Letters

Have you, I say, an answer of such fitness for all questions?

It must be an answer of most monstrous size that must fit all demands.

(All's Well That Ends Well, Act II,Scene II)

After reading John Conway's article on

finite simple groups I am prompted to share

two further monstrous observations of mine:

The form of the modular function

j(z) = e3(q)/qi~=1(1-qi)24, q = e 2~iz,

e(q) = 240~3(n)qn,

~3(n~ = d~n d3

suggests that jl/3(3z) may be a dimension

counting function connected with E 8 just as j(z) is for the Monster (or Friendly Giant)

simple sporadic group. That this is so has

been proved by Victor Kac [I].

There is, in the Monster, a class of involu-

tions which are centralized by a double

cover of the Baby Monster. If an element is the product of two such involutions then it

lies in one of nine conjugacy classes,

namely those for which [5,P.327] D<6:

~3 I

l a 2b 3a 4a 5a 6a 4b 2a

This graph is the affine Dynkin graph of

type E8" The numbers are the weights of the largest root. They are the multiplicities

associated with the singular fibres in an elliptic pencil (see [3]). They are also the dimensions of the irreducible represent-

ations of SL2(5), the largest of the finite one-dimensional groups of quaternions [2].

Centralizing a further involution in the

centralizer of a node element reveals the

remarkable property exemplified by the

5x21+8(A5xA5).2 by omitting structure the 5a node and adjacent edges, and 3x21+8.A 9 by similarly omitting the 3c node.

Lest this be thought coincidence, there are similar properties relating the graphs of

type E6 and E7 with centralizers of elements in the Monster, see [4].

I. Kac,V.G., An elucidation of "Infinite- Dimensional Algebras .... and the Very

Formula." E~) and the Cube Root of Strange

57 the Modular Invariant j. Adv. in Math. 35 (1980),264-273. 2. McKay,J., Caftan matrices, finite groups of quaternions, and Kleinian singularities. Proc. Amer. Math. Soc. 81 (1981),153-154. 3. Tate,J., An algorithm for determining the type of a singular fiber in an elliptic pencil. Modular Functions of One Variable IV, Springer-Verlag. LNM 476 (1975). 4. McKay,J., Graphs, singularities, and finite groups. Proc. Symp. Pure Math. Amer. Math. Soc. 37 (1980),183-186. 5. Conway,J.H. and Norton,S.P., Monstrous Moonshine. Bull. Lond. Math. Soc. 11 (1979),308-339.

John McKay Department of Computer Science Concordia University Montreal, Quebec

In the course of my research, I have sought in vain for a copy of

A.J.C. Cunningham and H.J. Woodall's "Factorisations of yn +_ I; y=2, 3, 5, 6, 7, I0, 11, 12 etc". Francis Hodgson, London (1925).

I wonder if one of your readers might let me have sight of their copy and if you could broadcast my appeal in the Mathematical Intelligencer.

This 24p book is not in the British Library or the National Union of Congress Library. Nor do Longmans, who absorbed Francis Hodgson, have a copy.

Guy Haworth 33 Alexandra Rd. Reading, Berks RGI United Kingdom

5PG

Cosgrove's fascinating biography of Ko Tan- Jen (M.I. 3 no.3) omits what was, if we can believe the reports, a very important period of Ko's work: his study of the theory of examples during a little-known visit to Germany. This is said to have led to his discovery of a particularly rich class of subobjects in the category of examples: a discovery which has influenced every branch of mathematics - the ubiquitous Ko unter- examples.

One reason for this discovery not being attributed to Ko could be that Cosgrove has been influenced by an alternative theory, one which I first heard from Graham Higman: that a counterexample is a homomorphic image of an example.

A discussion between the proponents of these two, equally plausible, etymologies would provide a most valuable contribution to the eponymy debate (M.I. 2 no.4 and M.I. 3 no.2).

C.A. Rowley The Open University Milton Keynes, U.K.

58 In a letter dated March 7, 1980, and pub- lished on pp.95-96 of the 1981 Mathematical Intelli~encer 14 mathematicians (Barucha- Reid et al), berate your magazine for protesting that a Fields medalist from Moscow was forbidden by the Soviet National Committee for Mathematics to attend the ICM at Helsinki. They defend this shameful act by citing other injustices which have nothing to do with the matter (the so-called Berufsverbot, which I happen to oppose, and the imprisonment of J.L. Massera, to whose defense, I dare say, I devoted more time and energy than Barucha-Reid et al). They des- cribed what happened in euphemistic terms. The Fields medalist was absent "because he was not included in the Soviet delegation" and anyway he was later alowed to visit Germany for 3 months (what overwhelming generosity). They tell us that "dissension" centers on the insistence that the USSR include in its "delegation" all invited speakers. This is a "delicate question" since the Soviets feel that "the decision as to who shall represent Soviet science abroad rests with the Soviet National Committee." And all this ends with a touching appeal "for our science and our world" to "eschew shrill confrontationalism."

Whom do Barucha-Reid et al. think they are kidding? They know that at every ICM in which the Soviets participated 30-50% of Soviet invited speakers were not permitted to come (and not just not included in the "delegation"). Jews are almost never per- mitted to attend an ICM, and neither are mathematicians who are politically suspect, or have the wrong kinds of friends, or have in any other way displeased the small clique which runs Soviet mathematics.

I wonder whether Barucha-Reid et al. approve of this. I wonder if they, in particular, approve of this and other manifestations of anti-semitism in Soviet mathematics. I hope this is not a "delicate question."

Lipman Bets Columbia University New York

Editorial Statement. Regarding this particular issue, the correspondence in these columns is now closed. But the debate will surely continue as it has in the past. We request readers wishing to respond to communicate directly with the indivi- duals concerned.

More e l e m e n t s

I would like to offer the following 'elements' to augment those given in Halmos' recent article (Math. Int. 3 (4) 1981,147- 153).

Flexibility. What happens inside you when you encounter ~17 ? Do you experience it as an object, a number with but one defining property, as an operation applied to 17, as the name of a Dedekind cut or Cauchy sequence? All of these are available, and you can adoptany one of them when you detect which is appropriate from the con- text. Such flexibility is the antithesis of the usual image of mathematics as the para- digmatic discipline of right/wrong answers and single perspective. People often talk of The Mathematics in some situation, yet it is precisely from variety of perspective/ interpretation that mathematical richness flows.

By their properties shall ye know them. The formalist notion that a point-A-n-d~-l-i-ne can be taken as simply names for things which are known only by their incidence properties revolutionized mathematical thinking. It is also a major stumbling block for students, and it crops up very frequently. Seeing ~17, ~, e, and i as names for numbers which are known only by their properties is not easy. Having been at some pains to provide a geo- metric, algebraic and/or calculus based sense of what ~17, ~ and e represent, i is often introduced simply by its formal alge- braic properties, perhaps augmented by the rather mystifying association with rotation. Trigonometric functions, the exponential function, F and Bessel functions are all names for entities which we know only by their properties. How many students ap- preciate this? Consider also a more complex geometric example (taken from C.Gattegno's Film I Circles in the Plane Educational Solution~.Y.)~

Imagine the family of circles in the plane passing through the origin and with centres on the x-axis. In what sense is the y-axis a member of this family? One mathematical story embeds the family in a projective plane. Imagining the family as a single circle with centre on the positive x-axis and increasing in radius, then the y-axis, then decreasing in radius on the left, in a 'smooth' sequence, can produce an impression that the y-axis belongs to the family. Such a mental imaging game provides some sort of sense of what a projective plane might be like, at least in one perspective. When it comes to sewing cross-caps on a sphere, however, it is very difficult for most people to imagine what is happening. Accep- ting cross-caps to be specified by their properties permits algebraic and even geo- metric calculations.

John H. Mason Faculty of Mathematics The Open University Milton Keynes MK7 6AA United Kingdom