I never had an honest job. In fact, I think that besides a few bucks–rather pesetas and Deutsch Mark–I made distributing leaflets and publications, all the money I have earned in my life has come from universities or assimilated institutions. In other words, all money spent in treats for my various cats, not to speak about my daughter’s Nutella, was paid to me because of doing math. I don’t think I got much Nutella as a kid, but whatever I got–my father liked Cadbury’s chocolate–was also paid with money my parents got for doing math. A very large proportion of the people I know also pay their bills with money earned in the same way.

Now, why do I say all that? Well, basically because something I find kind of surprising is how little interested are people in arguing that it is a good thing that they are paid to do math. Doctors, people flipping burgers in McDonalds, school teachers, geologists, lawyers, vets, policement, firemen, nurses, etc have no problem arguing why they should be paid for what they do. I have no problem to argue that they should, but why are mathematicians paid? Why does one feel that it is a travesty when positions or funding are cut? Why does one feel underpaid? I mean, why should one get paid at all? Why should any bit of the taxes of the guy flipping burgers get to pay a bunch of people doing math?

Mathematicians do almost never consider that question, and when they do the answers are not exactly satisfactory. I remember a guy in Germany who claimed that societies have always had something like the clergy, and university professors according to him are the version of the clergy in a post-religious society. What I don’t remember him ever explaining is how it is justified that the part of the clergy which has to do with 3-manifolds, not to speak about the cohomology of arithmetic groups, is better paid and has better conditions than the parts of the clergy dealing with sanskrit, music theory, literature, or the meaning of life.

Other mathematicians quote Hardy:

"The study of mathematics is, if an unprofitable, a perfectly harmless and innocent occupation." ― G.H. Hardy

Well, Hardy wrote that right after WW1, and then kind of doubled down on it during WW2. It goes without saying that at that point Turings work in Bletchley Park was a secret. The Hiroshima-Nagasaki byproduct of Hilbert’s sprectral theory lied still in the future. Evidently there are lots of other examples where math which was born as unprofitable, harmless and innocent, and then got a more profitable, harmfull and interested adult life. What about Itô calculus becoming via the Black Scholes equation one of the main ways to price financial derivatives? Markov chains, Furstenberg’s work on random walks on graphs, and Google’s crawlers anybody? But ok, I agree that in most of what we do, we are unprofitable, harmless and innocent. I agree that I could myself have done/do much more harm if I had gone into investment banking or were doing something like working on developing a cousin of ChatGPT. For what it is worth, I am pretty sure that the hyperbolic marimbas I recently wrote a paper on will not get profitably used by anybody in the next 1000 years, whether they mean harm or have the purest intentions. And yes, this argument, which can equally well be applied to other professions like painter or filologist carries some weight: Hitler’s dream was to be a painter and Goebel got a PhD in German literature. Still, I am not quite sure how the guy flipping burgers would react if he was told that his taxes go to pay for the treats for my cats because I don’t do anything harmful.

Other people, probably again quoting Hardy, argue that any form of society worth the name supports beautiful things, and that math is just a form of art. Well, I might agree, but that does not explain why society should finance more the mathematical form of art than others. I mean, math is, at best, a kind of art that only a small fraction of the population appreciates. Yes, Terry Tao or Cedric Villani might be kind of public figures, but they are not exactly Taylor Swift. And yes, Taylor Swift is dirt rich, but it is much harder to pay your bills being a mediocre musician, than a mediocre mathematician like me. But this applies in a much broader context. I mean, what do you think the being will answer if I ask her if she gets more eshtetical pleasure when I tell her something of math, or when she goes with a few of her skating friends to McDonalds? People making jewelry and selling it on street markets do beautiful things, but they don’t get civil servant status.

Then there are also people who go to the other extremum and tell you that we are paid because mathematics is useful and important. Sadly, I actually think that this is the real reason why we get paid. Sadly because in general it is not mathematicians who say that. Mathematicians–most mathematicians–know full well that they spend hours and hours trying to figure out things that are fundamentally useless. I mean, people doing data science talk about “the data manifold” and books with titles like Nonlinear Dimensionality Reduction bother to define words like “topology”, “Hausdorff”, and so on. So, manifolds are important, and I do manifolds right? Well, the issue is that the most sohpisticated manifolds that seem to appear on papers on data science written by data scientists are open sets in the sphere, the Möbious band, or what the call “the Swiss roll”, which is nothing other than a sheet of paper which has been rolled. I mean, if those are the manifolds which govern nature, one does not need to spend one’s time wondering about the hyperbolicty of mapping tori of diffeomorphisms of infinite type surfaces… It seems to me that mathematicians know full well that the research they do, the research for what they get hired, is as useful as playing chess or go–it is infinitely more interesting, but not more useful.

All of that was about what other people say. What do I think? Well, what I think is that for completely unexplainable reasons, the way mathematicians think is useful. Not by design, not that this is the reason why we do it, but it is. Mathematicians are very well trained to be given some information, accept it, and then start moving on from there. Mathematicians are well trained in extracting the important bits of information and simplifying the thought process. Mathematicians are very well trained to keep things separated and organised, to run in parallel different, but related, thought processes, keeping an eye on how they interact, but keeping them separate. Mathematicians understand exponential growth, proportions, and statistics. Mathematicians know how to make assumptions, work based on them, keeping in mind that they were assumptions. Mathematicians are trained to make sure that their conclussions pass the snif test. I don’t think that only mathematical training leads to those skills–working with high tension cables must also teach you to keep separated things that should not get mixed, I guess–, but mathematical training does. I doubt that you get those skills in medical training, while training as a historian or a journalist, or when getting and MBA. You surely get other skills, maybe more important ones, but not those.

Okay, doing math teaches some skills that can come handy, and so what? Mathematicians are often useless for all sorts of practical things. Incredibly many mathematicians are total space cadets, seemingly unable to not forget in their way to the store what they were going to get. So, those skills might be there, but one does not see them. I mean, 2 out of 3 times you go help a mathematician move, it is a complete fucking chaos. Still, I always thought that logistics is something for mathematicians. As proof you can see how mathematicians run the things they care about, their hobbies. No chaos and uselessness there, no forgetting to answer e-mails, no having ignored the issue of getting boxes to move, no booking flights out of the wrong city, and so on. Yes, in their regular lives mathematicians are often quite useless, but I stand to the claim that mathematical training gives useful skills. Not by design, but just because it happens to be the case.

It might look like an abrupt change of topic, but let me give, for a moment, an opinion I have about the teaching of religion: learning about the religions of the world as such is kind of totally useless, akin to learning about cuisines of the world as such. If you learn about Judaism, Christianity, Islam, Hinduism, and Buddhism, you will have learned about the trappings of all those religions, but I am not sure that you will actually believe that there are people who believe. Here is a proof: how much can you really imagine the ancient Greeks believing in Athena and Apollo? When it comes to religion, I think that the most important thing, at least if you are a non-believer, is to believe that people actually believe. Without that, how are you going to explain the enormous waste of building cathedrals while people lived in huts? How are you going to understand people fighting wars because of a word in the creed? A difference that eventually led to the incredibly different relations between church and state in Western and Eastern Europe. How are you going to understand the practical consequences of theological discussions around the question of whether Native Americans had souls or not? How are you going to come to terms with the fact that people are willing to blow themselves up, go bonzo, or, in their haste to bring closer the final judgment, support the state of Israel no matter what? Yes, you can reduce all of this to greed, or to power, or to some version of brainwashing, but that just means that you are discounting the possibility that these people actually had faith, and that that faith moved them to do things which we consider crazy, evil, or crazy and evil. If you want to learn about religion, you have to learn that for some people, faith is real. And you learn that when you actually see that some people have faith.

In the same way, I believe that to learn mathematical skills, you have to learn from people who do mathematics. It does not suffice to learn in an engineering school or as part of the economics curriculum. You should learn them from people who live in that world. You have to see the thought process in action. You have to see how arguments are built, how problems are approached. You have to see how things are deduced from first principles and how they are used. You have to see what is considered a correct argument and how it is written. Optimally, you have to do these things yourself. All of this is something we provide. We provide it when we teach, when we have project students, when we have master’s and PhD students, but we also provide the environment in which this is provided. In doing that, we provide skills that I believe are useful. Not that we use them profitably in our lives, or that we do anything with anybody’s profit in mind, but skills which, it seems, are useful for some fraction of society to have. And yes, when it comes to teaching the skills of mathematics, pure mathematics is the way to go, as Hardy said:

“One rather curious conclusion emerges: that pure mathematics is, on the whole, distinctly more useful than applied. ... For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.” — G.H. Hardy