### 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:

- David Mumford, Math and beauty and brain Areas,

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

- Michael Atiyah and Semir Zeki, The experience of mathematical beauty and its neural correlates.

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.

## 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.