, by exchanging something. It means that there are very, very few functions, as I say, defined by operators. There's one of the thing here that is a little bit strange. In fact, either you can look at it as when you exchange X for something else, either you can say you put back X here again, or you have this, so X is out, or you take away X for something else. There are really two ways of doing that. There are only two options. And then when you have two argument functions, there are very sticky values. This is one

1:32:30 kind of value. You can choose. It could be either that or x, but in distinguishing of which one of them. This is a value. And if you objectify, there's one element, perhaps that one. But in principle, this is what you get as a possibility, or a complement of that. So that's another possible, very few possible value, the complement of xy. And just to show What do you end up with, just with the reasoning?