At the ICM 2006 in Madrid I attended a lecture by Manin speaking about the different uses of mathematics, as models, theories, and metaphors. Of all the lectures I attended at that congress, this was the one that stuck out to me. It was obviously not a technical talk, but a philosophical one in the best sense of the term, namely, fuelled not by professional pedantry, but by a deep personal curiosity expressed in a very original and captivating way. A year later, a collection of Manin’s essays had been translated into English and handsomely published by the AMS under the title of ‘Mathematics as Metaphor.’ I got the book, read it with delight, as I had read previous books by him as a young man, and in fact I wrote a review of it which was published in 2010 in the EMS Newsletter – incidentally, a fact I had already forgotten this spring. However, I was alerted to it and learned that I had at its end expressed my regret that not more of his essays were available to readers not knowing Russian. Now my wish has been granted. That a wider collection had recently been published, I actually found out from Manin himself in what would turn out to be my last communication with him. I immediately got the book, published by the small French firm, Les Belles Letters, and thus containing French translations of his texts. The AMS version was about 200 pages, while this edition runs well over 500 pages, so one surmises that it is a very significant extension. On the other hand, a mere page count is a bit misleading because the pages of the first edition are larger than those in the latter one and also the font size employed is somewhat smaller. I estimate that the American edition sports about 3500 characters a page, and the French edition about 2000 characters, but still we are talking about a significant extension. My first intention was to single out what was new in the extended edition and concentrate on that, but I have decided to abandon that and treat it as a whole, fully independent of my first review.

We are talking about essays, not scientific articles, and there is of course a significant difference between the two. An essay is, like the terminology indicates, an attempt. Namely, an attempt to come to grips with a subject in a non-technical way using a meta-perspective. You should not write a scientific article if you are not an expert, but anyone is welcome to write an essay on any subject that occurs to them (they need not be published). In fact, any such attempt reminds me of the American diplomat George Kennan, who during his career wrote many dispatches from his various postings with scant hope that they would ever be read, but justifying his activity by claiming that he wrote in order to discover what he thought. This points to a crucial aspect of essay writing, namely, exploration. Karl Popper did not, unlike his colleagues in the Vienna Circle (disparaged by posterity as positivists) reject metaphysics, instead he was thinking of it as proto-science, potentially developing into one.

As indicated, most essays in general may be ignored (which does not necessarily mean that writing them is a useless activity); what makes Manin’s essays worth pondering is the originality of his mind and imagination, the precision of his formulations, all supported by his wide culture, and the boundless curiosity which made this culture possible. Essays should be classified as literature, and thus subjected to the demanding criteria such writing invites. Imagination requires obstacles to be circumvented in order to be properly stimulated; this is why, according to Hilbert, mathematics requires more imagination than poetry, or, as claimed by the biographer Peter Acroyd, the writing of a biography requires more imagination than the writing of a novel. But in this general frame there are different kinds of imaginations, the iron-clad laws of logic typically lead to frustration, while writing essays and fiction leaves you more liberty. Arguments need not to be watertight as long as they are exciting, and inconvenient facts can be ignored or simply made up, as typically in fiction; what matters are the ideas, which need not be technically developed. Thus, I cannot resist speculating that the writing of essays (and poetry?) gave Manin a relief from the rigors of mathematical work, but this does not necessarily mean that it should be thought of as a mere diversion – on the contrary, it was an essential component of his mathematical work, without which the latter may not have been possible. His essays are also more accessible to readers, provided they have the required temperament, than his purely mathematical work, although the charm of the latter derives much from being presented in an essayistic spirit (this is why the above-mentioned books made such an impression on my young mind).

The point of an essay is not only to profit the writer but also to inform and inspire the reader, this is why it is very hard for me not to elaborate on Manin’s essays, and to just present sober resumes; but then again, they are published and available for everyone to read and engage with in their own ways, so I hope that my taking of liberties can be excused as a kind of homage.

First, what is the nature of mathematics? This is a question that cannot be treated mathematically, but nevertheless must at least to some degree engage every serious mathematician, and even influence the way and why they persist in their obsessions. Manin himself is puzzled why mathematics engages him so much, yet without this potential skepticism in any way dampening his enthusiasm for the subject. Now there is a vulgar idea of mathematics, prevalent not so much among the general public as among philosophers and physicists and other concerned academics. Mathematics is, according to this view, seen as a game; you set up some axioms as rules and then apply logic to it and grind away. From this it does not take much to conclude that mathematics is just a matter of symbolic manipulation, and although its concepts do not have any real meaning (like vertices in a graph), it can still amazingly serve as a useful language and even tool in the study of the real world. The idea that mathematics is applied logic goes at least as far back as Frege and was further developed by his successors Russell and Wittgenstein. On the other hand, the American philosopher Charles Peirce claimed that the integers were more basic than logic, and that mathematicians had no need to study logic, they were anyway able to instinctively draw the necessary conclusions, on which mathematics rests and develops. The emphasis on logic has led to the dictum that mathematics is but a sequence of tautologies, which has been taken to heart by many. Any idea that has spread successfully must have some truth to it, so it is admittedly true that a large part of a mathematician’s everyday work may amount to a ceaseless manipulation of symbols. Manin cites, not without approval, the claim by Schopenhauer that when computation begins, thought ends. Mathematics is indeed a very special activity which delights in reasoning using long deductive chains and thereby coming up with true facts in a systematic way. We recall Leibniz’s exhortation, stop arguing let us calculate, hoping there would be a verbal calculus which would resolve human problems as neatly as celestial ones (for which calculus was once invented). Manin insists that the logical straight-jacket that mathematics is forced into is necessary – without it, it would degenerate, as anything to remain solid has to be contained. It is the possibility of falsification, that allows things to grow purposefully by pruning off false leads. It is also this that leads to the frustrations of mathematicians, by the presence of what which cannot be willed away. But for the serious mathematicians there is also something else to mathematics without which they would never pursue it. Mathematics involves more than a random walk in a logical configuration space. It requires thinking in a natural language, a thinking that is not in the nature of a computation in some generalized sense, but is meta-thinking whose mission is not to produce new facts, but to distinguish between the interesting and the fruitful, of coming up with new ideas and strategies. Without this meta-thinking mathematics would be a sterile subject indeed. In fact, what the serious mathematician aims for is the elusive goal of understanding, of seeing different pieces coming together, something which cannot be conveyed by mere mathematical formulations, just as little as ideas can be precisely formalized and expressed, at best only conveyed obliquely, and in this elusive vagueness lies their power. One important difference between a natural language and a formal artificial one is that the latter is precise, while the former is vague; as a result, the latter can be treated as a mathematical object. Being vague, natural languages have a recourse to forming metaphors, which, I never tire of pointing out, should never be taken literally, as they then become merely silly; while metaphors in formal languages have no choice but be taken literally. In a natural language nothing stops you from imagining the set of all sets (or the wish to have all ones wishes granted), but in a formal, strict logical setting one is forced to make explicit the different notions of ‘set’ involved and be forced to adopt a new word for one of them, such as ‘class.’ The Russell paradox does not affect natural languages, as they thrive on contradictions – in fact, languages evolved socially, meaning in particular that expressing truth is not necessarily the main purpose, rather deception; which incidentally ties up with Manin’s fascination with the ‘Trickster.’ Thus, the metaphorical idea of the diagonal argument when applied ‘literally’ (in the sense of rigidly logical) has interesting consequences. At the heart of Gödel’s argument, as Manin points out, is this partial embedding of the meta-language into a formal one on which it comments. Incidentally, there is much hype connected with Gödel’s theorem and Manin’s excellent presentation of it has as a purpose to demystify it. As he notes, the theorem has had marginal influence on mathematics as practiced.

What is mathematical intuition? Mathematical and logical concepts are anchored in a physical and hence tangible reality in the human mind. Numbers are in particular associated with the counting of physical objects, such as buttons and shells. One may talk about small numbers such as billions and trillions when they can so be concretely represented; but with the advent of the positional system of representing numbers one was able and hence seduced to write down huge numbers with millions of digits, numbers that in no way can be represented by the counting of physical objects of any kind, only of imagined objects of the mind, such as all possible books in Borges’ celebrated story. Let us call such numbers, numbers of the second kind, which for all practical matters can serve as (countable) infinities. Then of course there are numbers of the third kind, represented by those which need a number of second kind to count their digits, and we can proceed inductively, and the whole thing carries an uncanny analogue of Cantor’s hierarchy of infinities, except there are of course no precise boundaries between them, but the idea remains (one could of course impose precise demarcations, but that would be artificial and pointless). We are in the realm of natural language after all, where precision is not required. Of course, they are all finite, but even finite numbers can be unbelievably large and induce a sense of vertigo our usual congress with infinity does not involve. What is easier than suggesting infinity by a sequence of dots $1,2,3,…$ (you get the idea), but to really feel it, your imagination must be suitably stimulated by tangible intuition.

It is tempting to insert a slight digression here, touching upon Manin’s interest in Kolmogorov complexity. It is trivial to write down numbers of any kind by using specialized notation (or more generalized inductively-defined functions), but the generic number of, say, the third kind cannot be physically represented in, say, decimal form, which is the type of form that in general is the most efficient. So in what sense can we get our hands on them? How many ‘7’s are there in the decimal representation of a number of type $7_{7_{⋰}}$? Can any solution to this problem be feasibly described in any other sense than by the question itself? Maybe an interesting example of a totally uninteresting question.

Metaphors are important for human thinking, and Manin brings up the notion of the Turing Machine and the influence it had on logic. Yes, machines are tangible objects of the imagination and embody themselves logic in a palpable way – after all, their parts are connected in long chains of causes and effects, like the deductive chains in logical reasoning. Classically, they were represented by the sophisticated machinery of a clockwork; nowadays, we have the computer, although its machinery is not so much exhibited in its hardware, of which most users are blissfully ignorant, but in its software when the old tinkering with cogwheels has been replaced by letting the fingers dance on the keyboard instead, through writing computer codes. As David Mumford has pointed out, a mathematical proof and a computer program have much in common. Indeed, the word ‘mechanical’ is what we use in describing a mindless manipulation of objects subjected to inexorable laws outside our control.

Set theory was created by Cantor by taking infinities very literally as objects to be mathematically handled (but one may argue that infinite convergent sums actually involve a literal, not only potential infinity, and go back to antiquity – just think of Zeno). The uncountability of the reals is something most of us encounter in our teens, and it is usually considered as something rather metaphysical, apart from mainstream mathematics. However, without the negative aspect of the uncountability of the reals modern measure theory with its countable additivity would be impossible. For it to work, the setting has to be uncountable, and that uncountability could indeed be seen as the metaphysical setting of all those manipulations. It stands to reason that such a theory would have been developed sooner or later and then the uncountability of the reals would have been staring in our faces. Cantor’s hierarchy of infinities met a lot of resistance when it appeared, Manin reminds us, and also a lot of skepticism as it was developed. As it is based on human mathematical intuition involving the manipulation of physical objects, which has no longer any relevance, that ordinary expectations would come to grief is not surprising. What could be more natural than picking one object each from a collection of non-empty sets, but the Axiom of Choice has very counterintuitive consequences when applied in, say, an uncountable context, giving rise to the Banach–Tarski paradox, or the well-ordering of the reals. The fate of the continuum hypothesis is a case in point, the physical intuition was that here it was, a subset of the real line just in front of our eyes, it had to be true or not. But it turned out to be a question of mere convention, what rules are allowed or not in forming subsets. Thus, it degenerated to a formal game having no relation whatsoever to our conception of physical reality. The very notion of mathematical Platonism seemed to founder when exploring the transfinite world, where we seem at liberty to bend the rules at our discretion. ‘What did the paradoxes and problems of set theory have to do with the solidity of a bridge?’ – Ulam rhetorically asked, as reported by Rota. Our sense of the solidity of mathematics seems to be connected to tangible models, such as physical space to classical Euclidean geometry. The real line has for us an almost physical existence. But when it comes to models for set theory, the very notion of a set as a mental construct seems inextricable from a verbal description; but there is only a countable infinitude of such, and hence the existence of countable models even for uncountable sets (where there are two notions of cardinality, one extrinsic, and one intrinsic). Naively we think of all subsets existing of, say, the reals, but from a strict logical and formal point of view, only those which in principle can be described. This threatens, as noted, to indeed reduce mathematics to a game whose objects mean nothing (just like the chess pieces on a board). On the other hand, a piece of mathematics considered as a game has nevertheless some content as a game, and we can ask questions about it, such as its consistency, which we feel is a definite yes or no question, not contingent upon some axioms we introduce in the meta-game of investigation. According to Manin, it is as if we feel that the game itself, defined by its axiomatic rules, is a physical object, and systematically drawing all the conclusions is a physical activity anchored in the real world, no matter how unfeasible in practice; just as concluding that a Diophantine equation must have a solution or not by making an almost physical thought experiment of an infinite search. Manin’s attitude to set theory is pragmatic, as that of most mathematicians. He does not seem engaged in the classical controversies and refers to intuitionists and constructivists as somewhat neurotic. Set theory for Manin, like for most mathematicians, provides a convenient language of mathematics, as famously exemplified by Bourbaki. On a more existential level, Manin’s attitude to mathematical Platonism is ambivalent; he has described it as psychologically inescapable and intellectually indefensible. What is really meant by that can only be speculated upon. He stresses that his physically tangible intuition, especially when confirmed by mathematical applications to physics as a scientific discipline, makes him inclined to Platonism, an attitude made even more inescapable from his own experience as a mathematician, in particular when studying number theory; but as strong as those convictions may be, they are ultimately based on subjective experience. Of course intellectually Platonism is not amenable to any formal proof, as little as proofs of the existence of God pursued by the scholastics (the concerns of whom seem uncannily similar to those of set-theorists). But as Pascal famously noted ‘Le cœur a ses raisons que la raison ne connaît point.’

I would like to conclude this mathematical section with a nice toy example of Manin. Consider a finite set $X$ of $m$ elements. The power set $P(X)$ is naturally an $m$-dimensional vector space over the field $Z_{2}$ with $∅$ corresponding to the $0$. Its algebra of functions is given by the Boolean polynomials $Z_{2}[x_{1},x_{2},…,x_{m}](x_{1}+x_{1},x_{2}+x_{2},…,x_{m}+x_{m})$, thus any such polynomials can be written as a sum of monomials which are naturally identified by the elements (vectors) $x∈P(X)$, where, say, $(1,1,0,1)$ is identified with $x_{1}x_{2}x_{4}$ and $0$ with the trivial (constant) monomial $1$. Thus, given $x$ we have $x(y)=1$ iff $x⊂y$. The polynomials ($P$) are thus tautologically paired with the subsets $S$ of $P(X)$ by $P=∑_{x∈S}x$. But there is also another way of associating a subset to a polynomial, namely, to associate its zeroes. The fact is that every subset is given by the zeroes of a unique polynomial, so in particular $1$ is the only polynomial with an empty set of zeroes. To see this, we have to introduce $I=X$ (and note that $x+I$ is the complement of $x$ and the zeroes of $1+P$ make up the complement of the zeroes of the polynomial of $P$). Consider now the polynomial

We have $x(u)(x+I)(u+I)=1$ iff $u=x$; thus, for any set $S$ the polynomial

vanishes exactly on $S$ (if $S=∅$, then of course the polynomial is $1$). What is the point of this formal almost tautological game? Manin brings it up as a finite version of the Axiom of Choice: given a set of polynomials how do we pick an element in each of the sets they define, or show that the polynomial is $1$? Given the polynomial in canonical form (or any random form), this is not so easy in general: do we have to check all the elements of the vector space? This also leads to a particular instance the P/NP problem, an instance which, according to Manin, is intractable at the time.

Now I have not touched upon the section of mathematics and physics, which is greatly expanded, nor upon the essays on general topics from linguistics, Jungian psychology (of which Manin was charmed with many references in his works), art and poetry. Had I done so, the review would have been far too long, not only too long as it already is. Having thus failed to do full justice to the book, I hope that I have at least inspired a few readers to consult the master himself.

Yuri Manin, *Les Mathématiques comme Métaphore. Essais choisis*. Les Belles Lettres, 2021, 600 pages, Softcover ISBN 978-2-251-45172-5

## Cite this article

Ulf Persson, Book review: “Les Mathématiques comme Métaphore: Essais choisis” by Yuri Manin. Eur. Math. Soc. Mag. 129 (2023), pp. 58–61

DOI 10.4171/MAG/162