Tom Etter on alternative notions of equality & starting points for math

0:00 This talk has received LK's music in relation to a number of conversations we've had with Tom Hale over the last year, maybe even the last year. Eventually, the skeletons will turn out to be a different schema, a part of this sort of schema. But skeletons also record this information, right? Diagrams. Skeletons. What did Tom Hale and LK talk about? Well, they were two subjects. Key theory. Equality theory. What theory? Particularly about diagrams. Conflict. Equality theory. Particularly about globalization. Of course, the emotion of equality. And the academic foundation of mathematics. What do these conversations have to do with each other? Well, let's see. So let's begin by talking about diagrams. For example, I can use a climbing on vertex of a root. Indicate an algebraic product of x, y, and z, a, b, and the input, a, b, and put them together into trees to get their associations of products, so for example, a, b. Mutualities or inequalities occur in different categories. In all of mathematics, someone writes X equals Y and can only be expected to be Y and B, but do in some categories that don't have special notions of equality. If I know by nature one thing A equals another thing B, it says how two sets are equal exactly when they have the same numbers. S of A equals B is the A that X wants A and X wants B. So, for example, if you want to watch two of these lectures, you can watch two because they have the same numbers. Because one equals one. How many people are going to do it for one thing?

2:30 I can't recognize one thing by the definition that says that you have to find out the numbers. I'm not there. Because it's the same thing how it turns out. It's not the exact same thing. It's not the exact same thing. It's the same thing. I can only ask about the numbers? Yeah. I can't look inside. I can only select a number and sort it out. Okay. Or another thing. And so on and so forth. Steps can be created by collecting together entities from various mathematical domains. Two sets of equal steps may have the same numbers, but in order to discriminate the numbers one needs to know the equality category of each member. So far so good, but steps itself is often taken as a foundation, and then each mathematical entity is a step. It seems to me one notion of equality is the base, instead of one. I'm paramedically modular. I'm now kind of wandering into my own thought, which is kind of an argument of my own. Why do I need to think about what they're asking me? Um, well, just before we could, we didn't really sort of finish the sentence, so you've got to ask me. I said, well, it needs to be for two sets of people, two sets of people to answer the math question, right? Now, then, then probably we'll have to look inside and find out about the math question, right? The second system of, uh, you and me, is the second system of two arms of a guy asking a math question. How do we know it? It's not quite clear how humans decided on the math question. It's not sets anymore. Sets are often built with things which aren't sets. They can't be built with things which are sets. So, notionally, a set equality is based on the assumption that whatever you bring in, you already know what the equalities are. And they can be multiple. That's what I'm trying to make so far. So, already, when you deal with set theory as a practicing mathematician or a person, you are actually working with the equality categories which constitute different elements of the set. The mathematicians are purists. The mathematicians being purists have wanted to say that everything is built out of sets. The elements of set are sets. So is that. Set all the way down is a pure set. So, for example, if you build a set with one element by taking the set with 15 and the other set, and if the empty set has no members, and the set with 15 and the other set has a member, they're not equal, so you produce two points. And then you go around and put something else in, and now you take the two non-equal things and make a set out of them, and that has two elements. The states are not going to be ever different from the original set of one element, and then you create multiplicities. And having been able to bootstrap multiplicities, you can get numbers and all the other real things from that math. So at the level of informal, at this informal, constructive level, it seems like you can build mathematics on nothing, but by a set concept, and it's one-stage motion, one-person motion, you've got it. Well, I'm going to give you a little bit of a brief introduction to mathematics and mathematics theory. So, it's a sort of class, and you need some of the two classes to be equal by accounting for the classes.

5:00 The classes themselves are the classes, and eventually they write themselves. It's just going to be a split between the two classes. So, this is the way our J-class is made up, so it's kind of nice. And we have some physical people asking a lot of questions. Well, it's a concept. Look, look, how do you define things? Pick the actions to a group. If one group is 5, which is also a group. 5 is associated with this unit, which is this unit. The usual picture is the elements of the group are sets. Given two sets A and B in a group, A and B is another defined set. Thank you for watching. Yes, so a vehicle B will mean that a certain collection of abstract particles is outside of time, and that's why it's called a vehicle A. Or you might have some rules for explaining how to make these qualities happen, and it wouldn't necessarily be that's all of us can do. It's like, you lost a poem, two marks over another can't do nothing, so three can't do nothing, four can't do nothing, two can't do nothing, and those are the qualities. And those have not happened to explain in terms of who's the member of whom, except they're equal because our definitions of the language of the language have changed, and the language is the quality. So, uh, so we have to have a concrete level of learning order here in mathematics. We don't actually... Or, you might describe a group by saying that they have some element saying to me that they've got some tools that they can use to do these things. And then, of course, you'll be able to list in general what you think is going to be the next topology. We've got to write a whole bunch of creditors, but wider creditors, and say all those names of people as well, and then go back to this little language for this group, and apply those to this group of people, and do this with these, and that, and other things. So, while this is not exactly a habit concept, but one we'd like to describe to you, and we'd like to come up with some questions that you'd probably want to take. So it isn't really all sets. You can kind of convince yourself that it's all sets if you want to put them into practice, if you want to put them into practice. So what does Tom's point of view look like? If you can figure out how I should slow this down a little, I see I'm running out of time. So Tom's point of view is that let's not do sets, let's just divide our mathematics out of restorative logic so you can pursue logic.

7:30 So for example, if you have some product like G of P X Y, which might mean P is X times Y, then I have to tell you some things about what equality is going to be. A number of things, but not a huge number. For example, the first one says that you're allowed to substitute into the first one. So that says that if p is equal to x times y and q is equal to x times y, then p is equal to q. Or if p is equal to x times y and p is also equal to x pi times y, then x pi x. And so on. So you're going to have to say these things because they aren't given to you off the org. And then you have a notion of equal base structure and equality written in a logic over which you could write the skeleton of what it means to be true without using anything separate. And let's see. All right, I'm about to get to that. So what does it mean in the group, in this skeleton of the group, the group equality of x and y is that g1, g2, and g3 is x. And now you say there are two elements in this system. Now this could be some other algebraic system. This is a system of one binary operation, so this is applied to that. And then you were going to say that two elements are equal exactly when this predicate is true for all instances. And that's it. That predicate is in time for the notion of equality. That is the notion of equality being given by some consistent situation set. So it's done through language. And then you can look at the model.

10:00 What I find a little frustrating here is that all the models seem to be in different sets, although the one where I was talking about a presented group is not precisely the same, but it's hard to think out of the box, and Tom has talked himself out of the box about taking this abstract from the computer. So let's continue this example a little bit. So I can agree, having that notion, I can abbreviate the equality of x times y, which is y times x times y, but you understand that when you see x times y, it just means that it's less. And of course, there's also the ruptural term x times y, which would be already known. Imagine x times y, or s of x, y is the means of existence of the existence of s of z, and z is the x of y and z is the existence of z. And then the group axioms, take this one, composition, says that p is equal to x times y, and u is equal to x times y, and you need the associativity, take this one, right, p is equal to y times c, u is equal to x times v, so u is x times y times c, and it says, like that, and so on. So having the notions of equality and composition, write down the axioms for the group. So that's the skeleton for the first, written over the first order of logic, and that's the set. And you can read it on a certain bunch of problems in this one, and that's the same thing. Yeah? Well, how do you do that? How do you do that? Well, how do you do that? Well, our time has been thinking about it a long way since the end of this situation, that you are the, that you are the, and have foundations. And I have a system of my own, you know, I have a collection of little formalizations like this, each one of which is developed and those are not referring to much of the sector.

12:30 It's just a major part of those bits of language which define what it is. But the idea is that it may not be a common language, but that can be used in many ways. Now, what isn't obvious to me yet, because I haven't thought about it long enough, is maybe in some cases like we've seen, we've had some systems that are really different than the groups we're used to. Certainly there is groups given as sets where you can explain how they can combine the elements and get the reference, and then there are groups that are given by, say, generators of relations that I have to unfortunately go and turn any group defined by generators of relations into sets where I can see the total distribution. So I don't know. I can't point to that as a given. But in practice, because in practice, you don't think, oh, I have all these total installations and I'm multiplying these collections by each other, you don't think, I take this symbol and I bind with that symbol and I get this one. And so you're actually not working in a translated model. And it seems to me that these possibilities are now way down. Maybe we could use an arithmetical map. You already have instructions on how to do it. But in any case, you can take any hearing question and I'm kind of glad you asked to see what happened. So I'm going to put it in the book, so you can look up the book. You can put it in the book. You can put it in the book. You can put it in the book. You can put it in the book.

15:00 You can put it in the book. Step. All of this is part of the writing style of the book, but most of it is from the book itself. For example, up there, you can see that there are a variety of different types of gradients. There are two different types of gradients. There are those that are on the board, and those that are not on the board. There are those that are on the board, and those that are not on the board. Unless you make the equivalence, I'm just trying to figure out what you're talking about. But I don't really want to take more of the, for the sake of the relation between the elements, or all of the things that we talked about earlier. So, I'm trying to figure out what you're talking about. Thank you. I mean, there's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. There's a lot of things that you can do. Here, we put just what I can prove right now in the answer to the category, and we didn't found it in the sets, which is a distinct advantage except that the categories in this category should not be found in the sets, they were designed to. Now, whether it needs to be categorized is one and only thing, but conceptually, I think it's quite likely to understand that there's going to be a subset of non-catalogs. It means you lost the most information because you saw the relation to the number of elements, or more visible forces to be the object than the other. It means you lost the most information because you saw the relation to the number of elements, or more visible forces to be the object than the other.

17:30 I have no argument about set-through per se, which is not the use of it. It's just that set-through is used in mathematics, and this is actually what I want to talk about. Again, the idea that if you would start with logic in mathematics, how far do you go? And so there are lateral questions about how much of this logic you need. The category, first of all, the points are that one wants to use it to compare to the mathematical systems, and so why should it be found in any one or two? Of course, the objects aren't necessarily collections, or they might be collections that I hope to hear a little bit in the summer. There's going to be many things that can be focused on the final structure of the mathematical functions and some of the other issues. Anyway, I hope that gives some picture of this. Let me jump over to the top of that first distinction. Now I've selected two things. So this is the first distinction for you. The question is the following. Here, and I'm quoting. We write z, x equals y, x equals to y, as z. But what about, is x equal to z, as z? Let me read what it says. The same as predicate z, as equals y, is a very important feature that has no counterpart to the sort of quantum theory, which is that its outer variable is z, unlike the predicate symbol p. I can go inside the parentheses, but they ask whether z regards x as its equal. That is, is z of x equal to z true? The answer, yes or no, is blocked in the tip of the hat that's around our first distinction. Now I want this to be the answer. So let's look at an example. The bracket z here is whether two things are red will create one.

20:00 If they're red, then they're equal. If they're green, they're equal. If they're blue, they're equal. And if they aren't red, or green, or blue, then they're equal, because this predicate only discriminates red, green, and blue. So, B and U, not being red or blue, are equal. So, the discriminator Z distinguishes just four things, the red thing, the green thing, the blue thing, and everything else. Which of these things is Z? Since equality is neither colored nor material, z must be as much colored as anything else, but according to its own rights, this makes it the same as everything else. And then he has a whole lovely story. It makes it the same as everything else. It is the same as the number of three. It is the same as all numbers. It is the national debt. Do we think out of the Texas range? It is the Earth, the sun, space, time, truth, also, virtue, and vice. Should it catch even a fleeting glimpse of itself in the era of the stone and evil of the past? Oh, well, in fact, I mean, he goes on to show that there is a new story that we're going to have to work on again to drag it off to the possibility that we're going to have to do it. There's a very, almost, I mean, a paper that you've seen in the process of writing called Stereo Equality. Did you say serial equality or stereo equality? Stereo equality. Stereo equality. Stereo equality. Stereo equality. Stereo equality. You have to have at least two equalities in the system to get anywhere. So that's the posterior division of equality.

22:30 That's where the term comes from. I'll give you an example of this. So equality in the category of sets would be mono-equality. So equality in the category of sets, where you don't have this, would be the mono-equality. Lux-mono. Lux-mono, yeah. I'm starting to think about this. But even in itself. So the next paragraph here is, but such a glimpse is not a likely prospect, for the very modest conceit of our example, which is only a byproduct of color-crediting, all it conceives is the red thing, the green thing, the blue thing, that's the world to it, or at any rate the world that is present to it, while the rest of the world is absent, thus the first distinction, the distinction between what the scene presents and what it can actually distinguish. And itself is the distinction between presence, capacity, and running. It brings us back to our point of zero, so it's less than 400. And it brings us back in quite a different way. I mean, it is actually, by thinking about z, x equals to z as, it leads us to what I call the fundamental distinction between the items in the title of the web. In terms of the form. As I said, these are just some selections. Here's a theory of equality graph. This is showing how to build, this is a sketch of how to build theory of graphs on the basis of two common elements, qualities. And then this would be formalized. You might have grid bounds and grid graphs. You need two qualities here. Here's the transition that I helped a little bit by drawing the diagram. Here's a graph, and I'm not going to represent the graph right here. I'm making just the vertices here, and then I'm going to use the graph to draw the edges. So this is an edge, and this is an edge. This edge goes to the names of the media. And then if you were to cover this, the edge is just telling you something is going to be something else.

25:00 The two is connected to a certain sense. A, the concept is connected by an edge to it. So those are the two fundamental notions of equality that are contained in the graph when you look at it this way. There's the notion of being connected, and there's the notion of being equal, of just being equal. The vertices of the graph have some discrimination upon them, just as vertices. Then, on the other hand, some are connected to each other and some are not, and that's the other equality known as the graph. Two fundamental equalities described on the graph. And then, with the help of that, it goes on to formalize the sum of the graphs that's screwed on the graph. So it's quite graphic to see how you make graphs with two equalities and then you can come back to sets to see... It's quite clear that sets really mean two equalities to a thing about that way because a set can be described in a tree, a tree is a graph, so sets and trees are really the same subject and so since we can describe graphs, it doesn't mean it's too much of an equality between sets. So that's the situation. If you're thinking in terms of sets in the ordinary way, then there's that notion of membership, which is to be, which is to be, out of the space that you've got. I have to go from graphs to directed graphs. But you can go from graphs to directed graphs by putting some asymmetrical graph on the edge of the graph. The standard graph of the state constructed by graphs of graphs. And consequently, you can construct the one-way belonging relation in the 7th period out of these two notions that we followed by doing that graph the original term. Now, I think that he actually did this in a more abstract way than just going through graphs, and then he understood how to sort of go through graphs, and then the essence of the argument is just there.

27:30 But I think this is, for graph theory wise, quite a good example, because once you have graphs, you really have all the mathematics. So you see that you can get any mathematics on this two-to-one. Well, if you have graphs, you have sets. Or if you want to think of categories, a category is just some big diagram, right? Arrows, things between the arrows. Those are the objects, the arrows are the edges. And then there are some rules about when you get the edges wrong. So A equals B and B equals C, but it's also a given that there's an edge. And you see the category itself is just as good as that rule. So I could say things like, well, once you have graphs, yeah, as a language, I mean, I don't, I mean, I'm always trying to run into hurdles here. If I say, okay, I'm going to formalize this piece. And then, I'm going to count all the mathematics on my paper, so I don't have any reason to be informal about any of the mathematics, but maybe with some contortion, like, I mean, you might hand me some piece of mathematics that you have just made up, which really isn't easy to express in terms of graphs or categories or sets. You want them, right? In fact, I can give you an example of mock theory as one which requires a little bit of... Yeah, I mean, think of mock theory, for example, I can't... I've been learning for five minutes and I'm actually not good at this. But in mock theory, the elements that I like as commentaries are these guys, right? And of course I can say...

30:00 This is a graph, but let me relate an anecdote to show the conceptual difference. I said to Frank O'Rourke, who's gone on the graph for this, oh, I study knots. I mean, I've been studying knots for 40 years. They're really graphs. And he said, oh yeah, I discovered that back in 1940. I said, oh, really? I thought that was kind of strange. Because my first hand was on this graph. It was the beginning. Much important. What did you do, Frank? He said, well, he said, well, you draw these dots and we turn them into graphs. Like this. In fact, it's, in a certain sense, it's right. So, I mean, from the point of view of graphs, you can probably just pick any one of those vertices, or the edges, and maybe it's not so explicit over here. It's just something that we're trying to improve. So, of course, it wasn't exactly a huge advance for you to do that, but on the other hand, that's the difference. And over, but now if you jump back to the diagram itself, The diagram could be said to be an S-graph with extra-structured vertices, and now I have to start telling you what the extra-structure is, and I have to say, well, you know, I kind of did, but I'm going to do this on the crossing line, and the other one goes over, and you should think of this entire little arrangement of line, P, cut line, and so on, as a vertex diagram. And I have to go through a bunch of contortions in order to terminate the third one. It's not a terribly difficult contortion, but it is a contortion. And it started probing me a little farther about, well, what do you mean exactly by that? And I then started saying things like, well, you know, I don't need to set up points and do scales in this sort of way.

32:30 It's only this way out in here or across here. How do I make this discrimination? There's a whole bunch of proportionality. Whereas if you're regarding it as opinion itself, the notation that you're using, just like the integral of e to the x, then it is the notation that you're using and you know how to do it. And it's not acceptable to performance. So, any other questions about this theory that was brought up? Did you get an answer to what? Did you get an answer? You said, why are you worried about this? It is not really static. I am open to listening to what people think about this. I do think it kind of explains itself. I can say that the rather Australian way of thinking about mathematics and physics is still even in mathematics and physics. I also like to think that it's quite like mathematics. Yes, you're right. So there is this, there is a . But, in a certain sense, this is very close to the way that Thomas didn't think . Yes, the two notions of equality are connected or equal. I'll let that out. One thing is this, that if you think about the first station,

35:00 The general question is, how does the perfect picture make the face of the quantum measure, but the perfect picture is about having the world apart from the same and apart from the same, not sitting next to the same screen. So, how does that work? Well, we didn't actually talk about the perfect picture directly, but we have talked about it earlier. And you've rather seen it as a bit like this, on the 10% occurring in the air. Preparation and protection for this is not stock, and we would buy your head a lot of times, but I agree with you, all right? Now, this is big for the stock of mathematics. It just means that this records incredibly many things. So, this is UIJ, and this is VI and UI. So, it's tying the vectors into vectors and co-vectors into numbers. And when you state psi star, it's reversed. And now, if you multiply the psi star times psi, you put the diagram in like this because there are two B's. I've reversed the order of this so I can match them up. And then B has to get tied into U star and also gets tied into U. So the index here, I, becomes I here and I here as well, same index. And you're summing them against. So you can think of it that way. And then you can look at that and stand back and say, look, so that's what size-to-size is. And it required the preparation and the detection in the world in order to know how to make this happen, and then with the preparation and detection in place, the unitary operator is important. But if you unplug either of them from the circle, then the operators would cancel each other out.

37:30 And so this is the quantum picture of cutting the world in two. So, there's no question that, you know, there's no question that, you know, there's no That first distinction of talent and this notion of quantum distinction are somehow very tightly related, but I think there's much more to say than that. But by floating the notion of first distinction out of Boolean logic into Z, there's the possibility of then a reattraction of quantum mechanics and I'll have to worry about the fact that this whole way of thinking seems to be tied to Boolean logic. So that's another really strong advantage of this way of thinking. Your idea or was it Tom's idea? It was his idea. Oh, okay. So you were doing this as a favor for Tom or a favor for Peter? A favor for Peter. I'm sure that Tom was doing some very important mathematical things. He was able to talk intelligence into the body. He really is. So I got them together. We'll talk to Tom, and we'll read what there was before, so we can see the parts of it. And I have to say that I'm really, in fact, here, asking you two questions.

40:00 It's very interesting how, I don't know if this is far better than the previous one, excuse me, this is far, I don't know if this is better than the previous one, but I don't think it's the right one. The different kinds of equations that you can say, well, unfortunately, that idea has already changed. And what it actually meant, that's what I wanted to ask you. I have a question. There are tables that I can put in my pocket where I show all the presentations. I think there are late models. And so they add more, or very more effectively iterate all the lessons that are expressed in zero. And with that insight, I was wondering something late. How do I keep probabilities out of there earlier? So, I'm making that comment because I don't want to complicate your comments too late. So, that's the first part of the comment. The second, yeah, the second question you asked is, I've asked you this before,

42:30 but maybe you were busy at that time, but I didn't hear anything else about it. And I've seen that before, back to the Arlington space and the Franklin space, but I've never seen, I've never seen where the control line enters in here so that you could have additional problems. So I'd like to ask you in this forum, or you can answer later, how do you do a Franklin space with additional problems? There are two wires. Inclusively, it's just wires that you're taking notes from, right? And that's one state and the other state is Chris. Chris, don't touch me. Just, there is a problem. I'm not complying with any of you. Now, if you want, and of course, when you press the switch, it also goes through the algorithm. Thank you for your attention. In other words, it was likely to be a model for how changing the state of this would cause this to go from one place to another. And then if you do the same with topology, then at least I'd have sort of a mapping of how those things are related, especially to conditionality. And so that was sort of bridged back to conditionality, which... In a lot of the traditional systems, they're not linear. You know what I'm saying? So there's a lot of discussion about how to do that. Well, yeah, I think you should talk about what happened. There are a lot of different ways they existed. They didn't come out of their pocket. The change was in the description, so they were dropped. Right. They were dropped. Everything on the page has to be messed up. That's what you see. So that's the whole question again. How does it be formed?

45:00 Here's a little bit. This is... There are a number of different types of mathematics in the world of physics. The world of mathematics in particular is very complex. The world of mathematics in particular is very complex. The world of mathematics in particular is very complex. The world of mathematics in particular is very complex. The world of mathematics in particular is very complex. The world of mathematics in particular is very complex. As I've mentioned, and I've been certain with you, you know, if you're going to assume that we're going to have distinct things, let's say, if you are a student, if you've mentioned that we're not going to have the same thing, so what do you think? Yes, that's right. I think in some theory, you can do that. I'm curiously about the concept theory that the seal doesn't eat the fish even though it can put them in the same closet. Well, the seal didn't eat the fish. The three fish turned into one fish because of sex. Thank you for your attention.