Quantum Computability & Nonperiodic Tilings as Nonlocality

0:00 And the output comes out on the left, the universe algorithm, we feed in the original numbers here, and then we turn away, and we've got two numbers on the right. Of course this is a mathematical, one could take, and so on. The machine itself, let's say it's so-and-so, we create an idealization.

2:30 The machine itself is both finite, they take, there's a zero, particular type of a table, particular machine, you can go down the list. So what it does is it replaces each other's step by one, one, zero by one, and moves one step up, one step to the left, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one, one And Turing, of course, had to give us a long argument to explain why anything that you would like to call an algorithm can be coded in such a procedure. Now that's an obvious thing. One has to really persuade oneself it's reasonable to call that. We've got the Turing machine, called the Church-Turing thesis, which any of you who didn't read it will understand what it is. But now that we have such a precise concept, it's the Turing machine. I should say that in my description of the Turing machine, the thing either goes to the right or to the left, but you have to have something to tell you when the machine stops, otherwise you never know when you've got the answer, so we have certain, I have certain Turing's originals in this kind of way, the idea is that anything can be effective now, which is called the universal Turing machine.

5:00 Now one of the things Turing did was to show that you can in fact specify for one particular machine what imitates any other Turing machine. So that's what Turing machines. The way it imitates it is that you can read in a minute tape first, and that minute tape essentially reads that tape, and that minute tape pretends to read the minute tape is coded by a coding machine, and that minute tape is coded by a machine.

7:30 You can now code it and put it back into a number. So now every machine, for example, F, which we were talking about a moment ago, and Tu then acts on a tape, which is the number M followed by the number M, and gives you the result of the particular Tn acting on this procedure, or acting procedure, to code for the universal, to actually feed it on, to take the machine on itself. Well now, at the turn of the century, it's an algorithm. So clear enough, this is where you might find a few marks.

12:30 If you knew whether that machine stopped or not, if you knew it didn't stop, We're showing that there is no such H. There is no algorithm that's got the following machine.

15:00 Well, there's one plus, and then, first of all, the number n, the key number n. We want to make sure that what this means is that if Tn acting on n doesn't stop, if H exists, this is an entirely algorithmic operation, and this operation, Tk, Tk acting on n, So you might think from this that there are some very awkward, and in fact this doesn't tell you anything of the sort, the very procedure tells you the answer, if tn acts on n, if it stops, you know it's about tn, simply just, it's rather trivial, it's not much use, any use, this would only be of any use if you could prove one, but the thing is that whatever h you take, you can then

17:30 This machine, which I did before, I think we know the answer. The first one is wrong, it's Quorra, which is a possibility, therefore it must be what happens. Tk x times k doesn't stop. So, given any of these manufactured tapes, it's actually not the universal Turing machine you can quite easily use. The point I'm making here is that it doesn't know that this particular operation is aimed at something else.

20:00 It can be formalized by using all types of mathematical reasoning to a set of chemical rules. If you have stringed symbols and you then cease to worry about what they mean, what you need to begin with, if you formalize all the rules, the proof will be a proof, then you'll be smothered, you'll have some coding, you'll have to make an order, you'll have to make one, you'll have to make two. You consider things which we call propositional functions. So in your logical system, you would have certain strings of symbols which would represent if it's a proposition, a true or false statement. It depends on some number, for example, the number 2W, the sum of the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number, the number

22:30 You see here the start and end of the axioms, the point is to remember this. Now what you do is you consider a particular function, there doesn't exist any x such that pi of x is equal to p w of w, it's the same thing on itself, this means there is no proof of p, if w is the proposition function of the number w, that's now some function w says, but there's no proof of p w, w equals k, the reason it looks like that. The point is that this thing is something which isn't provable. This is not provable within the system because if it were, that's the simplest way of looking at it. It asserts that it's not. It asserts that itself is not provable. So if it were provable, it would be false. If it were provable, it would be false because it asserts it's not. So therefore it isn't provable. The point is that within the rules of procedure and the axioms of the system we've given, by its very statement of what it's trying to tell us, of course you might say, well, you've given the axioms wrong, but the point is that if you start with axioms, the whole point of formalizing that rule is that you choose axioms and rules of procedure which you believe are good ones. They're supposed to be valid mathematics. So it's got to be, you've got to believe that already, but the point is once you believe that, you can believe that.

25:00 If you've chosen your axioms well, then you must also believe that PKFK is true, and we've seen it by this argument here. We've seen it's true, but, nevertheless, it's not provable. The point I'm trying to make here is not so much that you can present a proposition which is not provable from the axioms, but you can actually state this within a mathematical structure, and what Goebbels showed, if your axioms are... Broad enough to include arithmetic and even particle theory. We can see this is true by some sort of insight, by really much the insight of the logical system that we started with is doing things which are valid operations. You might believe those, and you must believe the Baylor Proposition too. The Baylor Proposition. Alright, well this is meant to be an indication that, because we can see this, somehow you're doing something that the formal operations can't do. You know, people have argued about this on and off for a long time. Actually, I will just, not to come up with any others, but present an argument of this sort, which is supposed to indicate that we don't use algorithms when we think.

27:30 Well, I think I'd like to present the argument in a slightly different form, because what people would say, I mean, what you might say is, okay, suppose we have said algorithms. I'm using the word algorithm, and we're synonymous. The question is, does one use one, because you might say, well, If a brain operates algorithmically, then according to some logical system, then if you knew what that logical system was, and if you knew that that was a good system which only generated genuine truths, then, of course, the counter-argument is, why would you know? When mathematicians communicate with each other, they somehow convey something which is specific.

30:00 Which specific people may be using. It seems to be talking about some general algorithmic sequence which apply to humanity as a whole. All mathematicians. And in my text I do have to believe there is some logical system which we all use. But that logical system is so complicated, so obscure or something, that we can never understand why it works. We'll learn about that after. It's somehow the way you would get out of this. Here, it would be to say that the algorithm that we all use is so complicated and so obscure that we can never tell whether it works or not. Well, that seems to me to be quite the opposite of the way that we actually do mathematics, is to reduce things to simpler and simpler things. We don't bow down to some complicated algorithm. Now, there may be subtle steps. There may be people arguing in certain roundabout ways and so on. It's a good case. What we do when we do mathematics, procedure, is something that one seems to have to... You can do lots of things unconsciously. That part of thinking, which is consciously carried out, as I say, could it be questions about brains? If you go into physics, could it be that there's... It seems to me that this question is one which is not really one theoretical device known as a billiard ball computer. I'm talking about... Let's talk about new turning mechanics.

32:30 In terms of mechanics, people think about the Turing machine, but it's leading, but that's perhaps this is a device, a theoretical device, described by Fred Kuhn as topology. They show that you can actually imitate the action of a Turing machine using Illibro, which is a match, just by the Gs. I'm just giving you a particular instance of what we call a switch. The Illibro comes up here, and it's all going to have to work every time. It goes straight out there if there isn't one coming. If there's one coming in here, they hit each other and bounce like that. This one comes out as though it hadn't hit anything. It's actually the other ball. This one comes out of here and sits right here. So basically, it comes out of here if there is one here. It comes out of here if there's none. So by combining gadgets like this together, it's uncomfortable.

35:00 Turing machine and therefore principle. Turing machine is a long way around in a sense. It's not really a deterrent. It's a deterrent of mechanics with, you might say, walls that remain solid, and actually quantum mechanics, so it's all right. But the trouble with this sort of thing, certainly if there's energy in the balls, is that it has certain disabilities involved, and I've just indicated the sort of one you have in rigid ball mechanics. And imagine three balls coming together here, all equal, almost at the same time. It's a triple collision. Now, I'm going to hit just before C comes along, or if C and A hit just before the way they bounce out, it's completely different. It's a very, very subtle difference. You're going to encounter face-to-face, in which it's a nuisance.

37:30 If the volume weren't preserved, we could imagine it being smaller, and therefore pushing it down into a smaller region. Once it does this sort of thing, you can't really... This is all very hand-waving, turning mechanics. It's a bit difficult. But it seems to have to rely on funding mechanics. Let's say here, even solid walls here... Depending on quantum mechanics, you can't really discuss these questions within the framework, so in quantum mechanics, it only depends upon the procedures as it's used. Different people have different views. So this procedure is unitary evolution, or Schrodinger's equation, very much like classical physics, you have a state that evolves. But then if you actually want to use that state to tell you what's going to happen in the world, well you cannot at that point. Magnified to a macroscopic level, you have to bring in this other operation here, which is called state-of-the-art reduction and the collapse of the wave function, which is where the problem is this time.

40:00 It tends to get, I think, a slightly misleading impression about quantum theory, because there's a lot of discussions, you think they go well, since the quantum objects behave in some uncertain way. But that's completely untrue. Electrons behave according to unitary illusion. As long as you're at the quantum level, This operation U is a very precise way in which we can evolve. It's only when some power effects get magnified up to a maximum fit together, and if that's listed, the suggestion indicates how completely different these two processes are. U is deterministic, it's continuous, it's local in the configuration space. State factors are discontinuous and non-verbal. Sometimes people try to deduce this and have trouble with these procedures, because there are so many different ways. Two operations that you don't quite know when you should be using one, but you should use. Rough terms, you use this one when things get big enough. You use that when things are small in the sense of small energy and locality. So you have this famous experiment where what you have in this time that's going up in the picture here,

42:30 you have these two photons in their own state, and then you take observations of the two ends of the room, many metres apart. And these observations are polarisation measures. Each photon has a thing on its own, and you observe it, and it may have certain combinations between one thing and another, with the L-photon, and this is exactly which observation you observe beside.