## September 23, 2006

### Mathematical Kinds

#### Posted by David Corfield

I’ve just sent off a paper Mathematical Kinds, or Being Kind to Mathematics to appear in the journal Philosophica. The idea of the paper is to explore the extent to which the language of laws and natural kinds, so much a part of the philosophy of science, is also appropriate to mathematics. To give a fresh example of this phenomenon, let’s consider the classification of finite simple groups.

In 2001, Ronald Solomon wrote in his A Brief History of the Classification of Finite Simple Groups, Bulletin of the American Mathematical Society 38(3) 315-382:

Is there a completely new and revolutionary approach to the Classification waiting to be discovered? In Thompson’s eloquent article “Finite Non-Solvable Groups” written around 1982 [T4], he says:

“… the classification of finite simple groups is an exercise in taxonomy. This is obvious to the expert and to the uninitiated alike. To be sure, the exercise is of colossal length, but length is a concomitant of taxonomy. Those of us who have been engaged in this work are the intellectual confreres of Linnaeus. Not surprisingly, I wonder if a future Darwin will conceptualize and unify our hard won theorems. The great sticking point, though there are several, concerns the sporadic groups. I find it aesthetically repugnant to accept that these groups are mere anomalies…Possibly…The Origin of Groups remains to be written, along lines foreign to those of Linnean outlook.”

I doubt that any developments of the past two decades would change Thompson’s summary of the state of the field. We are still waiting and wondering. Are the finite simple groups, like the prime numbers, jewels strung on an as-yet invisible thread? And will this thread lead us out of the current labyrinthine proof to a radically new proof of the Classification Theorem? (p. 345)

[T4] J. G. Thompson, Finite non-solvable groups, pp. 1-12 in K. W. Gruenberg and J. E. Roseblade, Group Theory: Essays for Philip Hall, Academic Press, London, 1984.

1. There is a school of thought that sporadic groups are not groups, they are representatives of some wider class of objects, only finitely many of which have happened, by chance, to be groups.

2. Some “non-sporadic groups” are actually sporadic; they just happened to be isomorphic to groups in “classical series”. $PSL(3,4)$ is the most notorious example; it is actually $M_{21}$; its properties are truly pathological.

3. The general theory of “taxonomy” of finite objects described by relational languages is developed in model theory (Cherlin-Lachlan Theorem). It is based on CFSG. In that sense, CFSG is the mother of all taxonomies. This is one of the factors which contribute to the huge metamathematical importance of CFSG.”

In a follow-up message, he wrote:

I believe I mentioned to you Israel Gelfand’s prophecy:

Sporadic simple groups are not groups, they are objects from a still unknown infinite family, some number of which happened to be groups, just by chance.

A number of “classical” simple groups are in fact “sporadic” in a sense that they behave in an absolutely bizarre and pathological way. For example, $Alt(6)=PSL(2,9)$ and $PSL(3,4)$; acting on 9+1 = 10 and 16+4+1 = 21 points of their projective geometries, they appear in the stabilisers of points, $M_{10}$ and $M_{21}$ in the Mathieu groups $M_{11}$ and $M_{22}$. $PSL(3,4)$ probably holds the world record for the most bizarre Schur multiplier.

$M_{13}$ is an exciting object; maybe it provides a tantalising glance into some new possibilities, maybe not - much more research is needed.

For more on $M_{13}$, see The Mathieu group $M_{12}$ and its pseudogroup extension $M_{13}$.

See what the claim is? Finite simple sporadic groups do not form a natural kind. First, as they are currently defined they exclude close relations which are non-sporadic, i.e., which appear in the infinite families of the classification. Second, even with these additional groups they are part of a much larger kind, the ones appearing in the classification just ‘happening’ to be groups.

Posted at September 23, 2006 5:03 PM UTC

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### Re: Mathematical Kinds

Incidentally, Solomon won the Levi L. Conant prize for exposition for that article.

Posted by: Allen Knutson on September 24, 2006 12:07 AM | Permalink | Reply to this

### Re: Mathematical Kinds

The full list of prize winners is here. The prize is only for articles in the Notices or Bulletin of the AMS. Does anyone know of other prizes for exposition? Perhaps a competition for TWF to be entered for?

Posted by: David Corfield on September 24, 2006 11:19 AM | Permalink | Reply to this

### Re: Mathematical Kinds

I only just realised that you too were a winner of the Levi L. Conant Prize for your article with Terence Tao: Honeycombs and Sums of Hermitian Matrices.

Posted by: David Corfield on February 12, 2007 9:35 PM | Permalink | Reply to this

### Re: Mathematical Kinds

sporadic groups are not groups, they are representatives of some wider class of objects

Are there any hints towards the nature of this wider class of objects? What could they be? Groupoids, 2-groups, group objects internal to some unusual ambient category, maybe? Any hints?

Posted by: urs on September 25, 2006 12:30 PM | Permalink | Reply to this

### Re: Mathematical Kinds

I haven’t heard any hints, but you might expect that conceptual reformulation of the sporadic groups would go hand-in-hand with advances in understanding Monstrous Moonshine.

Posted by: David Corfield on September 25, 2006 12:51 PM | Permalink | Reply to this

### Re: Mathematical Kinds

might expect that conceptual reformulation of the sporadic groups would go hand-in-hand with advances in understanding Monstrous Moonshine.

I agree, that’s quite plausible. So maybe the answer is in a suitably categorical description of RCFT.

Posted by: urs on September 25, 2006 12:55 PM | Permalink | Reply to this

### Re: Mathematical Kinds

Gannon mentions the existence of 6 sporadics unrelated to the Monster and so presumably to conformal field theory:

There has been no interesting Moonshine rumoured for the remaining six sporadics (the pariahs $J_{1}, J_{3}, Ru, ON, Ly, J_{4}$). (p. 21)

But I see Andrew Chermak has lectured on Exotic loop spaces and sporadic groups:

The Dwyer-Wilkerson 2-adic loop space is a “2-local group”, by work of Levi and Oliver, and of Aschbacher and Chermak. Moreover, this object contains, as subgroups “at the prime 2”, the sporadic groups $Co_{3}$, $J_{2}$, and $O'N$. Since O’Nan’s is a “pariah” (i.e. not involved in the Monster) it may be of interest to have a context in which it lives in harmony with $Co_{3}$ and $J_{2}$, which are not pariahs.

Posted by: David Corfield on September 25, 2006 1:41 PM | Permalink | Reply to this

### Re: Mathematical Kinds

2-adic loop space is a “2-local group”

Phew. Sounds interesting.

I have a guess what a $p$-adic loop space would be.

What, though, is a 2-local group? (And is it “2-(local group)” or “(2-local) group”?)

Posted by: urs on September 25, 2006 3:18 PM | Permalink | Reply to this

### Re: Mathematical Kinds

It appears that:

A $p$-local finite group is an algebraic structure which includes two categories, a fusion system and a linking system, which mimic the fusion and linking categories of a finite group over one of its Sylow subgroups.

Returning to pariahs, and noting debates such as whether the Tits group should really count as the $27^{th}$ sporadic group as it’s not quite of Lie type, puts me mind of a paper David Bloor wrote called ‘Polyhedra and the Abominations of Leviticus’, British Journal for the History of Science (1978) 11: 245-72. This looked at some of the language used by the protagonists of Proofs and Refutations in light of Mary Douglas’ anthropological work on what falls foul of a classification. If memory serves, the Leviticus reference is to the kind of creatures that don’t fit in - e.g., fish without scales.

Posted by: David Corfield on September 25, 2006 8:39 PM | Permalink | Reply to this

### Re: Mathematical Kinds

The ‘$Ru$’ in the list is Rudvalis’s sporadic group. Now it appears that it too partakes of moonshine, here and here. The bibliography of the latter suggest that John Duncan will bring along Janko’s third group ($J_3$) to the party soon.

Posted by: David Corfield on November 14, 2006 3:00 PM | Permalink | Reply to this

### Re: Mathematical Kinds

The discussion continued in
Ars Mathematica. The author of Ars Mathematica wrote:

A couple of years ago, though, I came across a remark by Michael Aschbacher that made me rethink my view: the classification of finite simple groups is primarily an asymptotic result. Every sufficiently large finite simple group is either cyclic, alternating, or a group of Lie type.

Actually, there is a number of very specific results about finite simple groups and of asymptotic nature; Ashbacher’s thesis is much more than just a philosophy. One of such results is a paper by Larsen and Pink Finite Subgroups of Algebraic Groups. I quote the abstract:

Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of GLn over a field of any characteristic p possesses a subgroup of bounded index which is composed of finite simple groups of Lie type in characteristic p, a commutative group of order prime to p, and a p-group. While this statement can be deduced from the classification of finite simple groups, our proof is self-contained and uses methods only from algebraic geometry and the theory of linear algebraic groups. We believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics.

Posted by: Alexandre Borovik on September 28, 2006 10:37 PM | Permalink | Reply to this

### Re: Mathematical Kinds

Fascinating. But presumably you wouldn’t say that the whole story lies in this direction. What kind of construction was Gelfand pointing to when he noted the pathological behaviour of $PSL(3,4)$? How could it be explained? I took it that he didn’t think it was just down to chance.

Posted by: David Corfield on September 29, 2006 10:07 AM | Permalink | Reply to this

### Re: Mathematical Kinds

It was not Gelfand who noticed strange behaviour of PSL(3,4), it is a well-known folklore fact.

I still do not know what Gelfand meant when formulating his thesis about sporadic groups.

We now have a better understanding of what is a non-sporadic part of the classification of finite simple groups; the sporadic part is still very much a mystery.

Posted by: Alexandre Borovik on September 30, 2006 1:06 AM | Permalink | Reply to this

### Re: Mathematical Kinds

PSL(3,4) seems like stronger evidence that small numbers are badly behaved than evidence that sporadic simple groups are a special case of some unknown structure.

Posted by: Walt on September 30, 2006 4:59 AM | Permalink | Reply to this

### Re: Mathematical Kinds

Summing up, we might say there’s a range of opinion from:

• Asymptotic treatments give us the big picture of the regular structure of the infinite families of the classification. The strange properties of the sporadics are just an instance of bad behaviour in the small.

to

• Thompson, Solomon, Gelfand: We haven’t yet got a very good grasp on what’s going on with the sporadics. There should be a very rich story to tell which explains their strange behaviour, and which may put into association the sporadic groups with objects which aren’t even groups.

With no expertise in this area, my purpose is not to argue who is closer to the truth. It is to observe how similar perspectives on classification can be between mathematicians and natural scientists.

Posted by: David Corfield on September 30, 2006 11:08 AM | Permalink | Reply to this
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### Re: Mathematical Kinds

ought to examine the role of the Schur multiplier.
For the three sporadics M, B, F24’ the multipliers are
L*/L, L = root lattice for E8,7,6 respectively.

Generalizing Chevalley (Tohoku, 1955) seems worth
examining, at least for those with moonshine. The
approach through algebraic topology (see Hirzebruch,
Berger, Jung) is powerful and should pick up M.

Just how special M is, remains to be established.

A common construction for all simples, which is what is suggested, is distinct from their classification.

Too much of CFSG involves finite groups. New ideas
will emphasize more geometry.

Posted by: John mac on April 25, 2009 2:16 PM | Permalink | Reply to this

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