# Alan Turing: Mathematical Biologist?

## His biographer says Turing invented computer science, artificial intelligence, and mathematical biology

**Steven Cherry:** Hi, this is Steven Cherry for *IEEE Spectrum’*s “Techwise Conversations.”

As the history of the 20th century slowly gets set, like a resin that takes years to harden, one of its truly seminal figures is turning out to be Alan Turing. For example, the highest honor in the field of computing—its Nobel Prize, if you will—is the Turing Prize, for the man credited with inventing the modern computer.

Turing also single-handedly invented the field of artificial intelligence with a 1950 journal article that’s still read by philosophers. And he’s been recognized as having made probably the single greatest civilian contribution to the Allied victory over the Nazis in World War II, by leading the code breaking of German ciphers.

This June, the world will mark the 100th anniversary of Alan Turing’s birth with tributes of various sorts, magazine stories, journal articles, and at one time it looked like there was even going to be a biographical movie that was to star Leonardo DiCaprio, though it still hasn’t gone into production.

Most all of them, including the biopic, will rely on a single, definitive source, a 587-page biography titled *Alan Turing: The Enigma*, written by Andrew Hodges. The book, which is at once a mathematical, scholarly, and literary tour de force, was originally published in 1983 by Simon & Schuster, but a new centenary edition is coming out next month from Princeton University Press, with a new forward by Douglas Hofstadter.

Andrew Hodges is a tutorial fellow in mathematics at Oxford University’s Wadham College. He has a distinguished career in his own right as a mathematical physicist specializing in the study of space-time, but three decades ago he took two years out of his life to write Turing’s biography, and now I suspect almost as large a chunk is being giving over to the centenary. We’re fortunate in having him as our guest today; he joins us by phone from Oxford.

Andrew, welcome to the podcast.

**Andrew Hodges:** Thank you.

**Steven Cherry:** Andrew, Alan Turing’s life was so rich and complex, I hardly know where to begin. And I didn’t even mention his personal life and death, which were noteworthy in their own right. We’ll get to that in a bit. But let’s start by talking about Turing machines. And I think it’s a perfect example of how much we owe to Turing, and it was just one part, though it’s completely overshadowed the rest of his undecidability theorem. Maybe, actually, let’s start with that, the theorem, as much in layperson’s terms as you can.

**Andrew Hodges:** Well, the theorem that Alan Turing proved in 1936 was a negative thing, oddly enough. And it’s ironic, really, because what he did in the end was so constructive, but actually it was a negative thing. It was showing that there are some things you can describe but can’t achieve by a definite method or process or procedure. So that’s actually what he did. But in order to do that he had to do something rather constructive, which was to define what you meant by a procedure or a method or a mechanical process. And that’s what he achieved in a way that no one else had really thought of. And, well, what he did was to make a mathematical definition. That’s the kind of thing that mathematicians are very good at, formalizing something in very specific terms. And what it boils down to now is something that seems very easy: It’s just anything that can be done by a computer program. But of course it wasn’t easy then, because computers didn’t exist. And so what Alan Turing did was to think of the idea of a computer program before there were any computers or anyone had ever thought about computing in the modern way. And as a byproduct, that’s how, essentially, he invented the computer.

**Steven Cherry:** So this gets us to the Turing machine.

**Andrew Hodges:** Well, Turing’s formalism, in modern terms it’s not really terribly vital what Turing’s original picture was, but it was of an infinitely long or indefinitely long tape, a tape as long as you need, which is marked with squares, and then you imagine it being processed one square at a time. And now you can get software easily which shows you this on a screen, and that’s a very natural way, actually, to present it. But it’s not—one shouldn’t think that Turing imagined that you’d build a machine that would be working on one—one square at a time like this.

**Steven Cherry:** And the big significance of it is that basically no matter how complicated computers get in the real world—and they do get more complex every year—they’re all pretty much encapsulated by this very simple formalism.

**Andrew Hodges:** That’s right. Anything a computer does is equivalent to something that could be done in principle in this atomic formalism. And—but there are very much—more definite things than that that meant that Turing had invented the modern idea of computing. And the really vital thing, I think, is that he’d seen—it was absolutely essential for his proof of things, of this negative thing I was telling you about, the things that are some things you can’t formalize in a process like this—it was essential to that argument that you are able to think of the instructions or the program as being itself a form of data.

Now that’s a very radical idea. Well, it was radical then, anyway; of course, people now take it for granted. I mean, it’s easy now to see that a program that you feed into a computer is no different from feeding a whole string of numbers in. I mean people do it all the time now with data and downloading programs. They don’t know the difference—what’s data and what’s a program—it’s very hard to say. But in the 1930s, that seemed a very strange and radical thing to do, so it came very, very definitely out of his whole mathematical logic. And that sort of revolutionary, radical streak of saying that actually a program is just a form of data like—just as the numbers are—in fact isn’t really to do with numbers. It’s just any—any symbols you like. That’s the idea that really comes out of Turing’s 1936 work, and that was what was so constructive and new about—when he came to designing a computer in 1945.

**Steven Cherry:** That’s going to remind some of our listeners of Kurt Gödel’s famous incompleteness proof. Maybe you can just say a word about how they’re similar and different.

**Andrew Hodges:** Yeah, there’s a very—there’s a very important connection between them. What Gödel had done in 1931, which revolutionized logic, was to see that statements about numbers, theorems, or proofs of theorems could be encoded as numbers, which seemed—sort of seemed—crazy anyway. I mean it was completely—made a complete difference to logic. No one had thought of this before, and overnight, really, it changed the whole picture of what you could possibly achieve in logic. Well, Turing learned about this in 1935 from lectures at Cambridge, and then he learned that there was this outstanding problem that I mentioned, this question of whether there are things that you could describe but couldn’t do by formalism. And that’s why he had to design what a formalism actually is, what an algorithm is. But having done that, he was able to use Gödel’s idea in the following way: He encoded operations *on* numbers or symbols *as* symbols. It’s really the same idea.

**Steven Cherry:** Let’s talk about World War II and Turing’s role in it.

**Andrew Hodges:** Well, Alan Turing had an extremely amazing, dramatic role in the Second World War. He was the first major scientific figure—well, actually, the first scientific figure at all—to be brought into the British establishment, which happened in 1938. And thereafter he remained the chief scientific figure, which is most important because in tackling the German Enigma machine—which was the workaday thing that was used for the bulk of German military and naval messages—it was essential to go to a mathematical and scientific analysis [unintelligible]** **had to be done, rather on the linguistic, sort of guesswork-type things that had been all right before. So he was in the first position there, and he remained in that top position throughout. And he made absolutely crucial contributions very early in 19—and in war, and in late 1939, he saw this marvelous logical trick that you could use to overcome the great complexity, the Enigma, which had been introduced in the German military machine by the addition of this plugboard.

**Steven Cherry:** So the Germans had this device that was encoding their most important operational messages, things like where submarines would go and so forth. And Turing led this effort to basically crack the codes. And the British never let on that they were privy to these secret messages, but their ships could move in ways to avoid attack and so forth because they knew where the Germans were moving.

**Andrew Hodges:** I don’t want to get too technical about this, but this was an enormous enhancement of the commercial Enigma into something which really had quite a lot of security. But Turing spotted this logical weakness by actually an amazing trick and was able to devise a machine which could actually mechanize the logical sequence of ideas which you needed to exploit this loophole, this flaw in the design. I should say that Turing didn’t do this all on his own. A very important contribution came from Polish mathematicians, which were given to the British in 1939, and also his collaborator, another mathematician, Gordon Welchman, made an important contribution too. So this was absolutely a stunning contribution to the—to the war effort and something that then became extremely important when—indeed before the United States entered the war in 1941 as part of what the British war effort had to offer to Allied success.

**Steven Cherry:** Computer scientists and philosophers have something called—they call the Turing test. Now this comes from a 1950 article entitled “Computing Machinery and Intelligence.” What’s the Turing test?

**Andrew Hodges:** Well, the Turing test is really just Turing’s way of making—it’s his witty and interesting way and draws people into the problem in a nice manner, which people respond to very well. And it’s a problem of how you give an objective picture of what you even mean by artificial intelligence, what you would count as intelligence. It’s formalized a bit in terms of text messages in modern terms, so you’ve got to guess whether the originator of the messages you receive is a person or not a person. And his argument was that if you can’t tell the difference, then you’ve got to credit the originator with intelligence, and not knowing whether it’s a person or a machine. I mean lots of people have arguments about this, whether it’s a fair test or not, whether it’s a good guide to artificial intelligence and so forth. But it shows the scope of what he was thinking about. He wasn’t just thinking about this or that little trick that you can get a machine to do. He was seriously interested in the philosophical problem of what the brain and the mind are. I mean what it is—is about human consciousness, human thought—that makes it different from anything else that we can observe and whether that can be replicated in machines or not.

**Steven Cherry:** Turing was a pretty witty fellow, and I kind of wanted to turn to his personal life now. He was—Turing was gay at a time when it was impossible to be openly gay. There was one woman in his life, whom he met at Bletchley Park: Joan Clarke. Can you tell us about her?

**Andrew Hodges:** Yes. Well, he—Joan Clarke was a mathematician, and she came in as what in modern terms would be a research student in mathematics. And [she] came into the code-breaking work and was in the section on the naval Enigma problem that Alan Turing headed. This would be in 1940–1941. And yes, they were very friendly. And this, I suppose, I mean it’s hard to know exactly what went through his mind, but certainly I mean many gay men in that period would have gone into a conventional form of marriage for appearance’s sake. I mean, a lot of gay men did that. I mean, that was far from uncommon. And at one point, yes, he said, “Let’s do that.” And she said yes. But then he took it back almost immediately, explaining that the reason he couldn’t was because he was gay. And that was a very modern thing for that time, to be that explicit and that—and that definite about what was going on there. And so it was a great shame, I suppose, but they remained friendly. And he didn’t—didn’t choose that way forward again. I mean, after that period, he certainly was much more definite about his—his identity as a gay man. And took a much more and increasingly positive attitude to it, a very—really was much more in tune with modern ways of thinking than with the ideas of his own time. And that would be really a typical way of describing what he was like. He was ahead of his time in all the mathematical and scientific thought and what he saw as the vision of computing and how it fitted into science. But also he was very impatient with all the social conventions and what he would have called stuffiness and formality of the time. And in many ways, he just behaved much more like people behaved after 1968, I suppose, really. He was just sort of ahead of his time that way. This is quite shocking to other, more conventional people, but that’s pretty much what he was like.

**Steven Cherry:** I said it was impossible to be openly gay at the time. The laws in England really hadn’t changed that much from the days of Oscar Wilde. But I guess for his entire life Turing had a lot of trouble living by the conventional rules of society. And he did sort of come out, as we would say nowadays, even in the middle of the war.

**Andrew Hodges:** Oh, yes. He was quite—he was really—he was quite open about being gay to people he worked with at Bletchley. And the problem with—with security and vetting didn’t start until 1948, so that wouldn’t have been a problem in itself, you see. And there was sort of a Cambridge atmosphere there, which on the whole was a very protective—or could be anyway. And that’s what he—to some extent—he took advantage of. But he was very individual, and he was much more explicit about it than other people would have been. And there’s a modern sense: He didn’t like being dishonest. It was against the grain for him to be dishonest.

**Steven Cherry:** Now, so he was eventually arrested for homosexuality, and the authorities gave him a pretty draconian choice to make.

**Andrew Hodges:** Yes. Yes. He was treated very seriously, and he had, I suppose, an option, which he either accepted this, what was called “organotherapy,” which was in effect chemical castration, at least for a year, or else if he refused that—I don’t think he really seriously considered that option, because it would have destroyed his work—but the alternative would be to go to prison. And the law hadn’t changed at all since the days of Oscar Wilde, as every gay man in that period was well aware. And Turing referred to Oscar Wilde himself. The whole Oscar Wilde trial hung over the whole of that period very heavily. And it was almost, in fact, the same sort of scenario.

**Steven Cherry:** The therapy—I guess it was a sort of estrogen therapy, we might think of it today.

**Andrew Hodges:** Yes, that’s right. Yes.

**Steven Cherry:** Now, Andrew, when a great thinker’s life is cut short, it’s inevitable, I think, that we wonder what else they might have accomplished. And I know you have some thoughts about what Turing might have accomplished.

**Andrew Hodges:** Well, he had a great deal going in the immediate research work that he was involved in after 1950, when he turned away from the theory of computing and actually used the computer that had been built at Manchester as his own personal computer for his own scientific research work in the completely new field of mathematical biology that he started, something which is now hot stuff. I mean, this is now something which is in its own right a very large field, the only intersection of biology and mathematics. So in matters going from there, if he wanted to pursue that, he had his doctoral** **research students by that time, and so it was the beginnings of a modern research group with a lot of potential. But he had other interests as well. He had turned back to interests in some of the pure mathematical things he’d done but also had a renewed interest in theoretical physics, which had been actually one of his first interests as a teenager. And that is, again, something which he might well have taken up in the period after the mid-1950s, if he hadn’t taken this perhaps rather impulsive action.

**Steven Cherry:** I gather you think that he could have done some more in computing and really pushed computing more quickly than it ended up going.

**Andrew Hodges:** Well, what Turing could have done that he didn’t do is actually just simply publish more about what he knew and had put only in unpublished reports and internal reports and so forth on the computer work but didn’t publish in a journal article in the way that people would know and would have made his name if he’d done. He never really made anything of the fact that he, in 1936, had formulated computability. He’d formulated the idea of the universal machine. He’d seen that instructions are a form of data and that this is the crucial thing of serious computing. And that electronic computers as devised by, well, essentially by himself at the same time as von Neumann, that they embodied this idea. And he knew that they were universal machines, they would take over everything, and you can see how that’s just how that’s happened. But he didn’t actually do much to—to promulgate this. Certainly not to put his own name to it so that people really could see that he was the originator and the pioneer and the courageous person behind all this. He hid himself too much. So that’s one thing he could have done. And if he’d gathered together all the things that he poured out during that period into a coherent form and published it, well, that would have made a tremendous impact then—stamped his name on the whole emergent engineering discipline of computer science. So that’s something he didn’t do. I mean, he certainly didn’t do everything and didn’t do everything right, and that’s something one can—to some extent he’s been rescued from that by people like me digging up quite obscure pieces of paper and putting them together to see what actually went on. There is a connection between Turing’s interest in artificial intelligence—which essentially is the question of whether the brain could be modeled by an algorithm and therefore as a Turing machine—there is a connection between that and his interest in mathematical biology. And you can see that in things that he wrote when he was starting off on this new field, which were questions about how a brain grows, I mean, how it develops—how its nerve cells all proliferate, and so forth, in the embryo, and so on. And he didn’t actually follow that. He didn’t really take that up specifically in his theory, but that shows the kind of thought that he had behind him in moving from a abstract theory of computation into a quite technical applied mathematical description of patent formation in biological structure.

**Steven Cherry:**** **Andrew, the centenary of Turing’s birth is a pretty important event for those of us at the IEEE, and I want to thank you for taking the time to talk with us today.

**Andrew Hodges:** Thank you very much. It’s been a great pleasure to be with such distinguished company as your—your listeners. So I hope that they enjoy everything to do with this extraordinary story and the events of this year.

**Steven Cherry:** We’ve been speaking with mathematical physicist Andrew Hodges about the legacy of Alan Turing, the world’s first computer scientist. For *IEEE Spectrum*’s “Techwise Conversations,” I’m Steven Cherry.

**Announcer:** "Techwise Conversations" is sponsored by National Instruments.

*This interview was recorded 10 April 2012.
Segment producer: Jean Kumagai; audio engineer: Francesco Ferorelli*

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