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January 23, 2007

Two Café Owners Interviewed

Posted by David Corfield

As neither John nor Urs has announced it, readers might like to find out about their motivations for starting and running this blog in an interview they gave to Bruce Bartlett, available in written form and also as an MP3 file.

John commented:

I think all three of us - Urs, David and I - are pushing a new way of thinking: a very n-categorical way of thinking about a large bunch of ideas in math and physics. I’m very excited about this, because I can see how much potential it has. But we’re also simultaneously pushing a new idea of how to communicate ideas. And the combination is actually really, really interesting.

There’s an intriguing thought. If it weren’t n-categories which gripped us, would it make a difference to the way the blog works? Well, I can’t think of anything else which would allow us to talk about quantum gravity, logic and number theory in quite the same way. It seems to me at least as important a breakthrough as the burst of foundational activity in the decades around 1900.

Kenny Easwaran wrote:

It’s clear why other philosophers should care about notions of logic and basic arithmetic, and the possibility of knowledge of abstract objects. Maybe there’s reason for them to care about higher category theory, but I don’t think this has been made clear yet.

I answered that higher category theory showed that a prominent philosopher was wrong. I’m not sure what the rules are here. What else would I have to do, if this is not enough?

Posted at January 23, 2007 9:42 AM UTC

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16 Comments & 1 Trackback

Re: Two Café Owners Interviewed

If it weren’t nn-categories which gripped us, would it make a difference to the way the blog works?

Maybe there is indeed a correlation between

a) the desire to see the “big picture” in the background, to understand the natural nature of the objects of your research

b) the desire to extract the nn-categorical structure governing the objects of your interest

c) the desire to communicate ideas and share insights

Posted by: anonymous on January 23, 2007 10:25 AM | Permalink | Reply to this

Re: Two Café Owners Interviewed

Something that ties these together is a belief in the simplicity of the big picture. We love it when a complicated looking construction turns out to be a variant or categorification of some simple construction.

Of course, you are liable to see something as simple when you understand it, even if this took many months. ‘Simple’ seems to be used in both a subjective sense (how easily I can understand) and an objective one (sometimes measured by some complexity scale). Perhaps what we’re after are the first principles of a domain. And, if you follow Aristotle, there’s no need for these to be obvious to the layman. I wrote about this here.

Posted by: David Corfield on January 23, 2007 11:45 AM | Permalink | Reply to this

Re: Two Café Owners Interviewed

I wrote about this here.

There you paraphrase Hacking as saying that

[…] Lakatos is showing that as mathematics proceeds, if it is carried out properly with plenty of critical discussion, a point will be arrived at where the definitions are such that results will follow easily from them. A theory which was initially driven by (quasi)-empirical facts has become merely a collection of analytic statements, true by virtue of meaning.

Let me amplify a nice example of this phenomenon, which we talked about here recently:

More or less just by setting up his language in a more natural way than usual, Simon Willerton manages to ‘deflate’ the Freed-Hopkins-Teleman theorem to a triviality.

Or almost at least. He sets up a natural language and then demonstrates that internal to the (small) world of finite groups this makes the finite group version of FHK a triviality.

But it seems to me that the same language, internal to the world of Lie groups, does also pertain to the full FHK theorem.

Well, up to a technicality which I don’t have under control yet.

This is in fact also an example of what I would regard as a refinement of the statement quoted above, a refinement that you, David, seem to have in mind, too, if I understand you correctly:

A good ‘deflation’ of a mathematical subject sets up the language in such a way that either

a) everything becomes a triviality

b) or, if that’s not possible, that all mysteries become plausible and their proofs straightforward and at most tedious.

That’s maybe the slogan:

{mysteries} \to {straightforward (but possibly tedious)}

Posted by: urs on January 23, 2007 12:24 PM | Permalink | Reply to this

Re: Two Café Owners Interviewed

The key point though is what to make of the ‘triviality’. I take it that Hacking was wrong to call Lakatos a deflator. Deflation, in its financial sense, which I take to be Hacking’s, suggests a loss of value. I don’t believe Lakatos took the output of the dialectic of mathematical research to be in any sense of less value. For him it was more about getting definitions ‘right’. Indeed, I take Lakatos to be quite Greek in valuing the quest for first principles.

A true deflator is one who makes claims that mathematics is more-or-less capturable in some system of logic, without caring whether the content is properly framed, and as such is just a collection of tautologies.

Posted by: David Corfield on January 23, 2007 12:42 PM | Permalink | Reply to this

Re: Two Café Owners Interviewed

Deflation, in its financial sense, which I take to be Hacking’s, suggests a loss of value.

Ah, I see. I had taken “deflated” to be more like the opposite of “overblown”.

(That’s at least what I was thinking of when I used ‘deflate’ above.)

Posted by: urs on January 23, 2007 1:50 PM | Permalink | Reply to this

Re: Two Café Owners Interviewed

On the one hand, it’s just a question of semantics. But on the other, there’s a serious point lurking. ‘Deflation’ is a little dangerous as it its uses aren’t all that consistent: economically difficult phase when prices fall; removal of air from something which means it cannot be used but is easily stored; disappearance of euphoria on encountering some truth.

About ‘mysteries’, I would see a deflation as quite different from a solution. If someone claims there’s a mysterious light in the sky, deflation might arise from it being pointed out that the person’s glasses are misted up. Solution without deflation explains it in terms of solar particles.

The more serious side of this is that the kind of philosophical deflation of mathematics which sees it as a bunch of logic truths has had a very strong grip over the past century. The very notion of ‘getting concepts right’ is completely foreign to the largest part of the philosophy of mathematics.

Posted by: David Corfield on January 23, 2007 2:09 PM | Permalink | Reply to this

Re: Two Café Owners Interviewed

Hi,

I just wanted to mention my appreciation for the existence of blogs such as this one. I meant to speak up last time there was something posted about the blog itself, but I couldn’t work up the nerve.

I’m just a 16 year old with a rather keen interest in mathematics and related areas, and I’m glad to have places like this, especially so since I’m mostly teaching myself until I’m able to get into university. Although a fair bit, perhaps even most of the material posted here goes far over my head, I can usually find some interesting detail that doesn’t, and I always leave with plenty of ideas for new things to look into. As a whole, the discussion here makes for great inspiration to keep working, so that the more complicated stuff won’t be out of reach.

So, thanks!

Posted by: Anonymous Coward on January 23, 2007 12:06 PM | Permalink | Reply to this

Re: Two Café Owners Interviewed

I was talking with a few undergrads who came to the Joint Mathematics Meetings earlier this month, and I’ll tell you what I tell them: no you’re not going to understand half of what’s going on, but just by being here you’re ahead of the game.

Some people get this idea that learning higher mathematics proceeds like a Bourbaki text. It just ain’t so. There’s a lot of processing that goes on unconsciously, and just being around the dialogue starts the process. The same is the case for me: as far as higher categories go I’m a good amateur. I miss a fair bit of what goes on here, but I pick up what I can and just bask in the rest. When it’s time, the rest will snap into place.

Posted by: John Armstrong on January 23, 2007 3:27 PM | Permalink | Reply to this

Re: Two Café Owners Interviewed

I think it’s evident from my posting patterns that a hefty fraction of the discussions here go over my head. (Perhaps a better phrase might be “beside my head”: I’ve spent a longish amount of time studying topics which don’t arise here that often, so while the magnitude of the technical difficulty is something I can handle, it’s pointing in the wrong direction. But anyway…) More frequently than I had anticipated, however, things come along which I find not only comprehensible but deeply thought-provoking: A Plea to Save New Scientist or games, categories and surreal numbers.

I certainly wish I’d had a resource like this (or, more generally, like the wikiweb/blogosphere in general) when I was sixteen. Instead, I had to walk through three kilometers of snow, uphill both ways, just to send an e-mail to the Mersenne Prime Search discussion list.

Posted by: Blake Stacey on January 23, 2007 3:46 PM | Permalink | Reply to this

Re: Two Café Owners Interviewed

And since I mentioned New Scientist… .

Via Neil Gaiman, I just learned that New Scientist had uncritically swallowed a press release by the “Public Employees for Environmental Responsibility” which began as follows:

Grand Canyon National Park is not permitted to give an official estimate of the geologic age of its principal feature, due to pressure from Bush administration appointees.

As it turns out, this is a distortion of the facts (details at Bad Astronomy, eSkeptic and National Parks Traveler). One could discuss this incident endlessly, but instead I’ll just offer it for inclusion in the New Scientist file.

I apologize for the digression; my love of hyperlinks forced it upon me.

Posted by: Blake Stacey on January 23, 2007 9:35 PM | Permalink | Reply to this

Re: Two Café Owners Interviewed

Since you’re talking about the goal of the blog, a question about category theory (including it’s generalisations) which I’ve been meaning to ask for a while (and which I haven’t seen directly commented on anytime since inauguration). [My first degree was in maths but it’s been a long while so forgive any mis-rememberings of details in example in the following]

In maths there’s two ends of a spectrum: “understanding why and having a framework for things” and “concrete details”. So, for example take differential equations. At one end you’ve got understanding of existance/uniqueness/stability/computability/etc of solutions and how this varies for partial d.e.’s as dimension/form/etc varies, properties of systems of eqs, etc, proofs about other properties, etc. In a sense, this is how you “understand” differential equations. At the other end you’ve got “interesting details”: knowledge of special solutions to particular equations, how far you can push perturbation around a solution of a similar system, numerical methods of solution, etc. In a lot of ways this is just as intellectually challenging as other stuff and requires clarity of thought, but they aren’t giving you a glimpse of the “‘big picture’ in the background”.

So, the question is: is category theory essentially about the framework or is it also useful in working on “the details”? I notice that virtually all the posts seem to be about how you can “view thing X in a category-theory context as…”. One of the things that seemed interesting about (what little I followed about) the categorifying geometry threads was that you might end up taking the categorical framework you’d created inspired by existing geometry and “instantiated” part of that with new concrete objects - which we don’t already have a geometrical viewpoint on - and following through to thus come up with a geometrical viewpoint on something novel, which might be useful. But maybe that was me misunderstanding things.

I suspect there’s a trivial sense in which category theory applies to “the details”, in the sense that you do everything in terms of category theory. What I’m really asking is if category theory is efficacious (in some sense)for working with “the details”.

Posted by: dave tweed on January 23, 2007 2:01 PM | Permalink | Reply to this

Re: Two Café Owners Interviewed

As you mentioned differential equations, take a look at Sergiu Klainerman’s PDE as a Unified Subject. No category theory there, but a lovely piece.

As for category theory and details, it certainly can happen that when you want to develop a concept in a new area that there are many choices which may be made as to its details. Category theory often acts to narrow down this choice, and in some cases to force the choice upon you. Urs talks about a case of this phenomenon here.

It’s a little like having scaffolding to help you paint a large mural, rather than just tying your paintbrush to a long stick.

Posted by: David Corfield on January 24, 2007 7:59 AM | Permalink | Reply to this

Re: Two Café Owners Interviewed

It’s a little like having scaffolding to help you paint a large mural, rather than just tying your paintbrush to a long stick.

I like that one!

Posted by: urs on January 24, 2007 9:56 AM | Permalink | Reply to this

Re: Two Café Owners Interviewed

May I suggest replacing “deflated” by “distilled”, an action that shrinks while increasing value.

Off-topic: I introduced rings yesterday in a graduate algebra class, as one-object additive categories. Not yet sure how people liked that.

Posted by: Allen Knutson on January 23, 2007 4:55 PM | Permalink | Reply to this

Re: Two Café Owners Interviewed

I introduced rings yesterday in a graduate algebra class, as one-object additive categories. Not yet sure how people liked that.

I think that’s the viewpoint (on categories, at least) we need to bring up more often. Categories can be used for abstract nonsense, but there are also very down-to-earth uses like the category of matrices over a given ring.

Posted by: John Armstrong on January 23, 2007 6:33 PM | Permalink | Reply to this

The original n-category cafe

Lol - my favourite part in the interview is when Urs says:

Urs: And I remember one day, I was walking home in Hamburg, and I somehow felt dissatisfied. And then, I had this idea. I said, ‘why don’t we have a new blog where we discuss these things’? And I think, before I even went home, I went to an internet café - because I don’t have internet at home -

That was the original n-category cafe! Its just like Brougham bridge - where Hamilton had his moment of inspiration about quaternions, and scrawled it on the stones.

Perhaps in years to come, there will be a similar placque outside this internet cafe in Hamburg,

Here as he walked by on the ? of ?, Urs Schreiber in a flash of genius wrote the email which founded the n-category cafe.

Please take this with a pinch of salt!

But its fun to be at the n-category cafe. (YMCA?)

Posted by: Bruce Bartlett on January 23, 2007 5:24 PM | Permalink | Reply to this
Read the post Peering Through the Veil
Weblog: The n-Category Café
Excerpt: Twice in recent days I have confronted the possibility of experiencing a kind of alienation due to interviews. First, my co-author Darian Leader and I were interviewed by the New Scientist about our book Why Do People Get Ill?. A...
Tracked: January 27, 2007 10:53 AM

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