# September 6, 2003 {#week198}

I recently got back from a summer spent mostly in Hong Kong. It was
interesting being there. Since I wasn't there long, most of my
observations are pretty superficial. For example, they have a real
commitment to public transportation. Not only is there a wonderful
system of subways, ferries, buses and green minibuses where you can pay
for your ride using a cool high-tech "octopus card", the local gangs
run their own system of *red* minibuses. These don't run on fixed
schedules, and they don't take the octopus card, but they seem
perfectly safe, and they go places the others don't.

Another obvious feature is the casual attitude towards English, which is
still widely used, but plays second fiddle to Cantonese now that the
Brits have been kicked out. Menus feature strange items such as "mocked
eel" and "mocked shark fin soup", which bring to mind the unsettling
image of a cook ridiculing hapless sea creatures before cooking them.
Also, perfectly nice people wear T-shirts saying things that wouldn't
be wise where I come from, like

                             Lost Pig

or

                              I SEE 
                               WHY 
                             YOU SUCK

On a more serious note, it was interesting to see the effects of the
July 1st protest against Article 23 --- an obnoxious piece of security
legislation that Tung Chee-Hwa was trying to push through. About 8% of
the entire population went to this demonstration. It stopped or at least
delayed passage of the current version of this bill, and seems to have
invigorated the democracy movement. Time will tell if it leads to good
effects or just a crackdown of some sort. The police have placed a large
order for tear gas.

While in Hong Kong, I received a copy of a very interesting book:

1) David Corfield, _Towards a Philosophy of Real Mathematics_, Cambridge U. Press, Cambridge, 2003. More information and part of the book's introduction available at `http://www-users.york.ac.uk/~dc23/Towards.htm`

I should admit from the start that I'm completely biased in favor of
this book, because it has a whole chapter on one of my favorite
subjects: higher-dimensional algebra. Furthermore, Corfield cites me a
lot and says I deserve "lavish praise for the breadth and quality of my
exposition". How could I fail to recommend a book by so wise an author?

That said, what's really special about this book is that it shows a
philosopher struggling to grapple with modern mathematics as it's
actually carried out by its practitioners. This is what Corfield means
by "real" mathematics. Too many philosophers of mathematics seem stuck
in the early 20th century, when explicitly "foundational" questions ---
questions of how we can be certain of mathematical truths, or what
mathematical objects "really are" --- occupied some the best
mathematicians. These questions are fine and dandy, but by now we've
all heard plenty about them and not enough about other *equally*
interesting things. Alas, too many philosophers seem to regard
everything since Goedel's theorem as a kind of footnote to mathematics,
irrelevant to their loftier concerns (read: too difficult to learn).

Corfield neatly punctures this attitude. He calls for philosophers of
mathematics to follow modern philosophers of the natural sciences and
focus more on what practitioners actually do:

> \[...\] to the extent that we wish to emulate Lakatos and represent
> the discipline of mathematics as the growth of a form of knowledge, we
> are duty bound to study the means of production throughout its
> history. There is sufficient variation in these means to warrant the
> study of contemporary forms. The quaint hand-crafted tools used to
> probe the Euler conjecture in the early part of the nineteenth century
> studied by Lakatos in "Proofs and Refutations" have been supplanted
> by the industrial-scale machinery of algebraic topology developed
> since the 1930s.

He also tries to strip away the "foundationalist filter" that blinds
people into seeing philosophically interesting mathematics only in the
realms of logic and set theory:

> \[...\] Straight away, from simple inductive considerations, it
> should strike us as implausible that mathematicians dealing with
> number, function and space have produced nothing of philosophical
> significance in the past seventy years in view of their record over
> the previous three centuries. Implausible, that is, unless by some
> extraordinary event in the history of philosophy a way had been found
> to *filter*, so to speak, the findings of mathematicians working in
> core areas, so that even the transformations brought about by the
> development of category theory, which surfaced explicitly in 1940s
> algebraic topology, or the rise of non-commutative geometry over the
> past seventy years, are not deemed to merit philosophical attention.

To me, it's a breath of fresh air just to see a philosopher of
mathematics *mention* non-commutative geometry. So often they seem to
occupy an alternate universe in which mathematics stopped about a
hundred years ago! Elsewhere in the book we find interesting discussions
of Eilenberg-MacLane spaces, groupoids, the Ising model, and Monstrous
Moonshine. One gets the feeling that the author is someone we might meet
on the internet instead of the coffeehouses of fin-de-siecle Vienna, who
writes using a word processor instead of a fountain pen.

The book consists of chapters on loosely linked subjects, some of which
seem closer to "real mathematics" than others. The chapters on
"Communicating with automated theorem provers" and "Automated
conjecture formation" are mildly depressing, given how poor computers
are at spotting or proving truly interesting conjectures without lots of
help from humans --- at least so far. True, Corfield describes how in 1996
the automated theorem prover EQP was the first to crack the Robbins
conjecture. This states that a Boolean algebra is the same as a set
equipped with an commutative associative binary operation "$\mathrm{or}$"
together with a unary operation "$\mathrm{not}$" for which one mind-numbing axiom
holds, namely:
$$\operatorname{not}(\operatorname{not}(p \operatorname{or} q) \operatorname{or} \operatorname{not}(p \operatorname{or} \operatorname{not}(q)) = p$$
All the rest of Boolean logic is a consequence! But proving this seems
more like a virtuoso stunt than the sort of thing we working
mathematicians do for a living. This is actually part of Corfield's
point, but I find it a somewhat odd choice of topic, unless perhaps
philosophers need to be convinced that the business of mathematics is
still a mysterious process, not yet easily automated.

Apart from the one on higher-dimensional algebra, the chapters that make
me happiest are the ones on "The importance of mathematical
conceptualisation" and "The role of analogy in mathematics".

The first is a marvelous study of the so-called "conceptual approach"
in mathematics, which emphasizes verbal reasoning using broad principles
over calculations using symbol manipulation. Some people are fond of the
conceptual approach, while others regard it as "too abstract".
Corfield illustrates this split using the debate over "groupoids versus
groups", with the supporters of groupoids (including Grothendieck,
Brown and Connes) taking the conceptual high road, but others preferring
to stick with groups whenever possible. As a philosopher, Corfield
naturally leans towards the conceptual approach.

The second is all about analogies. Analogies are incredibly important in
mathematics. Some can be made completely precise and their content fully
captured by a theorem, but the "deep" ones, the truly fruitful ones,
are precisely those that resist complete encapsulation and only yield
their secrets a bit at a time. Corfield quotes Andre Weil, who describes
the phenomenon as only a Frenchman could --- even in translation, this
sounds like something straight out of Proust:

> As every mathematician knows, nothing is more fruitful than these
> obscure analogies, these indistinct reflections of one theory into
> another, these furtive caresses, these inexplicable disagreements;
> also nothing gives the researcher greater pleasure.

I actually doubt that *every* mathematician gets so turned on by
analogies, but many of the "architects" of mathematics do, and Weil
was one. Corfield examines various cases of analogy and studies how they
work: they serve not only to discover and prove results but also to
*justify* them --- that is, explain why they are interesting. He also
examines the amount of freedom one has in pushing forwards an analogy.
This is a nice concrete way to ponder the old question of how much of
math is a free human creation and how much is a matter of "cutting
along the grain" imposed by the subject matter.

The analogy he considers in most detail is a famous one between number
fields and function fields, going back at least to Dedekind and Kummer.
By a "number field", we mean something like the set of all numbers
$$a + b \sqrt{-5}$$
with $a,b$ rational. This is closed under addition, subtraction,
multiplication, and division by anything nonzero, and the usual laws
hold for these operations, so it forms a "field". By a "function
field", we mean something like the set of all rational functions in one
complex variable:
$$\frac{P(z)}{Q(z)}$$
with $P,Q$ polynomials. This is again a field under the usual operations
of addition, subtraction, multiplication and division.

Sitting inside a number field we always have something called the
"algebraic integers", which in the above example are the numbers
$$a + b \sqrt{-5}$$
with $a,b$ integers. These are closed under addition, subtraction,
multiplication but not division so they form a "commutative ring".
Similarly, sitting inside our function field we have the "algebraic
functions", which in the above example are the polynomials
$$P(z)$$
This is again a commutative ring.

So, an analogy exists. But the cool part is that there's a good
generalization of "prime numbers" in the algebraic integers of any
number field, invented by Kummer and called "prime ideals"... and
prime ideals in the algebraic functions of a function field have a nice
*geometrical* interpretation! In the example given above, they
correspond to points in the complex plane!

The analogy between number fields and function fields has been pushed to
yield all sorts of important results in number theory and algebraic
geometry. In Weil's hands it led to the theory of adeles and the Weil
conjectures. These in turn led to etale cohomology, Grothendieck's work
on topoi, and much more. And the underlying analogy is still far from
exhausted! But if we ever get it completely nailed down, then (in the
words of Weil):

> The day dawns when the illusion vanishes; intuition turns to
> certitude; the twin theories reveal their common source before
> disappearing; as the *Gita* teaches us, knowledge and indifference are
> attained at the same moment. Metaphysics has become mathematics, ready
> to form the material for a treatise whose icy beauty no longer has the
> power to move us.

Or something like that.

Anyway, I hope this book shows philosophers that modern mathematics
poses many interesting questions apart from the old "foundational"
ones. These questions can only be tackled after taking time to learn the
relevant math... but what could be more fun than that?! I also hope
this book shows mathematicians that having a well- informed and clever
philosopher around makes math into a more lively and self-aware
discipline.

(The same is true of physics, of course. I listed a few good
philosophers of physics in ["Week 190"](#week190).)

Someday I'd like to say more about the analogy between number fields
and function fields, because I'm starting to study this stuff with
James Dolan... but it will take a while before I know enough to say
anything interesting. So instead, let me say what's going on with spin
foam models of quantum gravity.

I've already talked about these in ["Week 113"](#week113),
["Week 120"](#week120), ["Week 128"](#week128) and
["Week 168"](#week168). The idea is to calculate the amplitude for
spacetime to have any particular geometry. An amplitude is just a
complex number, sort of the quantum version of a probability. If you
know how to calculate an amplitude for each spacetime, you can try to
compute the expectation value of any observable by averaging its value
over all possible geometries of spacetime, weighted by their amplitudes.
When you do this to answer questions about physics at large distances
scales, the amplitudes should almost cancel except for spacetimes that
come close to satisfying the equations of general relativity. This is
how quantum gravity should reduce to classical gravity at distance
scales much larger than the Planck length.

But in a spin foam model, a spacetime geometry is not described by
putting a metric on a manifold, as in general relativity. Instead, it's
described in a somewhat more "discrete" manner. Only at distances
substantially larger than the Planck length should it resemble a metric
on a manifold.

How do you describe a spacetime geometry in a spin foam model? Well,
first you take some $4$-dimensional manifold representing spacetime and
chop it into "$4$-simplices". A "$4$-simplex" is just the $4$-dimensional
analogue of a tetrahedron: it has 5 tetrahedral faces, 10 triangles, 10
edges and 5 vertices. Then, you label all the triangles in these
$4$-simplices by numbers. These describe the *areas* of the triangles.
Here the details depend on which spin foam model you're using. In the
Riemannian Barrett-Crane model, you label the triangles by spins
$j = 0, 1/2, 1, 3/2 \ldots$. But in the Lorentzian Barrett-Crane model, which should
be closer to the real world, you label them by arbitrary positive real
numbers. Either way, a spacetime chopped up into $4$-simplices labelled
with numbers is called a "spin foam".

To compute an amplitude for one of these spin foams, you first use the
labellings on the triangles and follow certain specific formulas to
calculate a complex number for each $4$-simplex, each tetrahedron, and
each triangle. Then you multiply all these numbers together to get the
amplitude!

In ["Week 170"](#week170), I mentioned some mysterious news about
the Barrett-Crane model. At the time --- this was back in August of 2001 -
my collaborators Dan Christensen and Greg Egan were using a
supercomputer to calculate the amplitudes for lots of spin foams. The
hard part was calculating the numbers for $4$-simplices, which are called
the "$10j$ symbols" since they depend on the labels of the 10 triangles.
They had come up with an efficient algorithm to compute these $10j$
symbols, at least in the Riemannian case. And using this, they found
that the $10j$ symbols were *not* coming out as an approximate calculation
by Barrett and Williams had predicted!

Barrett and Williams had done a "stationary phase approximation" to
argue that in the limit of a very large $4$-simplex, the $10j$ symbols were
asymptotically equal to something you'd predict from general
relativity. This seemed like a hint that the Barrett-Crane model really
did reduce to general relativity at large distance scales, as desired.

However, things actually work out quite differently! By now the
asymptotics of the $10j$ symbols are well understood, and they're *not*
given by the stationary phase approximation. If you want to see the
details, read these papers:

2) John C. Baez, J. Daniel Christensen and Greg Egan, "Asymptotics of $10j$ symbols", _Class. Quant. Grav._ **19** (2002) 6489--6513. Also available as [`gr-qc/0208010`](https://arxiv.org/abs/gr-qc/0208010).

3) John W. Barrett and Christopher M. Steele, "Asymptotics of relativistic spin networks", _Class. Quant. Grav._ **20** (2003) 1341--1362. Also available as [`gr-qc/0209023`](https://arxiv.org/abs/gr-qc/0209023).

4) Laurent Freidel and David Louapre, Asymptotics of $6j$ and $10j$
symbols, _Class. Quant. Grav._ **20** (2003) 1267--1294. Also available as [`hep-th/0209134`](https://arxiv.org/abs/hep-th/0209134).

The physical meaning of this fact is still quite mysterious. I could
tell you everyone's guesses, but I'm not sure it's worthwhile. Next
spring, Carlo Rovelli, Laurent Freidel and David Louapre are having a
conference on loop quantum gravity and spin foams in Marseille. Maybe
after that people will understand what's going on well enough for me to
try to explain it!

I'd like to wrap up with a few small comments about last Week. There I
said a bit about a $24$-element group called the "binary tetrahedral
group", a $24$-element group called $\mathrm{SL}(2,\mathbb{Z}/3)$, and the vertices of a
regular polytope in 4 dimensions called the "$24$-cell". The most
important fact is that these are all the same thing! And I've learned a
bit more about this thing from here:

5) Robert Coquereaux, "On the finite dimensional quantum group $H = M_3 + (M_{2|1}(\Lambda^2))_0$", available as [`hep-th/9610114`](https://arxiv.org/abs/hep-th/9610114) and at `http://www.cpt.univ-mrs.fr/~coque/articles_html/SU2qba/SU2qba.html`

Just to review: let's start with the group consisting of all the ways
you can rotate a regular tetrahedron and get it looking the same again.
You can achieve any even permutation of the 4 vertices using such a
rotation, so this group is the $12$-element group $A_4$ consisting of all
even permutations of 4 things --- see ["Week 155"](#week155). But
it's also a subgroup of the rotation group $\mathrm{SO}(3)$. So, its inverse image
under the double cover
$$\mathrm{SU}(2) \to \mathrm{SO}(3)$$
has 24 elements. This is called the "binary tetrahedral group".

As usual, the algebra of complex functions on this finite group is a
Hopf algebra. But the cool thing is, this Hopf algebra is closely
related to the quantum group ${U}_q(\mathfrak{sl}(2))$ when $q$ is a third root of unity
--- a quantum group used in Connes' work on particle physics because of
its relation to the Standard Model gauge group!

In short: the plot thickens.

I'm not really ready to describe this web of ideas in detail, so I'll
just paraphrase the abstract of Coquereaux's paper and urge you to
either read this paper or look at his website:

> We describe a few properties of the non-semisimple associative algebra
> $H = M_3 + (M_{2|1}(\Lambda^2))_0$, where $\Lambda^2$ is the Grassmann algebra
> with two generators. We show that $H$ is not only a finite dimensional
> algebra but also a (non-cocommutative) Hopf algebra, hence a "finite
> quantum group". By selecting a system of explicit generators, we show
> how it is related with the quantum enveloping algebra of ${U}_q(\mathfrak{sl}(2))$
> when the parameter $q$ is a cubic root of unity. We describe its
> indecomposable projective representations as well as the irreducible
> ones. We also comment about the relation between this object and the
> theory of modular representations of the group $\mathrm{SL}(2,\mathbb{Z}/3)$, i.e. the
> binary tetrahedral group. Finally, we briefly discuss its relation
> with the Lorentz group and, as already suggested by A. Connes, make a
> few comments about the possible use of this algebra in a modification
> of the Standard Model of particle physics (the unitary group of the
> semi-simple algebra associated with $H$ is $\mathrm{U}(3) x \mathrm{U}(2) x \mathrm{U}(1))$.

------------------------------------------------------------------------

**Addenda:** I got some interesting feedback from Martin Krieger and
Noam Elkies.

Martin Krieger writes:

> In the interchange on Corfield's book, and John Baez's discussion of
> it, there is a reference to Weil's Rosetta Stone analogy. The
> quotations come from a charming and deep letter Weil wrote in 1940 to
> his sister, Simone, from Bonne Nouvelle prison. In my book *Doing
> Mathematics: Convention, Subject, Calculation, Analogy* (World
> Scientific, 2003) that long letter is translated into English (see
> Appendix D, pp. 293-305). I also happen to have a discussion of the
> analogy, in chapter 5 (pp. 189-230), in connection with the Langlands
> program and with results in statistical mechanics of the Ising model.
>
> Martin Krieger\
> University of Southern California\
> Los Angeles CA 90089-0626\

You can now find Krieger's translation of this letter online, as long
as you register with the American Mathematical Society (it's free):

6) Martin H. Krieger, "A 1940 letter of Andre Weil on analogy in mathematics", _AMS Notices_ **52** (March 2005), 334--341. Available at `http://www.ams.org/notices/200503/200503-toc.html`

Noam Elkies writes:

> Hello again,
>
> You write:
>
> > \[...\]
>
> >I'd like to wrap up with a few small comments about last Week.
> >There I said a bit about a $24$-element group called the "binary 
> >tetrahedral group", a $24$-element group called $\mathrm{SL}(2,\mathbb{Z}/3)$, and 
> >the vertices of a regular polytope in 4 dimensions called the 
> >"$24$-cell".  The most important fact is that these are all the 
> >same thing!  And I've learned a bit more about this thing from 
> >here:
>
> > \[...\]
>
> Here's yet another way to see this: the $24$-cell is the subgroup of
> the unit quaternions (a.k.a. $\mathrm{SU}(2)$) consisting of the elements of norm
> $1$ in the Hurwitz quaternions --- the ring of quaternions obtained from
> the $\mathbb{Z}$-span of $\{1,i,j,k\}$ by plugging up the holes at $(1+i+j+k)/2$ and
> its $\langle 1,i,j,k\rangle$ translates. Call this ring $A$. Then this group maps
> injectively to $A/3A$, because for any $g,g'$ in the group $|g-g'|$ is
> at most $2$ so $g-g'$ is not in $3A$ unless $g=g'$. But for any odd prime $p$
> the $(\mathbb{Z}/p\mathbb{Z})$-algebra $A/pA$ is isomorphic with the algebra of $2\times2$
> matrices with entries in $\mathbb{Z}/p\mathbb{Z}$, with the quaternion norm identified
> with the determinant. So our $24$-element group injects into $\mathrm{SL}_2(\mathbb{Z}/3\mathbb{Z})$
> --- which is barely large enough to accommodate it. So the injection
> must be an isomorphism.
>
> Continuing a bit longer in this vein: this $24$-element group then
> injects into $\mathrm{SL}_2(\mathbb{Z}/p\mathbb{Z})$ for any odd prime $p$, but this injection is
> not an isomorphism once $p>3$. For instance, when $p=5$ the image has
> index 5 --- which, however, does give us a map from $\mathrm{SL}_2(\mathbb{Z}/5\mathbb{Z})$ to the
> symmetric group of order 5, using the action of $\mathrm{SL}_2(\mathbb{Z}/5\mathbb{Z})$ by
> conjugation on the 5 conjugates of the $24$-element group. This turns
> out to be one way to see the isomorphism of $\mathrm{PSL}_2(\mathbb{Z}/5\mathbb{Z})$ with the
> alternating group $A_5$.
>
> Likewise the octahedral and icosahedral groups $S_4$ and $A_5$ can be
> found in $\mathrm{PSL}_2(\mathbb{Z}/7\mathbb{Z})$ and $\mathrm{PSL}_2(\mathbb{Z}/11\mathbb{Z})$, which gives the permutation
> representations of those two groups on 7 and 11 letters respectively;
> and $A_5$ is also an index-$6$ subgroup of $\mathrm{PSL}_2(\mathbb{F}_9)$, which yields the
> identification of that group with $A_6$.
>
> NDE

------------------------------------------------------------------------

> *The enrapturing discoveries of our field systematically conceal, like
footprints erased in the sand, the analogical train of thought that is
the authentic life of mathematics*
> 
> --- Gian-Carlo Rota