Amir D. Aczel (1950–2015)

I really enjoyed this book. Now what are we going to call the numerals used in the West that seem to have evolved from several sources? Arabic-Indian-Cambodian? AIC? Who knows what other cultures may have had a hand in them? Africa south of the Mediterranean countries and their near neighbors, sunken Southeast Asia?

Aczel has two main issues that he is looking into, although at time I think that they get a bit in each other's way. One is the form of the letters, and the other is the concepts of zero and infinity. He has mostly limited himself to Eurasia, with a brief side trip to Egypt. He isn't interested in the Americas because the numerals that interest him developed independently, but he does mention that the Mayans had a numeral representing zero. Interesting in view of Aczel's argument that he thinks the concept of zero originated from south Asian religion and philosophy.

That wouldn't apply to the Mayan zero, but I don't know that we know enough about indigenous American religions to say if they had the same ideas, especially with the realization that South America was much more heavily populated and had a lot more cities than we supposed. The history of Africa, particularly outside of the Mediterranean rim, is also in need of a lot more study. Similar things often develop independently around the world in any case.

Aczel includes a lot of personal information, as he is recording his search for a lost artifact that was written about in the early 2oth century. Sometimes this can get tedious, as in a certain book where the author kept going off on unrelated tangents, and filling the reader in on personal trivia. Aczel led a very interesting life, and tells his story well, so I enjoyed this. Most of his tangents related to interesting fact about famous mathematicians that were interesting in their own right, or mathematical controversies such as the arguments about Set Theory, as well as other mathematical systems that used base 60 and base 20 instead of base 10.

His father, of Hungarian heritage, was a cruise ship captain of the S. S. Theodor Herzl, named for the Hungarian political theorist, and his steward, Laci, a Hungarian mathematics student who ran afoul of the Soviets, was one of the most influential people in Aczel's life, who served as an informal tutor and developed his interest in mathematics and numbers.

A Hungarian-French mathematician named George Cœdès (another Hungarian connection) was a language teacher who discovered that he had an uncanny ability to decipher ancient scripts, and spent much of his life in Cambodia. In the 1920s and 1930s, there was a bitter linguistic debate about whether the zero originated in eastern or western Eurasia. Cœdès published a paper in 1931 arguing that the oldest zero represented by a character was on a seventh-century Cambodian inscription. on a stone marked designated as K-127, Unfortunately, it had disappeared, and with the destruction wrought by the Khmer Rouge, possibly destroyed. Aczel made it his mission to find the stone, and declared that he would spend the rest of his life trying to find it if need be. I won't ruin the suspense.

This is where the distinction between representation and concept gets a little murky. Other people's had the concept of zero, without developing a character to represent it, so one might question its tie to philosophy. They often simply left a space to represent it, which probably worked well enough to represent 20 cows, but not 2,000 soldiers. Europe and Indian weren't the only thinkers in Eurasia. An Egyptologist, Alan Gardiner suggested that the nfr hieroglyph, found in the the eighteenth century BC/BCE represented zero, although under his requirements, Aczel might dismiss it as not leading to the zero used in Western nations today. The Cambodian zero in the inscription was a round depression in a stone, part of the number 605. The Indian zero was a circle, and it is easy to see how in writing, rather than chiseling, the two characters could be interchanged.

Aczel presents an eloquent, even moving, description of the importance of the representation, especially for dealing with large numbers. It permits the same ten digits to be used to represent numbers of any size. I felt very fortunate to be one of the heirs of this system.

One of my favorite chapters was Six, in which Aczel is explaining Indian philosophy in which something can simultaneously be one both true and untrue, as opposed to Aristotelian philosophy in which something is or is not. His example is that a cup of coffee with a very small amount of sugar in it could be said to be neither sweet or unsweet. A friend long ago pointed out that Aristotelian logic doesn't really allow for something becoming. It also highlights something that has always frustrated me about the English language: the difficulty of expressing neutrality or indifference. If someone asked me if I like so-and-so, and I say "no" they are likely to assume that I dislike them, and ask what I have against them, unless I explain that I have no strong feelings or use.a double negative: "I don't dislike them," or perhaps shrug my shoulders. Yes and no stand in opposition to each other.… (més)

Summary: The story of the Bourbaki, named after the greatest mathematician who never existed, who led a revolution in the emergence of the “new math,” introducing a new rigor into the field.

When I was in middle school, we were introduced to “the new math.” One of the things I was always curious about was why the first thing we did was learn about sets. I was reminded of this when I read this book, which explained why sets were foundational to the approach.

This is the story of Nicolas Bourbaki, who convened a group of mostly French mathematicians around him, creating a tremendously productive group that in its day revolutionized the practice and teaching of math. Aczel introduces us to the key figures in this group–Andrew Weil (who later solved Fermat’s Last Theorem and brother of philosopher Simone Weil), Laurent Swartz, Henri Cartan, Claude Chevally, Jean Delsarte and Jean Dieudonne. We are also introduced to Alexandre Grothendieck, perhaps the most brilliant and also eccentric of them.

The most striking thing we learn is that the group formed around a mathematical joke upon which Weil built. Nicolas Bourbaki never existed except as a made up identity that reflected the collective effort of this group to rehabilitate and revolutionize mathematics in France that had fallen into the backwaters of German mathematics and science. These mathematicians met regularly and forged a consensus on how math would be practiced and taught in France that resulted in the prolific production of texts, revolutionized not only math education throughout the world, but touched a variety of other disciplines. Their approach was founded on set theory. They emphasized math in the abstract, focusing on mathematical proofs and rigor.

They were trying to articulate the structure of mathematics and this led to interesting interactions with pioneering anthropologist Claude Levi-Strauss, child psychologist Jean Piaget, linguistic theorists, and even writers including Italo Calvino. Aczel traces how structuralism for a time replaced existentialism in philosophy until the turn to the post-modern.

During the war Weil fled to America and stayed there, and gradually, his influence in Bourbaki waned. In the early 1950’s Alexandre Grothendieck joined for a time. His brilliance both stimulated the work of the Bourbaki and led to his departure as he recognized the weakness of set theory as a basis for Bourbaki, trying and failing to convince them of the idea of categories. Grothendieck differed from the Bourbaki, preferring to work alone.

The parting spelled a turning point for both. While Bourbaki continued to have a spreading influence for a time, it was more on the basis of past work. Grothendieck went on to do innovative work for a time, and directing students into significant problems. He held a position at the IHES, a French version of the Institute for Advance Study. Then he became more engaged in political and environmental causes, and when his efforts failed in these areas, he retreated to the Pyrenees, where his whereabouts remained unknown. After this work was published, he died in 2014 in Saint-Girons, Ariège.

The title of this work is a bit of a puzzle. Apart from a chapter on cubism, Braque, and Picasso, and its connections to antecedents to the Bourbaki, this is not a book about artists, unless this is a contrasting reference to Grothendieck and Weil, which was opaque to this reader. I found the organization of the book a bit labyrinthine. Nevertheless, it was an intriguing account of a movement in mathematics I’d never heard of. It was fascinating to see how productive this group was for a period and yet how significant the human factors were in the ultimate fate of Bourbaki.… (més)