## October 20, 2015

### Four Tribes of Mathematicians

#### Posted by John Baez

Since category theorists love to talk about their peculiar role in the mathematics community, I thought you’d enjoy this blog article by David Mumford, which discusses four “tribes” of mathematicians with different motivations. I’ll quote just a bit, just to whet your appetite for the whole article:

The title refers to an “astonishing experimental investigation” of what your brain is doing when you experience mathematical beauty. This was carried out here:

But on to the four tribes….

I quote:

I think one can make a case for dividing mathematicians into several tribes depending on what most strongly drives them into their esoteric world. I like to call these tribes explorers, alchemists, wrestlers and detectives. Of course, many mathematicians move between tribes and some results are not cleanly part the property of one tribe.

• Explorers are people who ask – are there objects with such and such properties and if so, how many? They feel they are discovering what lies in some distant mathematical continent and, by dint of pure thought, shining a light and reporting back what lies out there. The most beautiful things for them are the wholly new objects that they discover (the phrase ‘bright shiny objects’ has been in vogue recently) and these are especially sought by a sub-tribe that I call Gem Collectors. Explorers have another sub-tribe that I call Mappers who want to describe these new continents by making some sort of map as opposed to a simple list of ‘sehenswürdigkeiten’.

• Alchemists, on the other hand, are those whose greatest excitement comes from finding connections between two areas of math that no one had previously seen as having anything to do with each other. This is like pouring the contents of one flask into another and – something amazing occurs, like an explosion!

• Wrestlers are those who are focussed on relative sizes and strengths of this or that object. They thrive not on equalities between numbers but on inequalities, what quantity can be estimated or bounded by what other quantity, and on asymptotic estimates of size or rate of growth. This tribe consists chiefly of analysts and integrals that measure the size of functions but people in every field get drawn in.

• Finally Detectives are those who doggedly pursue the most difficult, deep questions, seeking clues here and there, sure there is a trail somewhere, often searching for years or decades. These too have a sub-tribe that I call Strip Miners: these mathematicians are convinced that underneath the visible superficial layer, there is a whole hidden layer and that the superficial layer must be stripped off to solve the problem. The hidden layer is typically more abstract, not unlike the ‘deep structure’ pursued by syntactical linguists. Another sub-tribe are the Baptizers, people who name something new, making explicit a key object that has often been implicit earlier but whose significance is clearly seen only when it is formally defined and given a name.

The rest of the article gives examples of the four tribes, and it’s very fun to read — at least if you’re a mathematician!

What do you think you are? I suppose I’m mainly an alchemist, with a touch of detective: I can’t say I’ve ‘doggedly’ pursued the ‘most difficult, deep questions’, but I do enjoy slowly collecting clues to solve mysteries. I have gotten involved in wrestling — I’ve coauthored a few papers on analysis that systematically marshal inequalities to prove something hard — but it’s always been my coauthors who have done the really hard work: on my own, I quickly become tired and decide it’s more fun to think about something else.

Posted at October 20, 2015 1:06 AM UTC

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### Re: Four Tribes of Mathematicians

From the vantage point of physics, a fair portion of mathematics seems to pre-exist in nature. For example, the SU(3) symmetry one finds in the study of quarks and gluons is quite peculiar. Who would have guessed such Lie algebraic objects would so accurately describe the inner structure of protons and neutrons?

And here we are presently, pondering the quantum mechanical structure of spacetime, where noncommutative geometry seems to be highly applicable, in every approach to quantum gravity. If Riemannian differential geometry was the mathematical key to general relativity, why should we doubt the full power of noncommutative geometry (and beyond) in regards to quantum gravity?

Apparently, nature has made ample use of mathematics we have yet to imagine, or we’d surely be done with quantum gravity by now. The stakes are high, unlike any other moment in history, and it takes a mental framework that lives in all four tribes simultaneously to move forward.

Posted by: Metatron on October 20, 2015 5:32 AM | Permalink | Reply to this

### Re: Four Tribes of Mathematicians

I think two other tribes were left out:

• Drunks searching for keys under a streetlight: Those who start with some mathematical structure and then extend or generalize it without showing the extension is useful or relevant to anything else. These would fall under the Explorer tribe except they like to stay close to home.

• Users: Those who really don’t care how the properties of some structure are proved but just want to use them, and may only resort to proofs if they need something similar.

Too many mathematicians fall into the first tribe while many mathematical physicists (biologists, computer scientists, etc.) fall into the second.

Posted by: RodMcGuire on October 20, 2015 4:04 PM | Permalink | Reply to this

### Re: Four Tribes of Mathematicians

One activity I have been involved in since 1965 or so is to find mathematics to express an intuition.

After writing out many times the proof of the many base point van Kampen Theorem it seemed to me that the proof should generalise to dimension 2 if one had the right homotopical gadget. I drew many times a diagram of a square subdivided into little squares and saying: “Surely the big square should be the composition of the little squares!” Then in 1965 I came across Charles Ehresmann’s double categories.

So I spent 9 years trying, and failing, to define a homotopy double groupoid of a space. Then in 1974 Philip Higgins and I realised that to recover a theorem of JHC Whitehead on free crossed modules, we should probably need to work not for a space $X$, but for a space $X$ with a subspace $A$ and a set of base points $C$. This led, using maps of $I^2$ to $X$, to the definition of the double groupoid (with connection) $\rho(X,A,C)$, a symmetrical version of $\pi_2(X,A.C)$, and all went swimmingly (especially with Philip working on it, and after work with Chris Spencer!).

This success, after repeated failures, explains why I emphasise what one can do with structured spaces, and the use of cubical methods to represent “algebraic inverses to subdivision”.

Posted by: Ronnie Brown on July 28, 2016 10:12 PM | Permalink | Reply to this

### Re: Four Tribes of Mathematicians

Yes, did the original poster neglect applied studies as a tribe of its own? Or are applied mathematicians also divided among the 4 tribes in regard to their choice of routes the follow within the discipline of applied math?

Posted by: Mark Harder on August 20, 2016 4:21 AM | Permalink | Reply to this

### Re: Four Tribes of Mathematicians

I haven’t yet read the rest of Mumford’s article, but I can’t say that that brief classification speaks to me personally: I don’t really recognize myself in any of those descriptions.

To the extent that I am a mathematician at all, I’d say that most of my efforts go into trying to represent mathematics, to myself anyway, as simply and clearly as I can. (I am referring to an idealized version of myself.) I certainly couldn’t bring myself to say I am a ‘detective’ who works on the “most difficult, deep questions”; the closest thing I see there might be ‘strip miner’, but even that (in Mumford’s description) sounds way too grandiose – I would much sooner apply it to Grothendieck than to myself.

There ought to be a ‘tribe’ (or ‘guild’ might be a better word here) not of Gem Collectors but of Gem Cutters. It’s not that they dig into the earth and discover these beautiful specimens themselves necessarily, but someone else might bring these gems to the attention of the Gem Cutter, who carefully examines them and then proceeds to help bring out their inner beauty, carefully chiseling and polishing. That type of description might roughly apply to many who call themelves category theorists, although I still don’t see the metaphor as totally apt. Mainly I think of much categorical work in terms of finding the right architecture in which to house mathematics, ‘right’ in the sense of simplicity and generality and clarity, and more in the direction of spare elegance than rococo. But it’s hard finding good analogies for this type of work.

John, I’d agree that ‘alchemist’ is not at all a bad fit for you (and it also ties in well with your Wizard persona, from Usenet days of yore).

Lawvere might be a good candidate for ‘Baptizer’.

Posted by: Todd Trimble on October 21, 2015 2:27 PM | Permalink | Reply to this

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