Two Volumes….

My favorite form of writing (to read) is the essay. Books on single themes, no matter how well-written, invariably lapse into sections or passages that are redundant, plodding, or pedantic. The essay form is brief enough to be rich and scintillating from beginning to end, in the hands of a good craftsman. 

All this to say, that even though it’s only June I think I have already found my favorite book-of-the-year in Jim Holt’s “When Einstein Walked with Gödel,” a compendium of 20-years-worth of Holt essays. I don’t expect anything I see the rest of the year (though I could be wrong) to surpass the joy I’m getting from these beautiful pieces on physics, philosophy, culture, and abundant math. At 3/4 of the way through there hasn’t been a bad, boring, or weak essay yet, nor expected in the final 1/4. I’ve been marking my very favorite essays as I go along, but so many are now thusly-marked it’s not even worth noting them all. Wonderful descriptions of and anecdotes about great figures in the history of math/science; wonderful discussion of debates/controversies in the scientific/philosophical realm; wonderful, thought-provoking, often novel, commentaries and overviews. I’ve already touted this volume in various places, and can’t recommend it enough; readable and enjoyable by professionals and laypersons alike. 

Here are some more formal review links:

https://bookmarks.reviews/reviews/when-einstein-walked-with-godel-excursions-to-the-edge-of-thought/

…and also a review and interview with Holt here:

https://medium.com/@ja.alper20/when-einstein-walked-with-godel-3431e2349ef6

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Received an uncorrected review copy of Eugenia Cheng’s forthcoming (September?), “The Art of Logic In an Illogical World” awhile back. It’s Dr. Cheng’s third book and again an attempt to present somewhat abstract topics (previously category theory and infinity) to a general audience. Oddly, in each instance I’ve enjoyed one half of each of her books better than the other half; in this case it was the second half I enjoyed most — I won’t dwell on that, since your ‘mileage may vary,’ but mention it just so you know that if the first half doesn’t grab you, persevere, and the second half may be more rewarding.

Dr. Cheng’s topic this go-around is very timely and important, as it has to do with how people think, reason, form opinions, argue, etc. in this highly-polarized world we inhabit. Important to note that there is very little technical logic/symbolism in the volume, even when she is discussing elements of formal logic. Her tone/presentation is much more informal/casual, almost conversational — that has always been her writing style as she strives to reach a broad audience. In fact I would almost say that the title of the book could equally well be “The Art of Common Sense” because she is dealing with logic in such an informal, introductory way that it will often seem like just formalized common sense.

One of the main strengths of the book is that she employs very current issues or examples (often related to equality or feminism)  to illustrate her points throughout, making what could have been abstract or stodgy material, rather more pertinent and interesting.

Early in the book Dr. Cheng notes that she has sometimes been described as “pedantic” for her previous writing, and I think that is a fair warning of how some will view passages here as well (again, more-so the first half of the book). But then she consistently takes on subjects (like category theory, infinity, and now logic) that lend themselves to pedantry (despite her constant attempt at casualness… which I find a bit annoying at times, but may well appeal to her intended audience).

Dr. Cheng’s enthusiasm for her topics is unmistakeable; she already has a large fan-base and 3 books (and many articles and videos) under her belt — each of those volumes are good, but I suspect the most excellent works from this energetic, relatively young writer are yet to come, as she endeavors to spread the good news of mathematics AND logical thinking far and wide.

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