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October 1, 2006

The Consolation of n-Categories

Posted by David Corfield

As mentioned here, Good Life in the Scientific Revolution: Descartes, Pascal, Leibniz, and the Cultivation of Virtue by Matthew L. Jones should be well worth reading. Here’s the publisher’s blurb:

Amid the unrest, dislocation, and uncertainty of seventeenth-century Europe, readers seeking consolation and assurance turned to philosophical and scientific books that offered ways of conquering fears and training the mind - guidance for living a good life.

The Good Life in the Scientific Revolution presents a triptych showing how three key early modern scientists, René Descartes, Blaise Pascal, and Gottfried Leibniz, envisioned their new work as useful for cultivating virtue and for pursuing a good life. Their scientific and philosophical innovations stemmed in part from their understanding of mathematics and science as cognitive and spiritual exercises that could create a truer mental and spiritual nobility. In portraying the rich contexts surrounding Descartes’ geometry, Pascal’s arithmetical triangle, and Leibniz’s calculus, Matthew L. Jones argues that this drive for moral therapeutics guided important developments of early modern philosophy and the Scientific Revolution.

It’s worth pausing to reflect on how little mathematics we might have without the accompanying spiritual motivation - Pythagoreans, Athenian and Islamic mathematicians, and so on.

This association of mathematics and spiritual improvement was still alive in the nineteenth century. Christopher Phillips in Augustus De Morgan and the propagation of moral mathematics ( Studies In History and Philosophy of Science, Part A, 36(1), pp. 105-133), which I would classify as lying on the ‘genealogical’ wing, tells us how:

In the early nineteenth century, Henry Brougham endeavored to improve the moral character of England through the publication of educational texts. Soon after, Brougham helped form the Society for the Diffusion of Useful Knowledge to carry his plan of moral improvement to the people. Despite its goal of improving the nation’s moral character, the Society refused to publish any treatises on explicitly moral or religious topics. Brougham instead turned to a mathematician, Augustus De Morgan, to promote mathematics as a rational subject that could provide the link between the secular and religious worlds. Using specific examples gleaned from the treatises of the Society, this article explores both how mathematics was intended to promote the development of reason and morality and how mathematical content was shaped to fit this particular view of the usefulness of mathematics. In the course of these treatises De Morgan proposed a fundamentally new pedagogical approach, one which focused on the student and the role mathematics could play in moral education.

Elsewhere, MacIntyre speaks admiringly of a group of early nineteenth century Lancastrian cottage weavers who met to improve themselves after work by studying books, including Euclid, before their livelihoods were swept away by the coming of the factories.

In our secular age, we now wonder what is special about mathematics education, and discuss the importance for children of acquiring a new mode of thinking.

Posted at October 1, 2006 12:26 PM UTC

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28 Comments & 5 Trackbacks

Re: The Consolation of n-Categories

Well, I’ll repeat the question at the link. Does anyone have a reference on “Transformation Dialectics”, in the math teaching context?

Posted by: Dick Thompson on October 1, 2006 10:14 PM | Permalink | Reply to this

Re: The Consolation of n-Categories

Transformation Didactics. Borovik has a later post about it. Google is unusually stingy with the term. One paper describes it thus:

Approaches that emphasise the process of learning as an essential aspect within curriculum development are creating bodies of knowledge that are related to action-oriented curriculum initiatives. In generalised terms these can be characterised as elements of ‘transformation didactics’. Due to the new challenges this kind of expertise is needed to articulate the kind of connectivity that is needed between different content areas and contexts for learning.

Presumably, then, the pupil is transformed by the learning experience. It still seems a far cry from Plato’s eye of the soul being awakened (Republic, Book VII).

Posted by: David Corfield on October 2, 2006 8:46 AM | Permalink | Reply to this

Re: The Consolation of n-Categories

A nice example of “transformational didactics” is provided by the suggestion to replace, in undergraduate analysis courses, the classic Riemann integral by gauge integral ; see an open letter to authors of of calculus textbooks.

This does not mean that I support the proposal; but it sounds very, very interesting and its pedagogiacal merits deserve a careful assessment.

Posted by: Alexandre Borovik on October 2, 2006 11:40 AM | Permalink | Reply to this

Re: The Consolation of n-Categories

Google offers even less for “Transformational Didactics”. Could you explain to us what it’s about? Is it anything more than getting the students to gain a real understanding of a piece of mathematics, rather than just an ability to prove and calculate?

Posted by: David Corfield on October 2, 2006 11:56 AM | Permalink | Reply to this

Re: The Consolation of n-Categories

I put a bit more historic details in my post Auguste Comte on Didactic Transformation

Posted by: Alexandre Borovik on October 4, 2006 10:11 PM | Permalink | Reply to this

Re: The Consolation of n-Categories

Does John have something of the mystic in him? From his 30 September entry of his diary:

The material [categorifying everything associated to simple Lie groups and quantum groups] is getting really fascinating at this point, after several years of work. Enormously deep patterns. It’s like peering down into the foundations of the universe and seeing the mysterious machinery that makes it tick.

Posted by: David Corfield on October 2, 2006 12:47 PM | Permalink | Reply to this

Re: The Consolation of n-Categories

David writes:

Does John have something of the mystic in him?

Yes, of course I do! I’m something between a Pythagorean, a Taoist, and a rational empiricist. It’s a difficult stretch, but stretching exercises keep you flexible.

Posted by: John Baez on October 2, 2006 4:57 PM | Permalink | Reply to this

Re: The Consolation of n-Categories

So long as your taoism doesn’t stretch to teaching on the Klein 2-geometry thread by not being there:

Going to Visit the Daoist Master on Dai-tian Mountain But Not Finding Him. A translation of a poem by Li Bai (701-762 CE)


      A dog barks
amidst the sound of water;
      peach blossoms
hang heavy with dewdrops.

      In the deep forest
I glimpse a passing deer;
      the rushing brook
muffles the noonday bells.

      Wild bamboos
slice through the green mist;
      streams in flight
hang between emerald peaks.

      Nobody knows
where the master has gone—
      left to wonder,
I rest among some pines.

Posted by: David Corfield on October 2, 2006 5:47 PM | Permalink | Reply to this

Re: The Consolation of n-Categories

Nice poem! I’m certainly no Taoist master - I’m way too talkative. And indeed, just today I posted a remark on the Klein 2-geometry thread. But I do try to follow the advice of Prince Huei’s cook, and cut mathematical reality at its joints:

Prince Huei’s cook was cutting up a bullock. Every blow of his hand, every heave of his shoulders, every tread of his foot, every thrust of his knee, every whshh of rent flesh, every chhk of the chopper, was in perfect rhythm – like the dance of the Mulberry Grove, like the harmonious chords of Ching Shou.

“Well done!” cried the Prince. “Yours is skill indeed!”

“Sire,” replied the cook laying down his chopper, “I have always devoted myself to Tao, which is higher than mere skill. Falling back upon eternal principles, I glide through such great joints or cavities as there may be, according to the natural constitution of the animal. I do not even touch the convolutions of muscle and tendon, still less attempt to cut through large bones.

A good cook changes his chopper once a year – because he cuts. An ordinary cook, one a month – because he hacks. But I have had this chopper nineteen years, and although I have cut up many thousand bullocks, its edge is as if fresh from the whetstone. For at the joints there are always interstices, and the edge of a chopper being without thickness, it remains only to insert that which is without thickness into such an interstice. Indeed there is plenty of room for the blade to move about. It is thus that I have kept my chopper for nineteen years as though fresh from the whetstone.

This was written around 275 BC by Chuang Tse - not bad!

A more recent master of not “hacking” was Grothendieck:

Many mathematicians will choose a well-formulated problem and knock away at it, an approach that Grothendieck disliked. In a well-known passage of Recoltes et Semailles, he describes this approach as being comparable to cracking a nut with a hammer and chisel. What he prefers to do is to soften the shell slowly in water, or to leave it in the sun and the rain, and wait for the right moment when the nut opens naturally.

Posted by: John Baez on October 3, 2006 6:42 AM | Permalink | Reply to this

Re: The Consolation of n-Categories

A more recent master of not “hacking” was Grothendieck:

I’ve always loved that passage of Recoltes et semailles, especially as it tends to mirror my own style. There is amazingly little “hard” math in my work, which can be misinterpreted as trivial. Most of the effort is in coming up with the right question, which then all but answers itself.

I also carry around another apocryphal story; this one about a man who takes his car with its (seemingly) dead engine to the mechanic. The mechanic pops the hood glances in, taps some part with a mallet and the car starts right up. Presented with the bill for $100, the man is livid. “$100 for a little tap? Where on Earth do you get off charging like that?” The mechanic presents the man with an itemized version of the bill: “tapping the engine: $1. knowing where to tap: $99”

Unfortunately, this approach lacks in quantity what it posesses in quality. Hack at a problem and you can find a hundred little answers leading to a hundred little papers. Bide your time and you get a handful of true insights. Guess which path will get a freshly-minted Ph.D. a job first.

Posted by: John Armstrong on October 3, 2006 8:53 AM | Permalink | Reply to this

Re: The Consolation of n-Categories

I talked about the theme of trivialising by getting the concepts right on my old blog. The only hope for this work to be promoted is for senior mathematicians to speak up for it.

Do you find category theory helps you to carve out true insights?

Posted by: David Corfield on October 3, 2006 4:51 PM | Permalink | Reply to this

Re: The Consolation of n-Categories

Do you find category theory helps you to carve out true insights?

Indeed I do. My entire program is to extend knot theory invariants to monoidal functors on tangles. Category theory provides exactly the mathematical structure behind an analytic-synthetic approach to knot invariants: break the knot into its simplest pieces (generating tangles), calculate the functor on those pieces, and reassemble the answers. I’d even venture to say that category theory is behind analytic-synthetic approaches to any field of study.

In another way, higher category theory comes into play. Of course I’m dealing with monoidal categories of tangles, which we all know are “really” (weak) 2-categories. Then I can pay attention to isotopies rather than isotopy classes and I’ve got a monoidal 2-category, which is a 3-category. Better yet, a large part of my dissertation (being spun off into a paper soon) concerns tangles as sitting in the 2-category of cospans of (3,1)-manifolds (3-manifolds with embedded 1-manifolds), which then makes the fundamental object of study some sort of 4-category…

Categories are like fnords. Once you learn how to see them they’re everywhere.

Posted by: John Armstrong on October 3, 2006 5:14 PM | Permalink | Reply to this

Re: The Consolation of n-Categories

Why not also put the whole thesis up on the web, if you haven’t already? It sounds very close to the interests of one of the hosts (John Baez), and one of his students (Aaron Lauda). Do you need new kinds of algebraic object to form these invariants?

Posted by: David Corfield on October 4, 2006 9:45 AM | Permalink | Reply to this

Re: The Consolation of n-Categories

Why not also put the whole thesis up on the web, if you haven’t already?

Well, as I said, I’m turning it into papers as we speak. The first one completed and being submitted to JKTR is up on the arXiv.

It sounds very close to the interests of one of the hosts (John Baez)

Actually, truth be told, Dr. Baez was in the room at Union College’s conference last December when I was informed that the construction I’d invented I’d actually reinvented, and was called “cospan”. As for whether it’s in his interests.. well, he’s seen an early version of my talk, so we could just ask him.

Do you need new kinds of algebraic object to form these invariants?

Not as yet, but I’ve only done a handful of invariants yet. The newest I’ve had to use is this cospan construction, which as I’ve noted is just a new use for an old idea.

The paper linked above is about ways to turn the bracket into a functor – there’s a variety (even a family of varieties) of ways – to some category of R-modules. Basically, it turns out to have a very deep connection to bilinear forms and their asymmetries, which goes some way towards explaining why categorifications of the bracket have something to do with Serre functors, which are relatively new, but drawing that connection explicitly is for the future.

I really do mean to use the term “program” for my research, though. There are a lot of link invariants out there, and most remain to be turned into functors. Maybe a new structure will be needed to really grok the A-polynomials or the simplicial volumes of tangles.

Posted by: John Armstrong on October 4, 2006 1:00 PM | Permalink | Reply to this

Re: The Consolation of n-Categories

But nobody will get to see the idea of the whole program if you just publish a series of papers without an overview. You could do something like Jeff Giansiracusa has done.

Posted by: David Corfield on October 4, 2006 1:56 PM | Permalink | Reply to this

Re: The Consolation of n-Categories

That’s the third paper I have planned. I want a few examples out there to refer to when I write the general outline of the program.

The bracket paper, the cospan paper, then the overview.

Posted by: John Armstrong on October 4, 2006 2:19 PM | Permalink | Reply to this

Re: The Consolation of n-Categories

John Armstrong wrote:

As for whether it’s in his interests.. well, he’s seen an early version of my talk, so we could just ask him.

Hi! Yes, the stuff you’re doing is among my interests - but even more among James Dolan’s interests. It was Jim who first told me how the Alexander polynomial was related to cospans of groups - or better, cospans of quandles. It’s also fun to go up to higher dimensions, using n-categories of cospans of cospans of cospans… of quandles to get invariants of n-tangles.

Jim will never write his ideas down, so I’m glad you’re developing similar ones, and also bringing other invariants into the fold. There’s a big picture here waiting to be fully seen, that’s for sure.

By the way, there’s something Monty-Pythonesque about these span or cospan n-categories - the way they go span, span, span, span…

Posted by: John Baez on October 4, 2006 9:20 PM | Permalink | Reply to this

Re: The Consolation of n-Categories

Actually, I’m approaching the Alexander polynomial, or the module at least (haven’t gotten the generalization of the polynomial nailed down yet), directly from the topological definition on links. That is, use the linking number of a loop in the complement of the link with the link itself to give a homomorphism of the knot group to Z[1]. Unwind this infinite cyclic cover and look at its homology as a module over the group algebra of the group of deck transformations. How to run this over the cospans is the tricky bit.

But this is getting way too far afield from the original post. If we weren’t at opposite ends of the country we could take the discussion offline more easily, or bring Dolan into it to see how his ideas and mine mesh, but such is life.

[1] I still haven’t figured out the TeX on here, and Safari refuses to render MathML anyhow.

Posted by: John Armstrong on October 5, 2006 12:05 AM | Permalink | Reply to this

Re: The Consolation of n-Categories

To write something in TeX, just go to “Text Filters” on that little page where you compose your comment, and choose “itex to MathML with parbreaks”. Then type something like

$$\sqrt{\pi^\gamma + \int_0^{17} x dx} \in \mathbb{R} $$

and you’ll get

π γ+ 0 17xdx\sqrt{\pi^\gamma + \int_0^{17} x dx} \in \mathbb{R}

Or, at least we’ll get that - it won’t be much fun for you, if your browser can’t handle MathML. Firefox rules.

For more details, people should read the TeXnical FAQ. That’s the best place to ask questions, too! I’ll see them.

Posted by: John Baez on October 7, 2006 10:35 PM | Permalink | Reply to this

The Category Illuminati

Categories are like fnords. Once you learn how to see them they’re everywhere.

In fact, trying to identify underlying structures in math is indeed a lot like trying to identify a secret conspiracy.

Once you see the same pattern hidden in a couple of apparently unrelated contexts, you begin to wonder if you have stumbled upon footprints left by an unknown institution, which secretly governs all these fields.

As with conspiracy theories of society, you may be right - or you may be fooled by your imagination. But in math the chances to identitfy a secret conspiracy of underlying structure are considerably better than for human society, I’d say.

But sometimes people tend to see more pattern than there actually is. For instance, there is a meme which says that all of physics is governed by Clifford algebra. While there is something to this, I have seen evidence that some people are judging the conspiracy behind this Clifford pattern to be considerably more influential than it actually is.

On the other hand, there are also cases where a pattern is staring into people’s faces, but they keep fighting against it, for various reasons.

For instance, I recall being told that some theorists working on λ\lambda-calculus were dismayed by learning that it is just a part of cartesian closed category theory.

I guess it is hard to acknowledge that all you did your entire life was secretly controlled by the Illuminati.

Posted by: urs on October 4, 2006 12:39 PM | Permalink | Reply to this

Re: The Category Illuminati

As with conspiracy theories of society, you may be right - or you may be fooled by your imagination.

Jacques Lacan compared modern science to full-blown paranoia.

Posted by: David Corfield on October 4, 2006 1:47 PM | Permalink | Reply to this

Re: The Category Illuminati

Jacques Lacan compared modern science to full-blown paranoia.

Who you callin’ paranoid? Wait.. you’re one of Them, aren’t you?

Seriously, though, I’m not sure it’s paranoia. I’ve ben running into a bunch of young string theorists recently, and I’ve had an easier time getting an autistic child to listen to something he didn’t already believe. All physics is strings, and it’s fifteen minutes to Wopner.

Posted by: John Armstrong on October 4, 2006 2:02 PM | Permalink | Reply to this

Re: The Category Illuminati

Lacan’s thought was more about the epistemological stance of science itself, its disciplinary attitude to knowledge, rather than any individual scientist’s psychology, although someone like Newton should give us pause for thought.

It has been noted elsewhere how similar epistemologies are to forms of mental illness. For example, if someone genuinely lived as a Humean skeptic, refusing to believe that we could know anything other than through sense impressions, so that, for example, we could never know about the necessary operation of causes, that this would appear as madness. Hume himself worried that his philosophical reflections brought him close to madness.

Posted by: David Corfield on October 4, 2006 2:38 PM | Permalink | Reply to this

Re: The Category Illuminati

I’ve had an easier time getting an autistic child to listen to something he didn’t already believe.

While unfortunate, I find that in the majority of cases this happens actually the other way around: the physicist will find himself astounded by how the mathematician is refusing to communicate at all as long as the information reaches him in an unfamiliar formatting.

Posted by: urs on October 4, 2006 3:20 PM | Permalink | Reply to this

Re: The Category Illuminati

I mean direct quotes like, “This higher gauge theory stuff can only be interesting insofar as it is physics, which means it’s really string theory. Why bother with these messy categories when string theory already does it all for you?”

Mathematicians refuse to accept anything that’s not properly rigorously formatted, but I tend to find them listening to a lot of different ideas as long as they’re rigorous. There’s a particular breed of young string theorist which claims that everything is string theory with the vehemence that Wolfram claims everything is cellular automata, or that Lang claimed that everything was heat kernels.

Posted by: John Armstrong on October 4, 2006 7:23 PM | Permalink | Reply to this

Re: The Consolation of n-Categories

Urs wrote:

In fact, trying to identify underlying structures in math is indeed a lot like trying to identify a secret conspiracy.

Once you see the same pattern hidden in a couple of apparently unrelated contexts, you begin to wonder if you have stumbled upon footprints left by an unknown institution, which secretly governs all these fields.

Heh - that’s a fun way to put it!

Louis Crane compared it to digging up the fossil of an enormous dinosaur.

I sometimes say that math is the field where it most pays to be paranoid, because in math it’s really true that nothing is a coincidence.

However, this leads to a discussion of whether it’s really really true that nothing is a coincidence in math. People like Chaitin make a good case that there’s a lot of algorithmic randomness built into math, so that lots of things must be true “by coincidence”.

For example, if you treat the primes as randomly chosen with a density roughly like 1/log(nn), a simple calculation shows that Goldbach’s conjecture has a very high chance of being true after you’ve checked it for small even numbers - and you can make this chance as high as you like, by checking enough small even numbers. It’s possible that this is all there is to it.

But, you could try a similar probabilistic approach to Fermat’s last theorem, and you would miss out on the Taniyama-Shimura conjecture. In other words: there really is a cosmic conspiracy ruling out nontrivial integer solutions of x n+y n=z nx^n + y^n = z^n, but you have to dig extremely deep to see it.

The same kind of thing could be lurking beneath Goldbach’s conjecture, only so deep we can’t even sense it yet. There’s never any way to rule this out. After all, you can prove that if Goldbach’s theorem is true, you can never prove you can’t prove it!

The situation with the 26 sporadic finite simple groups is touch-and-go: we have some people here arguing that some of these exist “by chance”, while others are holding out for a deep “reason” we haven’t seen yet.

But I’m digressing a bit - the patterns Urs is talking about are quite different: they are deep “structural” patterns relating different branches of mathematics. Nobody has ever made a convincing case that these are “coincidental”. The only two choices I’ve heard are that 1) they reflect something deep about the mathematical universe, or 2) they reflect something deep about how we think.

Unfortunately, it’s hard to tell the difference between 1) and 2). It may always be hard - at least until we meet, build or breed nonhuman mathematicians.

Alas, right now the closest thing to a nonhuman mathematician is a African grey parrot named Alex who can count, read numbers and may have a primitive notion of zero. It’ll be a while before Alex learns category theory.

Posted by: John Baez on October 4, 2006 9:06 PM | Permalink | Reply to this

Re: The Consolation of n-Categories

The situation with the 26 sporadic finite simple groups is touch-and-go: we have some people here arguing that some of these exist “by chance”, while others are holding out for a deep “reason” we haven’t seen yet.

I do not see a contradiction between the two positions. I personally want to believe that

1. There are some deep, and still unknown reasons for the “sporadic thingies” to exist;

2. Some of these thingies happened to be groups - just by chance.

Posted by: Alexandre Borovik on October 14, 2006 5:43 PM | Permalink | Reply to this
Read the post On n-Transport: Universal Transition
Weblog: The n-Category Café
Excerpt: Paths in categories from universal transitions.
Tracked: October 6, 2006 5:15 PM
Read the post Classical vs Quantum Computation (Week 1)
Weblog: The n-Category Café
Excerpt: Types and operations; categories as theories; monoidal categories versus categories with finite products.
Tracked: October 6, 2006 8:57 PM

Re: The Consolation of n-Categories

On doing mathematics for spiritual reasons: I had one undergrad pure math prof who began every lecture with five minutes of chanting in order “to get us in a receptive mood” for the math, and several others who spoke of us finding proofs for already-proved theorems “for the good of our souls”. Although the language they used was (most likely) metaphorical, it is interesting that the metaphor was spiritual rather than instrumental.

Posted by: Peter McB. on October 7, 2006 7:52 PM | Permalink | Reply to this
Read the post Faith and Reason
Weblog: The n-Category Café
Excerpt: Polanyi on faith and reason
Tracked: June 21, 2007 9:47 AM
Read the post Mathematical Emotion
Weblog: The n-Category Café
Excerpt: Does mathematical writing express emotion?
Tracked: November 19, 2009 9:24 AM
Read the post the art of war
Weblog: the art of war
Excerpt: ...He wrote that . . .
Tracked: January 11, 2010 12:48 AM

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