### The History of n-Categories

#### Posted by David Corfield

I have to try to emulate Renaissance man over the next few weeks. I’m correcting the proofs for my book Why Do People Get Ill?, about which I am speaking at the Ilkley Literature Festival in October. I may be speaking to my colleagues at the Max Planck Institute about what I have learned on the geometry of statistical inference at a retreat in October. But there’s something yet more pressing to do which does at least concern the reason we’re meeting up in this Café. I must write my lecture for a workshop in Berlin held at the end of next week.Here’s my abstract:

**Why and how to write a history of higher-dimensional algebra**

In a recent paper ‘How Mathematicians May Fail to be Fully Rational’, I advocated the adoption in the philosophy of mathematics of Alasdair MacIntyre’s general notion of tradition-constituted enquiry. A central component of this notion requires of a rational tradition that it know the history of its successes and failures. This raises the question as to whether, were such a history to be written, it would fall foul of the criticism contemporary historians of mathematics have levelled at mathematicians’ histories that they are largely ‘Royal-road-to-me’ accounts. I shall address this question in the context of a research programme known as ‘higher-dimensional algebra’, and consider the charge mathematicians may make in return that historians are unable to treat research programmes which run for decades, supported by tens or hundreds of mathematicians from many countries and institutions.

I wrote a couple of posts about this subject at my old blog, here and here. Now I’d like to question John about his reasons for writing this history of higher-dimensional algebra with Aaron Lauda.

In the first of the above posts I mention Leo Corry’s idea that professional historians of mathematics now write a style of history very different from older styles, and those employed by mathematicians themselves. He characterises the difference as follows:

Science as Drama/Greek Tragedy

- a) We know what will happen: drama arises because we know that it will happen
- b) Human emotions, ideas, and behavior as products of, or responses to the unfolding of the human essence
- c) Universal elements of the human situation and fate

Science as Epic Theater (Brecht)

- a) “Things can happen this way, but they can also happen in a quite different way” (Walter Benjamin)
- b) Human emotions, ideas, and behavior as products of, or responses to, specific social situations
- c) Behavior people adopted in specific historical situations

To my mind a key difference is the historians’ emphasis in their histories that things could have turned out very differently, while the mathematicians tend to tell a story where we learn how the present has emerged out of the past, giving the impression that things were always going to turn out not very dissimilarly to the way they have, even if in retrospect the course was quite tortuous.

One response is to agree with this characterisation, and allow each to go about their respective businesses. Perhaps, like Ivor Grattan-Guinness, we might give the mathematicians’ work a different name, in his case ‘heritage’. But this solution doesn’t square with my MacIntyrean orientation which wants to use history from a contemporary perspective to question our current assumptions, while at the same time understanding that some modern constructions show why certain avenues in the past were doomed to fail.

So some questions for John, and anyone else tempted to write a history of their research area:

- Do you agree with the historians’ push to refuse to recognise this writing as history?
- Do you think that contingentist histories are doomed to overlook something of the truth of your discipline?
- What do you hope your history will achieve?

## Re: The history of n-categories

Hi, David - great to see you back! And I’m glad you’re asking me why I’m writing a history of n-categorical physics. It’s now my main project, and I’ve been working on it desultorily all day, hoping to gradually pick up speed as I get deeper into the project - that’s usually how it goes. Maybe talking about it will help. It’s late so this will be terse, but maybe it will keep the ball rolling.

I don’t mind that as long as they understand its importance within the mathematics, or at least don’t try to stop it or “pooh-pooh” it. We’re doing something different than what they do.

Most historians, even historians of mathematics, probably understand mathematics as poorly as mathematicians understand the discipline of history (even history of mathematics). I wouldn’t look to historians to understand the “truth” of my discipline. And that’s okay - they wouldn’t look to me to understand the truth of theirs!

I’m trying to teach people math and physics, and I’m trying to get them excited about a big project. There’s a certain way you can display the forwards momentum of the subject by presenting it in chronological order - its

telos. This is especially important for me in this particular subject - “n-categorical physics” - because I think it’s only just now becoming visible as a subject. I want people to see that there’s a magnificent painting here, only half-painted. I want to make them go ahead and paint the rest!As you can see, this is quite different than the sort of history that historians engage in.