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Mathematics without Apologies: Portrait of a Problematic Vocation

Michael Harris
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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This is a strange, sometimes interesting but often frustrating book, and certainly one that defies easy characterization. Although the book attempts to explain what mathematicians do and why they do it (the title, of course, is a reference to G. H. Hardy’s A Mathematician’s Apology), and although there are mathematical discussions intended for non-professionals, this is most certainly not a typical “mathematics for laymen” text, explaining deep ideas in overly simplistic ways. In fact, there is nothing about this book that could be described as “simplistic”. Harris writes in a very dense style, with frequent references to culture (literature, philosophy, history, and others) outside mathematics; this book frequently reads like a doctoral dissertation in English or philosophy, with an endless parade of quotes and references, rather than a book intended for a general audience. A reader should be prepared to work hard, and read large chunks of the text more than once.

We have not even gotten past the preface, for example, when the following statement is encountered: “I drew on my personal experience when it served to illustrate a larger point (when it was ideal-typical in the Weberian sense I try to mobilize in chapter 2.).” Not really knowing anything about Weber and therefore not really understanding just what the last half of this sentence meant, I dutifully turned to chapter 2 (“How I Acquired Charisma”), and read:

The ‘I’ of this chapter’s title is not the hateful ‘I’ of Blaise Pascal’s Pensees but rather the hypothetical ‘I’ of a Weberian ideal type. “Type of what?” Maybe we’ll know by the end of this book. In this chapter it’s the type associated with the possession of a specific degree of routinized charisma, with one curious feature: its subject crossed the threshold relatively late, granting him an unusual and, he likes to think, unbiased perspective of life on both sides of the charisma divide.

Well, that clarifies things. At least this paragraph appeared in a chapter, the title to which I understood; chapter 4 is entitled “Megaloprepeia” (italics in the original), a term that one encounters in that chapter after about twenty pages.

The opaqueness of the writing style is one of three general complaints I have with this book. The second is a self-congratulatory tone that I found rather off-putting. On page 38, for example, we find the paragraph

By granting me tenure at the age of twenty-seven, Brandeis University ratified my permanent admission to the community of mathematicians. Thus I was endowed with the routinized charisma symbolized by my institutional position and fulfilled the (rational-bureaucratic) obligations incumbent upon one enjoying the privileges befitting my charismatic status. These privileges included and still include regular invitations to research centers such as the IAS (where I began writing this book in 2011), the IHES, or the Tata Institute of Fundamental Research (TIFR) in Mumbai (Bombay); to specialized mathematics institutes like the Fields Institute in Toronto, the Mathematical Sciences Research Institute (MSRI) in Berkeley, or the Institut Henri Poincaré in Paris, where Cedric Villani is now director; or to speak about my work at picturesque locations like the Mathematisches Forschungsinstitut Oberwolfach, a conference center nestled high in the hills of the Black Forest in Germany, that for nearly seven decades has hosted weekly mathematics meetings.

This paragraph is just the beginning of a multi-page discussion of the author’s accomplishments. Other references to the author and his achievements appear throughout the text. These references are not, I believe, ameliorated by his statement that he only draws on his personal experience to illustrate a larger point (or, as he puts it, “a convenient focal point around which to organize the text”); the appearance of bragging is hardly excused by the disclaimer “I’m not intending to brag.” Perhaps I do the author an injustice, and bragging was genuinely not his intent, but if so, I think, he should have tried a little harder to prevent this sort of impression, perhaps by giving more thought to how these recitations of achievements would likely be interpreted by an average reader.

Finally, the book has a stream-of-consciousness feel to it, with the author jumping from one idea to the next, following no particular narrative pattern that I could discern. Chapter 4, for example, deals with the uses and abuses of mathematics in financial mathematics, with attention to the great financial crash of 2008; it is followed immediately by a chapter discussing the writing style in Thomas Pynchon’s Against the Day (the author characterizes Pynchon’s writing style as not just nonlinear, but quadratic; he notes that “you find a lot of hyperbolas” in the novel). Next up is a chapter (“Further Investigations on the Mind-Body Problem”) that looks at how mathematicians have been perceived (in, for example, film), and later on (chapter 8) the author muses about what constitutes a mathematical “trick” (this discussion is motivated by his own “tensor product trick”, about which, the author tells us, “I was the first to notice its relevance in the context of mutual interest” and which he has since applied “to solve problems in at least three completely different settings.” This chapter not only discusses “tricks” but also mathematics and popular music.

Then, in the very next chapter, the author goes into considerable detail about a dream he had “about the cohomology of unramified coverings of Drinfel’d upper half-spaces” and the insight it gave him in his work. (This dream is described earlier in the text as a “mystic vision” that “started me on the path to being bumped up a few rungs on the charisma ladder.”) You see my point, I hope: the impression I frequently got was that whatever editor Princeton University Press assigned to this book simply threw up his or her hands in defeat and allowed the author free reign. In this connection, I again quote the author, who points out that “even my modest level of charisma entitles me not only to say in public whatever nonsense comes into my head … but even to get it published.”

Issues like the ones described above can diminish the enjoyment of reading a book, but they can cause other problems as well. My concern is that books like this, with their self-congratulatory tone, may diminish rather than enhance people’s regard for academia in general and mathematics in particular. There are enough people out there with little regard for what university faculty do (witness the fact that a legislator in North Carolina actually proposed a bill that would require faculty in the UNC system, regardless of research expectations, to teach eight classes a year); the last thing we need is to provoke more hostility by suggesting that mathematicians are in the habit of daring the reader to try to understand their enlightened prose.

These defects in writing style notwithstanding, there are some interesting things to be found in the book, and also occasional flashes of humor. Readers of this column may have heard about Edward Frankel and the controversy caused by a movie that he wrote and starred in called Rites of Love and Math (here is a news article discussing this); there is an extensive discussion of that film here, accompanied by the amusing observation that “[r]elatively few professions are practiced even intermittently in the nude, and … Rites is likely to reopen the long overdue debate on whether mathematics… should be one of them”. Harris’ ruminations on the question of why one does mathematics also offer food for thought; he considers and rejects the “golden goose” argument that it is useful, and also analyzes G. H. Hardy’s answer that one does mathematics because it is beautiful, like an art form.

In addition, there is some actual mathematics here. Interspersed throughout the text are a series of chapters, “numbered” by Greek letters, in which the author explains “How to Explain Number Theory at a Dinner Party”. We are told that, at an actual dinner party attended by Harris, an actress asked him to explain “what is it you do in number theory, anyway?” The author gave a brief response but the amalgamation of these chapters, he says, “contains what should have been my answer.” Starting with the definition of a prime number, he gradually works his way all the way up to decidedly nontrivial things like elliptic curves and the Birch and Swinnerton-Dyer conjecture.

While I’m not at all sure that a layperson will likely get a great deal out of some of the later chapters, on the whole these were nicely done. The chapters end with a (mostly) imaginary conversation between a performance artist and a number theorist, in which both these people spend a lot of time quoting others (ranging from Chekhov to Marilyn Monroe) and in which the number theorist manages to apply Galois theory to Shakespeare (“So I’ll say that the Galois group of Hamlet has twenty-four elements, the possible ways to permute the characters without affecting the play.”). So one has to at least give the author credit for originality: I certainly have never heard Galois and Hamlet mentioned in the same sentence before.

Final verdict: a reader will have to make an individual determination as to whether the benefits of this book outweigh the author’s rhetorical excesses and heavy-handed writing style. Undoubtedly, some will say that they do, but I’m afraid that I can’t count myself in that group.

Mark Hunacek ( teaches mathematics at Iowa State University.

Preface ix
Acknowledgments xix
Part I 1
Chapter 1. Introduction: The Veil 3
Chapter 2. How I Acquired Charisma 7
Chapter α. How to Explain Number Theory at a Dinner Party 41
(First Session: Primes) 43
Chapter 3. Not Merely Good, True, and Beautiful 54
Chapter 4. Megaloprepeia 80
Chapter β. How to Explain Number Theory at a Dinner Party 109
(Second Session: Equations) 109
Bonus Chapter 5. An Automorphic Reading of Thomas Pynchon's Against the Day (Interrupted by Elliptical Reflections on Mason & Dixon) 128
Part II 139
Chapter 6. Further Investigations of the Mind-Body Problem 141
Chapter β.5. How to Explain Number Theory at a
Dinner Party 175
(Impromptu Minisession: Transcendental Numbers) 175
Chapter 7. The Habit of Clinging to an Ultimate Ground 181
Chapter 8. The Science of Tricks 222
Part III 257
Chapter γ. How to Explain Number Theory at a Dinner Party 259
(Third Session: Congruences) 259
Chapter 9. A Mathematical Dream and Its Interpretation 265
Chapter 10. No Apologies 279
Chapter δ. How to Explain Number Theory at a Dinner Party 311
(Fourth Session: Order and Randomness) 311
Afterword: The Veil of Maya 321
Notes 327
Bibliography 397
Index of Mathematicians 423
Subject Index 427