5th HLF – Lecture: Perspectives on Turing
Smale, Stephen
I have been inspired by Turing and his work and will discuss some thoughts coming from this inspiration. The opinions expressed in this video do not necessarily reflect the views of the Heidelberg Laureate Forum Foundation or any other person or associated institution involved in the making and distribution of the video.
 I should also introduce Stephen Smale. He received the Fields Medal in, when was that? It was early, it was in 1966, actually when I started my studies and he received the Fields Medal for his work in differential topology in particular and proving the Poirier conjecture for dimensions bigger or equal to five. So that was a kind of, when I started, one of the biggest result I could imagine. But at that time he was also a hero to me because of his engagement against the Vietnam War. That's a long time ago, but he's also known for that engagement. Later he turned his interest also to computer science and to complexity theory and his work with, about Turing. And that's what he is going to talk about, about Turing and his interest in it. Thank you. Thank you. Well, thanks very much. It's a great pleasure to be back in Heidelberg for this forum again. I'm going to talk about the question for me among other scientists. Well, with partly the goal of answering the question, who am I? Because, you know, as a mathematician, you know, I don't think that I have fit the label of applied mathematician at all. And I think conversely applied mathematics doesn't see me as an applied mathematician. On the other hand, I certainly am not a pure mathematician. So what am I? I think I am a mathematician, but I don't fit that dichotomy. And so this talk has a little bit to do with trying to see this kind of mathematics that I do, which has been inspired, especially by Turing. But also I'll talk a little bit about the influence of these other people who have inspired me, Newton and von Neumann and the Watson-Crick pair. So I'm bringing those in too. And this has to do with this question about my work. So often I'm asked about what some people might call models or mathematician of some subjects. And they say, they ask me, is it realistic? And I have a big problem with that question. For me, it's a question more of being idealistic. So I want to contrast this kind of different philosophy of idealism versus realism. So I'm not trying to make a realistic aspect of a certain subject, but I would like to idealize a lot when I develop that subject through my whole career, essentially. I'm thinking now of mathematics as embedded in the very large body of science and even outside of science. So mathematics for me has got this wide area of its relationships. OK, so let's see a little bit how this relates to Turing. And there are two examples I can think of. Turing has been a great help for me in theory of computation, but also in theory of morphogenesis. But both share the same aspect. Turing's theory of computation, I think, is great. It's inspiring. But is it realistic? I think most people might agree with me that it is not. It's idealizations. It's great because it idealized just a few key things about computation. Certainly in real life, computation is not unlimited. It's finite. And everything in real life is so finite, but a Turing machine is not. And it's a little bit more pervasive in applications of mathematics and physics and within mathematics. The notion of taking a limit as time goes to infinity is extremely insightful. But time doesn't go to infinity, at least in our lifetimes, so that this is another idealization but leads to great insights throughout math. So those are the kind of things I feel that idealization captures. OK, with Turing, this idea of a theory of computation, a theory which is measured with Turing more literally is a number of bits in a computation. Or in a version of computation that I've worked out with Lenard, Blum, Mike Schub, and also with Philippe Kuker, we use arithmetic operations. And so some aspects, many, some might argue most, but scientific computation where differential equations are involved, then it's the arithmetic operations that count. For example, I'm thinking just of the problem of measuring how long it will take for Newton's method to converge to a zero of a polynomial, a single polynomial of a real variable, how to understand Newton's method as a method of computation. For that, I think Turing does not give insights to that problem. But in other aspects of science and life, the Turing bit model is exactly the right form for that. OK, but so the point I want to make is that Turing theory of computation idealizes a lot. And it's important for success. The whole theory, for example, of NP completeness is based on infinite input size. Again, idealization, a big idealization. But still, the theory of NP completeness gives us a lot of understanding and ideas about what can't be computable. So these idealizations are great. But I think to say it's Turing realistic, I'd say no. OK. But it's not just Turing. I'm thinking also of Newton. Newton's laws of physics, his differential equations of mechanics, there is that an idealization. I would say that's it, except for maybe the astronomical aspects. It's a gross idealization. For example, it took 100 years after Newton to integrate in friction. And if you're going to apply laws of motion and so on, Newton's laws to phenomena on Earth, you need to really take into account friction. And Newton did not. And that took 100 years before it was incorporated. So what Newton did was idealize a way of friction when he talked about the apple falling. Now, a similar thing is going on with Von Neumann. Von Neumann's idealizations and his models of quantum mechanics. See, I think Newton, the Von Neumann, did something that's not appreciated so well. But his work on the foundations of quantum mechanics, he introduced the notion of Hilbert space. The concept of Hilbert space was first really elucidated by Von Neumann in his work on quantum mechanics. I think oftentimes these days, that's not recognized. They will say Von Neumann made the definition of Hilbert space. But that undercuts the importance of the introduction of the concept of a Hilbert space. But that certainly was a huge idealization of what was going on in quantum mechanics, but extraordinarily useful and not too many decades later. Because then it led to the whole theory of questions of operators, bunches of operators on Hilbert space. So that idealization, again, was the key thing. And one shouldn't ask, it's not so useful to ask the question, was Von Neumann's foundations was it realistic? OK. And the other main example I have, since I'm working in biology these days, I paying a lot of attention to some kind of foundations of modern biology, especially Watson and Crick. So Watson and Crick and their monumental discovery of DNA, that was people oftentimes don't appreciate what a great idealization that was. See, the theory of DNA according to Watson and Crick does not have anything to do with proteins. But the real DNA is discovered a couple of decades later, is wound around a car formed by proteins, called the histone proteins. And those structures are chromatin. So they are absolutely critical to get this deeper picture, more detailed picture of what is DNA. It's this spiral of the helix around a car of proteins which dictates so much to the physics and chemistry of DNA. That's called chromatin, that core. And again, this was a huge idealization. And moreover, this is a phenomenon that one sees, maybe not so much with Newton, but the other examples. Watson and Crick, their work was not addressed to biologists. When I was a student, biology was divided into two parts, botany and zoology. And they had biologists as a whole, almost all, had no knowledge of chemistry. And Watson and Crick published in maybe the magazine, Nature. And it was addressed to chemists. And I think it took a long time before the biologists really appreciated it or could read it. I think even the chemists took a while, except for maybe Pauling. So anyway, these idealizations, I think, are crucial. And my own work goes around these gross idealizations. So I'm not trying to make realistic models. And even when it comes to predictions, I think predictions are OK. But all of the examples I've mentioned did not make predictions. And in fact, the models I gave them hardly even had laboratory experiments. These pioneers did not use laboratory experiments. They're own, at least. But they used data that had been accumulated before them. And that's what I think is great. But I'm using the data from biology. There's such a vast amount of data in biology. So my perspective is to use that data, but not raw data. That data from biology has been digested and incorporated into articles by biologists to see a lot of down-to-earth phenomena in biology, where the data is behind it. But it's, I might say, second order. Because the output of the data is in these papers, biologists, right? And that's what I depend on a lot. And biology is to use that kind of data that's been accumulated. And the same, I think, goes for Newton also. He used the data accumulated by Galileo, Copernicus, and Kepler before him. Mostly. I don't think he really was doing much experiments himself. But he used that data to put together a very beautiful, unified picture of physics and mechanics. Anyway, those are my inspirations. And I try to follow a little bit in those footsteps. OK, so maybe I should give a little bit of illustration of what I'm saying in terms of my current work in biology. So I've written, let's see if I can use these things. I've written papers with in vika. This isn't too dark. Rajapaxi. OK, so this is three joint papers we have published. And we're working on one now. So the one I'm working on with indica is the following. I want to understand the heart. I want to understand how the heart beats. And so this is an old problem. Beating in unison is a nickname of the problem. Why does the heart beat in unison? Unison means how do all these myocytes in the heart, all these cells in the heart, called myocytes, part of the populations of cells in the heart, maybe half. They each have their own rhythm, independent. But somehow they get together and cooperate to obtain a full beating heart. So the heart beat is associated to the coordinated action of these myocytes, these cells in the heart. So what we want to do is to give some kind of a deeper understanding of that phenomena through mathematics. And again, a big part of that is due to the work of Turing. Turing has an article as a article written maybe as the last paper before he died on morphogenesis. And he's got some differential equations in there that he used and maybe people after him to understand stripes on a zebra and so on. And so we're using development of those equations. Actually, it wasn't too many years after that paper that I wrote a paper using the same system of Turing's equations for some other understanding. Turing's equations were also big idealization. He assumed they're all linear, which I don't think you can get too far in biology using only linear differential equations. And what happened was OK for what he was doing because he was looking at the limit of an infinite number of cells and in this infinite limit he got partial differential equations. And those partial differential equations, he could see these patterns in animals. And that has become a big subject ever since. But I think most biologists did not appreciate Turing and with good reason because morphogenesis is something that's involving what's happening with just a few cells, one, two, three, embryogenesis. That's where the new animals are formed starting with just a few cells and when you take the continuum limit of cells to get the PDE, that's not useful. And that's why I think the biologists have not paid attention to that paper. And I think that's right. So I think morphogenesis involves what happens just on the cell cycle where one cell is turned into two cells, one perhaps differentiated and you begin to see morphogenesis in those terms. Okay, so for this problem of beating in the heart, this is part of a huge area of investigation going back 100 years to Huygens but especially developed by Arthur Winfrey, the biology of time, his book. And it has a lot to do with synchronization. For example, when people clap for very long, they start clapping in unison. They synchronize. And there are many, many other phenomena where you see this synchronization effect of a lot of people doing the same thing but where it's not synchronized, then it becomes synchronized. And that's a beautiful and big subject. Okay, so for me, there is a background to this. There was some work I did with a Dika. There are three papers of a Dika that I can refer you to. But one paper had to do with the tissues. Tissues aren't like an organ of the body. It could be heart but we were thinking more like the liver. And to understand how the liver functions, so we were working on the problem of the function of a tissue. Now the tissues are made up of cells, all of the same cell type but they're a priori not necessarily coordinated. But what's going on in tissues and the heart too are proteins. Proteins play a big role in function, the main role in fact. So cell type, that means cell types are those that are working in a single tissue. One finds the basic proteins for the function of that tissue. So what we did in that paper it's published a few months ago was to see how that would work, how they would synchronize with those proteins to be able to function together in a tissue, could be the liver or anywhere. And that relied, that work relied on earlier work. And this was the beginning of a big sequence of idealizations. The earlier work was to construct in the genome. So we had the genome in each cell, the set of genes. And those genes are all listed, they're all known, biologists know them all. But how they work, they work together. So one has to study how these proteins are working in the cell. And that's going to be given by a system of differential equations which goes back even before Watson-Crick. Differential equations which regulate either mRNA or proteins. It's not, they're not too much different. Okay, those are the differential equations we constructed in a paper just a year or two ago. All these references you can find out on the internet. So what we did was to construct ordinary differential equations for the system of gene activity in a single cell. So it's the differential equations on the values of the output of this gene network. So it's based on the notion of transcription factor which is a central gene which controls other genes. So you make the nodes of a network of computer science into the genes themselves. And each node you associate a concentration say of proteins, so it'll be a positive real number. Okay, then on that we want to have this differential equation on the whole set of 20,000 genes. Okay, so this goes back a long way. But what we did was to formalize that which was very useful to have for us later to have this formal picture of a dynamical system or ordinary differential equation on this 20,000 dimensional space. And so we started up with building a graph, a network, graph in the sense of computer science so that each node corresponded to a gene. And we would get edges, directed edges, when one transcription factor is controlling a target gene. There are about 10,000 transcription factors or few, not much over a thousand transcription factors, but they can target a number of genes. And when they target a gene, they can elevate the activity or initiate that activity of the target gene. And you can write the differential equations which are quite old, go back 100 years, hill equations which say how that works. And what we did was to put all that together in the network and from the network then we're able to get a single system of differential equations which tell us how the genes regulate each other, mainly from going from the transcription factors to the target genes. So elements of that go away, way back, but scientists had never done anything systematic or something I would say maybe bold or wrong to write down the differential equations for the whole genome in this way. So that's what we did. And we were the first to say that this was a vast idealization. And I think to a certain extent, biologists could understand that most, but not all. We were idealizing drastically. I mean, we recognized in our model some inputs from outside of the cell, signals coming from outside of the cell and some other things. But one thing that came out of this, maybe the same paper or the next paper, was this notion of hard wiring of the genome. And I think this didn't shock biologists, but they never formalized it. But we were able to formalize the notion of hard wiring. So let me explain a bit about that. Hard wiring has to do with the original idea that every cell in every human and every organ consists of the same genes. It's universal. So this is well understood by biologists. But what we said more is true. The system of equations is hard wired. So that essentially in every cell and every human, you have the same differential equations operating on the genes. OK, one has to do that with a little care. Because some tissues, some of these edges will be silenced. So one has to take that into account. They're silenced by chromatin, essentially. So one has to be a little careful about just how extensive this goes. But the idea is in the same tissue, same cell type now, they will all be silent together or active together. That's what makes up the characteristic of a tissue. OK, so what? Let's see, I guess I don't have too much time. So what we did is to use this hard-wiring hypothesis. And we did it in two ways in our paper that was published. We made this second very drastic idealization of what those equations look like. We said in each cell there's going to be a single basin equilibrium. So all the equations have solutions, almost all, which will go to that equilibria. So it's a global basin. And the basin is going to have then a description in terms of the variables, which would be the proteins. So that equilibrium describes the composition of the proteins, which fits because in the cell type all the proteins have the same composition, the same ratios. They're all the same, the mouths are the same. As you go over cells of the same cell type or in a single organism, like the liver. So we made that hypothesis. Again, drastic idealization. That began to disturb some biologists, not others. But I think it was consistent with the known biology very much. And part of what we're doing is trying to give some suggestions of which make different parts of biology consistent. And give some kind of insights into the workings of biology in a broader sense. And so this is what we were trying to do, I think with some success. So that was a hypothesis we made. And then we proved the theorem using these equations of Turing extended to these nonlinear equations, which we needed to describe the equilibria, describe what's going on in each cell. So we extended, yeah, so what we did then was to get some kind of theorem with some kind of new hypothesis, we call it monotonicity. It wasn't an old hypothesis of monotonicity. It was a new one, which we expressed mathematically. And with that hypothesis on the actions of these cells, then we were able to get, by using the Turing equations, which took on to account diffusion between the cells, we were able to get this coordination of function. So we got the function of this tissue described now by all the cells working together. OK, yeah, so then more recently, this suggested to me at first that we could do the same thing to get this problem of beating in unison for describing the heart. And actually, somebody came from Hong Kong, a young woman, Lynn Lu, to work with me. She was an undergraduate in Hong Kong. But so in June, she came to work with me on some problem. And I suggested to her that she work on the genome of the heart, of the heart muscles. So I learned a lot over one month of talking to her about this problem that she did a lot of research. I did too, but that gave me some picture where there are two proteins that are fundamental for the heartbeat, actin and myosin. And so then we were able to use some computation to see that this fit together pretty well with Turing's equation where you had diffusion given by transport about two membranes. So you get a Laplacian defined by a matrix, which is a, what's it called? I guess a neighborhood matrix, adjacency matrix, essentially done by transport of the proteins through membranes. So nearby these myocytes, these cells in the heart, they will have a membrane type diffusion affecting them. And you can put all that together to get a nice Laplacian defined. And that Laplacian has an averaging effect then on the proteins. And one has to be careful because our counter examples, we found counter examples, but with this bottle of tenacity hypothesis, we were able to get this coordination. And then that was, oh, when you had an equilibrium, not for the heartbeat, we have to make a little broader hypothesis on the dynamics. It's going to be now a stable periodic solution to represent the heartbeat. And each myocyte does have each of these cells in the heart. They do have such a rhythm that's known. In fact, a lot of these things were known biologically and so on, but they hadn't been put together very well mathematically, not at all. So there was some appreciation that maybe the fusion between the myocytes could give rise to the heartbeat, but it was actually using those Turing equations that we were able to, and this is our work in progress, see the things I would say, we can see the end of the tunnel and we are beginning to have a theory emerge which will give some kind of model of the heart being in rhythm. Thank you very much. Thank you.
 I should also introduce Stephen Smale.  He received the Fields Medal in, when was that?  It was early, it was in 1966, actually when I started my studies  and he received the Fields Medal for his work  in differential topology in particular  and proving the Poirier conjecture  for dimensions bigger or equal to five.  So that was a kind of, when I started,  one of the biggest result I could imagine.  But at that time he was also a hero to me because  of his engagement against the Vietnam War.  That's a long time ago, but he's also known for that engagement.  Later he turned his interest also to computer science  and to complexity theory and his work with, about Turing.  And that's what he is going to talk about,  about Turing and his interest in it.  Thank you.  Thank you.  Well, thanks very much.  It's a great pleasure to be back in Heidelberg for this forum again.  I'm going to talk about the question for me among other scientists.  Well, with partly the goal of answering the question, who am I?  Because, you know, as a mathematician, you know,  I don't think that I have fit the label  of applied mathematician at all.  And I think conversely applied mathematics doesn't see me  as an applied mathematician.  On the other hand, I certainly am not a pure mathematician.  So what am I?  I think I am a mathematician, but I don't fit that dichotomy.  And so this talk has a little bit to do with trying to see this kind  of mathematics that I do, which has been inspired,  especially by Turing.  But also I'll talk a little bit about the influence of these other people  who have inspired me, Newton and von Neumann and the Watson-Crick pair.  So I'm bringing those in too.  And this has to do with this question about my work.  So often I'm asked about what some people might call models  or mathematician of some subjects.  And they say, they ask me, is it realistic?  And I have a big problem with that question.  For me, it's a question more of being idealistic.  So I want to contrast this kind of different philosophy  of idealism versus realism.  So I'm not trying to make a realistic aspect of a certain subject,  but I would like to idealize a lot when I develop that subject  through my whole career, essentially.  I'm thinking now of mathematics as embedded in the very large body  of science and even outside of science.  So mathematics for me has got this wide area of its relationships.  OK, so let's see a little bit how this relates to Turing.  And there are two examples I can think of.  Turing has been a great help for me in theory of computation,  but also in theory of morphogenesis.  But both share the same aspect.  Turing's theory of computation, I think, is great.  It's inspiring.  But is it realistic?  I think most people might agree with me that it is not.  It's idealizations.  It's great because it idealized just a few key things about computation.  Certainly in real life, computation is not unlimited.  It's finite.  And everything in real life is so finite,  but a Turing machine is not.  And it's a little bit more pervasive in applications  of mathematics and physics and within mathematics.  The notion of taking a limit as time goes to infinity  is extremely insightful.  But time doesn't go to infinity, at least in our lifetimes,  so that this is another idealization  but leads to great insights throughout math.  So those are the kind of things I feel that idealization captures.  OK, with Turing, this idea of a theory of computation,  a theory which is measured with Turing more literally  is a number of bits in a computation.  Or in a version of computation that I've worked out  with Lenard, Blum, Mike Schub, and also with Philippe Kuker,  we use arithmetic operations.  And so some aspects, many, some might argue most,  but scientific computation where differential equations are involved,  then it's the arithmetic operations that count.  For example, I'm thinking just of the problem of measuring  how long it will take for Newton's method to converge  to a zero of a polynomial, a single polynomial of a real variable,  how to understand Newton's method as a method of computation.  For that, I think Turing does not give insights to that problem.  But in other aspects of science and life,  the Turing bit model is exactly the right form for that.  OK, but so the point I want to make is that Turing theory of computation  idealizes a lot.  And it's important for success.  The whole theory, for example, of NP completeness  is based on infinite input size.  Again, idealization, a big idealization.  But still, the theory of NP completeness  gives us a lot of understanding and ideas about what can't be computable.  So these idealizations are great.  But I think to say it's Turing realistic, I'd say no.  OK.  But it's not just Turing.  I'm thinking also of Newton.  Newton's laws of physics, his differential equations of mechanics,  there is that an idealization.  I would say that's it, except for maybe the astronomical aspects.  It's a gross idealization.  For example, it took 100 years after Newton to integrate in friction.  And if you're going to apply laws of motion and so on,  Newton's laws to phenomena on Earth,  you need to really take into account friction.  And Newton did not.  And that took 100 years before it was incorporated.  So what Newton did was idealize a way of friction  when he talked about the apple falling.  Now, a similar thing is going on with Von Neumann.  Von Neumann's idealizations and his models of quantum mechanics.  See, I think Newton, the Von Neumann, did something that's not  appreciated so well.  But his work on the foundations of quantum mechanics,  he introduced the notion of Hilbert space.  The concept of Hilbert space was first really elucidated  by Von Neumann in his work on quantum mechanics.  I think oftentimes these days, that's not recognized.  They will say Von Neumann made the definition of Hilbert space.  But that undercuts the importance of the introduction  of the concept of a Hilbert space.  But that certainly was a huge idealization  of what was going on in quantum mechanics,  but extraordinarily useful and not too many decades later.  Because then it led to the whole theory of questions  of operators, bunches of operators on Hilbert space.  So that idealization, again, was the key thing.  And one shouldn't ask, it's not so useful to ask the question,  was Von Neumann's foundations was it realistic?  OK.  And the other main example I have,  since I'm working in biology these days,  I paying a lot of attention to some kind of foundations  of modern biology, especially Watson and Crick.  So Watson and Crick and their monumental discovery of DNA,  that was people oftentimes don't appreciate what  a great idealization that was.  See, the theory of DNA according to Watson and Crick  does not have anything to do with proteins.  But the real DNA is discovered a couple of decades later,  is wound around a car formed by proteins,  called the histone proteins.  And those structures are chromatin.  So they are absolutely critical to get this deeper picture,  more detailed picture of what is DNA.  It's this spiral of the helix around a car of proteins  which dictates so much to the physics and chemistry of DNA.  That's called chromatin, that core.  And again, this was a huge idealization.  And moreover, this is a phenomenon that one sees,  maybe not so much with Newton, but the other examples.  Watson and Crick, their work was not addressed to biologists.  When I was a student, biology was divided into two parts,  botany and zoology.  And they had biologists as a whole, almost all,  had no knowledge of chemistry.  And Watson and Crick published in maybe the magazine, Nature.  And it was addressed to chemists.  And I think it took a long time before the biologists  really appreciated it or could read it.  I think even the chemists took a while, except for maybe  Pauling.  So anyway, these idealizations, I think, are crucial.  And my own work goes around these gross idealizations.  So I'm not trying to make realistic models.  And even when it comes to predictions,  I think predictions are OK.  But all of the examples I've mentioned  did not make predictions.  And in fact, the models I gave them  hardly even had laboratory experiments.  These pioneers did not use laboratory experiments.  They're own, at least.  But they used data that had been accumulated before them.  And that's what I think is great.  But I'm using the data from biology.  There's such a vast amount of data in biology.  So my perspective is to use that data, but not raw data.  That data from biology has been digested and incorporated  into articles by biologists to see a lot of down-to-earth  phenomena in biology, where the data is behind it.  But it's, I might say, second order.  Because the output of the data is in these papers,  biologists, right?  And that's what I depend on a lot.  And biology is to use that kind of data that's been accumulated.  And the same, I think, goes for Newton also.  He used the data accumulated by Galileo, Copernicus,  and Kepler before him.  Mostly.  I don't think he really was doing much experiments himself.  But he used that data to put together  a very beautiful, unified picture of physics and mechanics.  Anyway, those are my inspirations.  And I try to follow a little bit in those footsteps.  OK, so maybe I should give a little bit of illustration  of what I'm saying in terms of my current work in biology.  So I've written, let's see if I can use these things.  I've written papers with in vika.  This isn't too dark.  Rajapaxi.  OK, so this is three joint papers we have published.  And we're working on one now.  So the one I'm working on with indica is the following.  I want to understand the heart.  I want to understand how the heart beats.  And so this is an old problem.  Beating in unison is a nickname of the problem.  Why does the heart beat in unison?  Unison means how do all these myocytes in the heart,  all these cells in the heart, called myocytes,  part of the populations of cells in the heart, maybe half.  They each have their own rhythm, independent.  But somehow they get together and cooperate  to obtain a full beating heart.  So the heart beat is associated to the coordinated action  of these myocytes, these cells in the heart.  So what we want to do is to give some kind  of a deeper understanding of that phenomena  through mathematics.  And again, a big part of that is due to the work of Turing.  Turing has an article as a article written maybe  as the last paper before he died on morphogenesis.  And he's got some differential equations in there that he used  and maybe people after him to understand stripes  on a zebra and so on.  And so we're using development of those equations.  Actually, it wasn't too many years after that paper  that I wrote a paper using the same system of Turing's equations  for some other understanding.  Turing's equations were also big idealization.  He assumed they're all linear, which I don't think you can get  too far in biology using only linear differential equations.  And what happened was OK for what he was doing  because he was looking at the limit of an infinite number  of cells and in this infinite limit he got partial differential  equations.  And those partial differential equations,  he could see these patterns in animals.  And that has become a big subject ever since.  But I think most biologists did not appreciate Turing  and with good reason because morphogenesis is something  that's involving what's happening with just a few cells,  one, two, three, embryogenesis.  That's where the new animals are formed starting  with just a few cells and when you take the continuum limit  of cells to get the PDE, that's not useful.  And that's why I think the biologists have not paid attention  to that paper.  And I think that's right.  So I think morphogenesis involves what happens just  on the cell cycle where one cell is turned into two cells,  one perhaps differentiated and you begin  to see morphogenesis in those terms.  Okay, so for this problem of beating in the heart,  this is part of a huge area of investigation going back 100 years  to Huygens but especially developed by Arthur Winfrey,  the biology of time, his book.  And it has a lot to do with synchronization.  For example, when people clap for very long,  they start clapping in unison.  They synchronize.  And there are many, many other phenomena  where you see this synchronization effect of a lot  of people doing the same thing but where it's not synchronized,  then it becomes synchronized.  And that's a beautiful and big subject.  Okay, so for me, there is a background to this.  There was some work I did with a Dika.  There are three papers of a Dika that I can refer you to.  But one paper had to do with the tissues.  Tissues aren't like an organ of the body.  It could be heart but we were thinking more like the liver.  And to understand how the liver functions, so we were working  on the problem of the function of a tissue.  Now the tissues are made up of cells, all of the same cell type  but they're a priori not necessarily coordinated.  But what's going on in tissues and the heart too are proteins.  Proteins play a big role in function, the main role in fact.  So cell type, that means cell types are those that are working  in a single tissue.  One finds the basic proteins for the function of that tissue.  So what we did in that paper it's published a few months ago was  to see how that would work, how they would synchronize  with those proteins to be able to function together  in a tissue, could be the liver or anywhere.  And that relied, that work relied on earlier work.  And this was the beginning of a big sequence of idealizations.  The earlier work was to construct in the genome.  So we had the genome in each cell, the set of genes.  And those genes are all listed, they're all known,  biologists know them all.  But how they work, they work together.  So one has to study how these proteins are working in the cell.  And that's going to be given by a system  of differential equations which goes back even before Watson-Crick.  Differential equations which regulate either mRNA or proteins.  It's not, they're not too much different.  Okay, those are the differential equations we constructed  in a paper just a year or two ago.  All these references you can find out on the internet.  So what we did was to construct ordinary differential equations  for the system of gene activity in a single cell.  So it's the differential equations on the values of the output  of this gene network.  So it's based on the notion of transcription factor  which is a central gene which controls other genes.  So you make the nodes of a network of computer science  into the genes themselves.  And each node you associate a concentration say of proteins,  so it'll be a positive real number.  Okay, then on that we want to have this differential equation  on the whole set of 20,000 genes.  Okay, so this goes back a long way.  But what we did was to formalize that which was very useful  to have for us later to have this formal picture  of a dynamical system or ordinary differential equation  on this 20,000 dimensional space.  And so we started up with building a graph, a network,  graph in the sense of computer science  so that each node corresponded to a gene.  And we would get edges, directed edges,  when one transcription factor is controlling a target gene.  There are about 10,000 transcription factors or few,  not much over a thousand transcription factors,  but they can target a number of genes.  And when they target a gene, they can elevate the activity  or initiate that activity of the target gene.  And you can write the differential equations  which are quite old, go back 100 years,  hill equations which say how that works.  And what we did was to put all that together in the network  and from the network then we're able to get a single system  of differential equations which tell us  how the genes regulate each other,  mainly from going from the transcription factors  to the target genes.  So elements of that go away, way back,  but scientists had never done anything systematic  or something I would say maybe bold or wrong  to write down the differential equations  for the whole genome in this way.  So that's what we did.  And we were the first to say that this was a vast idealization.  And I think to a certain extent,  biologists could understand that most, but not all.  We were idealizing drastically.  I mean, we recognized in our model some inputs  from outside of the cell, signals coming from outside  of the cell and some other things.  But one thing that came out of this, maybe the same paper  or the next paper, was this notion of hard wiring  of the genome.  And I think this didn't shock biologists,  but they never formalized it.  But we were able to formalize the notion of hard wiring.  So let me explain a bit about that.  Hard wiring has to do with the original idea that every cell  in every human and every organ consists of the same genes.  It's universal.  So this is well understood by biologists.  But what we said more is true.  The system of equations is hard wired.  So that essentially in every cell and every human,  you have the same differential equations operating on the genes.  OK, one has to do that with a little care.  Because some tissues, some of these edges  will be silenced.  So one has to take that into account.  They're silenced by chromatin, essentially.  So one has to be a little careful about just  how extensive this goes.  But the idea is in the same tissue, same cell type now,  they will all be silent together or active together.  That's what makes up the characteristic of a tissue.  OK, so what?  Let's see, I guess I don't have too much time.  So what we did is to use this hard-wiring hypothesis.  And we did it in two ways in our paper that was published.  We made this second very drastic idealization  of what those equations look like.  We said in each cell there's going  to be a single basin equilibrium.  So all the equations have solutions, almost all,  which will go to that equilibria.  So it's a global basin.  And the basin is going to have then a description in terms  of the variables, which would be the proteins.  So that equilibrium describes the composition of the proteins,  which fits because in the cell type all the proteins have the  same composition, the same ratios.  They're all the same, the mouths are the same.  As you go over cells of the same cell type  or in a single organism, like the liver.  So we made that hypothesis.  Again, drastic idealization.  That began to disturb some biologists, not others.  But I think it was consistent with the known biology very much.  And part of what we're doing is trying to give some suggestions  of which make different parts of biology consistent.  And give some kind of insights into the workings of biology  in a broader sense.  And so this is what we were trying to do,  I think with some success.  So that was a hypothesis we made.  And then we proved the theorem using these equations  of Turing extended to these nonlinear equations,  which we needed to describe the equilibria,  describe what's going on in each cell.  So we extended, yeah, so what we did then was to get some kind  of theorem with some kind of new hypothesis,  we call it monotonicity.  It wasn't an old hypothesis of monotonicity.  It was a new one, which we expressed mathematically.  And with that hypothesis on the actions of these cells,  then we were able to get, by using the Turing equations,  which took on to account diffusion between the cells,  we were able to get this coordination of function.  So we got the function of this tissue described now  by all the cells working together.  OK, yeah, so then more recently,  this suggested to me at first that we could do the same thing  to get this problem of beating in unison  for describing the heart.  And actually, somebody came from Hong Kong, a young woman,  Lynn Lu, to work with me.  She was an undergraduate in Hong Kong.  But so in June, she came to work with me on some problem.  And I suggested to her that she work on the genome of the heart,  of the heart muscles.  So I learned a lot over one month of talking to her  about this problem that she did a lot of research.  I did too, but that gave me some picture  where there are two proteins that are fundamental  for the heartbeat, actin and myosin.  And so then we were able to use some computation to see  that this fit together pretty well with Turing's equation  where you had diffusion given by transport  about two membranes.  So you get a Laplacian defined by a matrix,  which is a, what's it called?  I guess a neighborhood matrix, adjacency matrix,  essentially done by transport of the proteins  through membranes.  So nearby these myocytes, these cells in the heart,  they will have a membrane type diffusion affecting them.  And you can put all that together to get a nice Laplacian defined.  And that Laplacian has an averaging effect then  on the proteins.  And one has to be careful because our counter examples,  we found counter examples, but with this bottle  of tenacity hypothesis, we were able to get this coordination.  And then that was, oh, when you had an equilibrium,  not for the heartbeat, we have to make a little broader hypothesis  on the dynamics.  It's going to be now a stable periodic solution  to represent the heartbeat.  And each myocyte does have each of these cells in the heart.  They do have such a rhythm that's known.  In fact, a lot of these things were known biologically  and so on, but they hadn't been put together very well  mathematically, not at all.  So there was some appreciation that maybe the fusion  between the myocytes could give rise to the heartbeat,  but it was actually using those Turing equations that we were able  to, and this is our work in progress, see the things I would say,  we can see the end of the tunnel and we are beginning  to have a theory emerge which will give some kind of model  of the heart being in rhythm.  Thank you very much.  Thank you.
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I should also introduce Stephen Smale. He received the Fields Medal in, when was that? It was early, it was in 1966, actually when I started my studies and he received the Fields Medal for his work in differential topology in particular and proving the Poirier conjecture for dimensions bigger or equal to five. So that was a kind of, when I started, one of the biggest result I could imagine. But at that time he was also a hero to me because of his engagement against the Vietnam War. That's a long time ago, but he's also known for that engagement. Later he turned his interest also to computer science and to complexity theory and his work with, about Turing. And that's what he is going to talk about, about Turing and his interest in it. Thank you. Thank you. Well, thanks very much. It's a great pleasure to be back in Heidelberg for this forum again. I'm going to talk about the question for me among other scientists. Well, with partly the goal of answering the question, who am I? Because, you know, as a mathematician, you know, I don't think that I have fit the label of applied mathematician at all. And I think conversely applied mathematics doesn't see me as an applied mathematician. On the other hand, I certainly am not a pure mathematician. So what am I? I think I am a mathematician, but I don't fit that dichotomy. And so this talk has a little bit to do with trying to see this kind of mathematics that I do, which has been inspired, especially by Turing. But also I'll talk a little bit about the influence of these other people who have inspired me, Newton and von Neumann and the Watson-Crick pair. So I'm bringing those in too. And this has to do with this question about my work. So often I'm asked about what some people might call models or mathematician of some subjects. And they say, they ask me, is it realistic? And I have a big problem with that question. For me, it's a question more of being idealistic. So I want to contrast this kind of different philosophy of idealism versus realism. So I'm not trying to make a realistic aspect of a certain subject, but I would like to idealize a lot when I develop that subject through my whole career, essentially. I'm thinking now of mathematics as embedded in the very large body of science and even outside of science. So mathematics for me has got this wide area of its relationships. OK, so let's see a little bit how this relates to Turing. And there are two examples I can think of. Turing has been a great help for me in theory of computation, but also in theory of morphogenesis. But both share the same aspect. Turing's theory of computation, I think, is great. It's inspiring. But is it realistic? I think most people might agree with me that it is not. It's idealizations. It's great because it idealized just a few key things about computation. Certainly in real life, computation is not unlimited. It's finite. And everything in real life is so finite, but a Turing machine is not. And it's a little bit more pervasive in applications of mathematics and physics and within mathematics. The notion of taking a limit as time goes to infinity is extremely insightful. But time doesn't go to infinity, at least in our lifetimes, so that this is another idealization but leads to great insights throughout math. So those are the kind of things I feel that idealization captures. OK, with Turing, this idea of a theory of computation, a theory which is measured with Turing more literally is a number of bits in a computation. Or in a version of computation that I've worked out with Lenard, Blum, Mike Schub, and also with Philippe Kuker, we use arithmetic operations. And so some aspects, many, some might argue most, but scientific computation where differential equations are involved, then it's the arithmetic operations that count. For example, I'm thinking just of the problem of measuring how long it will take for Newton's method to converge to a zero of a polynomial, a single polynomial of a real variable, how to understand Newton's method as a method of computation. For that, I think Turing does not give insights to that problem. But in other aspects of science and life, the Turing bit model is exactly the right form for that. OK, but so the point I want to make is that Turing theory of computation idealizes a lot. And it's important for success. The whole theory, for example, of NP completeness is based on infinite input size. Again, idealization, a big idealization. But still, the theory of NP completeness gives us a lot of understanding and ideas about what can't be computable. So these idealizations are great. But I think to say it's Turing realistic, I'd say no. OK. But it's not just Turing. I'm thinking also of Newton. Newton's laws of physics, his differential equations of mechanics, there is that an idealization. I would say that's it, except for maybe the astronomical aspects. It's a gross idealization. For example, it took 100 years after Newton to integrate in friction. And if you're going to apply laws of motion and so on, Newton's laws to phenomena on Earth, you need to really take into account friction. And Newton did not. And that took 100 years before it was incorporated. So what Newton did was idealize a way of friction when he talked about the apple falling. Now, a similar thing is going on with Von Neumann. Von Neumann's idealizations and his models of quantum mechanics. See, I think Newton, the Von Neumann, did something that's not appreciated so well. But his work on the foundations of quantum mechanics, he introduced the notion of Hilbert space. The concept of Hilbert space was first really elucidated by Von Neumann in his work on quantum mechanics. I think oftentimes these days, that's not recognized. They will say Von Neumann made the definition of Hilbert space. But that undercuts the importance of the introduction of the concept of a Hilbert space. But that certainly was a huge idealization of what was going on in quantum mechanics, but extraordinarily useful and not too many decades later. Because then it led to the whole theory of questions of operators, bunches of operators on Hilbert space. So that idealization, again, was the key thing. And one shouldn't ask, it's not so useful to ask the question, was Von Neumann's foundations was it realistic? OK. And the other main example I have, since I'm working in biology these days, I paying a lot of attention to some kind of foundations of modern biology, especially Watson and Crick. So Watson and Crick and their monumental discovery of DNA, that was people oftentimes don't appreciate what a great idealization that was. See, the theory of DNA according to Watson and Crick does not have anything to do with proteins. But the real DNA is discovered a couple of decades later, is wound around a car formed by proteins, called the histone proteins. And those structures are chromatin. So they are absolutely critical to get this deeper picture, more detailed picture of what is DNA. It's this spiral of the helix around a car of proteins which dictates so much to the physics and chemistry of DNA. That's called chromatin, that core. And again, this was a huge idealization. And moreover, this is a phenomenon that one sees, maybe not so much with Newton, but the other examples. Watson and Crick, their work was not addressed to biologists. When I was a student, biology was divided into two parts, botany and zoology. And they had biologists as a whole, almost all, had no knowledge of chemistry. And Watson and Crick published in maybe the magazine, Nature. And it was addressed to chemists. And I think it took a long time before the biologists really appreciated it or could read it. I think even the chemists took a while, except for maybe Pauling. So anyway, these idealizations, I think, are crucial. And my own work goes around these gross idealizations. So I'm not trying to make realistic models. And even when it comes to predictions, I think predictions are OK. But all of the examples I've mentioned did not make predictions. And in fact, the models I gave them hardly even had laboratory experiments. These pioneers did not use laboratory experiments. They're own, at least. But they used data that had been accumulated before them. And that's what I think is great. But I'm using the data from biology. There's such a vast amount of data in biology. So my perspective is to use that data, but not raw data. That data from biology has been digested and incorporated into articles by biologists to see a lot of down-to-earth phenomena in biology, where the data is behind it. But it's, I might say, second order. Because the output of the data is in these papers, biologists, right? And that's what I depend on a lot. And biology is to use that kind of data that's been accumulated. And the same, I think, goes for Newton also. He used the data accumulated by Galileo, Copernicus, and Kepler before him. Mostly. I don't think he really was doing much experiments himself. But he used that data to put together a very beautiful, unified picture of physics and mechanics. Anyway, those are my inspirations. And I try to follow a little bit in those footsteps. OK, so maybe I should give a little bit of illustration of what I'm saying in terms of my current work in biology. So I've written, let's see if I can use these things. I've written papers with in vika. This isn't too dark. Rajapaxi. OK, so this is three joint papers we have published. And we're working on one now. So the one I'm working on with indica is the following. I want to understand the heart. I want to understand how the heart beats. And so this is an old problem. Beating in unison is a nickname of the problem. Why does the heart beat in unison? Unison means how do all these myocytes in the heart, all these cells in the heart, called myocytes, part of the populations of cells in the heart, maybe half. They each have their own rhythm, independent. But somehow they get together and cooperate to obtain a full beating heart. So the heart beat is associated to the coordinated action of these myocytes, these cells in the heart. So what we want to do is to give some kind of a deeper understanding of that phenomena through mathematics. And again, a big part of that is due to the work of Turing. Turing has an article as a article written maybe as the last paper before he died on morphogenesis. And he's got some differential equations in there that he used and maybe people after him to understand stripes on a zebra and so on. And so we're using development of those equations. Actually, it wasn't too many years after that paper that I wrote a paper using the same system of Turing's equations for some other understanding. Turing's equations were also big idealization. He assumed they're all linear, which I don't think you can get too far in biology using only linear differential equations. And what happened was OK for what he was doing because he was looking at the limit of an infinite number of cells and in this infinite limit he got partial differential equations. And those partial differential equations, he could see these patterns in animals. And that has become a big subject ever since. But I think most biologists did not appreciate Turing and with good reason because morphogenesis is something that's involving what's happening with just a few cells, one, two, three, embryogenesis. That's where the new animals are formed starting with just a few cells and when you take the continuum limit of cells to get the PDE, that's not useful. And that's why I think the biologists have not paid attention to that paper. And I think that's right. So I think morphogenesis involves what happens just on the cell cycle where one cell is turned into two cells, one perhaps differentiated and you begin to see morphogenesis in those terms. Okay, so for this problem of beating in the heart, this is part of a huge area of investigation going back 100 years to Huygens but especially developed by Arthur Winfrey, the biology of time, his book. And it has a lot to do with synchronization. For example, when people clap for very long, they start clapping in unison. They synchronize. And there are many, many other phenomena where you see this synchronization effect of a lot of people doing the same thing but where it's not synchronized, then it becomes synchronized. And that's a beautiful and big subject. Okay, so for me, there is a background to this. There was some work I did with a Dika. There are three papers of a Dika that I can refer you to. But one paper had to do with the tissues. Tissues aren't like an organ of the body. It could be heart but we were thinking more like the liver. And to understand how the liver functions, so we were working on the problem of the function of a tissue. Now the tissues are made up of cells, all of the same cell type but they're a priori not necessarily coordinated. But what's going on in tissues and the heart too are proteins. Proteins play a big role in function, the main role in fact. So cell type, that means cell types are those that are working in a single tissue. One finds the basic proteins for the function of that tissue. So what we did in that paper it's published a few months ago was to see how that would work, how they would synchronize with those proteins to be able to function together in a tissue, could be the liver or anywhere. And that relied, that work relied on earlier work. And this was the beginning of a big sequence of idealizations. The earlier work was to construct in the genome. So we had the genome in each cell, the set of genes. And those genes are all listed, they're all known, biologists know them all. But how they work, they work together. So one has to study how these proteins are working in the cell. And that's going to be given by a system of differential equations which goes back even before Watson-Crick. Differential equations which regulate either mRNA or proteins. It's not, they're not too much different. Okay, those are the differential equations we constructed in a paper just a year or two ago. All these references you can find out on the internet. So what we did was to construct ordinary differential equations for the system of gene activity in a single cell. So it's the differential equations on the values of the output of this gene network. So it's based on the notion of transcription factor which is a central gene which controls other genes. So you make the nodes of a network of computer science into the genes themselves. And each node you associate a concentration say of proteins, so it'll be a positive real number. Okay, then on that we want to have this differential equation on the whole set of 20,000 genes. Okay, so this goes back a long way. But what we did was to formalize that which was very useful to have for us later to have this formal picture of a dynamical system or ordinary differential equation on this 20,000 dimensional space. And so we started up with building a graph, a network, graph in the sense of computer science so that each node corresponded to a gene. And we would get edges, directed edges, when one transcription factor is controlling a target gene. There are about 10,000 transcription factors or few, not much over a thousand transcription factors, but they can target a number of genes. And when they target a gene, they can elevate the activity or initiate that activity of the target gene. And you can write the differential equations which are quite old, go back 100 years, hill equations which say how that works. And what we did was to put all that together in the network and from the network then we're able to get a single system of differential equations which tell us how the genes regulate each other, mainly from going from the transcription factors to the target genes. So elements of that go away, way back, but scientists had never done anything systematic or something I would say maybe bold or wrong to write down the differential equations for the whole genome in this way. So that's what we did. And we were the first to say that this was a vast idealization. And I think to a certain extent, biologists could understand that most, but not all. We were idealizing drastically. I mean, we recognized in our model some inputs from outside of the cell, signals coming from outside of the cell and some other things. But one thing that came out of this, maybe the same paper or the next paper, was this notion of hard wiring of the genome. And I think this didn't shock biologists, but they never formalized it. But we were able to formalize the notion of hard wiring. So let me explain a bit about that. Hard wiring has to do with the original idea that every cell in every human and every organ consists of the same genes. It's universal. So this is well understood by biologists. But what we said more is true. The system of equations is hard wired. So that essentially in every cell and every human, you have the same differential equations operating on the genes. OK, one has to do that with a little care. Because some tissues, some of these edges will be silenced. So one has to take that into account. They're silenced by chromatin, essentially. So one has to be a little careful about just how extensive this goes. But the idea is in the same tissue, same cell type now, they will all be silent together or active together. That's what makes up the characteristic of a tissue. OK, so what? Let's see, I guess I don't have too much time. So what we did is to use this hard-wiring hypothesis. And we did it in two ways in our paper that was published. We made this second very drastic idealization of what those equations look like. We said in each cell there's going to be a single basin equilibrium. So all the equations have solutions, almost all, which will go to that equilibria. So it's a global basin. And the basin is going to have then a description in terms of the variables, which would be the proteins. So that equilibrium describes the composition of the proteins, which fits because in the cell type all the proteins have the same composition, the same ratios. They're all the same, the mouths are the same. As you go over cells of the same cell type or in a single organism, like the liver. So we made that hypothesis. Again, drastic idealization. That began to disturb some biologists, not others. But I think it was consistent with the known biology very much. And part of what we're doing is trying to give some suggestions of which make different parts of biology consistent. And give some kind of insights into the workings of biology in a broader sense. And so this is what we were trying to do, I think with some success. So that was a hypothesis we made. And then we proved the theorem using these equations of Turing extended to these nonlinear equations, which we needed to describe the equilibria, describe what's going on in each cell. So we extended, yeah, so what we did then was to get some kind of theorem with some kind of new hypothesis, we call it monotonicity. It wasn't an old hypothesis of monotonicity. It was a new one, which we expressed mathematically. And with that hypothesis on the actions of these cells, then we were able to get, by using the Turing equations, which took on to account diffusion between the cells, we were able to get this coordination of function. So we got the function of this tissue described now by all the cells working together. OK, yeah, so then more recently, this suggested to me at first that we could do the same thing to get this problem of beating in unison for describing the heart. And actually, somebody came from Hong Kong, a young woman, Lynn Lu, to work with me. She was an undergraduate in Hong Kong. But so in June, she came to work with me on some problem. And I suggested to her that she work on the genome of the heart, of the heart muscles. So I learned a lot over one month of talking to her about this problem that she did a lot of research. I did too, but that gave me some picture where there are two proteins that are fundamental for the heartbeat, actin and myosin. And so then we were able to use some computation to see that this fit together pretty well with Turing's equation where you had diffusion given by transport about two membranes. So you get a Laplacian defined by a matrix, which is a, what's it called? I guess a neighborhood matrix, adjacency matrix, essentially done by transport of the proteins through membranes. So nearby these myocytes, these cells in the heart, they will have a membrane type diffusion affecting them. And you can put all that together to get a nice Laplacian defined. And that Laplacian has an averaging effect then on the proteins. And one has to be careful because our counter examples, we found counter examples, but with this bottle of tenacity hypothesis, we were able to get this coordination. And then that was, oh, when you had an equilibrium, not for the heartbeat, we have to make a little broader hypothesis on the dynamics. It's going to be now a stable periodic solution to represent the heartbeat. And each myocyte does have each of these cells in the heart. They do have such a rhythm that's known. In fact, a lot of these things were known biologically and so on, but they hadn't been put together very well mathematically, not at all. So there was some appreciation that maybe the fusion between the myocytes could give rise to the heartbeat, but it was actually using those Turing equations that we were able to, and this is our work in progress, see the things I would say, we can see the end of the tunnel and we are beginning to have a theory emerge which will give some kind of model of the heart being in rhythm. Thank you very much. Thank you.
to complexity theory and his work with, about that I do, which has been inspired, especially by infinite number of cells and in this infinite biologists did not appreciate Turing and with cell type but they're a priori not necessarily of the cell, signals coming from outside of the will have a membrane type diffusion affecting