by Rob Kirby
The following is a transcript of a recorded interview that took place on 8 December, 2019. Tom Scanlon, Rob Kirby and Celebratio Mathematica’s Managing Editor, Sheila Newbery, spoke with Lou van den Dries at the Shattuck Hotel in Berkeley, California on the eve of the Julia Robinson Centennial Symposium held at MSRI. Van den Dries was as an invited speaker at the Symposium, which was organized by Hélène Barcelo, Tom Scanlon and Carol Wood.
Rob: Let’s start with your full name.
Lou: Well, my first name is really Laurentius. Normally it’s Lou, written as L-o-u, which is a bit strange.
Rob: Then you have some more names…
Lou: Yeah, OK. That’s true. Laurentius Petrus Dignus van den Dries.
Rob: So, is there a story behind that name?
Lou: Not really. Van den Dries is originally a Flemish name.
Rob: It sounds very Latin. Very Roman.
Lou: By family tradition and as the second of six I was named after my grandfather on my mother’s side (Laurens), my father (Pieter), and my oldest uncle on my mother’s side (Dingenis). The Latinization is a Roman Catholic habit.
Rob: So, what’s your birth date?
Lou: May 26, 1951.
Rob: And where were you born?
Lou: I was born somewhere in the Netherlands, not in a city, on a farm, in a certain area that is below sea level, which is not uncommon in the Netherlands, but not near any town.
Rob: OK. North? South? East? West? Just roughly…
Lou: The Northeastern quarter of the Netherlands.
Rob: What sort of farm? A dairy farm or…
Lou: No, a farm where you grow products like potatoes, wheat, sugar beets, flax, and peas.
Rob: Your father was a farmer. Was his father a farmer?
Lou: As far as I know, as far as could trace it back, they all have been farmers.
Rob: On the same farm? Does that go way back?
Lou: Well, my parents are from Zeeland, in the southwest of the Netherlands, and grew up on farms there which had existed for centuries. The farm I grew up on was a new one, dating from 1949, in an entirely different part of the country.
Rob: Did your mother…?
Lou: Also. Same kind of background. That family did own a farm. It’s very much a farming family from both sides.
Rob: Do you have brothers or sisters?
Lou: Yes, we were six.
Rob: Same as Lenstra.
Lou: Oh!
Rob: Lenstra, I believe, has five siblings, also.
Lou: Oh I see. Huh.
Rob: So, which one were you?
Lou: Second. I was one year after the first born.
Rob: How many boys? Girls?
Lou: Two girls. Four boys.
Rob: And the girls were which…?
Lou: OK, the third one, that is my oldest sister, and then the very youngest one, the sixth one, was also a girl.
Rob: OK, is there anything interesting about them? Did they have unusual careers — or something of interest that you want to mention?
Lou: Well, let’s see, my oldest brother was basically the first one in my family who had higher education. He became an agricultural engineer. He worked a lot in Africa. In fact he has a degree in tropical engineering. And he still goes there a lot. And other tropical areas in the world.
My youngest brother, the fifth one, he took over the farm. He’s the farmer. And that’s also where I go back, still, most years.
Rob: Alright. Your schooling — did you just go to a regular…?
Lou: Yeah. So the nearest village, Ens, was about five kilometers from our farm. We biked every day to school and back. That was just a primary school — a six-year primary school. And then after that, I went to the local lycée, well lyceum, a sort of a high school in Emmeloord. At that time, that was a school that if you finished it with success, you could go to the university. So, that was about ten kilometers away. Also, I went there usually by bike — and back.
Rob: And so then, where did you go to college?
Lou: Utrecht. And that is in the middle of the country, the middle of the Netherlands.
Rob: So, did you go in 1969? When you were eighteen? Or earlier?
Lou: Yeah, exactly. I went there in 1969. In high school, or lyceum, as we called it, I did become very interested in mathematics. And in fact one of the books that got me hooked was this little book called From Zero to Infinity by Constance Reid [e14], which of course I only found out later was the sister of Julia Robinson, when I was already here in the US.
Rob: Did you ever meet her? Constance Reid?
Lou: No, but I met Julia Robinson and Raphael Robinson. I was an assistant professor at Stanford for a few years, and so at that time I did meet some people in Berkeley, including Julia Robinson and Raphael Robinson.
Rob: I knew Constance Reid a little better than I knew the Robinsons. She was a lively lady!
Lou: Anyway, I certainly remember that book; it made mathematics look very interesting to me.
Rob: Did you have calculus in lyceum?
Lou: Oh, yeah, it certainly included some differentiation and integration, yes. It didn’t go very far. We did learn some things.
Rob: So then, as an undergraduate at Utrecht you were a math major?
Lou: Yes. In the beginning you had to choose — you couldn’t just be a math major. You had to choose so-called minors as well, and I do remember I had physics and astronomy as minors, but I was somehow able after a year or two to focus on math.
Rob: When did you turn to logic?
Lou: In the first two years or so I took a course in mathematical logic, and I thought, “This is a subject where you start from scratch and you can still get some nontrivial things!” And the book we used was Shoenfield’s [e2], I remember. Actually, that book is quite a good book. I think now it would actually be considered a little on the hard side for undergraduates to learn it from that.
Rob: Were there any particularly professors that were important to you?
Lou: Well, back in high school, I participated in this mathematical Olympiad, and I was second in the Netherlands. When we went to the Hague to get our prize, Freudenthal was present, and probably had chosen the books that we were given. And this also influenced my choice of University, since Freudenthal was a professor in Utrecht and I did indeed take a course from him in the first two years, called “Structuren”, which was quite nice.
In college — we call it simply university — one course that really did have an influence on me was a course on several complex variables that I took from van der Put. He was a coauthor, for example, with Michael Singer. Singer–van der Put is a well known book on differential algebra [e12]. He gave a nice course which I took in my third year as an undergraduate and I and some other student [ Tom Vorst] were appointed as note-takers. So we really had to learn it. Although some of it went over my head — like Dolbeault cohomology, I remember. But I don’t remember very much of it.
Rob: I think it went over my head, too.
Lou: Čech cohomology, and so on. But what I did learn are things like Weierstrass preparation and that has always served me very well since then.
Rob: Did you stay at Utrecht?
Lou: Yeah. It was fairly common at that time to stay at the same university and continue there.
Rob: Who was your advisor?
Lou: Van Dalen. So he was from this Dutch tradition of intuitionism. I certainly had courses in that subject but I decided for myself, rather quickly, that this was not the way to do math. So I simply worked on my own. It was perhaps more common then than it is now. Nowadays, graduate students typically work in the expertise of their advisor, but working through Shoenfield, there was this chapter on model theory and I thought, “Hey, this is the way to use logic in mathematics.” In a way that is creative. So, I got hooked on that. But basically it was self-study.
Of course the topologist Brouwer was also the father of intuitionism.
Rob: My reaction to it was that it was like doing mathematics with your hand tied behind your back.
Lou: Yeah. This was also sort of my reaction.
Rob: Maybe this is also why nobody does it!
Lou: Well, one could say that if you prove something in an intuitionistic way, you often get more information out of it. But of course, classical mathematicians also know that if you prove something in a constructive way, often that simply means that you have proved more. Right? But I didn’t see why one had to have an intuitionistic philosophy behind that, where one could simply be interested in better results or more constructive results anyway.
Tom: Well, does this have an influence on the way you do mathematics. I find that your work is much more constructive than other people working in our field.
Lou: Well, I doubt it. I somehow doubt that being exposed to intuitionism had a big influence on me. I think I am able to reason intuitionistically if I want, but model theory is also a way to extract constructive information that is not overtly there, but which, by model theory, you somehow know how to extract. So I definitely am interested in extra information that one can get this way. But I don’t think that came from being exposed to intuitionism.
Rob: So you really wrote your thesis pretty much on your own.
Lou: Right, yeah, well, of course I did have contacts in my years as a graduate student — I did get in touch with people like Angus Macintyre and [Gregory] Cherlin, whom I met at meetings in Belgium.
Rob: I was at UCLA in the 60s, and met Abraham Robinson. Is he…?
Lou: Yes, of course, his way of doing model theory certainly had a very big influence on me, because I found out later the exercises I had been doing in model theory from that book of Shoenfield’s really came from his work. And then later on I read some of his things, like this nice little book on complete theories — that was an eye opener for me.
Tom: Did you read that before you wrote your thesis?
Lou: Yes.
Tom: I actually looked at your thesis [1] a few years ago. My student Will Johnson was trying to look for examples of multivalued fields, and especially the section about multivalued fields and multiple orders looks like it could have been taken almost directly from Robinson’s book.
Lou: Well, of course Robinson didn’t consider multiple-valued fields, but I mean it was certainly the first time that someone considered valued fields from a model theoretic perspective. Yes. So I’m pretty sure I did read that book — I did read much of it when I was a graduate student before finishing my thesis. I do remember thinking of possible thesis topics. I had to, you know, invent some topic that I could work on, and at some point I decided, “OK, let’s look what happens if you have fields with more than just one valuation.” And I figured out enough to say at some point to Macintyre when I met him, “I think I figured out that the theory of fields with more than one valuation has a model companion.” And I asked, “Do you think that would be interesting?” And he said, “Oh, yes, that would be very interesting to write out.” So I decided this will be part of my thesis. Right.
Rob: Is Macintyre still alive?
Lou: Sure, otherwise I’d have heard about it! He’s about ten years older than I am and I’m — sixty…
Rob: Sixty-eight.
[laughter]
Lou: Yeah. Nowadays I always have to think. Am I sixty-seven or sixty-eight or sixty-nine…? It’s one of those three.
Rob: So where was your first — where did you go when you got your PhD? Did you go to Stanford?
Lou: No, I was a postdoc at Yale University with Macintyre from 1979–1981.
Rob: I thought Macintyre was in England. But he moved or something…?
Lou: Before I was a graduate student even, Van Dalen had actually contacted Robinson who was then at Yale University, and said, “I have a student who is very interested in your work, and has learned about it, and could he be a graduate student?” and, you know — because that certainly would have been better than working on my own. Or more efficient — and Robinson actually replied, “Yes, that would be fine.” At that time, I think these things were done without too much bureaucracy. In fact, what was actually planned was that I was going to be there as a graduate student, but then he wrote back half a year later — that was actually the time that he got very ill. And so then I decided to stay in Utrecht, and he said, “I cannot take any students anymore”. That was also the time Macintyre had gone there as an associate professor, And so I think I still should have gone there, but anyway. I ended up there anyway as a Gibbs instructor.
Rob: And after two years there…?
Lou: Then I was at Stanford for a few years. And I was also a year at the Institute for Advanced Study. And then in 1986, I was hired by the University of Illinois.
Rob: So, I think I remember that Berkeley made you an offer somewhere in there…
Lou: No, that didn’t happen. I think they did consider me at one point for a position. When I was at Stanford, I definitely came here to give a talk that was sort of considered as a kind of interview talk, and I think they did consider me, but I don’t think they made me — well, I would have known if….
Rob: I just remember your name from those days.
Lou: Aha.
Rob: You probably chose a different field.
Lou: Well, I do know that when I came for this talk, I did not actually talk about my own work. I had no idea that this was supposed to be what you should do. I just talked about something that I found interesting and learned about. It might have been about Denef’s work or something like that. Of course, later on I actually got to work with Denef on related issues, but at that time, I was still more or less familiarizing myself with some of his work, so I talked about that but not about my own work, which would have been probably — at that time I was already considering things like \( o \)-minimality. Right, yes.
Rob: Could you go back and say the person whose work you talked about…the name is…?
Lou: Jan Denef. He is a Belgian mathematician, with whom I did some work later on.
Tom: And, so was the introduction of Weierstrass division your idea? Did it come from the classic…?
Lou: Yeah, so I had introduced this \( o \)-minimality business — I mean I didn’t call it \( o \)-minimality, but I had introduced some things that developed into \( o \)-minimality [3]. In particular, I also observed at the time that this theory of subanalytic sets — if you just change the definition from subanalytic to what I called globally subanalytic, then the reals with that is an \( o \)-minimal structure. So then it was natural to start thinking about “what about the \( p \)-adics?” Right? And in fact there was this question by Serre in an article that he had written [e5] that seemed — that it might be solvable if you had a reasonable theory of \( p \)-adic subanalytic sets. I actually started to think about that, and I was on leave at the Max Planck Institute in Germany in 1986, before I took up my job in Illinois, and I had some idea that went into the right direction and I actually gave a talk about it in Belgium, where Denef was present. Of course I knew Denef already. In fact, we had already written something related to Artin approximation, so I knew him quite well. Then, a week later, after that talk, we basically both had realized independently how to use Weierstrass preparation and division in such a way that you could develop a theory of \( p \)-adic subanalytic sets. This involved a somewhat tricky use of Weierstrass. And then we wrote the paper [4]. It was a very nice collaboration. It did answer Serre’s question on Poincaré series of \( p \)-adic analytic varieties.
Tom: And had you worked out the similar argument with the reals beforehand? Or is that after…?
Lou: No, then we decided, OK, now that it worked so well for the \( p \)-adic, let’s go back to the reals and see if we get this quantifier elimination that we had for the \( p \)-adics. Could we also get it for the reals, because the theory of subanalytic sets developed by Hironaka and Gabrielov did not actually give you quantifier elimination. It was model completeness basically, what you get. And then we did that for the reals in a way similar to what we did for the \( p \)-adics. But in the \( p \)-adics, it’s a little simpler. So it was nice that we did it there first and then went back to the reals. But in some sense it went from the reals — basically, with the realization that the reals with the globally subanalytic sets form an \( o \)-minimal structure — to the \( p \)-adics, and then back again to the reals.
Rob: Is there a version for the complex numbers?
Lou: Well, for the model theorist, the complex numbers are simpler than the reals. The complex numbers are interpretable in the reals, as we say, so I don’t think there is a special theory of subanalytic sets over the complex numbers.
Rob: So, the mathematics you’re talking about here is mostly during your first six years, ten years…?
Lou: Well, what I was just talking about — this collaboration with Denef happened in 1986, when I was on leave. I had some sort of semester off. I was at the Max Planck Institute. I think I was already hired by UIUC. But, for example, these ideas about \( o \)-minimality, I got in 1982, so that was a few years earlier.
Tom: And it ends all the work that you did on ultraproducts of polynomial rings and faithful flatness.
Lou: Oh yeah, that was in my thesis.
Tom: Why is that — OK, I don’t know who your coauthor Schmidt is.
Lou: Oh yeah, Karsten Schmidt.
Tom: How did it come about that you were working with him?
Lou: He was from Kiel (Germany) and got in touch about these bounds when I published something from my thesis about it in the proceedings of a meeting. We then collaborated on getting further bounds of this kind, considering also cases where they don’t exist [2]. He went on to publish some papers of his own on these matters. Schoutens took up these things as well, and so did Aschenbrenner (when he was my student) in another direction, polynomial rings over the integers, where it turns out there are bounds of a different nature [e13].
Anyway, in the last chapter of my thesis, I wanted to understand these complicated algorithmic results of Seidenberg and…what’s her name? They always mention two people, right? Grete Hermann and Seidenberg. At the time I didn’t like algorithms. I wanted to understand from a more conceptual point of view. Now I think algorithms are actually — I’m getting better at it, perhaps, or have more appreciation for it, but at that time, I wanted to understand in a more conceptual way how these bounds came about. After all, they mean that certain things are of a first-order nature. And of course my interest was also piqued by Abraham Robinson’s list of problems. He stated some of these things as open problems or partially open problems. Then I noticed that faithful flatness and some other well known commutative algebra facts could be used to get these bounds in a very cheap way.
Rob: Was Abraham Robinson’s list of problems published?
Lou: Yes. He gave an address as President of the ASL — an address leaving this position. He mentioned a number of problems and described them in a very attractive way, and they were published in, I think, the Journal of Symbolic Logic.1 They’re certainly in his collected papers or something.
Rob: Actually, let me interrupt just so I don’t forget. Is there unpublished work — your notes, something of that sort — that would be valuable to know about…?
Tom: What about your work on co-logic. Was it ever published?
Lou: No. With Macintyre and [Gregory] Cherlin, the three of us wrote a fairly long paper on some sort of co-model theory for Galois groups or profinite groups where everything is sort of dual. Later on I realized it’s just a form of many-sorted first-order logic. But at the time, I somehow did not realize that.
Tom: Ah.
Lou: So that paper was never published because we couldn’t really agree on how to finish it, how to put it into final shape. But we distributed it to people and people picked up on it and so we decided, ok, what the hell, let’s just have it in the public domain in this way. Sometimes I still get requests for a copy, yeah.
Rob: And you’d allow us to include that…?
Lou: Well, if Macintyre and Cherlin are OK with it.
Rob: Sure, of course we’ll ask them.
Lou: I think I must have a copy somewhere.
Tom: I have a photocopy. It’s an actual physical photocopy of a copy.
Lou: You can probably see that different parts are in different fonts.
Rob: Yeah, it may not be perfect, by definition, but still it’s of historical interest.
Lou: Right. Something that I was some time ago planning to publish but never got around to was a problem that Serre once posed about Hilbertian fields. I’ve actually given a few talks about it, but never got around to writing the solution down, until about a year ago I heard someone else talk about a similar thing, and I mentioned this. What’s his name again…this guy in Dresden?
Tom: Arno Fehm?
Lou: Fehm, yeah! I said, “This looks very much like the kind of thing I was thinking about — I don’t know, it must have been about twenty years ago, and I still have notes about it. I think I should send it to you to see if there’s some overlap.” And it turns out, that he had indeed also proved these things — but in a different way and published something about it [e15]. I should have been quicker with that.
Rob: There’s a good place in Celebratio Mathematica for things like this, unpublished work…because it’s part of the historical record. It shows something you were thinking about. It’s part of your oeuvre.
Lou: I don’t know. In fact, I do remember you [Tom Scanlon] were at the talk that I once gave about this in Urbana, and I think you asked me to send you the slides.
Tom: Huh. I must have those, too! Which theorem is this?
Lou: Yeah, what was the problem again…? It was something about…Ah, yeah. So, Serre had defined some notion of Hilbertian variety, right — sort of a generalization of Hilbertian fields satisfying the Hilbert irreducibility theorem, which means that a field of rational functions in one variable over it would be the…or the affine line would be an Hilbertian variety in his sense. And his question was whether the product of two Hilbertian varieties is Hilbertian. And I saw that one could use some arguments of Roquette’s, or one could adapt — modify, generalize — arguments of Roquette’s to indeed answer this question positively. And I wrote it down. On one or two occasions I gave a talk about it, and then I sort of forgot about it. About a year ago or two years ago, it turned out that someone else had in the meantime also done this.2
Tom: I wanted also to ask about some other work. There are two directions I wanted to talk about, and they’ll each one take a long time. One would be about the work you did with Dave [Marker] and Angus [Macintyre] on power series models…and connections to the quantifier elimination [5], [6] .
Lou: Oh yeah.
Tom: Was that the origin of your thinking about transseries and the twenty-year project?
Lou: Yes. Right, so let me see. I have to go back in time now. Before Alex Wilkie had proved the \( o \)-minimality of the real field of exponentiation [e10], Dahn and Göring had written a short paper [e6] suggesting a concrete, nonstandard model of this theory, which consisted of logarithmic exponential series over the reals. So, after this work of Alex’s, we started to think about whether perhaps we could indeed prove that this was an elementary extension of the real exponential field, and then we did…Well, it’s a little complicated again because — first, I did some things with Chris Miller where we basically adapted techniques of Alex’s and we did prove that the reals with restricted analytic functions and exponentiation is model complete, and \( o \)-minimal as a consequence.
I think that only gave model completeness, and then right after that — yeah, what is the story exactly? The three of us — David, Angus and I — were basically thinking about this independently. We had already been talking and I think we decided to collaborate, and then we found a better way of proving this result which in addition gave a quantifier elimination and the fact that the field of logarithmic-exponential series is an elementary extension of the real exponential field. And then, of course, I got also very interested in this structure as a differential field, which is a far more complicated object. So then I started to work on that around the same time as Matthias Aschenbrenner arrived as a graduate student in Urbana, and together we did some things, and then a few years later we found out that Joris van der Hoeven had actually written a thesis about this differential field [e11]. We got in touch with him and the three of us (Matthias, Joris and I) decided to take it on as a sort of project to really understand the elementary theory of this differential field of transseries. So that has been my main occupation for twenty-five years now. In fact, I will talk about it tomorrow.
We did get somewhere — we published a book about it a few years ago [7], but since then we have actually gotten some further results, which I think are also very interesting.
In some sense one can connect it to some original things that Julia Robinson did. That’s how I get to talk about this tomorrow.
Rob: I was just going to ask you that question.
Tom: Well, so do you think there’s any chance that you could describe that structure as a difference field? It has many different structures as a difference field…
Lou: Yeah. Well, it’s certainly in the back of our minds to look at that also. Someone who has ideas about that would be Joris. But I must say I haven’t really gotten far enough into that to have an opinion about it. There are so many difference operators there. What is the right — well, I guess maybe just the one that sends \( x \) to \( x + 1 \) would be the most natural one. Or the one that sends \( x \) to \( e^x \). Either one would be very natural to consider. Well, actually these automorphisms do play even a role in our own work. For example, to show that certain things are not definable in our structure. But we basically have only been using it so far in this way. We have not really investigated the model theory of this as a difference field. But it’s certainly something that at some point, you know, people should do it.
Rob: [To Tom] You had another topic…?
Tom: Well, actually, it ends up being the same topic.
Lou: Right.
Tom: You know, one kind of flits…
Lou: Yeah, one thing leads to another. Going back to Julia Robinson and this book by her sister, From Zero to Infinity… At that time — that was back before I started college — I really found number theory a fascinating subject; I should really get into that. And of course, as one gets older it gets harder and harder to branch out in such a direction where there are such brilliant people already involved in connecting model theory to number theory.
Tom: But you did the work on the algebraic numbers.
Lou: Yes, right. That was nice.
Tom: And I remember there being some small talk you gave about complexity bounds for computing greatest common divisors…
Lou: Ah yeah. Those were sort of excursions, which were very nice. One sometimes needs to take a break from bigger projects to do something that —
Rob: To feel like you accomplished something!
Lou: Yeah, for example, this transseries thing that has been going on for twenty years — for the first fifteen years we made very little progress. I did feel the need sometimes to do something where you could actually show someone that you had done something.
Rob: Yeah, it’s true that as I’ve heard you talk about things that I don’t know hardly anything about — but the Weierstrass preparation theorem and things of that sort, I haven’t heard anything that sounded like Julia Robinson’s work. But there is a connection there. Not between Weierstrass, but between…
Lou: Well, the thing is that Weierstrass preparation and division can be used to prove things that we call model completeness, which means that all definable sets are existentially definable. And this notion of an existential definable set plays a big role in Julia Robinson’s work — but in another way, so to say. I mean, model completeness means that all definable sets are existentially definable while in Julia Robinson’s work, the existentially definable sets turn out to be the ones that are recursively enumerable. Of course that means that for Hilbert’s tenth problem there is no decision procedure for solving Diophantine equations, so that goes in the opposite direction, so to say.
Rob: Can you say what existentially definable is in twenty-five words or less?
Lou: OK. A set of tuples, of numbers, is existentially definable if it’s the set of parameters for which a system of equations is solvable. So we have a system of equations with parameters in them — you have unknowns and parameters. And then the set of parameters for which it is solvable is existentially definable. The set of parameters for which a system is solvable. That is what we call an existentially definable set.
Tom: If you were to write it in terms of the logical formulas, you’re saying there exists such and such that defines it.
Lou: So, there is this paper by Julia Robinson in 1952 [e1] as I will mention tomorrow in my talk because I think it’s a seminal paper in this whole area that led to the negative solution of Hilbert’s tenth problem — well, in the sense that there is no algorithm. But it also has a positive meaning, namely, that the existentially definable sets are exactly the recursively enumerable sets.
Rob: For my own education in the field of model theory, who are the early giants?
Lou: Well, Tarski and Robinson, I would say. And Maltsev. Those, I think, are usually mentioned as early model theorists. After that there are many people who have made big contributions to the subject. But maybe it’s too early to…
Rob: Tarski was a model theorist but really he did more parts of logic also…?
Lou: Who?
Tom: Tarski? I think he invented the term.
Rob: And by “Robinson” here you mean Abraham Robinson.
Lou: Right, right. I guess Julia Robinson isn’t counted as a model theorist, but somehow her work, because it involved definability, it’s in some sense pretty close in spirit — or part of it. And also with Raphael Robinson…they both took the notion of a definable set quite seriously and got some very nice results.
Tom: So you decided not to name people who are still alive on your list of important model theorists.
Rob: He’s being very careful! [laughter]
Tom: So there’s one person in particular who is still alive —
Lou: Shelah?
Tom. Yeah.
Lou: Oh, yeah, of course. I can safely call him a giant. Even his work is gigantic.
Tom: But the approach that he takes to model theory is very different from the one you take. Can you see places where what he’s done or his approach to classification theory or stability theory or anything has influenced your work, or you’ve responded to it in some way?
Lou: In some sense, when I grew up as a model theorist, I grew up sort of isolated from what turned out to be the mainstream of model theory, namely, this work of Shelah. In fact, I remember even Shelah giving talks in Belgium that I attended and I thought, “Well, is this model theory?” [laughter] “Why would anyone be interested in knowing how many models a theory has in a certain cardinality?” At that time I did not appreciate that at all. Of course when people talk about these things all the time around you, you are exposed to so much of it that at some point you realize, “Oh, well, hmmm.” I guess there are some people like Cherlin, Pillay, and then especially of course Hrushovski… When these people started to… You realize, “Oh wow! This is really a fantastic subject!” Or a very big development, which I completely missed out on. So at some point of course you do get influenced by these things. But I basically never — I did not grow up in this style of model theory. The things I’m doing for example with Matthias and Joris, we do pay attention to issues like NIP — whether a structure has NIP, right? Which is a notion that came out of this whole development.
Rob: NIP is…?
Lou: NIP stands for the nonindependence property, which is a very robust property that structures or theories can have and has many consequences. But it’s typically a notion that was invented in the school of Shelah and pretty far from Robinson’s way of thinking. But of course we learn about these things and pay attention to it, but I was never inclined, for example, to work on this as a pure notion by itself.
Rob: So is it fair to say that the sort of model theory that you’ve done has more connections to the rest of mathematics than Shelah’s work?
Lou: Not any more. [laughter] No, at the time that I heard first about Shelah’s work, I thought, “This will never make any difference to mathematics.” But I was totally wrong about it. It’s just that at the time, the way he talked about this or the way everyone talked about it at the time was, “Well, we want to know how many models a theory can have in a certain cardinality,” and I had no sense that this was an important question. Of course now we know that somehow investigating this you actually investigate very robust properties of theories, but this was not how people talked about it at the time. That came later.
Sheila: Is it relevant to ask how did you come to realize the importance of the question? At what point and how did you come to that realization?
Lou: Yeah, just being exposed to people talking about it. Also partly it was \( o \)-minimality, right? I invented this notion which turned out to be — later people called it \( o \)-minimality, and I asked people like Pillay, “Why do you call it \( o \)-minimality?” And he said, “Oh, it’s analogous to strong minimality.” And strong minimality is an earlier notion that also came up in this pure model theory trend that I had never really paid attention to. But then I suddenly understood why this was such a strong, powerful notion. Because I saw it is analogous to \( o \)-minimal; then it must be good! [laughter]
Rob: What does the letter \( O \) — where does that name come from?
Lou: Order.
Rob: Order minimality?
Lou: Yeah, order minimal. So, yeah, this is something that Pillay and Steinhorn and Julia Knight — they wrote [e8] (see also [e7]). So after I gave some talks about this, which Anand Pillay and Steinhorn attended, they made this into a definition: they called it \( o \)-minimal. Which stands for order minimal, and thought of it as analogous to an existing notion called strongly minimal, which existed already in model theory. So that’s also when I realized, “Aha, there is really a whole industry there that one should really pay attention to.”
I’ve always been most interested in understanding rather concrete structures, where one of the main things you want is, for example, quantifier elimination or model completeness. It means that you already can understand a lot about it, and these are also often the main tools that you need to prove other properties like, you know, NIP or stability or whatever.
I wish that I had paid attention to these things earlier but there are lots of things to do anyway. You don’t have to know everything.
Rob: This fellow — Sela. Is he connected at all to — ?
Lou: Oh, the group theory!
Tom: Zlil Sela?
Lou: Zlil Sela. Well, he’s more a group theorist but did solve a very important problem of Tarski. An older problem of Tarski which goes back to the 1940s, trying to understand model-theoretically the theory of free groups. And that is what — how do you pronounce his name?
Tom: Zlil Sela.
Lou: That is what Zlil Sela managed to do, which is very deep work, I believe.
Rob: It is, to some extent, model theory…?
Lou: Yeah, definitely. Right. Although he’s not really a model theorist, but it has been picked up by model theorists. Definitely. The very hard work he did was basically…he did it from scratch, so to say. I guess he was exposed to some logic. After all, the problem that Tarski stated wasa posed in terms of — as a logic problem.
Sheila: I want to take a step back for a minute and pin down the time when you learned about the analogousness of \( o \)-minimality to strong minimality. When was this brought to your attention?
Lou: OK, let me see. That was the mid 80s. But the Shelah revolution already started fifteen years earlier and a lot of model theorists immediately caught on to that. Not me. [laughter]
Rob: Well, the Shelah thing was in the late sixties, wasn’t it?
Lou: The end of the 60s, yeah.
Tom: He wrote his PhD thesis in 1968…? 1969?
Rob: Yeah.
Tom: But the book [Classification theory and the number of nonisomorphic models] came out in the early 70s.3
Rob: Because it seems like UCLA and Berkeley — I went to Berkeley in 1971 — and both places were trying to hire him, with no success.
Sheila: I just wanted to pin down the question about the Hilbertian varieties — when the product of two Hilbertian varieties are Hilbertian. When did you think of that? And then I know that Arno Fehm ended up proving it or publishing his result. When do you think your work was?
Lou: OK. I remember this was fairly early when I arrived at UIUC in 1986, so probably the late 80s or the early 90s. I remember there was a little book — or lecture notes — that was published by Serre [e9] or written by a student, where he started to talk about Hilbert’s irreducibility theorem, and so on, and generalized this notion of Hilbertian fields, and I saw these questions there. I was aware that there was this treatment of Hilbertian fields by Roquette and I thought, “Hey, maybe these techniques can help answer this question.” I thought a bit about it and I realized indeed that that was the case. I could answer these questions using these ideas. Then, I wrote them down but somehow never got to the point of writing it down as a publication. I guess there are always so many things that you want to do that sometimes you simply don’t get to put it in a form that you’re really happy with. Anyway, it’s not a big deal.
Rob: How do you spell Roquette?
Lou: R-o-q-u-e-t-t-e. This is a number theorist in Heidelberg, who actually worked with Robinson (Abraham Robinson) in the early 1970s. And who gave a nonstandard treatment of Hilbert’s irreducibility theorem [e3], which, at the time, I read carefully and liked a lot. So I learned something from that.
Sheila: Can I ask a general question? If you were going to do something with your mind and your talent and not think of yourself as being “too old” to begin something new, what would it be? What would you plunge into?
Lou: Ouf! Sometimes I have some ideas and every time it’s a different thing. [laughter]
Sheila: That’s a fair answer.
Lou: I mean, I could very well imagine starting over and trying to become a number theorist, for example. I mean, one of the things that has always frustrated logicians is that we have Gödel’s incompleteness theorem, which says that if you have a little bit of arithmetic in your theory, then you get all kinds of annoying things. There is no algorithm to decide whether things are true and so on. But, on the other hand, number theorists who work, for example, with global fields have a very important axiom which is called the product formula, right? The problem for logicians is that it’s not a first-order statement. Or not in a natural way, at least. But now there is a new — a sort of variant of first-order logic called continuous logic by which it is actually possible to formulate the product formula as a single axiom. And so Hrushovski and Itaï Ben Yaacov have been exploring this in a way that sounds very promising to me. And so this would be a way to combine model theory with number theory in a new way. I think it sounds quite fascinating. So if I had enough time, I can well imagine that I would like to get into that.
But, you know, the projects that I am already involved in are sufficiently interesting that I will probably stick with them. To learn new techniques and so on — at least at this… Well, I think I’m pretty slow anyway, so when you get older you get even slower!
Rob: I didn’t ask whether you have any hobbies… Music?
Lou: Well, no, just simple things like reading — especially history, for example. I like to read history.
Rob: Any particular time?
Lou: Well, basically, any time. As long as it’s not too recent. [laughter]
Tom: What’s wrong with recent history?
Lou: Well, because you live it anyway, so… I mean, the things that have happened since I was born, somehow it doesn’t sound so exciting to me.
I just went to the bookstore and I got these. This is a book on Kaiser Wilhelm II. This is the book by Hilary Mantel called A Place of Greater Safety. This is about the French Revolution.
Sheila: But is this…this is, uh, fiction!
Lou: Yes, this is a novel. But it’s historical fiction! That’s good enough. [laughter] And this is a book by Julien Benda called The Treason of the Intellectuals. I’ve always heard that this is a good book to read.
Rob: I used to read those sorts of things when I was much younger.
Lou: Somehow one has the feeling that he [Wilhelm II] might have been a figure a bit like Trump.
Sheila: You chose this in English!
Lou: Yes, I should have bought it in…I think I actually have the French version but the book was so badly organized that when I saw it here in English and I looked at it, I thought, “Oh, this is much more readable.”
Rob: So how many languages do you speak?
Lou: Well, basically, just Dutch and English and a bit of German. I can read French, but not really speak it.
Rob: It’s much easier to read —
Lou: Yes, especially mathematical French is quite easy in my opinion. And of course I learned Latin and Greek but never got any fluency in that.
Rob: So, you still got that!
Lou: Yeah, well, I was in a so-called gymnasium. The lyceum that I went to had two parts: one direction where you had to take classical languages and one which did not require it, and for some reason, the part that required classical languages was supposed to be the more prestigious part. And so you were basically sort of steered in that direction if your grades were high enough, but I never caught on to Greek and Latin very much, except history. Classical history I do like.
Sheila: Tacitus?
Lou: Yes, of course. I did read some Tacitus. But the books that we had to read — of course we started with De Bello Gallico in the second grade of the gymnasium, and then after that we had to go through Livius. The whole history of the city of Rome from the beginning. It was always too hard, You could really only do it with a dictionary. I never got enough facility with it.
Rob: My father took four years of Greek and Latin because he expected to become a minister and that’s what you did.
Lou: Oh yeah.
Rob: I took two years of Latin in high school and it didn’t take at all. But my nephew is only 26. He took four years of Latin in high school and liked it, so it just depends on…
Lou: Yeah, I mean, sure, there were people who were good in it, but I definitely was not.
Rob: My father always knew the derivation of words better than anyone else in the room from his four years of Greek and Latin.
Lou: Of course the Greek alphabet is still quite useful for mathematicians. [laughter] In fact, sometimes you need more than the two alphabets.
