A student favourite who loves mathematical conversations

With his calm and ability to communicate in ways that evoke enthusiasm about mathematical phenomena, Gunnar Berg has become a legend among students.

When he held a lunch lecture arranged by the Uppsala Union of Engineering and Science Students, UTN, the seats filled up quickly and the union had to arrange an extra lecture for all the disappointed souls who were turned away at the door. So it does not come as a surprise that Gunnar Berg has now been awarded with UTN’s pedagogical prize. Meet Gunnar in a conversation about everything from mathematics to God.

We are heading towards the break room where I will interview Gunnar Berg and I feel the same joy I used to feel about his lectures when I studied mathematics in Uppsala. Even before the interview, we start engaging in a conversation about mathematics. On the way to the break room, Gunnar tells me he is currently thinking about implication.

“How can a false statement imply a true one? The basic concepts are not always obvious," he says, adding that he often goes back and thinks about the basics. It is easy, he notes, to forget how difficult it can be for students to understand new things when encountering them for the first time.

 “It's the same with induction. What you must prove is not claim a, but the claim that if a is true, then b is true. When you understand that, it’s very easy. But if you’re teaching it, you must realize that this is a really difficult assertion.”

I am thinking that this ability to understand the student's perspective is a shining example of what makes Gunnar so appreciated as a teacher, but when I ask him how he felt when he got UTN: s pedagogical prize he says he was surprised.

 Why?

 “To begin with, I don’t teach as much as I used to. Then there are so many great teachers here at the Department. I believe mathematics as a subject encourages us to make an effort and explain things in ways that you don’t do within other disciplines. Mathematics exists to a large extent in conversation.”

What's the best thing about teaching math?

 “It’s a tangible way of experiencing when something suddenly makes sense. Very often, you try out a variety of routes, and after much work there’s a flash of light and suddenly everything’s clear. Another thing I like is that you keep improving your own understanding. You’re never done. Not even when it comes to the most basic concepts, for example the issue about implication.” 

But do you really understand, or do you just accept or get used to the concepts?

 “No, I think there’s a difference there. You may recall what von Neumann said: mathematics is not to be understood, you get used to it by keep on doing it. And yes, that’s correct, you can adopt concepts and move on that way. But true understanding is deeper,” he says and continues:

“Take, for example, such a thing as -1 times -1 equals 1. It’s something you have to embrace in school, probably without any kind of argument. And it's okay, it’s something you work with and you’re meticulous and eventually you make almost no mistakes. But once you understand that there’s a reason for this connection, that there is a reasoning which is general, that there are rules governing, something that applies to many other things as well, then it all falls within a different framework. And you’ve gained a deeper understanding.”

Although Gunnar has a love for teaching, it was not school that sparked his interest in mathematics. The arithmetic exercises bored him and he even failed math class one year. It was only when he started to read popular scientific books about mathematics in his spare time that his interest was awakened. The books showed there was a whole world beyond calculations, a world of ideas, fascinating people and historical progress.

His interest in the historical developments of ideas followed Gunnar into his professional life. After defending his thesis in mathematics, he studied history of ideas while working and got as far as being admitted to doctoral studies. But he decided not to continue on this track as he realized it could be diluting his knowledge.

 “For a while, I conducted research on the history of physics in the 20th century, especially Einstein's general theory of relativity. I remember once when I was at a conference, swimming in a pool in Pittsburgh along with one of the major authorities in the field. "Aren't you spreading yourself awfully thin?" he said. He was right. I was doing too much. If you’re that much involved, it’s difficult to reach true depth in any area.”

Gunnar mentions Archilochos’ fable about the fox and the hedgehog, where the fox has mastered many things while the hedgehog has mastered only one major thing.

 “Isaiah Berlin brought it up when he described Tolstoy's philosophy. Tolstoy happens to be a fox while Dostoyevsky is a hedgehog. I myself am a bit too foxlike,” he says, explaining that this trait is also apparent at his house. He keeps a pile of books at the bed side because he does not know which book he might want to read. One evening, he may want to read a certain book, only to put it aside the next evening and reach for another. He describes his reading as mood-dependent.

 “It’s the same thing when I'm watching TV. If somebody mentions Iceland, I immediately get up and bring out Njal’s Saga. These things wet my appetite because I keep getting so many associations. It’s hopeless. But okay. That’s also a way of living.”

 Can’t it also be a strength?

 “You have to get to a place where you can take advantage of it. In teaching, it’s quite useful to be somewhat of a fox as you must be able to adapt to different situations.”

 Do you have any educational philosophy?

 “For me I think it’s about having an equal fascination for the subject as for my students, for the teaching situation. And be interested in communication. And to think about it all the time. But I can’t say I have some kind of theoretical framework for teaching,” he says, and returns to the importance of conversation.

 “That’s why I think it’s so much fun when the students come in and ask questions. You also learn a lot by talking to them. Oh, that’s another way of thinking about it; that was particularly difficult, and so on. This dialogue, interaction, it’s among the most important there is.”

Conversations and ideas seem to be central in Gunnar's way of practicing his profession. He says that already as a high school student he was attracted to the most imaginative part of mathematics that stretches beyond the limits of arithmetic, moving towards strange geometries, topology and weird numbers. He was especially fascinated by the 1800's.

 “Something happens in the early 1800's,which is much more sophisticated.”

 What happens?

 “Well, it’s hard to say, but much of it has to do with the development of algebra. Abstract structures appear and we acquire ways of reasoning that are very general. Mathematicians such as Abel, Gauss and Galois represent that in some sense. I think this is the most exhilarating century in the history of mathematics. We still find ourselves in the aftermath of the 1800's. And it's also fun that in modern mathematics you will find references to books and articles from the 1800's. Somehow history is alive.”

What do you think the mathematical development will look like in the future? Will we reach a point where it all ends?

 “I find it hard to imagine. But there are times in history when people thought that was the case. In the late 1700's, if I’m not mistaken, there were a couple of people saying we’ve got it, the analysis is complete, we’ll be able to solve all problems, there’s nothing more to discover. But then comes the 1800's, new geometry rises and modern mathematics begins.”

 It sounds plausible that there will always be mathematics that we aren’t yet aware of. But do you think there is a limit to how much mathematics a human can understand during a lifetime?

 “Sometimes, after a lot of ifs and buts, you find the right way of formulating something. And then you’ve reached a smoother way of dealing with the concepts and you’ve acquired a better theory, which makes it easier because you don't have to walk such a long and crooked path again. You’ve found shortcuts. “

The conversation is just as much fun as I’ve expected. We are moving fast between various exciting angles on mathematics and dwell for a moment on the metaphysics of numbers. Gunnar tells me about Plato's theory that the idea the world is the real world and says he probably does not consider himself a Platonist.

 “If we were to meet a different civilization from outer space, I suspect they would have something very much reminding us of our numbers.”

 “Here we get into the question of existence. What exists really? In the 1800's, as soon as you discussed something without running into contradictions, it was considered of existence. But in that case we could apply this to discussions about theology. Does God exist? I think it is pretty easy to come up with contradictions when you start talking about God. It usually is.”

One can probably make up a god which isn’t contradictory. If you make one up, it exists.

 “The question is whether there is any point in talking about that god at all then. Doesn’t it seem like a most diluted logical god that will not have any effect on life at all? As soon as he or she begins to affect our human existential conditions, I think it will lead to contradictions.”

We are discussing the human contradictory nature for a while and end up in the conclusion that the human being cannot be pinned down by mathematically logical reasoning.

 “Thank God,” Gunnar adds with a chuckle.

text: Alma Kirlic

translation: Anneli Björkman

Live: In Uppsala

In my spare time: I exercise, spend time in nature, watch birds, read, travel now and then.

Hidden talent: (laughs) It's so hidden, I can't see it myself.

Currently reading: 10 books at least. But above all, Richard II by Shakespeare. And I relax with my favourite crime writer Edmund Crispin. He's fun, but I've never heard of anyone who’s read any of his books. I immerse myself in them to escape the problems of existence. His language is also amusing.

Listen to: Youtube. A lot of classical music, everything from the 12^th century until today. A listener’s scope for pleasure. My dad was a musician and played at the Opera. He didn't like Wagner because of the long performances, but when he finished playing, he started to listen to Wagner. I somewhat follow in his footsteps. The point with Youtube is that everything is there. You get a feel for which soloists are active. I also listen to Dan Andersson and such things, or Thåström where he sings Bellman. It depends on what mood I'm in and what I do. When I'm working, Chopin is great.

Memorable moment at work: I think all days are memorable in their own way. I’ve never had a bad day. Of course there have been days when I feel crummy, but work has always been a pleasure. What makes it memorable are all the collegial conversations and students asking questions. The exchange is the prerequisite.