Avner Ash and Michael J. Barany wrote very flattering reviews of MWA in Harvard Magazine and MathSciNet, respectively. Some choice excerpts have been added to the Reviews page of this blog. I’m particularly pleased that Barany chose to call MWA a “work of cultural criticism” and wonder why none of the other reviewers saw the book in that way.
Although Ash and Barany both belong to réseaux of which I am also a member, they did not hesitate to point out what they saw as flaws in the book. This post will be devoted to airing their criticisms, which are comprehensible as well as legitimate. A future post will deal with illegitimate or incomprehensible complaints about MWA. Finally, as promised, I will explain what I think is really wrong with the book.
Ash writes
Granting the book’s pleasures and insights—there is rarely a dull moment— Harris’s writing is at times choppy, jumping from one level of discourse to another. It can be hard to follow the nuances and consequences and connections among the ideas in their rapid flow.
That is a very kind way of putting it. Chapter 8 — in keeping with its subject matter (The Science of Tricks) — was designed to be read as a series of sudden transitions that are supposed to be experienced as surprises that in retrospect are seen as inevitable. Chapter 4 — in keeping with the author’s feelings about its subject matter — has some violent and shocking language, but the first drafts (which will not be revealed here) were considerably more violent. Chapter 10 consists of two narratives, the first about Hardy’s aestheticism, the second about the contemporary insistence on utility.
Much of the apparent (or real) choppiness, in other words, is the result of conscious authorial decisions. But it’s hardly up to the author to comment on the wisdom of these decisions. Barany puts it a different way (my emphasis):
Harris is emphatic that non-mathematicians have had important and insightful things to say about the discipline, but also that they have been limited to a significant degree by the kinds of public representations mathematicians have made about themselves. His expositions thus mingle an eclectic mix of ideas and source materials from a wide range of traditions, juxtaposing them largely on the basis of their thematic suggestiveness in a way that can sometimes seem unmotivated. The results are frequently striking, but often depend on the reader’s ability to place references in context.
That’s a fair description of the intention as well as of the method, and Barany is also being kind: on more than one occasion the “thematic suggestiveness” amounted to little more than the author’s feeling that the sentences flowed well, and that nothing as inconsequential as thematic coherence should be allowed to disturb their flow. This may be responsible for what Mark Hunacek, writing in the American Mathematical Monthly, calls the book’s “stream-of-consciousness feel,”
with the author jumping from one idea to the next, following no particular narrative pattern that I could discern.
In view of what can be said (and what I will eventually be saying) about some of Hunacek’s other complaints about the book, one might wonder why he feels it is to his credit to broadcast his lack of discernment. But where stylistic choices are concerned, either they work or they don’t, and the reader’s judgment is final.
Barany saves his sharpest criticism for substance rather than style. He writes
For all Harris packs into Mathematics without apologies, the book feels at the same time very unfinished.
He tempers his judgment by attributing part of the fault to the Princeton University Press editorial process, but elsewhere in the review he makes the author’s responsibility clear for a book that is “fascinating and frequently frustrating”:
his answers typically fall short of his provocations… Some of the impression of incompleteness owes to Harris’s style and approach, which avowedly favors provocations over syntheses.
I have two things to add to this comment. First, Barany is absolutely right that the author made a point of avoiding syntheses. To have done otherwise would have been to usurp the prerogatives of the historians, philosophers, and sociologists ‘to whom the author apologizes in the preface. The reality principle reminds us, owever, that there is a flip side to this appearance of modesty and restraint: provocation is easy, but history, philosophy, and sociology are hard work. A historian may well find it frustrating that, as a practitioner, I am entitled
not only to say in public whatever nonsense comes into my head … but even to get it published
(as I write on pp. 39-40 — a sentence Hunacek quoted but completely failed to understand, by the way). How would specialists react if a historian of mathematics, in the interest of provocation, proposed an account of the goals of (for example) the Langlands program at odds with the explicit goals of its practitioners, while claiming insufficient expertise to provide more than “thematic suggestiveness” as justification. It would be worse than lèse-majesté! But what makes the practitioners more majestic than the interpreters who have worked hard to acquire a professional understanding of the historical and intellectual context?
But now I’m anticipating a future post, so I’ll just thank Ash and Barany again for their sympathetic, insightful, and excessively flattering readings.
Mathematics without Apologies, by Michael Harris 
Thanks for this post, which among other things promotes the very worthy principle that neither the book itself nor its initial reviews should have the last word. I’d add here a little elaboration to another line from my review, which may have seemed a bit of a throw-away in context. I wrote: “Indeed, confronting and working around obscurities seems very much a part of the mathematical life Harris aims to convey, as is the pleasure of finding insight in unexpected places.”
Critical comments in this genre of review often fall into one of two types, either wishing that the author had written the book differently, or advising (warning?) readers to read the book differently than they might otherwise do. There are comments of both types in my write-up, and some of the comments discussed in this post that might look more like the former type may be better understood as the latter, as advice on how to read a very unusual book rather than a wish that the book had been more ordinary.
Part of the implicit advice in my review is to try to savor the questions without worrying that they’re mostly unanswered (or only partially answered). But even the remarks about how frustrating the book sometimes is should be understood as mostly reader-oriented. An unusual virtue of MWA is how the author delves into the affective experience of being a professional mathematician, an experience full of frustration and anxiety as well as insight and pleasure. Other aspects noted in the review, like how much the book often relies on readers to put claims in context, and how it demands that readers puzzle through the inevitable portions where they don’t have such context near-to-hand, are also regular parts of the professional mathematical experience. So, as a consequence of some of the author’s decisions about the writing process, MWA has a number of what you might call second-order lessons about the mathematical life, gained by indirect showing rather than direct showing or telling. In my experience, these second-order lessons were very hard to appreciate in the moment, when trying to make sense of the book while reading. But they certainly add to MWA’s value and importance, and they’re the type of lesson that reviewers can especially help readers to recognize.
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