Finally I can begin to fulfill the promise I made last August to point out a few of the things that are really wrong with Mathematics without Apologies. The diagram reproduced above is taken from an article Langlands wrote when the editor invited him to contribute an article on mathematical beauty to the volume cited above, with the title The Many Faces of Beauty. (In the summer of 2011 I was invited to contribute an article on mathematical beauty to an issue of the Portland-based literary journal Tin House with the title Beauty. So Langlands and I have at least that in common.) The diagram lists “the names of some of the better-known creators of the concepts” that contributed to the solution by Andrew Wiles of Fermat’s Last Theorem, which Langlands chooses as the starting point of his account of the theory of algebraic equations and, ultimately, automorphic forms.
One will have noticed that all the names belong to men, practically all of them European. This is a problematic feature of the hierarchical nature of mathematics, but it’s not my topic today. The question instead is Who speaks for mathematics? which (it seems to me) is at least implicit in Piper Harron’s identification of the oppressiveness of mathematical hierarchy. Much to my regret, MWA did not break with the convention of quoting the reflections of Giants and Supergiants and those most visible among our contemporary colleagues, the people whose names appear in lists and diagrams like the one copied above. Thus the question Who speaks for mathematics? is answered by pointing to those who occupy the most prominent positions in the (oppressive?) hierarchy.
This is in part due to the unsystematic nature of my research and in part due to structural features of the hierarchy, which I emphasize on p. 39, in what I have already identified as the key passage in Chapter 2:
We’ll see throughout the book quotations by Giants and Supergiants in which they conflate their own private opinions and feelings with the norms and values of mathematical research, seemingly unaware that the latter might benefit from more systematic examination. One of the premises of this chapter is that the generous licence granted hieratic figures is of epistemological as well as ethical import.
My own experiments with the expression of what appear to be my private opinions resemble this model only superficially and only because they conform to the prevailing model for writing about mathematics. My friend’s point was that even my modest level of charisma entitles me not only to say in public whatever nonsense comes into my head…
In other words, it’s much easier to be quoted if you have published your thoughts in the first place, and it’s much easier to get your thoughts published if you are identified as a consequential mathematician. I don’t know how to overcome this (possibly oppressive) characteristic of the mathematical hierarchy, and it’s one of the main reasons I am hoping sociologists will pay closer attention to mathematics.
Having said that, I should dispel any notion that Langlands took advantage of his mathematical eminence, in the article from which Diagram B is taken, to write “whatever nonsense” came into his head. On the contrary, while the modesty of his intentions is evident throughout, there is no nonsense but rather a good deal of profound and unconventional thinking about the nature of our vocation. So I will take the risk of promoting the (oppressive?) hierarchy once again and encourage you all to read the article, if you have not done so already; and I will quote a few of its more memorable passages.
On his own limitations and uncertainties:
I learned, as I became a mathematician, too many of the wrong things and too few of the right things. Only slowly and inadequately, over the years, have I understood, in any meaningful sense, what the penetrating insights of the past were. Even less frequently have I discovered anything serious on my own. Although I certainly have reflected often, and with all the resources at my disposal, on the possibilities for the future, I am still full of uncertainties.
On the effects of (what MWA calls) charisma:
Because of the often fortuitous composition of the faculty of the more popular graduate schools, some extremely technical aspects are familiar to many people, others known to almost none. This is inevitable.
On Proposition 78 of Book X of Euclid’s Elements:
This is a complicated statement that needs explanation. Even after its meaning is clear, one is at first astonished that any rational individual could find the statement of interest. This was also the response of the eminent sixteenth century Flemish mathematician Simon Stevin…
On the beauty of Galois theory:
What cannot be sufficiently emphasized in a conference on aesthetics and in a lecture on mathematics and beauty is that whatever beauty the symmetries expressed by these correspondences have, it is not visual. The examples described in the context of cyclotomy will have revealed this.
On cooperation — and charisma! — in mathematics:
Like the Church, but in contrast to the arts, mathematics is a joint effort. The joint effort may be, as with the influence of one mathematician on those who follow, realized over time and between different generations — and it is this that seems to me the more edifying — but it may also be simultaneous, a result, for better or worse, of competition or cooperation. Both are instinctive and not always pernicious but they are also given at present too much encouragement: cooperation by the nature of the current financial support; competition by prizes and other attempts of mathematicians to draw attention to themselves and to mathematics.