Non-commutative geometry / Panel discussion, incl. G Longo, FW Lawvere, A Peruzzi (contd.)

0:00 ...trying to explain first the motivation for me for organising a computer conference. Why could I have that job? One of his sayings was entitled, we know what quantum space is. Of course you know somebody with a title like that.

45:00 So after the talk, he was quite a nice guy. And then we went at dinner and he told me. And at dinner at some point I mentioned to him this bad fact. He didn't say anything. He left. And he came back from his office, took papers, and what was absolutely amazing, he didn't even know about this very fact, but he had thought of abstract philosophies in quantum grammar. This is the way things should not be common. In other words, he had found a very amazing fact that, you see, we are talking about time. Time, it is something which has graphically evolved. In fact, you know, first there was Newtonian time. It's very simple. The cosmic rays which meet the upper atmosphere have neurons at 20 km from each other and these neurons only live enough time with their upper life time to live for, to go, they go at the speed of light. So you can say normally they cannot hit the ground, but they do hit the ground because of special voltage. And of course this change is mind blowing. You cannot think about it because the Newtonian idea of time is so much ingrained in us that we cannot. So then you go to general relativity. In general relativity you can say it's a cross in a space. That's true. But what you find out is that there is no time.

47:30 Because when you write down the Einstein equation, you find that the Hamiltonian is what is called Wittgenstein. So there is no time. Now, this guy comes up with a dream, it's a very deep dream. And he has found the following fact. But the correct idea of time should be like this, that there should be some thermodynamical state of the universe and that any such thermodynamical state of the universe would create its own time. We had written some semi-classical equations for how the time would be created from the state. There are the semi-classical equations which create the evolution of the earth. So then, you know, we were astounded. We write and we wrote very complicated, which has not been understood at all. But I mean, we sort of got the absolute conviction that things should be turned around. That means that if time passes here, it's because we are in the 3D plane of radiation. It's a purely thermodynamical effect which is eroding us and which is coming from this thermal path. It's because we are in the thermal path that time is passing. I have one remark and one question. The remark is about what you say about proving things in mathematics. It's amazing that computing is a very different world. Although it shares many techniques and many mathematical approaches, there is something getting very interesting to me. What is the C.O.M. in mathematics or in the physical view, mathematics for physics and so on? It's sort of a general statement that gives you the power of understanding many things at a time. Okay? And this is what the theorem is. It gives you power to do things, and this is what you call a theorem. So somebody, I don't know who it was, made some harm in the subject by calling theorem any sentence that is proven in logic. It's not exactly what we mean even in logic. So what is the analog of that in computing? We start having these things now. For example, I take my portable phone and I call my wife. This is a theorem. This is our... This is exactly what I would like to... the theorem that I can call... It's really interesting because there is just no way to write that with... How do you tell that on the phone? If you can't tell... If you cannot tell it on the phone, it's not really language. It is because of the necessity of these things that category theory is important. The point is the style of language is that mathematics somehow is able to deal with more formulae and computing is not.

52:30 I think that Bernard pointed out that it's very important, very important, because sometimes some people say to do mathematics you need imagination, and they need to say a few things, because you can say somebody will be a mathematician and so on, and he has a lot of imagination, but of course... Imagination is a much more precise notion in mathematics, and I think Bernard put his finger on, I think, something extremely important, which is related to the discussion just now, that's why I mentioned it, which is roughly speaking the following, you see, if you write symbols like the ones which are on the blackboard, they are good examples. For me, at the moment, they are devoid of confidence, and a mathematician is precisely a person who, when he is given a sheet of paper, is full of problems. That's it.

55:00 He's able to create in his brain images. But it's not a question of freedom of imagination. No. He's able to create the right images that will allow this to function and will allow him to get amazing consequences of this and that, just because his brain functions in such a way that he can associate with these sort of silent and stupid letters. Sounds interesting. And so when we talk about imagination, I think for the lay people it's very confusing because they think it's just a question of being imaginative. It's not at all. It's a very, very precise and controlled process on which we depend. I mean, we depend with major will because if you are presented with a subject that you don't know, this subject will be completely silent and dead. Until, precisely, you have created in your brain these images that are out of function. So, I don't know about that. So that's why, you know, if you write something like that, for me it's simple. I just want to make one remark. One thing I think is important about geometry is that it's immediately meaningful. And perhaps that's why we like so much, we view it as a paradigm of meaning. It may be wrong, but yes, it is. I think we should think about it. If all our perception of the world was by electric waves, perhaps mathematics would be better. I just want to add one more thing. Hilbert made a program to know if you have a good look at, I think, the logical deduction. There is one problem that I think he did analyze, and I think which is more or less what you are looking for. Imagine you have a computer, and this computer produces theorems in the degraded sense that you have here.

57:30 Then we shall have a real problem, which is among these myriads of statements, which ones are meaningful, which ones are meaningless. And to me, if you want, this is the question. It's not how we create a proof and so on. No. It is what is the meaning. And whether or not there is a meeting, then your philosophical stand will play a major role. If you are a formalist, you will say they are all the same thing. If you are, what I am, a Platonist, then you should believe that in fact somewhere out there, there is some sort of archaic mathematical narrative, some primitive mathematical narrative that we are trying to explore. A little bit of that reality, or it doesn't matter, or it's interesting. Now, I believe that, you know, after a bit there, there is a major issue. And this issue is, can you put an evaluation function, like in chess, playing, and so on, which tells me, okay, by comparing computer programs, this is meaningful, this is very good, okay? How do you do that? I think it's the same in computing. If you go to FNAC, you can buy a software that measures most of its meaning. It's an absolute essential to know the difference. Fine. But OK. I would like to make a vague evaluation function. Not as critical. But anyway, it's about that.

1:00:00 Convincing hundreds.

1:02:30 There are actually dimensions in Newtonian space, in all of those. We talk about dimensions, one in two dimensions, or a category theory is very successful in that it tells us an awful lot via its dimensions. And dimensions are quite clear. We share these dimensions. It takes a little while.

1:07:30 They are addressing the following question. Imagine I take some axiomatic system, which contains enough to talk about our experiences. And then, okay, so I take this axiom. And then I take other states, which I know. Then I take its negation, and I add it as new action. This I find very puzzling for the formalist, because you have a non-formalistic system, but it contains the positive. No, no, but I was just thinking about this. No, no, I understand. But what I mean is that, you know, if you want to cling to something, you have to... There is some saying, a question, which is the... It's not only practical.