Review of "Mathematica" by David Bessis

The elephant in the room is human understanding

From David Bessis, Mathematica. Hint: rotate 90°counterclockwise.

While writing Mathematics without Apologies I occasionally felt a sudden urge to type the word elephant. Not, I must add, just to leave it marauding incongruously in the middle of a sentence — like, well, the proverbial elephant in the room — but to surround it with something like a natural habitat. And it was with a distinct sense of relief that I finally did insert an elephant in the narrative (in Chapter 7; and then a second one for good measure in the endnotes to Chapter 8).

Then I began to spot elephants in all the books I read about mathematics for the general public. When I started counting I realized that, whatever the reason for my compulsion, it was one I shared with the authors of most of the books I read. You can find a list of 12 such authors in a blogpost written March 24, 2015. Since I have been thinking recently about the recently departed Yuri Ivanovich Manin, whose influence on the mathematics of my lifetime was and remains both massive and multifarious — though not otherwise like an elephant — I decided to test my theory by looking at his book Mathematics as Metaphor. Curiously, the word слон, Russian for elephant, appears only in the Russian version, in a chapter that was omitted from the later English translation:

Фасады развалин отделены от редких туристов решетками, как слоны в зоопарке.

which means

“The facades of the ruins are kept separate by bars from the rare tourists, like elephants in the zoo.”

The ruins in question are on the Palatine Hill, quite some distance from the Vatican, where Manin wrote these lines — in a chapter entitled “Vatican, Autumn 1996” — during the visit in which he was inducted in the Pontifical Academy of Sciences. Neither the elephant nor the Palatine is mentioned again. The only sensible explanation for the appearance of this line in the middle of Manin’s notes on his stay is that he succumbed while in Rome to the pachydermic compulsion I noted above¹. But the allusion does serve as a parable for the theme of Manin’s brief chapter; and since this theme is relevant to today’s topic I will return to it at the appropriate time.

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Photo taken in Paris Metro June 3, 2022

                                     A book of awakening and emancipation

David Bessis’ book Mathematica stands out, even compared to the books in my March 2015 list, for the profusion of its elephants. If elephants were likes™ then Bessis’s book would be by far the most popular of all. As you can see in the photo above, the book was the subject of a comprehensive advertising campaign — there were three identical posters along that same bannister — and was reviewed extensively in the press; and it was and remains popular² among its target audience. It is nevertheless an excellent book, thoroughly original, and although this "book that changes our way of seeing the world" (to quote the ad) was only published a year ago it is already overdue for an English translation. More to the point, it is highly appropriate for this newsletter, as a celebration of the kind of mathematics that no one working in AI is trying to emulate.

The book’s leitmotif, the contention to which Bessis returns by many different routes, is that, whether you realize it or not, you, the reader, are already an extraordinarily accomplished mathematician. Bessis calls it “a book of awakening and emancipation”³ from the misconception that those who are not born with a special talent should stay away from mathematics. On the contrary, any reader can appreciate “the true pleasure of mathematics, the one you feel the day when suddenly you realize that you can see the stars in your head, though you had never seen them before.”⁴

Understanding mathematics is seeing and feeling, it’s traveling along a secret path that brings us back to the mental plasticity we had as children.⁵

It’s especially gratifying to find such sentences in a book written by a mathematician who can’t be accused of hostility to computing or AI. Bessis left mathematical research some years ago to found a startup to apply deep learning algorithms to marketing.⁶ "AI is changing the world and we’re proud to be pioneers," we read on Bessis's company's site; but also "AI is built by humans for humans." And this old-fashioned humanism is crucial to what Bessis has to say about mathematics.

When a human being is faced with a mathematical text, the aim is not to read from the first line to the last, as a robot would. The aim is to grasp “the thoughts between the lines” … Mathematical texts are written by humans, for humans. Without our ability to give them meaning, without “the thoughts between the lines,” there would be no mathematical texts.⁷

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                               Mathematics is a sensual, carnal experience

This is the attitude Bessis attributes to Bill Thurston “and all mathematicians.” He continues in a vein likely to puzzle much of the Lean community and anyone who adheres to the Central Dogma:

For Thurston and for all mathematicians, mathematics is a sensual, carnal experience situated upstream from language. Logical formalism is at the heart of the apparatus that makes this experience possible. Mathematics books are unreadable but we need them. They are a tool that allows us to share in writing the true mathematics, the only one that really counts: the secret mathematics, the one that is in our head.⁸

Surveillance capitalism, as you have probably read, wants to peek at everything that is in our head; but I don’t think that’s why DeepMind and MetaAI are investing in mathematics. I have speculated more than once on their real motivations; for now I just wish to express my profound conviction that they do not envisage the mathematics that survives Silicon Valley’s disruption as a secret, sensual, carnal experience.

More than publications or official works, the mathematician’s great creation is the intuition that is the accomplishment of an entire life.⁹

Angel investors want to know what is the use of that great creation that is irritatingly inalienable from its possessor. Can the accomplishment of an entire life be auctioned on the market of Non Fungible Tokens?

Thurston is a model for Bessis, as he has been for Silicon Reckoner, and for those fortunate enough to have worked with him, like Benson Farb:

Bill changed our idea of what it means to “encounter” and “interact with” a
mathematical object. The phrase “I understand X” has taken a whole new meaning. Mathematical symbols and even pictures are not sufficient for true understanding, especially in geometry and topology. We must strive to live somehow inside the objects we study…¹⁰

“Mathematics loses a dimension with his death,” Farb added, and this is true, but Bessis, who wants to stress how mathematics is accessible to every reader, sees lessons in Thurston’s attitude that can benefit anyone.

To understand something “morally” means to be able to give oneself an intuitive explanation of, and to name, the reason something is true, the “moral” we need to retain.… No moral can be derived from logical reasoning.¹¹

Bessis also takes inspiration from Descartes, whose Discourse on Method he reads as an outline of a “System 3” that escapes the sterile opposition between the “System 1” (immediate intuition) and “System 2” (mechanical reasoning) of Daniel Kahneman’s Thinking Fast and Slow.¹² Descartes' earlier Rules for the Direction of the Mind serves to justify Bessis’s claim that

Mathematics is a spiritual awakening. It teaches us to recognize the physical sensation that must guide us on the path of knowledge. Until we have personally encountered that crystalline and translucide form of truth, it is impossible for us to understand the meaning of the words “clear” and “distinct” [as used by Descartes, M. H.]… and, according to [Descartes], it is impossible to embark upon a genuine approach to knowledge.

Grothendieck, “a great yogi who invented his own meditation technique,”¹³ provides Bessis with a third model of thinking mathematically. In his unclassifiable 1000-page manuscript Récoltes et sémailles, just published in 2021, Grothendieck refers to this model as “listening to the voice of things” [être à l’écoute de la voix des choses], the attitude of the “child” who has the “privilege” of “discovery” and who is “not yet afraid to be wrong”:

Fearing error and fearing truth are one and the same thing. The one who is afraid to be wrong will be incapable of discovery.¹⁴

Or, as Mephistopheles told the Homunculus (Faust II, 7847)

Wenn du nicht irrst, kommst du nicht zu Verstand.

Even the Devil knows, and wants all creatures to know, that Verstand (understanding) is more precious than Vernunft (reasoning). Nevertheless, it’s fair to ask whether Mephistopheles and Grothendieck aren’t precisely describing the principles of reinforcement learning, and whether Bessis’s own professional commitment to deep learning doesn’t ultimately put him on the side of the robots. We’ll return to this question in a moment, but first I need to address the possible charge that this review relies excessively on quotations from a handful of charismatic figures.

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                                         The mind of the mathematical mean

This week’s goal is not to construct a coherent argument in support of a precise thesis, the one expressed in the numerous quotations from Mathematica and from the mathematicians who inspired Bessis. It is rather to use these quotations to illustrate the mindset of what I would never call the “typical” mathematician, but which is typical nonetheless. The inherent danger in any attempt to account for mathematics by means of sociology or history — or for that matter journalism — is that you can only quote those who have already spoken or who have been willing to speak. So to refer to “what mathematicians believe” is to promote the fallacy that there is such a “what.” Philosophers have no right to point to a few texts and claim “The norm is this” — but neither do mathematicians. We quote Thurston and Grothendieck because we have their words. Thurston, in particular, gave the most sustained articulation of the centrality of understanding in the practice and lives of mathematicians, in one of the most-quoted articles on the values of working mathematicians, which incidentally explains as well as anyone why the Central Dogma is what he calls “a polite fiction.”¹⁵

The standard of correctness and completeness necessary to get a computer program to work at all is a couple of orders of magnitude higher than the mathematical community’s standard of valid proofs. [However, o]ur system is quite good at producing reliable theorems…. It’s just that the reliability does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas.

Still, this is just one man’s opinion, and there is no scientific means to determine whether or not Thurston’s values are really widely shared in the profession. The fact that this text continues to be quoted by mathematicians, however, and that articles expressing a contrary opinion are quoted (if at all) by philosophers (and not even by the philosophers I know), is a good reason to admit that Thurston’s article was a legitimate expression of the ethos of the mathematical community when he wrote it and that it remains so today.

Bessis quotes Thurston because he helps him put his own values into words, and also because the mathematics for which Thurston is best known is, or is at least said to be, based on a geometric intuition that exceeds what the rest of us imagine to be possible. We have seen that understanding as a “sensual and carnal experience” is not on anybody’s drawing board in Silicon Valley. It is the aim of Mathematica to convince the reader that the sensual and carnal experience of mathematics is already an “adventure,” as his subtitle proclaims, “to the heart of ourselves.”

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                                                       Le concept d’éléphantitude

That’s the real title of a section of Chapter 19, 3 pages after the elephant sketch. And Chapter 18 is entitled “The elephant in the room,” by which Bessis means that everyone wants to ignore the implications of the “problem with rationality,” epitomized by

…the contrast between our incapacity to give a 100% rigorous definition of the concept of elephant and the obviousness with which this concept presents itself to our intuition.¹⁶

Both chapters should be read with caution. On a superficial reading their message appears to be that the concept of “elephantness,” like the “concept of a cat” that Google’s computers invented in 2012, “emerges”

[t]hanks to the neural networks of deep learning… The network recognizes the elephant although it doesn’t have its definition,

as Bessis explained in a Le Monde interview¹⁷ in January 2022. But Bessis recognizes in the same interview that deep learning provides, contra BAyA, an "operant metaphor" for concept formation, a "pleasure… that should not be scorned":

To understand the nature of our intelligence and the mechanism of our thinking, deep learning algorithms furnish the best metaphor I know.¹⁸

“Best,” that is, for Bessis’s purpose, which is “awakening and emancipation” of human understanding:

What’s really at stake in mathematics is human understanding… mathematics… extend our intuitive understanding of the world around us… The mathematics that you understand augment reality and add a magic layer of intelligibility. They make you hyperlucid.¹⁹

That was the 13th time you read the word understand or understanding in this essay. Though this is perfectly consistent with the point I made in this earlier post, that mathematicians aim at understanding where software engineers are content to emulate reasoning in search of an ersatz understanding™, I was hardly cherry-picking quotations from Mathematica. Practically every page contains the word understand, or intuition, or imagination. Reasoning is not absent but it is clearly not the goal. Bessis calls mathematics “the science of imagination”; he records his astonishment at realizing "upon maturing" that he could use mathematics to develop his intuition, rather than the other way around.²⁰

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                                                  Truth cannot contradict truth

I have only shared half of the striking quotations I underlined in Bessis’s book, but Substack warns that I’m “near email length limit.” I can only recommend that you read it yourselves, because I want to return briefly to Manin’s report on his visit to the Vatican.

Manin’s short text is mainly a reflection on the Catholic Church position on scientific rationality. “Truth cannot contradict truth,” John Paul II — Manin’s host in 2012 — had written in 1996; the first “truth” is that of the theory of evolution and the second that of Scripture. Manin is bemused by the Pope’s apparent endorsement of an epistemological dualism, which admits evolution but places the “spiritual soul” beyond “any kind of rational discourse.”²¹ The elephant behind bars may then be a parable for the church, the tourists the scientists in the Pontifical Academy. But in a book entitled "Mathematics as Metaphor" perhaps it's the other way round?

1. It’s not strange at all. In his position I would have been thinking about Hannibal; wouldn’t you?

2. Amazon.fr, where Mathematica has 4.5 stars, calls it “#1 Meilleure vente dans Encyclopédies et dictionnaires des mathématiques” which is not at all a category to which the book belongs.

3. David Bessis, Mathematica, Editions du Seuil (2022), p. 331.

4. Ibid., p. 111-2.

5. Bessis, quotation from book jacket.

6. From its website: “Tinyclues was founded in 2010 on a mathematical hunch: B2C marketing databases contain sufficient amounts of implicit information to transform the way marketers interact with their customers, and a new class of algorithms, based on Deep Learning, holds the power to unlock this potential.”

7. Bessis, p. 72.

8. Bessis, p. 76.

9. Bessis, p. 17

10. Benson Farb, “On Being Thurstonized,” Notices of the AMS, Jan. 2016.

11. Bessis, p. 73.

12. Bessis, Chapter 14, entitled “A martial art.”

13. Bessis, p. 88.

14. A. Grothendieck, Récoltes et Sémailles, Gallimard (2021), 5.1, 5.2.

15. To be completely precise, Thurston uses the expression “polite fiction” in the next paragraph, referring to issues with axiomatic foundations. He also predicts that interactive proof checkers “will become part of the standard mathematician’s working environment” but repeats that “the humanly understandable and humanly checkable proofs that we actually do are what is most important to us.”

16. Bessis, p. 283. Elephants make appearances in other chapters as well, notably in chapter 12.

17. D. Larousserie, “David Bessis, mathématicien adepte du yoga mental,” Le Monde, January 22, 2023.

18. Bessis, p. 304.

19. Bessis, pp. 316-7.

20. Bessis, p. 187, p. 315.

21. …если дух не является эмерджентным свойством материи, то невозможно не только его экспериментальное изучение, но и никакой рациональный дискурс о нем.