Mathematics - Beauty vs Utility

Last year I met with mathematician Cedric Villani. He's somewhat of a celebrity, especially in his native France and the world of mathematics.

Professor Villani won the Field's Medal in 2010 and is director of Poincare Institute in Paris.

I've gradually been editing our interview into a series of videos - you can see them all on Numberphile's Cedric Villani Playlist.

The latest video is a discussion which started when I asked about a magic square on his blackboard... But the conversation then veered towards the beauty vs utility of mathematics?

Should mathematics be done for the sake of it - or should it be useful to society?

We also discussed why many mathematicians were considered poor at "outreach".

You can watch the interview at this video link. Or I have put a slightly edited transcription below...

BRADY: I noticed one other thing, I noticed on your board you have a magic square.

VILLANI: Correct.

BRADY: Why is that there today?

VILLANI: You know I spend part of my life doing interviews and sometimes you have to vary things, you know, change a bit from the usual. And so in that interview this was to recall my initial interest in mathematics. When I was a kid, this was the kind of thing that I loved doing - magic squares. I think it’s one of the oldest mathematic memories that I have, doing large magic squares.  

A child can learn how to do this. And I often explain about this in the schools - it gives you some sense of mystery and power of mathematics and algorithmics.

And there you are Brady, see, it's magic.

BRADY: So basically this is to entertain journalists?  

VILLANI: That's correct. But a bit more than that.  In this particular case the interview was intended for children. If I tell the children “you know mathematics is so good of the GDP because it its so good for technology” and so on - not sure they will like it. But if I tell them when I was a kid, at your age, I loved to do this kind of figures and it was kind of making me dizzy in a way to see the properties of the numbers... This was my start - and then it became my job.

BRADY: How do you find striking the balance? Sometimes when you're talking to people about mathematics you have to talk about things like GDP and the important role it plays in society... And other times you just want to show someone a magic square and just excite them - is that not enough, exciting them? Does it frustrate you that you’re always having to come up with these justifications for mathematics?

VILLANI: You have to justify. Part of the reason is there has been some excess in the past, I guess, in insisting on mathematics for itself - mathematics for the the beauty of it - and somehow we have to insist also on the other side now.
But it's okay, you have to balance. Every time I say something about the utility of mathematics, I will balance it with something about its beauty.  And vice versa.

BRADY: It’s unusual to hear a mathematician talk like that, you sound almost half politician. You think that's important? Pure mathematics is not enough these days, you've got to sing for your supper?

VILLANI: In front of you is not just a mathematician, I'm director of an institute, already six years. For the past four years I have been supporting a project for expanding my institution and managed to raise 13-14 million Euros for that project. This can only be achieved if you know how to talk to society; talking to businessmen, talking to bankers, talking to politicians, talking to school masters, talking to everybody. If you don't manage to talk the way they understand, you will not become their friend.

BRADY: But when you were younger man doing some of the work you're most famous for - most awarded for - were you thinking like that then? Or were you just doing it for the pleasure and was it just about mathematics then?

VILLANI: You know when I chose my subject, Boltzmann Equation, my adviser wrote for me the Boltzmann Equation, okay one variant of the Boltzmann Equation, I asked is it useful and who used it? And for me at that time this was an important question. To know that it was important in the modelling of rarefied gases, important for construction of shuttles or planes or simulation of what goes on around the wing of the plane and so on. It was important for me. And later when I worked on it, I discovered I really loved it for the beauty of the study, that the more I was going into it and the more it was a world opening for me. It was a curiosity that was driving me then. But to get into it I first had to be convinced of its utility. There's no shame in something being useful.

Somehow it's a little bit of poison that Hardy has instilled in our minds, making us feeling ashamed of what is useful. These were other days, other times and different context. Now we should not be ashamed of what is useful and even understand that when it is useful it is even more beautiful.

There has to be some useful and some useless. Sometimes the useful becomes useless, sometimes the useless will become useful. There has to be room for all kinds of mathematicians and it's a big illusion to think that there is a real recipe to decide what has to be funded and what should not. You know you have to allow some uncertainty in the system. If you try to predict, you lose the most interesting parts.

BRADY: Do you think mathematicians are good at outreach. Do you think mathematicians are good at explaining themselves?

VILLANI: In general mathematicians are not so good at outreach. And there are reasons for that. Some reasons are related to the field itself; because we go into such details. You know even between ourselves, mathematicians, it is difficult to communicate. It's so hard to communicate between mathematicians of different specialties. To talk to the outer public, the outer audience, is an even greater challenge. You have really get to the point, understand what you want to share, and so on. 

Another reason is when you are into mathematics you have been so high on the scale of complexity of reasoning that you are living in some kind of altered reality. You think that everybody on the street is able to understand complicated reasoning and so on. And you get very frustrated when you discover that's not the case.

And then there's a third reason, I guess. Which has to do with, how do you say, the business model of mathematics so to speak. The physicists, the chemists, the biologists - they are often in contact with the industry. Or the people who run big teams, they have to negotiate budgets, fundings, with the big pharmacy companies, or with the big sponsors, otherwise they don't have enough funding for the experiments and so on. And so they their learn to --- their rhetoric and speech and so on.

It struck me when I first attended the conferences of Nobel Prizes how good were some of the people, Nobel Laureates, in physics and biology, to present themselves. And I attributed that to the fact that they had to do it already many times for the funding in their career to people who were really out of their world, and it was very important to convince those people to trust them.

BRADY: And historically mathematicians don't need as much funding?

VILLANI: Historically we don't need the companies, historically no such need for support. I need the companies now because I am working on this outreach interface. We are soon opening some kind of museum of mathematics - soon means in three years or something. This will be impossible without the contribution of the industry, the economic world, so I had to learn to talk to them otherwise my projects could not have come into life.