...a companion blog to "Math-Frolic," specifically for interviews, book reviews, weekly-linkfests, and longer posts or commentary than usually found at the Math-Frolic site.

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"Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." ---Bertrand Russell (1907) Rob Gluck

"I have come to believe, though very reluctantly, that it [mathematics] consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-legged animal is an animal." ---Bertrand Russell (1957)

******************************************************************** Rob Gluck

Wednesday, May 20, 2015

Category Theory, Not Quite Fully Cooked


"How To Bake Pi" by Eugenia Cheng

[In Britain, this book goes under the title, "Cakes, Custard and Category Theory," a title I prefer!] 

An intrinsic hazard of reviewing media, be it movies, TV, plays, music, books etc., is that 'expectations' play a significant role in any judgment.
I like Eugenia Cheng's new book, "How To Bake Pi," and recommend it, and believe it will be on my year-end top 10 list... B-B-BUT I suspect it won't be in the upper tier of that list. It didn't quite live up to the high expectations I had for it, in a year (that isn't even half-over) with many excellent popular math volumes already out.

Very oddly, Cheng's book appeared almost the same week as Jim Henle's "The Proof of the Pudding," which I reviewed HERE -- two books sharing the unusual approach of combining mathematics and kitchen recipes! Other than the analogies to cooking, there is little similarity between the volumes however, and to my surprise, I actually enjoyed the Henle offering more, though Cheng's effort is more substantive, serious, and covers matters I specifically wanted to learn about.

"How To Bake Pi" is divided into two broad parts, 1) on what mathematics is, and 2) on category theory. Structurally, it reminds me of Paul Lockhart's "Measurement," which was also divided in two parts, and in both instances the first part is the easier, faster read, while the second is a heavier slog (for Lockhart it was "size and shape" largely on geometry, versus "time and space" largely on calculus).

"...Bake Pi" begins, simply enough, trying to explain "what math is," and in the process, demolish preconceptions that many hold. Cheng focuses on "abstraction" and "generalization" as key elements of mathematics. The author does a good job of explaining "abstraction," which involves taking away all the "clutter" or non-essential components of an idea or problem, and reducing it to bare necessities.
She further notes that mathematics is "different" from "science," where evidence is key. Instead, in math, "logic" is what drives thinking forward. She goes on to talk about "principles," and "process," leading to "generalization" and eventually "axiomatization." With analogies and examples Cheng does a good job of walking the reader through this garden of mathematical features.
Similar to Jim Henle's book, each chapter here begins with a kitchen recipe of some sort -- I didn't find Cheng's recipes as mouth-watering as Henle's, but their real purpose is to make some point via analogy that readers can relate to.

I enjoyed the whole first half of this book (especially the wrap-up eighth chapter), as Cheng delineates what mathematics is really about, for those who have the misperception of math as just numbers, computations, and memorization. I'm not fully comfortable though, with her conclusion that while "life is hard," "math is easy." I think math (and specifically, abstraction) is genuinely hard for many individuals (and that is regardless of how it is presented). It can make folks feel more frustrated or inferior if they repeatedly hear that 'math is easy' while they continue to struggle with it. I prefer Paul Lockhart's take when he writes in his book "Measurement," that:
[math is] "very hard work... Be prepared to struggle, both intellectually and creatively. The truth is, I don't know of any human activity as demanding of one's imagination, intuition, and ingenuity."

[There is also a greater sense of accomplishment, even exhilaration, for kids succeeding in math when they think it challenging rather than 'easy' -- somewhere there must be a middle-ground between scaring them by saying it's hard, and frustrating them by saying it's easy!]

Chapter 9, the beginning of Part 2 of Cheng's volume, starts off innocently enough, explaining that "category theory," which began as a study of topology, is "the mathematics of mathematics" or what can be called "metamath." The focus is on relationships and structure, but this is where the book begins faltering a bit.  Despite Cheng's earnest efforts, I still did not come away from Part 2 feeling a deep grasp of just what category theory is, how it is used, or what it's main differences/advantages over set theory are, though possibly a second-reading will help clarify what a first reading left fuzzy. I was also hoping to learn more about the connections between category theory and "type theory" or "homotopy type theory (HoTT), but these go unmentioned in the book. Perhaps I hoped for more than is even possible in a volume aimed at a general audience. [In a coincidental stroke of timing, Quanta Magazine just published a fine introductory piece to some of these latter topics:
https://www.quantamagazine.org/20150519-will-computers-redefine-the-roots-of-math/ ]

Another reviewer of the book (at MAA) writes, "I can't help but feel the target audience for this book is very small (in particular I can't think of a specific person I would give it to as a gift)..." I think that's overly-harsh, but I do understand the sentiment -- actually the "target audience" is quite broad, but I'm less certain how well-illuminated and satisfied readers will be on the central topic of the volume: category theory (...yet, there may be no other introductory books on the subject to compete with it).

In fact while making my way through Part 2 of the volume, I wondered whether a different publisher/editor might have fashioned a better result. Could Princeton University Press (my favorite publisher) have rendered a sharper edition of this volume than did Basic Books, the actual publisher (Basic publishes many good popular math volumes, but I usually feel their presentation is a notch below Princeton).
Or, alternatively, I wondered how an explanation of category theory from two of my favorite math explicators Steven Strogatz or Keith Devlin (if they could even expound on the subject) might have differed from/improved upon Cheng's effort.
But Cheng is clearly passionate about her subject, loves teaching, and I do hope will give us additional popular math offerings in her future (...and for now hers may be the best introduction to category theory available).

So by all means consider this book, and especially so if you've been waiting for a primer on category theory to come along... just don't presume that that murky topic will be made crystal clear by the time you finish the volume.

Natalie Angier's favorable review of the book is here:
https://theamericanscholar.org/a-taste-for-higher-math/#.VPjByIY8LCQ

...also worth noting that Cheng participates in a YouTube channel that further reviews category theory:
https://www.youtube.com/user/TheCatsters/featured


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