I.  A few days ago MWA was mentioned on MathOverflow in connection with a discussion introduced by Ryo with the following comment:

Many distinguished mathematicians, at some point of their career, collected their thoughts on mathematics (its aesthetic, purposes, methods, etc.) and on the work of a mathematician in written form.

Ryo invited users “to create a “big-list” of the kind of works described in the above blockquote.”  A few users obliged by proposing entries for a list that, for the moment, could hardly be called “big” (but maybe a “big-list” doesn’t have to be physically big?).

Do you notice anything wrong with this list?  Or maybe the question should be:  do you notice anything peculiar about the list, as it stands at present?  (Take another look before reading ahead to see the answer.)

If you haven’t found the answer yet, please look back to this (meta)-discussion and to these observations by Izabella Laba, all dating to early 2011.  That happens to be exactly when I began mining MathOverflow as a source of information about implicit assumptions about hierarchy and charisma (routinized and otherwise) in mathematics, with specific reference to this discussion.

A lot could be said about the MathOverflow discussion of the relative scarcity of women among active participants:  for example, that it comes across as a sincere effort to address a genuine problem but that the effort was hampered by the absence of reliable empirical data and even more by the participants’ inability to agree on an appropriate methodology to begin asking questions that would lead to evaluate the scope of the problem (or even to realize that such a methodology is a necessary starting point); or that it was marred by what seems to me a misplaced display of tolerance of the defense of mathematical cultures where, to quote one user, “there are no such insubstantialy [sic] strong feminist views in women communities.”  But I will just recommend that readers look over the comments and read Izabella Laba’s post carefully, and decide for themselves whether or not they have anything to do with the “big-list” problem.

Many women mathematicians (including two mentioned in this post) have written and continue to write books for a broad public that speak about mathematics; the items on the “big-list“, as I understand it, are meant to speak for mathematics, in the name of mathematics.  I assume some MathOverflow users are familiar with essays (though not books) written in the genre of “distinguished women mathematicians speaking in the name of women in mathematics,” which is a respected and necessary genre but apparently not considered worthy of inclusion in the  “big-list“.

Maybe some readers of this blog will want to remedy this omission.  [UPDATE:  it has already been done on MathOverflow.]  This post is concerned with what is wrong with the book to which the blog is dedicated; the question raised by the omission is whether, while purporting to speak in the name of mathematics as such, and indeed going so far as to announce the intention of doing so in its title, Mathematics without Apologies may instead be speaking in the name of a specifically male (not to say laddish) vision of mathematics.  I will not go so far as to invoke standpoint epistemologies and instead mention what may be a typical example.  Chapter 8 devotes a long discussion to the use of words like philosophy, strategy, and tools as metaphors for mathematical practice, and suggests a parallel with the trifunctional theory Georges Dumézil developed in his analysis of Indo-European mythology.  But the trifunctional theory clearly reflects a society organized around the leading roles of men (in spiritual, military, and productive matters).  Could MWA have focused on metaphors corresponding to female roles in the traditional division of labor?  Indeed it could have explored how proofs are put together from ingredients; the cleaning lemmas that appear in fields as diverse as quantum computing, combinatorics, and singularity theory;  and the domesticity of the Weyl chamber.

The suggestion is definitely NOT that it would somehow be more respectful to women mathematicians to stress that diverse sexually stereotyped metaphors occasionally appear on the dreary landscape of mathematical prose.  It is rather to acknowledge that the standpoint underlying the choices made in the writing MWA may well not be that of the neutral voice speaking in the name of mathematics itself but is rather overdetermined by a history of male domination in mathematics that parallel the domination in the broader society.  At the very least this undermines any claim that the author has (except in Chapter 9) systematically avoided introducing autobiographical elements except where they can be construed as representative of the mathematician in its purity that is ideal-typical, and presumably neutral with respect to gender as well as any other natural or social category.

II.  Much evidence is presented in MWA to support the contention that pure mathematicians pursue our vocation in search of pleasure — or, as Stefaan Vaes explained to the Queen of Belgium, omdat wij dit graag doen.  In the interest of honesty, however, more attention should have been given to mathematics as a source of pain, which may take the form of anxiety, dread, grief, or even despair.  I would guess that nearly all the mathematicians reading this blog know perfectly well what I have in mind, and that the impossibility of conveying to a non-mathematician what it entails makes the experience all the more painful.

MWA records the exhilarating moment when, in the words of Marie-France Vignéras, “one direction becomes more dense, or more luminous.”  But frustration may be the more universal experience of mathematicians.  It’s common enough for mathematicians to report that most of their working time is spent getting nowhere.  Occasionally something about the mathematician’s experience is so painful that it can only be relieved by suicide.  The word “suicide” occurs exactly once in MWA (not counting the ritual hara-kiri performed in the film Rites of Love and Math) but I have been aware of suicide among mathematicians and students of mathematics since my earliest days as a student.  Four of my fellow students had killed themselves by the time I had obtained my Ph.D.  I didn’t know them well enough to understand their motivations, but at the time it was assumed that they gave up out of a sense of failure. Two of them had been forced to take jobs in places where they did not want to live, with no prospect of leaving unless they left mathematics.  Their suicides may have made them martyrs, but not madmen — the two roles typically reserved for mathematical characters in popular culture.

In a peculiar way, the incidence of suicide among mathematicians — I don’t know how it compares to other professions, but it is certainly not negligible — confirms the model proposed by MWA that mathematics is a relaxed field.  Some may feel that mathematicians are being immature when we insist on the pleasure principle at the expense of the reality principle; but frankly, what’s so great about reality when it consists in subordinating one’s freedom of action to projects that outsiders are able to assign to us simply because they are Powerful Beings, and whose goals we may find profoundly distasteful?  The other day a graduate student in the common room was saying that he wanted his life to be organized in such a way that his work was his hobby.  A second student found that perfectly natural.  That is the purest possible expression of mathematics as a relaxed field, and nothing is more characteristic of the vocation than the willingness of its practitioners to suffer extraordinary levels of frustration, even pain, rather than submit to the reality principle that reigns elsewhere.

(Future posts will treat some of the many other things that are wrong with MWA but that have been neglected by the occasional hostile reviewer.)