## I don’t have to prove theorems to be a mathematician

I’m giving a talk at the Joint Mathematics Meeting on Thursday (it’s a 30 minute talk that starts at 11:20am, in Room 2 of the Upper Level of the San Diego Conference Center, I hope you come!).

I have to distill the talk from an hour-long talk I gave recently in the Stony Brook math department, which was stimulating.

Thinking about that talk brought something up for me that I think I want to address before the next talk. Namely, at the beginning of the talk I was explaining the title, “How Mathematics is Used Outside of Academia,” and I mentioned that most mathematicians that leave academia end up doing modeling.

I can’t remember the exact exchange, but I referred to myself at some point in this discussion as a mathematician outside of academia, at which point someone in the audience expressed incredulity:

him: Really? Are you still a mathematician? Do you *prove theorems*?

me: No, I don’t prove theorems any longer, now that I am a modeler… (confused look)

At the moment I didn’t have a good response to this, because he was using a different definition of “mathematician” than I was. For some reason he thought a mathematician must prove theorems.

I don’t think so. I had a conversation about this after my talk with Bob Beals, who was in the audience and who taught many years ago at the math summer program I did last summer. After getting his Ph.D. in math, Bob worked for the spooks, and now he works for RenTech. So he knows a lot about doing math outside academia too, and I liked his perspective on this question.

Namely, he wanted to look at the question through the lens of “grunt work”, which is to say all of the actual work that goes into a “result.”

As a mathematician, of course, you don’t simply sit around all day proving theorems. Actually you spend most of your time working through examples to get a feel for the terrain, and thinking up simple ways to do what seems like hard things, and trying out ideas that fail, and going down paths that are dry. If you’re lucky, then at the end of a long journey like this, you will have a theorem.

The same basic thing happens in modeling. You spend lots of time with the data, getting to know it, and then trying out certain approaches, which sometimes, or often, end up giving you nothing interesting, and half the time you realize you were expecting the wrong thing so you have to change it entirely. In the end you may end up with a model which is useful. If you’re lucky.

There’s a lot of grunt work in both endeavors, and there’s a lot of hard thinking along the way, lots of ways for you to fool yourself that you’ve got something when you haven’t. Perhaps in modeling it’s easier to lie, which is a big difference indeed. But if you’re an honest modeler then I claim the difference in the process of getting an interesting and important result is not that different.

And, I claim, I am still being a mathematician while I’m doing it.

By the way, the person who (I believe) asked that question is also outside academia (at IBM). You should have seen him dancing up a storm at the winter party.

Wish I’d been there! I love parties that have dancing.

Very interesting. As a student and aspiring mathematician I’ve found that it can be really difficult to articulate what that means to people that don’t study math; that definition of mathematician as someone that proves things has been one thing that I’ve stumbled over, particularly when talking about math-folk outside of academia.

Part of what makes a mathematician is the material she engages in. Presumably some theorem or theory is involved, there is some kind of logic being used, and there is some way to explain the work to other mathematicians (not an exhaustive list). I am just saying this because grunt work as you describe it presumably applies to other disciplines.

Although the feeling of never getting anywhere for a long time which you describe may be unique to mathematics. I’m not sure.

I agree that being a mathematician is far more than proving theorems; I suspect you’ll agree that making interesting conjectures (like Taniyama), profound definitions (Grothendieck) and even computing important objects (e.g. Fokko du Cloux with the representations of E8) is being a mathematician.

However, whilst the process of modelling may in some ways resemble proving theorems, I feel that doing mathematics means doing something novel and abstract and it is abstraction that is lacking from modelling concrete data sets. To me the difference is as stark as that of being a moral philosopher with being a policeman dealing with criminals.

I’m afraid that’s just wrong. The modeling novelty that has brought you google searches and movie recommendations is very real. Just because it matters to normal people doesn’t make it unoriginal.

Meh, I do a lot of modelling, but I don’t have a math degree nor do I consider myself a mathematician (nor do I think that I should).

It seems that you’re arbitrarily extending boundaries of what you call being a mathematician. You come to modelling from a math background, I come from a physics background – am I therefore being a physicist (this is the extent of the logic that I see in your post)?

You’re a modeller – deal with it, you don’t need to call it math to make it cool, original or interesting. My 2c anyway.

I disagree. Think about the case of Newton. His goal was to model some physical processes, not do math for the sake of math. Yet, in the process of developing his models he invented calculus. By your understanding Newton would not be a mathematician, but his pioneering of calculus makes him one.

Maybe having a model that works without an explanation of why or how you got it is not mathematics. On the other hand, the Birch Swinnerton-Dyer conjecture was discovered with the help of machine computation.

Mathematics is about understanding things, not about proving theorems — that said, proving theorems turns out to be a spectacularly useful mechanism for understanding the things that mathematics sets out to understand. I think what you’re doing now is in spirit no different from what applied math has always been; you don’t prove theorems, but you probably care that theorems exist.

If you’re thinking abstractly, then you’re being a mathematician. This is the whole point of modeling. There’s some real result that you’re going for, but you have to think of the data under the lenses of many different abstractions. You’re doing math. You’re a mathematician.

If you subscribe to the Curry-Howard correspondence, all programs are proofs. As I suppose that you code quite a lot, you proved many theorems, and you are still a mathematician.

QED ;-)

For those interested : http://en.wikipedia.org/wiki/Curry–Howard_correspondence

Eight days ago somebody launched a discussion thread in the Friends of MAA group on LinkedIn entitled “Who can realistically call themselves a mathematician and be considered a serious mathematician?” Here’s what I wrote:

“I go with the following fill-in-the-blank, SAT-style analogy to define mathematician:

music : musician

math : _____________

I’ve never wondered what a musician is, so this does the job for me.

Edward’s mention of a mathemusician above caught my eye. Since I am both a professional trucker and amateur mathematician (my LinkedIn page gives the details), for many years now I have called myself a mathematrucker.”

Looking forward to the talk Thursday.

Hey, mathematrucker! I’m looking forward to meeting you at last!

Cathy

I’m looking forward to meeting you too!

It seems to me that to define what a mathematician is and does, you first have to define at least roughly what mathematics is. Most people seem to point to a certain existing body of knowledge, but I think that is highly misleading. Mathematics is *not* just that; it is also the craft of creating new knowledge. But what distinguishes this craft from other areas of human endeavor? After much thought about this, I decided that mathematics is more or less the craft of creating new knowledge from old using basically two fundamental tools: logical deduction and abstraction.

So a mathematician is someone who practices the craft of mathematics, as defined above, in any kind of setting. In some sense, you are always proving theorems or at least lemmas but in more practical settings it makes little sense to articulate what you are doing in that way.

This comment resonated in that my response “What is a mathematician?” came primarily from what the person creates. Knitters create knitting; musicians create music; mathematicians create mathematics. Artists create art. Art historians and art appreciators do something different. Being a math appreciator or a math historian is interesting, but not the same as a mathematician. If modellers are creating kinds of math then they’re mathematicians. If modellers are creating applications of mathematics they are applied mathematicians. If modellers are simply applying pre-created models to data sets that are not novel, then they are practitioners of mathematics…. are they mathematicians? Hm.

Well, do we call the guy or gal who plays Christmas carols on the piano at the Menards store over December a musician? Maybe your answer to that clarifies the answer re: mathematicians.

Sorry, but I didn’t say it well enough. When I said “creating new knowledge from old”, I did *not* mean necessarily “new mathematics”. Only a few people are lucky enough to spend their lives creating new mathematics from old. But many people, including mathbabe as well as most of the people commenting here, are creating new insights, understanding, and knowledge in areas *other* than mathematics but using the craft of mathematics.

I would add that I think the mathematics community devotes too little attention to the craft of using mathematics in other areas of endeavor and too much to the craft of creating new math (i.e., theorem proving). There is this ridiculous belief that the former is “less rigorous” and therefore somehow less worthy than the latter. In particular, I think secondary school and university mathematics education should be primarily focused on training students in the *craft* of using mathematics (which despite what everyone believes requires solid skills in rigorous logical reasoning), rather whatever it is we’re teaching now, which seems to be based on the theorem-proving view of mathematics but has by now been heavily watered down.

Even within research mathematics proofs and theorems are overrated. Mathematics as a science would benefit hugely if other mathematical activities were considered serious mathematical research, besides proving new theorems. Some such activities include communicating mathematics, integrating the many different mathematical disciplines, participating in mathematical communities (JMM as well as Mathoverflow), writing and maintaining mathematical software, formalizing mathematics and, certainly, mathematical modeling. These are not “service” activities but part of research, and they are fundamental to the continued success of mathematics as a science. We should give them proper academic recognition. Unfortunately, the anachronistic view that a mathematician is someone who has managed to attach his/her name to a theorem is still deeply ingrained in the self-image of the mathematical community.

Some time ago, I wrote a blog post about this here:

http://blog.felixbreuer.net/2012/02/27/beyondtheorems.html

In my view, mathematics is all about creating, studying, and interpreting models. Proving theorems is just one way to do that.

If you spend a lot of time working with models, does that make you an agent? Or a craftswoman? :)

(Hope to see you this weekend — maybe we can dust off our duets…!)

woohoo!

Meh. As someone who has a Math Ph.D and does modeling in industry, I find this whole topic really odd. Does it matter in any sense? Who wants to create these artificial boundaries? This type of categorical thinking is exactly what causes academic pigeonholing.

My PhD is in mathematics. I work for a statistical software company. I spend most of my days programming a computer. I also write and blog. Am I a mathematician? A statistician? A computer programmer? A writer? Yes, yes, yes, and yes! No one complains if someone is an actor, a dancer, AND a singer. Why should someone complain if you are a mathematician AND a modeler AND a writer AND…. Roles are not exclusive!

Amen brother.

If someone is modeling weather systems, and using weather data, and using pre-existing mathematical tools and concepts to do so, then that person’s discoveries will be about weather and weather data and the relationships found therein. Why call these discoveries about weather “mathematics?” It just seems like a misclassification. To the extent that new mathematical principles are developed in the course of this work, then that would be mathematical discovery (e.g Newton inventing calculus while trying to solve physics problems). If someone invents new math (or independently rediscovers old math) then that person is “doing” math and acting as a mathematician.