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What is a mathematician?

I recently got a new set of business cards printed. I wanted to keep them as simple as possible, and have them communicate only the important information – my name and contact details. But then came the question of what to put underneath my name. I don’t currently have a full time job anywhere, but I do spend pretty much all my time working freelance as a maths communicator, talking to people about maths and popularising the subject. I also do loads of maths related things in my spare time, including running a Maths Jam, and I’m still in the process of writing a paper based on the work I did in my PhD thesis, which I finished last summer. How do you sum that up? I recall recently our own Peter Rowlett struggled similarly when filling in the corresponding field on his census form.

This got me thinking about the word ‘mathematician’, and what it means. I’ve heard people say in the past that they’re not really a mathematician, even though they were studying maths or used it regularly in their work. I recently asked around for opinions on the topic, via Twitter and at Maths Jam, and found that there are a few people who feel that the term doesn’t apply to them, and that it should be reserved for people who are proving theorems and doing research. Others felt that its use is as a job title – to describe what you do.

In his essay ‘Mathematics and Mathematicians’, Jean Dieudonné (one of the Bourbaki) defines a mathematician as ‘someone who has published the proof of at least one non-trivial theorem.’ Lord Kelvin was reported to have written the integral of $e^{-x^2}$ on the blackboard, and then said ‘ A mathematician is one to whom that is as obvious as that twice two makes four is to you. Liouville was a mathematician’.

Commonly, a lack of confidence in one’s own ability results in not feeling worthy of the title – one person answered, ‘If I call myself a mathematician, then I’m equally qualified to call myself a musician and a cricketer’. Students currently studying the subject are maybe allowed to be called ‘mathematician’, in the same way that someone studying medicine can be called a ‘medic’.

Interestingly though, quite a few people share an opinion closer to my own. I think the word should be used to apply to anyone who thinks like a mathematician. One person said, ‘Being a mathematician is about the way my brain works. It’s a massive part of my personality.’ This requires, I guess, some basic level of mathematical training and knowledge – perhaps an undergrad degree, or an equivalent amount of experience gained through other means. One Maths Jam attendee was adamant that anyone who is interested enough in maths, outside of their work, to attend a Maths Jam regularly, is allowed to call themselves a mathematician. (For anyone unaware, Maths Jam is a monthly pub gathering for maths enthusiasts, usually resulting in games of SET, pictorial proofs and endless puzzling, and covering maths ranging from modular origami to modular forms.) In his book ‘On Mathematics and Mathematicians‘, Moritz quotes Novalis as saying ‘The real mathematician is an enthusiast per se. Without enthusiasm no mathematics.’

In Episode 41 of the Math/Maths Podcast, a weekly maths news/issues roundup, which is hosted at, Peter (of this site) and his American colleague Samuel Hansen discussed this issue, (listen here from around 11:30 minutes) and Samuel’s quite nice observation was that if you learn mathematics in a class (even at university), you don’t count as a mathematician until you’ve done maths independently of your learning, outside of a classroom. This seems to work pretty well as a definition for me, since it doesn’t matter what context you do the maths in – at a Maths Jam, or working on an engineering problem, or in a research department.

And does being a mathematician last forever? If it’s a way of thinking, then surely yes. If it’s a job title, then does it continue beyond the point where you switch careers and become a postman? And if you can be a mathematician by simply being interested in it and Jamming in the pub with other maths fans, then you should surely be able to use it as long as your interest survives.

Another thought I’d like to throw in is this: if you reserve the term ‘mathematicians’ for hardcore researchers and academics, you leave out a lot of people. If that were the definition of a mathematician, then most people in the world would probably never meet someone who fits that title. As someone who works in outreach, it seems to me that this is counterproductive, especially when maths is already seen as ‘difficult’ and somehow other-worldly. If more people described themselves as mathematicians, even just in their Twitter profile, maybe people would start to question the stereotype. I am often told ‘You don’t look like a mathematician’. What does that mean!? Obviously, if you are asked what your job is, you answer that honestly – but if you’re trying to list adjectives that apply to you, or describe yourself in N words, then throw in mathematician. Anyone can be an artist, right? Maybe not a good one, but that doesn’t necessarily matter. You’re not claiming to be a virtuoso if you say you play the violin.

One respondent summed this up nicely – he didn’t feel that, as someone who’s ‘never proved a theorem’, he can really call himself a mathematician. He adds, ‘I know this is wrong, [and] unhelpful in the cause of promoting maths.’ It’s obviously a personal choice, and if you don’t think you can call yourself that, then you don’t have to. But I feel that the term ‘mathematician’ is empowering. It’s something to be proud of. Citizen science is a term for scientific research conducted, in whole or in part, by amateur or nonprofessional scientists. Since the bird counts dating back to 1900, projects like these and the more modern Galaxy Zoo and SETI Live mean that everyone can contribute to scientific research. While there are sadly few equivalent projects for maths (GIMPS, and the recent Turing’s Sunflowers project at Manchester Science Festival – although feel free to correct me if you know of others), surely this means that anyone can be a scientist. So can anyone who wants it hard enough be a mathematician?

I’m not necessarily going to come to any kind of conclusion here, particularly because what I’m trying to debate here is essentially the definition of a word. Wikipedia just says that a mathematician is ‘someone with an extensive knowledge of mathematics’, which sidesteps the question of their occupation or the level to which they practise; but it does later use the phrase ‘Mathematicians do research’ (although this of course is only a one-way implication). just goes with ‘an expert or specialist in mathematics’. Maybe this does exclude amateurs and non-professionals but let’s be honest – who uses words only to mean what the dictionary says these days anyway? Any mathematician knows that sometimes we employ definitions for their use in the context of whatever we’re doing at the time.

And here’s what I eventually went with on my business card.

Katie's business card

18 Responses to “What is a mathematician?”

  1. Peter Rowlett Peter Rowlett

    I think there is a massive problem if I put “mathematician” on my Twitter profile that those who do research in mathematics would (not unreasonably) say “no you’re not”. But as an active member of the Institute of Mathematics and its Applications (IMA) I am aware that people don’t join because they don’t think of themselves as associated with mathematics (even after a maths degree), yet the IMA welcomes a much broader church than the research mathematician and many people who received a mathematical training or just have a strong interest will enjoy IMA membership. This is why I like the variable definition given by the Science Council.

  2. Avatar Joseph Symons-Smyth

    From ‘Jonathan Strange & Mr. Norrel’, by Susannah Clarke:

    “Your second task is to take a message to all the magicians in England. Do you understand me?”
    “Oh yes! But…”
    “But what?”
    “But there is only one.”
    “There is only one magician, sir. Now that you are here, only one magician remains in England.”
    Strange seemed to consider this for a moment. “My pupils,” he said. “My pupils are magicians. All the men and women who ever wanted to be Norrell’s pupils are magicians. Childermass is another. Segundus another. Honeyfoot. The subscribers to the magical journals. The members of the old societies. England is full of magicians. Hundreds! Thousands perhaps! Norrell refused them. Norrell denied them. Norrell silenced them. But they are magicians nonetheless.”

  3. Avatar Sue VanHattum

    I wrote a paper for the Journal of Humanistic Mathematics, titled On Doing Mathematics. Mainly it’s about my problem-solving process (on a particular problem). But I started by mentioning that the Mathematics Genealogy Project defines a mathematician as someone with a PhD in math. I don’t have one, so I don’t count, according to them. But I do think of myself as a mathematician.

    • Avatar John McKenna

      I consider myself to be an amateur mathematician. I don’t have a PhD, and no one is paying me to do this. I do it because it’s fun. I don’t have the experience or skill of the professionals, and haven’t done anything remotely original. But it still feels like the real thing.

      The description in your paper of how you went about solving that problem is very familiar. I’ve been through the same process too, and I suspect that most of the beautiful proofs we read in textbooks came from the same place.

      First you flail around randomly, trying anything that feels like it might help. Then you hit on something that does, and slowly make progress towards your goal, following many dead ends along the way.

      Once you’ve reached the end, you can look back and see the clear direct path. “It’s so obvious. How could I have missed that?”

      The trouble is that the beautiful polished end result is the only thing that most students ever see. Their own work, messy and rambling, full of false starts, is nothing like it. Sometimes I see a proof and think “I’d never have thought like that. I’ll never be a real mathematician” (I do know better, but the thought is still there).

      • Avatar Christian Perfect

        Tim Gowers and others have been trying to articulate these kinds of problem-solving methods in the Tricki. There’s also Pólya’s book How to Solve It, which I haven’t read.

        Are you a “mathematician” if you use these methods to solve problems? Is “mathematician” a way of doing things rather than a job title?

  4. Colin Beveridge Colin Beveridge

    I’d say that a mathematician is anyone who describes him/herself as a mathematician. Which makes it sound a bit like MathsJam needs an n-step programme.

  5. Avatar Thalinda

    I think that mathematician is an old fashioned word, that only appears in movies and stories, I have a good knowledge of maths and when I am not teaching I spend my time solving or writing maths problems but I would never call myself a mathematician.

  6. Avatar Dave

    I understand the desire to call yourself a mathematician. However, I urge some caution. I studied English as an undergrad, but have a friend who calls himself a mathematician. It sounds very pretentious, even though he does think mathematically, and even has a strong math component in his work. (Note: plumbers have a strong math component to their work, too, yet are not mathematician-plumbers.) I would say one would require a non-trivial publication or be actively involved in trying to solve a non-trivial problem to be a mathematician. And don’t be afraid to call yourself an amateur mathematician if you are not being paid to tackle those non-trivial problems. Drop the amateur when you publish.

    • Avatar Joseph

      You say it sounds pretentious, but you preface that by saying “I studied English as an undergrad..”, as if that gives you grounds for making pronouncements on how people use words?
      Much too much importance is given by people to tangible things like degrees and publishing and jobs.
      I play the violin. I have done since I was a small child, and by now I’m pretty good at it. I didn’t take a degree in music, I’ve never published any compositions, and it’s not how I make my living. But I’m certainly a musician.

  7. Avatar Steve Hewson

    I love this thread. Despite having a maths PhD and worked as a mathematics research fellow and now spend a huge portion of my time thinking about mathematics I still worry about this. I have agonised for years over whether to describe myself as a mathematician. In the end, I came to the conclusion that a multiplicity of self-categorisations are useful and sometimes describing myself as a mathematician is more useful than not doing so.

    However, the ‘way of mind’ mentioned in the thread appeals to me. I suspect that only a mathematician would spend so much time worrying about into which category of being he or she should be put. I therefore propose the definition: “A mathematician is a person who worries about whether or not they are a mathematician.”

    (p.s. The mathematician in me is desperate to tighten up this definition, but for the sake of humour I’ll leave it as it is)

  8. Avatar Rebeka Fox

    Musicians are fickle creatures. Doing a little trig and calling oneself a mathematician is like playing three chords and calling yourself a musician. Maybe that worked for Kurt Cobain, but mathematics tends to be a little more sophisticated. There is nothing wrong with enjoying math and not being a mathematician. There is nothing wrong with being an Amateur Mathematician either. Most people however, in the course of a day, who bestow unto themselves the privileged title of mathematician haven’t the foggiest what that means–even in the slightest. A plumber is a tradesman, I was a tradesman having been a machinist/millwright. I also consider myself an amateur mathematician because I do not have the refinement of a professional–but I have a broad knowledge of the subject, I can manipulate all varieties of mathematical objects, and am capable of devising unique solutions to the problems I am faced with. In everyday I do not see why an amateur mathematician should not be able to abbreviate this to simply mathematician where if one were to be speaking with a professional that could be elaborated on! Apologies for my lengthy comment . . .

  9. Colin Beveridge Colin Beveridge

    There’s a John Hegley poem about his first poem:

    I wrote my first poem when I was ten
    It was about my den
    And I said “It’s just like a real poem, Miss!”
    And Miss said “It is a real poem, John!”
    I’ve been a poet since then.

    A poet is someone who does poetry. A mathematician is someone who does maths.

    I think an inclusive definition is helpful for breaking down the boundary between boffins (there, Peter, I said it) and the rest of the world.

  10. Avatar Harry Miller

    Anyone who is interested in Mathematics, and is willing to study Mathematics for their own leisure should be considered a Mathematician. I’ll give an example in Sports, like Football (English), if your a player in League Two and still play for a team then your considered a footballer regardless of your ability or skill, your still a footballer by definition and practice.

    Anyone can be Mathematician, there is just varying abilities, skills and knowledge and people like to study specific areas. I enjoy Mathematics and beginning to have a growing interest in Geometry, Topology and Graph Theory. I don’t really like Calculus though, kinda boring in my opinion.

  11. Avatar Nancy

    When I was 5, numbers were never shown to me.
    As a 1st grader, I was called to the blackboard to solve a 1+1 problem.
    I stood there facing the blackboard, indexing my fingers, while the class roared in laughter.
    Well, I am grown now. And, Yes, I developed an anxiety with numbers, where I could not see them in my head. However, I learned to forgive and forget. Thank goodness for that.
    Imagine, my excitement to see a mathematical 1+1 problem in College Algebra as:
    a + b = c
    Wait! It gets better, how about the number: 2 with an exponent of zero equaling to 1.
    Somehow, my brain began to see the mathematical problems, and their solutions were so easy to see. Amazing!
    Yes! A mathematician was realized, just like that!


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