"

**Mathematics Without Apologies**" by Michael Harris....

"

*Insofar as the present book is about anything, it is about how it feels to live a mathematician's double life: one life within this framework of professional autonomy, answerable only to our colleagues, and the other life in the world at large.*"

-- Michael Harris

"

*...the only way somebody can be a scientist is that somehow their personality gets frozen at an early age... at the playful stage.*"

-- Andrei Kolmogorov (as quoted in Harris's book)

For starters, I'll say that this will be among my

**favorite**math volumes for all of 2015... to which I'll add, your mileage may vary! (it won't suit everyone's taste). This is a volume that almost defies adequate review, so vast, variegated, and tangled is its content. It is as odd, eclectic, even intractable a piece of mathematical writing as I have ever come across. Peculiar, dense, trenchant, hopscotchy, sometimes turgid, verbose, hard-to-follow, even inscrutable... but also interesting, rich, quirky, creative, original, a tad enigmatic, unpredictable, curious, cerebral and thought-provoking, sprawling across the landscape of ideas and interests. Gregory Chaitin, in his blurb for the book, writes, "

*Mathematical high culture collides with pop culture and all hell breaks loose! Harris takes us on a wild ride -- never a dull moment!*" I couldn't agree more (and how often can you say 'all hell breaks loose' in a math book!). A press release for it states that it is for "

*intellectually curious readers*" -- that is a colossal understatement! The same press release calls the volume "post-post-modern" -- I understand where the description comes from, but as someone who isn't a fan of post-modernism I prefer to avoid that terminology (even though it clearly fits some passages in the book); but, if not post-modern, it's certainly post-1950s! ;-)

The author says at some point that this is a book he always felt needed to be written... and only when no one else did it, did he take the task upon himself. And he writes: "

*My original aim in writing this book was to suggest new and more plausible answers to the 'why' question; but since it's pointless to say why one does something without saying what that something is, much of the book is devoted to the 'what' question*[what is mathematics]."

I'll say up front that much of the technical math in this book (which is not the bulk of the content, but it does exist) is beyond my ability to even judge, though given the author's credentials I certainly assume it credible and valid. I'll be dealing here more with the style/approach/presentation than with the technical aspects.

There is a lot of focus in these pages on Galois, Grothendieck, and Robert Langlands as initiators of grand, sweeping programs in mathematics. I mention that just to indicate the level of mathematics that is often addressed in these pages, even though the discussion is more general, and intended for a broader audience. This is

*not*the typical popular math book written at the level of high school or early college math. Even without including a lot of technical, computational mathematics, Harris's subjects are deep and abstract and, depending on your background, may be difficult for a lay reader to follow. I like that sort of challenge, but some may find it off-putting, if they get lost in Harris's commentary or occasional technical jargon.

The book ranges all over the place: math, literature, science, psychology, history, music, culture, film... it's all here in bites. I'm sure the author had some idea how it was organized in his own mind, but I can't offer a clue! And that's not said as a criticism, just a warning to readers, and for me it gave the book a rollicking, unpredictable style that made it spirited. The writing is sometimes less-than-scintillating... because the subject matter is sometimes less-than-scintillating. It is a strange mix of dense, heavy, at times convoluted or rambly stuff, that is yet made scintillating and engaging by its very originality.

So even though I sometimes found myself lost reading these pages, it was lost in an exciting, challenging way... not in a dark, foreboding forest, so much as in a sunny, expansive, amusement-filled park!

Along the way, the author intersperses conversations between an imagined "performing artist" and a "number theorist" to make various points -- it is reminiscent of a technique used by Douglas Hofstadter in the past. I think Harris does an even better (less tedious) job of it than Hofstadter did, and doesn't

*over*due it as Hofstadter may have.

If you like puzzles, equations, computations, proofs, etc., this is NOT the math book for you. It's hard to describe WHO this book IS for!... certainly someone more interested in the contemplation of, and philosophy underlying, math, and its role in society, more than in doing or learning mathematics. On the other hand, only those with an already deep background in math will fully appreciate the ideas Harris eagerly romps about in.

I've tried to think of any other book I've read with a similar feel to this volume. Perhaps William Byers' "

**How Mathematicians Think**," but only very slightly so. Harris references Jeremy Gray a number of times, so I wonder if Gray's volume, "

**Plato's Ghost**," might bear some similarity, but I've not read it and can't say. The writing style reminds me a tad of David Foster Wallace (who also gets an occasional nod in the volume), but that too is a stretch. The author cites Thomas Pynchon from time to time, and mentions Pynchon's "

*non-linear*" writing style -- again, I've not read Pynchon myself, but "non-linear" or "Pynchon-esque" just may be an apt descriptor for Harris's prose. Finally, there's a slight similarity here to Ed Frenkel's "

**Love and Math**," but only in so much as both volumes turn some attention to the Langlands program, and both touch on subjects well outside the perimeter of mathematics, but Harris scampers MUCH farther afield than Frenkel. Essentially, I'm doubtful this work can be readily compared to any other math volume.

And I'd love to have been a fly-on-the-wall for the editorial sessions that produced this book! I could be completely wrong, but I imagine much give-and-take, with the editor requesting lots of changes, and the author unwilling to assent to them. To my ear, the writing just seems less edited, less clean, more disjointed, than what's customary in a popular math book -- it's more like the unvarnished author coming through. I also imagine the author making changes/additions right up to the very last moment, as this volume could

*never*really be finished. ...But these are all just idle (and perhaps wrong) impressions on my part. Again, not intended as criticism, but rather a sign of the uniqueness of this multifaceted effort.

I'll only touch upon a few highlights from the volume:

Early on, somewhat oddly, the author discusses "charisma" (and competition) in mathematics -- both in the field of math study, as well as in the practitioners who do math. The discussion revolves around the "sociology" of mathematics, which, unlike either the logic, philosophy, or computations of math, doesn't usually get wide play. Harris also ends up touching on the Elsevier boycott that was spearheaded by Fields Medalist Timothy Gowers, and the "

*fascinating questions about the possibility of reconciling the goals of science with the material organization of society.*"

Chapter 4 presents an interesting treatment of the 2008 financial collapse, the Black-Scholes equation, and failures of mathematicians... there's a good reason that economics has been called "the dismal science" and this chapter exposes it.

Chapter 6 on the "mind-body problem" covers a lot of mostly non-mathematical ground. Harris spends some significant time discussing Ed Frenkel's independent film "

*Rites of Love and Math*." Though both men work on the Langlands program, Harris actually spends more time considering, not Frenkel's mathematical work, but rather his popular short film that replaces the geeky mathematician stereotype with a loving (even nude!) mathematician in an uncompromising search for truth. The chapter feels like something written by a devotee of the humanities, moreso than the product of an academic mathematician.

There are a couple of chapters focused on number theory (Harris's main field), parts of which may be hard to follow without a firm grounding in that subject, but I think I found chapters 7 and 8 the most difficult to follow of the more general chapters. One interesting segment in chapter 7 though, recounts the use of drugs by many who were successful mathematicians (Erdös and his amphetamines being a classic case). And then, Harris writes, "

*Pharmaceutical enhancement may be redundant if mathematics is itself the drug*." Next he quotes several mathematicians who make reference to how doing mathematics is like being on a drug. It's an interesting notion that helps explain the addictive quality that math has for so many practitioners, that non-mathematicians often find inexplicable.

In chapter 8 he looks at "tricks" in mathematics, but these aren't the simple or quick tricks you might learn in a YouTube mathy presentation, but much higher-level or abstract tricks that may not be easily grasped. Later in the chapter the always-interesting relationship between math and music is discussed.

Chapter 10 is on the "math-is-beauty" and "math's-unreasonable-effectiveness-in-science" themes. Even in dealing with such oft-treated matter, Harris's presentation is fresh and thoughtful. And it is also in this chapter that he comes full circle back to G.H. Hardy's "

**A Mathematician's Apology**," in order to reiterate that he makes NO apologies for the beauty and usefulness of pure mathematics in our lives.

I've barely scratched the surface here of this volume's contents -- I won't apologize for that or for the large chunks that fell beyond my comprehension. Nor can I fully communicate the oddity and freshness of this uncommon effort. Along with a regular index, at the end there is a separate "

*Index of Mathematicians*" of those mentioned by name in the text -- this index includes well over

**250**individuals within a book body of 325 pages! And as for others (non-mathematicians) who get more than a passing mention in the book, here are some of those names: Aquinas, Goethe, Huizinga, Kant, Kuhn, Mehrtens, Pynchon, Wittgenstein... to give a sense of the breadth of material here (what the publisher, Princeton University Press, calls "

*a ridiculously diverse assortment of... sources*"). Indeed! Additionally, there are ~70 pages of 'notes,' and a wide-ranging 20+ page bibliography -- much of it technical (indeed, I was surprised that several "popular" math works I looked for weren't even included).

I've read through this book once... which doesn't do it justice -- it's the sort of volume you can pick up, randomly turn to any page, begin reading, and gain more insight than was bestowed by the first read-through. The professional mathematician will no doubt get more from these pages than the lay reader can... and, probably find more to disagree with as well. Having said that, I also suspect people will, Rorschach-like, read into many of Harris's passages what they think or want him to mean, rather than what he intended. Doug Hofstadter has written in the past, that despite the praise (and awards) lavished upon his first book, "

**Gödel, Escher, Bach**," most reviewers got the book

*wrong*. The same may happen for much of Harris's content.

Still, I love that this volume reveals a mathematician who breaks out of the nerdy math stereotype to talk philosophy, humanities, arts, culture, literature, and other subjects too often missing from the public's image of the professional mathematician. I can appreciate Harris's discussion... even when I don't fully comprehend it, because I value seeing the inner mathematician turned loose for public consumption. I don't recommend you all read this book once... I recommend you read it, indeed savor it, 2-3 times. It will certainly be re-appearing on the Christmas list I compile at the end of the year for readers. It stands, perhaps, in a category of its own.

Peter Woit reviewed the volume here:

**http://www.math.columbia.edu/~woit/wordpress/?p=7479**

...and there is an interview with the author here:

**http://blog.press.princeton.edu/2015/02/04/qa-with-michael-harris/**

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