Noncommutativity at the foundations of discrete physics

0:00 So, um, the title, uh, remains the same. Not community, city, or foundation. So, in the kind of sort of hierarchy, one compares bodies to the next one. And, uh, then you can discriminate among any vectors. There's a couple of questions. Why matrices? Uh, with vectors, you have more possibilities. There's lists, ordered lists of, uh, states. And, uh, depending on the motion, you can discriminate from those. And I wanted to back up a little bit and make some remarks just because they're interesting from the point of view of thinking about this basis, thinking basically about these things. And also because of the thought of yesterday, about what is the relationship between this way of thinking about discrimination and the background laws of science and technology. What do you mean by discrimination? What do I mean by discriminating discrimination? Well, I don't have a definition of discrimination. Discrimination is something, is an action that's performed by someone in making a distinction. The major distinction is... It's quite circular, and the only way to get out of it is by experience. It looks like a good batch. Pardon me? It looks like a good batch. Pardon me? It looks like a good batch. Pardon me? It looks like a good batch. Pardon me? It looks like a good batch. Pardon me? It looks like a good batch. Pardon me? It looks like a good batch. Pardon me? It looks like a good batch. Pardon me? It looks like a good batch. Pardon me? It looks like a good batch. Pardon me? It looks like a good batch. Pardon me? It looks like a good batch. Pardon me? It looks like a good batch. Pardon me? It looks like a good batch. Pardon me? It looks like a good batch. Pardon me? It looks like a good batch. Pardon me?

2:30 It looks like a good batch. Pardon me? Now, there's another issue. There's another issue, which I said, because that tends to be my bias, that discrimination is a discrimination made by someone. I do take that point of view. So that means that if there isn't any deterrent, there isn't any discrimination, and the world could be any better. But, on the other hand, if you believe in a wonderful place in the external world, then it must have lots of discriminations. Or else you believe in a wonderful, external world intended to protect those in the nations in it, except the ones that we make. I'm going to go on with the first round, though, because I don't think we're going to be able to do it. I'm going to go on with the first round, though, because I don't think we're going to be able to do it. I'm going to go on with the first round, though, because I don't think we're going to be able to do it. I'm going to go on with the first round, though, because I don't think we're going to be able to do it. I'm going to go on with the first round, though, because I don't think we're going to be able to do it. I'm going to go on with the first round, though, because I don't think we're going to be able to do it. So, it's the framework of ordered lists of states. Now, what about touching the globe? Spetsagram is a diagrammatic notation, and that's another of the motives for bringing it in, because diagrammatics is quite interesting. We can do it there, and then we can have a piece of mind as well. So, for Spetsagram, there's this mark, which stands for distinction. And, of course, I also draw some curve, like this, to indicate some distinction. But the mark is the one that's distributed. And then there's a multiplicity of interpretations that goes on here that's really actually quite interesting.

5:00 If you use that mark to indicate the impact of crossing the boundary of some distinction, which will take you to the first distinction, which at one point would have to go into the first, and keep working until you go to the second. And then, if A denotes the state from the inside, then A would mark over it and denotes the state from the outside, so it's a B, so A cross. On the other hand, you get something different if you start throwing away excess baggage, excess computational baggage. If you said, well, I will take my mark literally and let one side be unmarked, the other be marked, then you notice what happens here. Now the inside is unlike the outside, it's marked by B, and if you cross from the unmarked state, you get B, and if you cross from the unmarked state, B, but now equality is suggesting to you that B would be taken to be equal to the unmarked state, and then you would be access to one thing, the name. There are also a number of different forms of mathematics that can be used in the field of mathematics, such as algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, That's almost all. But this condensation of symbols that could be regarded as an act or as an act is very significant and the diagrammatic notation has to hold on to that and think about it. I find that there's still quite a few things about this that we haven't reviewed. Not enough time to think about it.

7:30 So that's right, but the point is that the discrimination and the operational discrimination at a certain level are not the same. And that's important. I think it's very important to stay open with that particular domain, that there's always something that people don't know, and that this guy can add information, Oh yeah, that's right. I mean, if you start to think about how this particular kind of thing is happening in science on mathematics, yeah, I'll identify that in my mind, or I might take this as a projection. There can be a multiplicity of things out there. I'm going to take a break for a second, but I'll get to you in a minute. I'm stoked with this comment about both of them for the remaining duration. Here's the distinction, and the outside is labeled with its name. But on the other hand, in this form of representation at least, you don't really need to put the name on the outside because you know it's outside. So again, by removing the excessive notation, you don't have to put the mark on the outside. The distinction itself could be taken to be the mark itself, and it's not just the mark that's being taken. And so we find that the market, next to itself, could be regarded as giving the same information, and that's another equation that we can understand.

10:00 Now, these of course look very similar to boolean expressions like AA and Z, and so on. Now, how is this, now we're trying to figure out how to answer it. Locals, as in topology. Oh, locals. The point is, these are just sort of very basic ideas that are encoded in this notation in a way that doesn't make sense and also suggests to one that we could... In any case, what's the translation between that and what we're doing? I just wanted to mention it because, you see, XOR, if you were to write it out in terms of the Boolean operations of Andover, it's a little complicated. But XOR is very naturally expressed in terms of the system directly. Anyway, divide, define, A is B, B is A, B is unknown. Remember, XOR is superimposing the two marked states, and then if you wanted to translate two zeroes to ones, you'd mark one, zero, eight, and then you might write some vectors with marks like this.

12:30 And at this level, I'm not saying anything very interesting. I think what's interesting is when you start playing with zeros and ones and thinking about discriminations, to remember this story or to construct some other story for yourself about what you meant by this. So, I'm ending this section by a question. Why shouldn't we translate loss of time and really algebra and complex sort of hierarchical questions into questions? Well, there are a number of issues I just wanted to remind people to read it. The epistemology, of course, and in particular, the meaning of these, I was sliding the meaning of these books in a number of different directions in the last few minutes, and then we'll talk about the skeletons and how they're supposed to be an entire question in this place. Then there is, as I said, the notion of distinction not as a separation but as a joining. It's very inherent. So if you're thinking about the combinatorial hierarchy or those ideas as something fundamental about how the world comes into being, then you're talking about a world that was joined and somehow became unjoined, but is able to be joined and doesn't want to just end up just getting more and more distinct. And then, finally, there's the matter of time and process, which comes in, naturally, in this image, in this law of performance, by thinking about a subverted distinction, that is, by thinking about what would happen if there were a tunnel between the inside and the outside of the distinction, if you weren't so distinct after it. If you had such a tunnel that was self-referenced, then it would be possible to feed back from the outside into the inside, or that a form would sit inside its own computational space.

15:00 And if so, then you're looking at something which is beyond moving and watching, quite explicitly, because it would remark to them that this is some kind of a person. So there are different exits from this. One is topological, of course. They add topological structures to space to take care of that. Another is temporal, because you can think of this philosophy as a mathematical form. And then, of course, there's the medley of tunes that can be played by different kinds of sub-levels and forms in the book. So time arises as a way of folding the paradigms of sub-levels. And I want to talk about time later, not necessarily at this point. When you do that, I don't want to try to define any kind of topology business, but the point about any topology is that you can then deal with boundaries. So, in a simplest instance, you might think of a boundary that you've drawn here instead of... that's part of the crystallology, right? You want to draw something like this to indicate a set. And then that line of the boundary which exists as an actual entity in the topology sense of the plane doesn't exist in your logic yet. There is no boundary there, it's just an indication of the difference between the inside and the outside. But if you wanted to add a little topology, then you can have the boundary in there. The boundary could be invariant across, for example, or other variations of that. But the point I wanted to make is that, the real point I wanted to make is that time, process, and topology come up immediately. So that's the end of the law of components. Now what's the back and comons? I don't know. We're interested in the structure of discriminative law of sets, like this one, which I wrote the concept of the zeroes and ones. And this is the vectors here.

17:30 And this isn't a minimal one or anything, it's just one that might be produced by some process like this program universe where you start with one thing and you either discriminate something from something else or you get something new from discriminating from anybody. I can put it in, and on the other hand, if you can't do that, then you're going to end up with a row of dots randomly going back and forth. If you go wrong for a while, you will reduce some discriminatory sets, and that process has been studied. These systems that we're looking at are commutative and associative. You can also multiply them just as well as add them, multiply them in a different way. I think crossing a bound. Crossing a bound. And of course, A and A is equal to A. And you also have discrimination in terms of multiplication, right, but it's not, it's a little awkward. And A and B equals A and, not A and not, really, but as far as it's pointing out, yesterday you might want to use it to project out what's working for you.

20:00 So you want to add multiplication to the system. And you also want to think, at least in my point of view, completely naturally at this point. There was a little unnatural perhaps at the very beginning to say a very wordless statement, and so that any given collection was a response to all the other permutations. And so now at this point you can feel glad it wasn't. So that's the first instance of arriving at that theme of non-mutability. And let's look at how this might play out symbolically. So here's a vector A1 to A n, and here's a permutation in the symmetric form of n letters. And sigma 1 through sigma n is my permutation, so what I mean by that is, I guess sigma might be equal to 3, 1, 2, which means that we can pair with 1, 2, 3, and reorder 3, 1, 2, alright? So sigma 1 is 3, sigma 2 is 1, and sigma 2 is 2. So then you reorder these according to the permutation, and that's the actual permutation of the vector. And I'll have sigma applied to the vector as a beat up of sigma, and if I have another vector, then I could consider w times sigma applied to b, which is w times b to the sigma. It's good sometimes to get the sigma out of the way. And so the claim is that certainly that combination of vectors, taking one and multiplying it by another, all that is natural, and here's a little fact of life, that if you think of multiplying another vector by one of these and applying W to a vector and applying sigma to that, it's the same as taking W and permuting it. So what I'm really saying is that if you were to apply this, if you've got any vector here, the question is what happens when you multiply them, right?

22:30 And multiplying by the vector sigma applied to W is the same. I'm sorry, multiplying by W and then multiplying by sigma is the same. The reason is obvious, although I think it's a little confusing, so let's just look. If I multiply by a, b, and then I apply sigma, I get b, y, and a, s. And that's b, a times y, s. So in general, you have this algebraic structure that you have a vector times a permutation, and I put the permutation in the back to remind me of this permutation. You put the permutation on the other side, and the vector gets permuted in the process. Well, by now we've actually reached matrix algebra, but you might not know it. You can consider, of course, sum of the vectors, and the sums are going to be in the form of a vector times a permutation. Since I can permute a permutation and a vector in my way, I can put the permutations all the way on the vector. So if I'm thinking of doing permutations and having some vectors in my way, I'll put all the permutations over on the right of the vector. And then I can add some of these together. What kind of an algebra am I getting? So I'm talking about this algebra. The algebra consisting of sums over elements in the symmetric group at the given size, vectors of that size, vector indexed by that element in the symmetric group, times the permutation, that's the general element in this algebra. You can multiply two such things together and by the rule that I just gave you, you can multiply the individual elements together to get another one of the same type. So here's an algebra, a kind of proof algebra of vector coefficients. And here's the fact that A sum n contains the usual algebra and mathematical matrices, and let's see how this works. Here's a 2x2 matrix, and I will write it as the diagonal matrix AB and the anti-diagonal matrix BC, and then I'll take my diagonal matrices by definition to be vectors.

25:00 Because if you multiply two diagonal matrices by one another by the usual rules of matrix multiplication, it's the same as the order that implies multiplication vectors, which are typically multiplication vectors. On the other hand, the BC over here is a diagonal matrix multiplied by a permutation matrix. And so I'll call this one sigma. So while sigma is a general permutation, it also represents... And so now we've written this matrix as a sum of a vector and another vector times sigma, and that writes the two vector matrices in the algebra A2, and that shows you that the algebra A2 Yeah, that's right. So I have some vector, some, yeah, so I have this vector algebra, and the vector algebra is the one where I add, and you can add four times and multiply four times. And then there's some range of underlying elements, A, B, C, and D, and so on. It could even be a number. The coefficients could be in C2, which is where we started, or they could be complex numbers, or they could be some non-negative degree, it all depends on where you want to start. They could be matrices. They could themselves be matrices, yes. So, we think this is a way to produce new algorithms. For ordinary matrix algebra, you might think you want to take C2, or real numbers. Sorry, I'm speaking very badly. It's not clear to me that there's anything else other than matrix algebra. Wait, there's A sub m and there's matrix algebra. I mean, given the linear algebra you can think about, it's matrix representation.

27:30 Oh, well, for two by two it's equal, but I'll show you what happens in terms. Now the original idea that I was thinking about in relation to this actually goes back to this process of where this came from, which is why I'm studying it. Because I was trying to think of the matrix as a kind of two-dimensional discrete wave form. That is, it was this, I think, wrong pattern. We started drawing a pattern that made this two-dimensional pattern, and somehow the matrices freeze on that pattern, just like in the 0-1-0-1 solution to the paradox, it freezes to 0-1-1-0, and so the thing with the matrices is that it freezes, and... And then I wanted to dissolve them into one or two. But in any case... Having brought this forth, we can then articulate matrix algebra two by two this way, right? Here's alpha plus beta sigma, alpha being xy, if you don't have it, you can keep that one a lot of the time. Alpha bar would be alpha to the sigma, where r is yx and alpha is xy. And if you have another one, then you multiply it, and it's like an overpriced number, and that's not what it's supposed to do. It's a point. It's a point. So, you can think of even two by two matrices as kind of like a complex number system. Here's a vector of epsilon, which I'll call epsilon 1 minus 1 squared is equal to 1 minus 1, and epsilon to the sigma of epsilon bar is equal to minus 1. So if I let iota be equal to epsilon times sigma, that would become like iota squared is equal to minus 1, and that would be equal to the absolute value of epsilon sigma epsilon sigma, putting sigma past epsilon turns it into epsilon bar. x1 bar times x1 is just 1. x1 squared, I've got groups.

30:00 x1 times x1 bar is 1, minus 1, 1, plus 1, minus 1. The system, the reason why i squared is equal to minus 1 is because it multiplies the vectors. 1 minus 1 minus 1 equals 1. So there is I, and this isn't an accident, this is something we're familiar with, because if you look at the matrices of the form a, a, b, minus b, then you get a plus b times epsilon-6, and b minus b, b times epsilon-6, so that's the usual two-by-two matrix representation. So complex numbers of two-by-two matrices of a certain form in the whole set of two-by-two matrices is a kind of generalization. What happens with higher-order matrices? Well, let's look at a catagory, and we'll see what the general catagory is. I take a 3x3 matrix, and I write it as a sum of bits, each of which is supported by a permutation matrix, and the sixth permutation matrix is the matrices one on one. And if I do that, I find that every entry in this matrix is repeated twice. We can use an exercise to see that, in general, for an n-by-n matrix, every n-3 can do the same thing, repeating n-1, n-4, n-times. So, in this case, you see that we resolve the original matrix into some vectors times the system of equations. Now... How do we solve the matrix? Well, what are these things, right? I'm taking a permutation matrix, it's a matrix of zeros and ones, so that every one occupies a unique row and column. And so for each one of them, it's also the set of things that you need to compute and determine. And then, the reason why, and now you see why I said contain... Because in this form of matching matrices to elements in A3, we get a special set of elements in A3 because of the interrelationship between the vectors.

32:30 It's not just A3. If I take an arbitrary set of six vectors, then they will not fit together into this pattern. Well, yeah, I mean, that's the whole point. If I have a vector times one of the sigmas...