# What would happen if there were no math?

## In conversation with mathematician Sir Martin Hairer

SIR MARTIN HAIRER SMILES FROM THE COMPUTER AND SAYS HELLO: Our Skype connection ends in a study in London full of bookshelves. The native Austrian Hairer teaches in this city as a professor at Imperial College. Since 2017 he has held the mathematics chair “Probabilities and Stochastic Analysis”. Hairer, 45, did his doctorate in Geneva (physics and mathematics) and has lived in England with his wife Xue-Mei Li, also a mathematician, since 2002. Hairer named his second home Knight Commander of the Order of the British Empire in 2019.

Why? Because he is a distinguished mathematician. In 2014, Hairer was awarded the Fields Medal by the International Mathematical Union, which is equivalent to a Nobel Prize. Last September, Hairer was finally awarded the highly endowed Breakthrough Prize for 2021: three million dollars in prize money for contributions to the theory of stochastic analysis, "in particular the theory of regularity structures in stochastic partial differential equations". According to the laudation on the relevant homepage, Hairer can use it to solve equations that describe random processes, "from fluctuating share prices to the movement of sugar in a cup of tea". A relaxed visit to the world of mathematics.

*TERRA MATER: The movement of sugar in tea? That sounds a bit strange.*

MARTIN HAIRER: The jury may have meant milk in the tea

(laughs heartily, not for the last time). There is nothing in the original text about sugar.

*Please explain your specialty to us.*

I can try to explain to you what stochastic partial differential equations are. So: Differential equations describe simple mechanical systems, for example the planetary orbits. Partial differential equations, in turn, describe similar systems, but instead of a system that is described by finitely many numbers, one has one that is described by a continuum. Like tea in a cup: do I want to describe it

I need to know the speed of the tea at every point in the cup. In the case of stochastic partial differential equations, chance also comes into play - in our case, the behavior of the milk poured into the tea. My research is on these mathematical objects.

*How does the world benefit from your formulas?*

As pure mathematicians, we are primarily interested in understanding the world, or: We want to describe these models mathematically. It is not about drawing consequences for the world from it.

*How about an illuminating example?*

Let's take a magnet: if you heat it up, there is a critical point, the Curie temperature, at which its magnetic field collapses. The closer you get to this temperature, the smaller the magnetic field becomes, but it does not become uniformly smaller, it starts to fluctuate. How these fluctuations behave is described by stochastic partial differential equations.

*So if chance plays a role in an equation, then you come into play ...*

That's where you come into play as a probability theorist, yes. Using the magnet as an example: The point is not just to make a prediction. Physicists make predictions. Mathematicians, on the other hand, want to understand why this is so.

*Do we need more physicists or mathematicians?*

Both. It's a very human drive to try to gain a deeper understanding of something and not just make predictions. The crux of the matter is: As soon as you have an interesting physical question, it automatically leads to interesting math.

*How often does the following happen: You research area A, but in the end the findings in area B are useful?*

Early 20th century mathematician Godfrey Harold Hardy did a lot of math, but he was a number theorist. He was once asked why he liked number theory so much. His answer: "Number theory is so far removed from the world that I'm sure people will never use it for any nonsense." That was his logic, but without this type of number theory there would be practically no internet.

*No internet without Hardy: What would we be missing if your formulas didn't exist?*

It is far too early to know. Hardy taught in 1920, and now it's 100 years later. Very often abstract mathematics at the limit of knowledge is typically useless because one has not yet found the usefulness.

*A forester plants a tree, but it wasn't until 50 years later that there was a trunk that could be used to make a table. How does it feel to think in one's discipline over such long periods of time?*

We mathematicians concern ourselves with this very little. We are really only interested in mathematics, we want to understand what no one has understood before. We are similar to nuclear physicists: If they are now researching something at CERN, it is of practically no use. The main benefit is to understand nature ...

*... and not whether there is now the Higgs boson.*

Yes. On the other hand, nuclear physicists have to build these gigantic billions of machines. This benefits the technology because they have to build instruments that are much more precise than the ones they currently have. It's similar in mathematics: you don't know beforehand which parts of historical mathematical research will be useful.

*Asked in an amateur way: can the future be calculated?*

If I believe in quantum mechanics - and there are more reasons to believe in quantum mechanics than not to believe in it - then I must also believe that it is impossible to accurately predict the future. In quantum mechanics, practically everything is always random: Even if I had a complete knowledge of the universe and a giant computer that sits outside the universe and could calculate everything infinitely quickly, I could only calculate probabilities. There is simply chance - and this can only be calculated in the sense that one can calculate probabilities.

*Astrophysicist Mario Livio asks, “Is God a mathematician?” And it takes an entire book to leave this question unanswered.*

I'm not going to try to answer that question (laughs).

*My question is anyway: Are mathematical formulas invented or discovered?*

That depends on the formulas.

*Let's stick with your area of expertise.*

You could ask a writer the same question. Does the person describe: Did he invent or discover this? Some formulas are invented, some are discovered because you start somewhere and don't yet know where you will end up. Like a writer: he starts with a rough character, and that character develops as you write.

*The Iranian mathematician Maryam Mirzakhani, who received the Fields Medal with you in 2014, said something similar about her work: "It's like writing a novel."*

To write a good math article, you have to somehow come up with a story. Of course the language is not the same, but the structure: which parts are told? How do you present what?

*Music critic and mathematician Edward Rothstein drew parallels between music and mathematics in a book. Music is something familiar to you too: is there some form of aesthetics in a mathematical formula for you?*

For me, beauty is in the thought process, in the arguments, in the formulation or in logic.

*For me, the formula for the circular area, A = r2π, is an elegant beauty ...*

With a short formula, I can agree. But in modern math, it's not often that you have a short, elegant formula that stands on its own.

*Maybe the days of simple formulas are over?*

Naturally. The mathematical objects one deals with today are much more complicated to describe. These are no longer simple geometric objects like circles or squares. It is rare to discover new formulas that have the aesthetic level of Euler's formula, e to the power of πi + 1 = 0, which combines all the major mathematical constants in one short formula.

*To Rothstein again. In contrast to mathematics, music can be understood intuitively, even if one cannot read notes. Or can mathematics be understood intuitively after all?*

It may be a little more difficult. But there are already nontrivial mathematical proofs that can certainly be understood intuitively by students - secondary school level, high school level. An example that comes to mind is Georg Cantor's proof that there are practically more real numbers than whole numbers.

*When you were a boy: How did your father, who is also a mathematician, introduce you to the magic of numbers?*

I don't think math has much to do with numbers, math has to do with logical arguments. Numbers play a role, of course, and are often used to describe mathematical objects, but they are not the main part of mathematics.

*Pardon: How did you become a logician?*

(Thoughts.) As a child, I was very impressed by the way my father explained the four-color problem to me. It works like this: If you take a map - countries of any shape - and you now want to color these countries so that two of the same color never collide: How many colors do you need? If you think about it a little and play around a bit, you see: there are at least four. In addition, it is no longer so clear whether it is possible to draw a map where you have to use five colors; it has since been proven that it is indeed impossible. This is a classic math problem that has nothing to do with numbers.

*With this example, did your father make math palatable to you?*

Exactly. My father also explained to me, "If you can prove it, you would be the first person in the world to find out." The problem was still unproven at the time. That impressed me: a problem that somehow sounds simple, and there is no weird counterexample with 30,000 countries.

*People now write letters to the editor. I found one in the Guardian - on the occasion of the Breakthrough Prize - who urged the British government to turn to you when it comes to Brexit and Corona to teach them more sense and efficiency.*

I didn't see the letter

(laughs).

Should mathematicians get involved more to make politics more sensible?

A good mathematician is not necessarily a good politician. You need to think logically, yes, but ultimately as a politician you have to make decisions with incomplete information. I think, as a mathematician, there might be the risk that you never make a decision because you always have the impression that you don't really understand the problem and that information is still missing.

*Could you solve Corona in a mathematical equation?*

This is another problem that cannot be packed into an elegant, short formula. As a mathematician you can develop models for the epidemic, you can try to predict how it will develop - people do that too. In terms of mathematics, these models are relatively primitive - the challenges of Corona are huge, but they are not mathematical challenges.

*Which problems does mathematics want to solve in the future?*

There are tons of math problems, for example in number theory. They have the advantage that they are very difficult, but easy to explain. In number theory there is the problem that every even number can be written down as the sum of two prime numbers. They haven't found an even number that doesn't work, but there is no proof. This classic problem has existed for almost 300 years.

*Why not type that into a supercomputer that has the solution after a week?*

Supercomputers can only test and try out. I don't think there is any mathematical proof where a computer somehow contributes to the logic. Mathematical proofs can be written down in a way that computers can understand, but a human has to find the proof.

*A conciliatory outlook into the future ...*

Artificial intelligence is not very intelligent at the moment. It works extremely well for certain problems ...

*... but not in higher mathematics.*

Not even in the "lower" one.

*There is an anecdote that colleagues tell about you: Your findings are so clever that aliens must have whispered to you.*

(Laughs heartily) I would say that there was a lot of luck involved. You think about something and at some point something happens to you by chance. I can't explain why I came up with something.

*It is charming that you owe a prize like the Breakthrough Prize to chance.*

When it comes to awarding mathematical prizes, luck plays a pretty big role. There are mathematicians who are better than others, but there are many very good mathematicians. The difference between these is partly really the luck of thinking about the right problem at the right moment, when there is also a solution that is somehow simple enough.

*The Breakthrough Prize is endowed with three million dollars. What are you doing with it?*

My wife and I moved to London three years ago and haven't bought a house or apartment yet.

*Do you understand that the money is tax-free?*

Yes. However, I wrote to the tax office to verify this and have not yet received an answer.

*The conversation with the mathematician Sr Martin Hairer first appeared in Terra Mater 1/2021.*

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