How to be Right

An AI is always right, but for humans it's not so easy!

Consider the alternative

Armchair philosophers of mathematics might be surprised, if they bothered to look, at how infrequently mathematicians refer to "truth" in qualifying our work.  To say that a mathematician has proved a "true" theorem is a sign of minimal confidence, like saying that an architect designs structures that don’t fall down.  What may be more surprising is how frequently mathematicians use the word "right," or "correct," in praising a colleague's work.  Being able to prove true theorems, at least in a fully formalized language, is the degree of competence that artificial intelligence may well learn to detect. "Human-level" mathematical reasoning, in contrast, depends on knowing when and how to detect what mathematicians mean by right.

A few years ago Peter Scholze allowed me to share some comments that illustrate just how important it is to get things right. 

What I care most about are definitions. For one thing, humans describe mathematics through language, and, as always, we need sharp words in order to articulate our ideas clearly. (For example, for a long time, I had some idea of the concept of diamonds. But only when I came up with a good name could I really start to think about it, let alone communicate it to others. Finding the name took several months (or even a year?). Then it took another two or three years to finally write down the correct definition (among many close variants). The essential difficulty in writing “Etale cohomology of diamonds” was (by far) not giving the proofs, but finding the definitions.) But even beyond mere language, we perceive mathematical nature through the lenses given by definitions, and it is critical that the definitions put the essential points into focus.

Unfortunately, it is impossible to find the right definitions by pure thought; one needs to detect the correct problems where progress will require the isolation of a new key concept.

I quoted these lines briefly in an earlier essay in order to stress that "writing down definitions is neither a logical nor a rhetorical act: it is a creative act."  Today my purpose is different:  I will argue that, appearances notwithstanding, "right" and "correct," as Scholze uses the words, are objective judgments — adapted, however, to the time, the circumstance, and the expectation.  Even without knowing what Scholze means by diamond it's clear from what he writes that he has some concrete rightness in mind, and this is so whether his "mathematical nature" is preexisting and platonistic or whether it's something he creates in the process of writing his definitions. 

              Rightness as seen by mathematicians, social historians, and robots

A few years earlier I had pointed to Scholze's perfectoid geometry as an example of "the right theory" in my contribution to a book entitled "What Is a Mathematical Concept?"  I also intended right to be understood as an objective judgment, and not just a report on the way mathematicians talk.  The editors allowed me to make this claim but were not entirely convinced.  Another reader felt that such judgments are inevitably "theory-laden," and lurking in the background there's always the social historian's concern that "rightness" is the outcome of a struggle for power and influence, in this case the power of a young (mostly) boys' network.

More recent developments have confirmed that, if anything, my article understated the rightness of Scholze's perfectoid geometry and its subsequent elaborations.  Scholze's own paper, on a problem I had tried and failed to solve, was a landmark application of his new geometry.  His long paper with Fargues, in resolving a big piece of the local Langlands program, reframed the program's main conjecture as a consequence of a new categorical conjecture that could not even be formulated without the definitions he spent years writing down.  And I'm sure Scholze himself didn't anticipate that his innovations would be so widely adopted so quickly, to the point that the methods he initially developed to solve one specific open question in arithmetic geometry have become indispensable for literally hundreds of specialists in the entire area, and a signpost marking a generational barrier, as one of my colleagues gently suggested a few years ago (slightly modified):

tu devrais faire attention, la centaine de jeunes comprend parfaitement les maths qui sont dans ton article et ils voient clairement que tu ne connais pas le sujet (dont le référé). Si j'étais toi je me concentrerais sur des projets sympas qui me plaisent et, tranquillement sans me mettre la pression, j'en profiterais pour me faire plaisir plutôt que de chercher à avoir je ne sais combien de projets en simultané. A partir d'un certain âge et un certain niveau dans la communauté mathématique comme le tien, autant en profiter :)

At this point, in other words, Scholze's perspective has been so widely adopted as to make its rightness a fait accompli.  Social historians are therefore within their rights in asking whether the success of Scholze's approach to p-adic geometry isn't due as much to extraneous considerations (that remain to be determined) as to factors internal to the field.  Scholze is a cool guy, he established his standing early by proving some cool theorems, he was trained at one of the élite research centers, and the rest may just be an effect of herd mentality.  An equally cool and well-connected mathematician might have led the herd no less successfully in a very different direction.

I already argued against that kind of relativism in my "Concept" article, by explaining how Scholze's approach met a set of very specific criteria that successive generations of specialists had set as conditions for a desirable (i.e., right) theory of p-adic geometry, and ticked off a few extra boxes that no one had anticipated.  But the topic is unnecessarily technical for my purposes here.  Instead, I'll try try to address this question by looking at a different and (initially) more intuitively accessible example:  the problem of counting in geometry.

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                               Counting from antiquity to the 19th century

"When I considered what people generally want in calculating," al-Khwārizmī wrote on the first page of his treatise on algebra, "I found that it always is a number."   Thirteen centuries later that's still true to a remarkable extent.  In my field at least, the search for the correct definition has, at least for decades, been motivated by the wish to find systematic ways to count things, or to show that two ways to express a number give the same answer.  This often involves hundreds or thousands of pages of preliminary material, made to order to create the language in which to express an elaborate or even frankly bizarre answer to what is at root a very simple question.  Conventional set-theoretic foundations are stretched or set aside if they interfere with the search. 

I can explain what I mean by going back to the beginning.  Two lines in the plane intersect in one point, unless they are parallel, in which case they don't intersect at all.  Euclid didn't put it quite in this way but two of his Postulates are concerned with just this issue.  Why should parallel lines be an exception?  In Poncelet's projective geometry they are not:  points are added at infinity in just such a way as to guarantee that any pair of lines meets in exactly one point.  Is this the right way to look at intersections of lines?  Poncelet — cited at the beginning of Jeremy Gray's Plato's Ghost as a precursor of "mathematical modernism" — apparently thought so.  Lorraine Daston writes¹ that, in a text from 1822, Poncelet

…introduced the principle of continuity as the new method of demonstration: figures derived from each other by continuous change share each other’s properties. If, for example, figures degenerate by a convergence of vertices (e.g. a hexagon into a pentagon), then any property of the original figure will have a suitably altered analogue in the degenerate figure.

According to Daston, synthetic geometers like Poncelet aimed to "prevent metaphysical difficulties," and indeed Poncelet had given the title Considerations philosophiques et techniques sur le principe de continuite dans les lois géométriques to the 1818-19 notebook in which he worked out his principle of continuity. 

The principle of continuity dictates that, if you move two lines in the plane, the number of intersections is always one.  Bézout's theorem took this principle one step further:  if C and D are curves in the plan of degree m and n, respectively, then they intersect in exactly mn points.  Daston writes that Poncelet "recommended that in the interests of rigor mathematicians practice synthetic methods in which they never lose contact with objects of sense," but Bézout tells you that the circles of radius 1 and 2 around the origin in the plane intersect in 4 points,² while your senses tell you they don't intersect at all.  Even this very simple example requires your senses to be triply enhanced:  by working in the projective plane, by using complex rather than real coordinates, and by counting intersections with multiplicity.   The fundamental theorem of algebra, which asserts that a polynomial of degree n has n roots, is another example of Bézout's theorem, but again the roots may be complex and have to be counted with multiplicity.

One gets the impression that Richard Wesley Hamming was on the mark when he wrote³ that

…mathematics is not simply laying down some arbitrary postulates and then making deductions, it is much more; you start with some of the things you want and you try to find the postulates to support them!

Hamming could have written that this is how human mathematicians proceed when they are looking for the right postulates; Scholze's correct definitions have the postulates built in.  The quotation is taken from the section of Carlo Cellucci's The Making of Mathematics that cites five mathematicians making the same point.  "A naive non-mathematician," to quote Reuben Hersh — a computer scientist who is trying to build a "human-level" robot, for example — might look into Euclid, or into a philosophy book, and conclude that things go the other way around. Mathematicians reading this, on the other hand, can verify Hamming’s observation empirically, based on their own experience. 

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                                                 Counting in the 20th century

History seems to be teaching us that "human-level mathematical reasoning" is as concerned with metaphysics as with logic.  Mathematics in the 20th century extended Poncelet's principle of continuity to the case that both Euclid and Poncelet missed, when the two lines coincide.  To handle this case one needs to rethink the notion of intersection in order to count one intersection when a line meets itself.  Bézout's theorem is proved by algebra; algebraic geometers before the middle of the 20th century used a combination of algebra and geometry to count intersections of projective varieties in higher dimension; and Jean-Pierre Serre’s Tor-formula applied methods of homological algebra to get numbers that satisfy the principle of continuity in great generality.  Mid 20th century geometers introduced a variety of increasingly exotic techniques — Chow's moving lemma, Fulton's deformation to the normal cone, and algebraic K-theory, among others — in the quest for something that deserved to be called "intersection theory" because it was based on the right definitions. 

The principle of continuity is reflected in names of two of the methods just mentioned, but remember that the goal is still to get a number, as it was at the time of al-Khwārizmī.  In the process, however, new goals have emerged that somewhat obscure the origins of the theory in the wish to calculate a number.  Intersection theory is one aspect of the theory of algebraic cycles.  This is a branch of mathematics where the ratio of theorems to conjectures is distressingly small, but the conjectures in this area are deemed so important that two of them were included in 2000 in the list of Clay Millenium Problems.⁴  But there are lessons to be learned even if we limit our attention to problems of counting points.  It can be argued that the successive innovations in the foundations of algebraic geometry were constructions of increasing complexity motivated by just such problems.

The prime example is Weil's proof of the Riemann Hypothesis for algebraic function fields, which can be expressed as a formula for the number of points on a (projective) curve over a finite field.  Weil didn't care so much to count the points exactly (although this has proved useful in subsequent applications, including in cryptography) as to prove optimal estimates for the number of points.  It's important to bear in mind that Weil knew the answer, even knew how to prove it, long before he could write down a proof.  The proof proceeds counting the intersections of two curves in an algebraic surface by means of topology, which is a term encompassing principles of continuity in general.  The problem is that topology, as understood at the time, doesn't apply over finite fields.  In his Foundations of Algebraic Geometry and several other books written around the same time, Weil developed a substitute based purely on algebra. 

Weil's foundations were adequate for the problem at hand, but his conjectures anticipated a much more flexible point-counting formalism that would imitate the known results of topology — specifically the Lefschetz fixed-point formula — but over finite fields.  Serre took a first step with his theory based on sheaves, still in many respects the right theory for algebraic geometry.  But algebraic geometers today are systematically trained in the formalism developed in the 1960s by Grothendieck and his school.  Grothendieck's foundations established their rightness quickly and in numerous ways, none more decisive than providing the means to prove Weil's conjectures.  Here again the statement of the Lefschetz fixed-point formula was known in advance, as was the argument by which it was to be applied to (all but the last of) the Weil conjectures; but even after finding the right definitions it took literally thousands of pages⁵ to put everything together.

              It's the 21st century and we're still looking for the Right way to count

The habit of promoting the right way to do this or that by producing 1000-page treatises has become so ingrained, at least in the branches of mathematics of interest to me, that no one is surprised to see several new ones offered for sale every year in our favorite publishers’ online catalogues. Some of us feel obliged actually to buy the physical books although, as Aristotle points out⁶ “of things constituted by nature some are bodies and magnitudes, some possess body and magnitude, and some are principles of things which possess these,” and furthermore "body alone among magnitudes can be complete," and it's much harder to squeeze a body or a magnitude than a principle into a bookshelf that also possesses body and magnitude. And besides, how many of us have the time to read several 1000-page treatises per year? A colleague who has made very productive use of some of this material (and who has produced several 1000-page treatises of his own) confided that his method is to write what he needs to be right, hoping it's consistent with the formalism in the treatise he wishes to quote, and then confirms his surmise with the author of the treatise in question. This is not so different from what most of us do even when quoting short papers, which are often ambiguous or imprecise or simply hard to read.

I know for a fact that this behavior drives certain colleagues up the wall, and the only way they know to climb back down is to promote formalization as a remedy for reliance on the fragile memories of mortal mathematicians. Silicon Valley promotes the idea that every problem has a technological solution, a message these colleagues are eager to hear. The problem in this case is that the ground on which they hope to build is constantly shifting. Among the 1000-page treatises I have in mind here are Jacob Lurie’s books with the word “Higher” in the title.⁷ These books respond to, and provide satisfying solutions to, longstanding concerns in topology and category theory, but the timeliness of this work, and the (relatively) rapid adoption of the ∞-categorical language outside its natural topological habitat, have as much, if not more, to do with the old concern with finding the right way to count, yet again, as with the author's initial motivations.

This is explained in a remarkably readable 9-year old lecture by Gabriele Vezzosi, one of the pioneers in applying the “Higher” methods to create new foundations, for counting as well as for other purposes, in the form of derived algebraic geometry (DAG). One of the motivations for DAG is cited early in Vezzosi’s slides:

Hidden smoothness philosophy (Kontsevich): singular moduli spaces are truncations of ‘true’ moduli spaces which are smooth (in some sense) ; good intersection theory.

The word “true” is decidedly not the philosopher’s term; here it refers exactly to what we have been calling “right.” And “good intersection theory” exactly means that DAG provides the (provisionally) right way to count. In retrospect, the intersection theory of Fulton’s eponymous book was certainly right for the 20th century but it’s no longer quite right enough for the counting problems that require “hidden smoothness.”

In particular, we now have to believe that the moduli spaces that provided so much satisfaction to so many 20th century geometers were the wrong ones. So we have to ask: was this the case all along? Back in the 20th century, when Grothendieck and Mumford asked the question “What are moduli?” and pondered “the meaning to be given to the word ‘classify’”⁸, were the constructions they offered in answer to these questions no more than miserable truncated substitutes for “true” moduli spaces, in the same way that the images venerated in Catholic and Orthodox churches around the world are merely symbolic representations of the "true cross"?

Euclid’s problem of the intersection of two lines acquires a new meaning in DAG. When the two lines coincide, the intersection is a derived object, which can be seen as the “true” intersection, of which the original line is just a truncation. Those so inclined, in other words, can trace DAG back beyond Serre’s Tor-formula to Euclid. A less ideologically driven historian will note that an early version of derived moduli spaces appeared in a 1994 article of Kontsevich aimed at counting things, specifically “to formulate rigorously and to check predictions in enumerative geometry⁹ of curves following from Mirror Symmetry.” The problems arose in the setting of string theory, and it’s significant that Kontsevich posted his article in the arXiv’s “high energy physics” preprint server. You'll have to ask the high energy physicists why they want to count these particular quantities; for a mathematician the "formulate rigorously" part means committing to a process, stretching over decades and thousands of pages, of substituting "true" objects for truncated objects. The original purpose of DAG was just to count things that were already known in the old-fashioned "truncated" algebraic geometry; but the language of DAG has gradually so completely infiltrated adjacent areas of mathematics, including my own, that it marks yet another generational barrier, invisible except for the discomfort expressed at seminar talks on the faces of those caught on the barrier's wrong side.

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Rightness in mathematics is a generational construct, and generations are how humans measure history. Each generation has the impression that it is finally getting things right. We think the new notions are like improved microscopes that allow us to see what was previously hidden; but they also create a barrier between us and old ways of looking, if only because the latter cease to be taught.

Robots can be taught to teach themselves to win at chess or go, but winning is not the measure of rightness in mathematics. Will the obsession with “true” of a sizeable population of outside meddlers, along with a minority among mathematicians, erect a barrier beyond which mathematics can no longer perceive right?

I will reframe that question four weeks from now as one of the challenges I will be posing to the mathematical AI community.

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1. "The Physicalist Tradition in Early Nineteenth Century French Geometry," Stud. Hist. Phil. Sci., Vol. 17, No. 3, pp. 269- 295, (1986).

2. Exercise: Find them.

3. Mathematics on a distant planet. The American Mathematical Monthly 105: 640–650.

4. The Birch-Swinnerton-Dyer Conjecture and the Hodge Conjecture; and here it's customary to underline the importance by mentioning that a million dollar prize will be awarded for the solution either of these problems.  I haven't checked whether the prize will go to the AI or its programmer if the winner is a robot.  The point of today's essay is to try to suggest warm-up exercises to programmers aiming for one of these prizes.  For example:  starting with Euclid's postulates, the AI is to come up with the notion of intersections at infinity and the principle of continuity.  Unlike AlphaZero, the AI doesn't have to be entirely self-taught, but it's not allowed to read anything that wasn't available when Poncelet got started.

A second warm-up exercise:  starting only with what was known through Hasse's proof of the Riemann Hypothesis for elliptic curves, develop foundations of algebraic geometry adequate for a proof of the Riemann Hypothesis for all curves over finite fields discussed below.  Extra points to be awarded for a spontaneous proof of the Weil Conjectures.

5. Deligne reduced this to a few hundred pages in SGA 4 1/2.  One can hope that someone will eventually do the same for the homotopy-theoretic methods that have been introduced to solve 21st century counting problems.

6. “On the Heavens,” second sentence.

7. But there are also less exotic titles, like this book and its companion volume by Mœglin and Waldspurger.

8. David Mumford, Geometric Invariant Theory, Chapter 5, § 1.

9. Elsewhere in this paper Kontsevich calls them “counting problems.”