## September 4, 2006

### 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:

1. Do you agree with the historians’ push to refuse to recognise this writing as history?
2. Do you think that contingentist histories are doomed to overlook something of the truth of your discipline?
3. What do you hope your history will achieve?
Posted at September 4, 2006 3:46 PM UTC

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### 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.

Do you agree with the historians’ push to refuse to recognise this writing as history?

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.

Do you think that contingentist histories are doomed to overlook something of the truth of your discipline?

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!

What do you hope your history will achieve?

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.

Posted by: John Baez on September 4, 2006 6:03 PM | Permalink | Reply to this

### Re: The history of n-categories

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.

Aha! Your using that glorious word telos. This is at the heart of the clash with the historians. This is certainly not to say that there isn’t a large range of different purposes in writing a history. A fair chunk of history of science now looks at science in the context of the society supporting it, e.g., the study of 19th century British science throws a very illuminating light on 19th century Britain. And this kind of work will have no whiff of teleology, aside from writing about how the historical participants used teleological conceptions. Typically, we don’t think anymore of a political entity such as Britain as having a telos, though once many did.

But what happens when we come to look at the history of a discipline for its own sake, in our case the history of mathematics. To adopt a historical position is to adopt a philosophy. Many historians now write as though they don’t believe mathematics has a telos. You clearly do. This is what makes me think that there can’t just be a happy settlement. To the extent that they deny a telos they are not telling the truth of your discipline.

This isn’t to say you can’t learn from historians’ histories. They might throw into question what you had taken to be successes of earlier stages of your program.

Posted by: David Corfield on September 5, 2006 10:35 AM | Permalink | Reply to this

### Re: The history of n-categories

David wrote:

Many historians now write as though they don’t believe mathematics has a telos. You clearly do.

Regardless of whether “mathematics” has a telos, I damn sure do. Any mathematician must. A mathematician surveys the mathematics known so far and looks for things left to do. This means seeing mathematics as an unfinished project, and seeing what steps one can take to carry it on.

An unambitious mathematician may look at just a small corner of mathematics before doing this. Perhaps his or her thesis advisor computed the elliptic cohomology of $\mathrm{SU}(2)$. This suggests trying to do it for $\mathrm{SU}(3)$, and maybe the higher $\mathrm{SU}(n)$’s. That might take a lifetime.

An ambitious mathematician will try to survey as much mathematics as possible and try to find patterns in it which have not been worked through to completion. The grander the patterns, the better!

So, we might say that to be a successful mathematician involves investing the subject with a telos.

But mathematics is a communal effort. If you have lots of people collaborating, talking to each other, reading each others papers, teaching, hiring and promoting each other, each one viewing mathematics as a project with goals, surely these goals will not all be utterly distinct. They will wind up agreeing to some extent.

Isn’t that enough, then, for us say that “mathematics” has goals?

Surely we don’t need to fuss over whether some entity called “mathematics”, distinct from the community of mathematicians and floating over their heads somehow, has goals. If the mathematical community has goals, that should be enough.

Or is that really what the fuss is about - whether mathematics has intrinsic goals that would survive even if every mathematician died in a nuclear blast? This seems more like a question for philosophers than the historians you’re talking about.

Posted by: John Baez on September 6, 2006 4:26 AM | Permalink | Reply to this

### Re: The History of n-Categories

I’m not sure you can separate a philosophical question about a practice from that practice. A practitioner may choose not to think about it, may accept the community’s position, may not even be aware of it, but the way the issue is answered or ignored will make a difference to that practice.

What swept into history of science over the past 50 years was the Nietzchean idea that one should look primarily to power relations. This can be done in subtle or blatant ways, but an indication is the asking of questions such as: Who gains from this decision? What are they trying to achieve by presenting this as ‘natural’? How and why are they defining the boundaries of their discpline? Typical ways of reacting to such historical decisions are to unmask them, to show what was really at stake. In its crudest form this might reveal some political advantage of the ruling class.

So a feature of practioners’ histories which rankles is an unquestioned belief that they are on some long-term project with clear disciplinary boundaries, and that the majority of decisions of the practice through its history were taken for what will always be seen as good reasons.

Against this image, disciplinary boundaries are to be challenged, e.g., accountancy in 4th century BC Piraeus is as much mathematics as Euclid. Decisions could have gone very differently, e.g., the adoption of Cauchy’s form of the calculus wasn’t decided on rational grounds.

Now, histories vary greatly in their adoption of Foucauldian/Nietzschean thinking. Some as I mentioned in a previous comment are more interested in the light thrown on the society which supported the practice than the practice itself. I would also say that historians have provided a very useful service reacting to the practitioners’ use of generally inaccurate and anachronistic narratives as a means to justify current positions.

The question is whether true narratives can be written which accept that a discipline such as mathematics is the movement towards a more adequate understanding of its subject matter. Such narratives may portray decisions made in the past as incorrect, and could suggest why incorrect decisions were taken. They could also include an account of current weaknesses of a program. An account of the program written a decade later could change its mind about what it had seen as the correctness of a decision. Historical findings could change practitioners’ current conceptions. All this is discussed in my paper How Mathematicians May Fail To Be Fully Rational, which I must say hasn’t received much of a reaction yet.

Posted by: David Corfield on September 6, 2006 12:14 PM | Permalink | Reply to this

### Re: The History of n-Categories

Can we lure you into posting your lecture notes on “Why and how to write a history of higher-dimensional algebra” when you’ve written them?

Posted by: John Baez on September 11, 2006 3:27 AM | Permalink | Reply to this