Reverse-engineering Poincaré, two years later

An inconclusive second attempt to take up Kantor's challenge

In November and December 2021 I devoted a series of three posts, under the title “What is human-level mathematical reasoning,” to the challenge posed earlier that year by my former Paris colleague Jean-Michel Kantor:

Can a computer come up with the idea of the Poincaré conjecture?

The third of these posts was devoted to “Reverse engineering Poincaré.” Last July I received a message from Henry Wilton, a geometric group theorist, which gently pointed out that

A couple of mathematical errors have crept into your analysis which, it seems to me, make some of your inferences invalid.

One of them concerns my reliance on Bartocci’s claim that Poincaré’s homology sphere is “the only possible example” that contradicts the initial form of his conjecture. Wilton explains that this is inaccurate, at least in the sense in which I interpreted Bartocci.

More serious are Wilton’s critique of my attempt to respond to Kantor’s challenge in a way that appears to support Kantor’s skepticism regarding the possibility that Klara 3.0 might spontaneously formulate the Poincaré Conjecture.

Most of this is quoted from the introduction I added to the December 2021 post immediately upon receiving Wilton’s message. My primary objective in this post is to propose a different response to Kantor’s challenge while taking Wilton’s objections into account. A secondary objective is to respond to a challenge posed by AG, also known as Alexander Gamburd. While Kantor argues that machines will never match the kind of creative imagination exemplified by Poincaré’s formulation of his eponymous Conjecture, AG’s challenge is to find reasons to believe that machines will not inevitably replace mathematicians. Here is a quotation from one of AG’s comments on this website:

If it would take for aleph-0 a decade or two (and this strikes me as a conservative estimate) to "work through all of the exercises in the Springer-Verlag Graduate Texts in Mathematics series", I personally doubt it would take much longer subsequently for aleph-4 to be able to submit a competitive paper to, say, Inventiones Mathematicae. 

AG is referring to the hypothetical (א(0 (my system sees a Hebrew character and starts writing right to left) from Venkatesh’s essay to appear in the forthcoming special issue of the Bulletin of the American Mathematical Society, which I quote again:

Our starting point is to imagine that א(0) teaches itself high school and college mathematics and works its way through all of the exercises in the Springer-Verlag Graduate Texts in Mathematics series. The next morning, it is let loose upon the world—mathematicians download its children and run them with our own computing resources. What happens next—in the subsequent decade, say?

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                            Setting aside the computational complexity argument

My speculation stopped with aleph-3, an Artificial Topologist named Klara 3.0. (I have since learned that I misgendered the Klaras, each of whom uses they/them pronouns.) My response to Kantor’s challenge had two aspects. On the one hand, an attempt to reverse engineer Poincaré appeared to run up against practical limits arising both in logic and in computational complexity, and I rashly opined that

the argument from computational complexity [shows convincingly] that deep learning, no matter how deep, will never match the unfathomable insight of a Poincaré.

On the other hand, I asserted that this opinion is beside the point, because, to quote Thurston,

…the important things in mathematics … are somehow alive; they live in people’s brains, they live in communities of people talking to each other.

Wilton’s message, I think, invalidated the argument from complexity, at least in my formulation. I am no longer quite so convinced by what I wrote about such arguments at the time, and I am left to try to turn Thurston’s view into a meaningful alternative as well as a response to AG.

With Wilton’s permission, I am quoting his message to me verbatim:

I hope you don’t mind if I share some comments on the mathematics that arises in your December 2021 posts about reverse-engineering the Poincaré conjecture.  A couple of mathematical errors have crept into your analysis which, it seems to me, make some of your inferences invalid.

First, Bartocci, whom you quote in your 14 December post, is wrong when he writes:

“… we know now that he [Poincaré] stumbled upon the only possible example [the Poincaré homology sphere].”

In fact, there are infinitely many 3-dimensional homology spheres: Max Dehn constructed an explicit infinite family in 1910, only six years after Poincaré formulated his conjecture. Bartocci is perhaps thinking of the fact that the Poincaré homology sphere is the only spherical example. But any homology sphere would be enough to refute Poincaré’s first version of his conjecture, and Dehn’s work shows that the mathematical tools of the time were quite capable of constructing further examples.

As I already wrote in July, after receiving Wilton’s message, my reliance on Bartocci as a source for the history of Poincaré’s conjecture was simply wrong. In itself, this factual error is only marginally connected to Kantor’s challenge; but Wilton goes on to argue that the statistical ease of constructing homology spheres invalidates my attempt to draw theoretical conclusions about the likelihood of a machine coming up with the Poincaré Conjecture:

Not only are homology spheres easy to construct, they are extremely common in the Hodgson—Weeks census of low-complexity examples (see Dunfield & Thurston, 2003), and also arise in certain random constructions with high probability (see Dunfield & Thurston, 2006). So your inference (in your 28 December post) that a computer would be unlikely to ”formulate the Poincaré Conjecture purely by analyzing data” appears to rest on a fallacy.

I return to this important observation below, in connection with Kantor’s challenge. Wilton continues with a (very diplomatic) refutation of my attempt at a more sophisticated statistical argument:

A few more points of your analysis in the same post also appear to be questionable. You cite the unsolvability of the word problem for finitely presented groups as evidence that it is

“strictly impossible for any algorithm based on conventional logic to know whether or not a randomly generated simplicial manifold is in fact simply connected.” (*) 

This deduction isn’t valid, because worst-case lower bounds (like the unsolvability of the word problem) are beside the point for naturally occurring, low-complexity examples. There are plenty of simple and reliable computational techniques — “rigorous heuristics”, if you like — that work most of the time, and that are sufficient to decide simply connectedness for large families of low-complexity or randomly constructed examples.

To illustrate the point, let me quickly describe these rigorous heuristics. To check that a simplicial 3-manifold is simply connected, write down a presentation for the fundamental group (using the Seifert—van Kampen theorem) and then greedily look for consequences of the relations that prove that every generator is trivial. To certify that a 3-manifold is NOT simply connected, greedily search for a non-trivial homomorphism to some finite symmetric group. Both of these procedures work very well in practice, and crucially — contrary to (*) — allow a simple-minded human (or machine) to check whether or not 3-manifolds are simply connected.

Dunfield and Thurston, in the papers that I cited above, used something very like these techniques to perform their analyses of the census and of random 3-manifolds. They were in no way hindered by the general unsolvability of the word problem. Rigorous heuristics were enough.

It’s been fun to read your appreciations of Thurston’s expositions of mathematics as a social subject. Thurston was also a great pioneer of the use of computers to visualise and explore abstract mathematical objects, so there’s a tension in your appeal to his ideas that I’d be interested to see you address.

Anyway, I hope these comments are helpful for you. I share your admiration of Poincaré’s extraordinary creativity (also of Thurston’s), which is why I think it’s important for his achievements to be put into the correct context.

Wilton’s message looks to me like a very serious effort to take up Kantor’s challenge, and I will be very interested to know what Kantor himself thinks of these arguments. But it is necessary to add that Wilton is not responding directly to Kantor’s question but rather to my choice to reframe the challenge in terms of my understanding of the difficulties of generalizing abstract principles — in this case, the relation between simple connectedness and homeomorphism — on the basis of random undirected search. Briefly, Wilton is arguing that general considerations of computational complexity or unsolvability are irrelevant to the formulation of a general conjecture. This is because, as demonstrated by the work of Dunfield and Thurston — with which, not being a topologist, I was understandably unfamiliar — it is surprisingly easy to generate a sufficient quantity relevant data by random computation to provide the grounds for conjectures as far-reaching as that of Poincaré.

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                                                  Must Klara 3.0 tell stories?

In view of Wilton’s convincing arguments, Klara 3.0 and I must go back to the drawing board. Let us take a close look at this board, which Kantor has assigned to us. Remember that the challenge is to use this board to draw a convincing cartoon that looks like a positive answer to Kantor’s question, in a series of steps illustrated by successive panels; the challenge raised by AG is to show that this is unlikely.¹

ChatGPT apologized for not being able to draw any more than I can, but did offer a verbal description of its contents, which I reproduce, slightly edited:

Panel 1: A confused-looking Klara 3.0 stands in front of a chalkboard filled with complex mathematical equations and diagrams.

Panel 2: Klara 3.0 starts erasing and rearranging equations, trying to make sense of them. They² look determined but puzzled.

Panel 3: Klara 3.0 suddenly has an "Aha!" moment and starts connecting various pieces of the puzzle. Their face lights up with excitement.

Panel 4: Klara 3.0 continues to work feverishly on the chalkboard, writing down the key elements of the Poincaré Conjecture and drawing a simplified representation of a 3D object being deformed.

Panel 5: Klara 3.0 proudly presents the completed Poincaré Conjecture with a triumphant expression, and there's a small audience clapping in the background, acknowledging the achievement.

ChatGPT doesn’t say anything useful about Kantor’s challenge but assumes the answer to his question has the structure of a story — specifically, a “how-to” or “how-it-could-happen” story. I have to assume this is also what Kantor expects; what other form could an answer take? And now we’re at risk of getting lost in the depths of Klara 3.0’s circuits. Any electromechanical process unfolds in time, but does it necessarily follow that it can be described in narrative form?

Picture Klara 3.0 standing in front of a chalkboard filled with high school and college mathematics and the solutions to all of the exercises in the late 19th century equivalent of the Springer-Verlag series. How can we fill in the three middle panels?

I am willing to reverse engineer the third and fourth panels by following Wilton’s hint.³ The third panel’s “Aha!” moment consists in generating examples as in the work of Dunfield and Thurston. Panel four depicts the calculations, using the greedy search algorithms to which Wilton refers.⁴ An appropriate statistical analysis of the results of these calculations then spits out the Poincaré Conjecture in panel five.

I don’t see how Kantor can object to any of the panels presented thus far. So all it takes to conclude our response to his challenge, and to wait for AG to turn to us with a triumphant expression — “you see, we are doomed after all” — is to fill in the missing second panel, to make the transition from a contemplation of the sum total of contemporary mathematics to the mechanical work of generating examples and the statistical process of detecting underlying patterns in these examples, such as the pattern summarized by the Poincaré Conjecture.

There may be a path to the triumphant unveiling of the Conjecture not based on random generation of examples followed by their statistical analysis — abductive reasoning, in the terminology of C. S. Peirce, the reasoning from calculated effects to the most plausible cause — and thus there may be a response to Kantor’s challenge that has nothing to do with this scenario, and that does not require the kind of transition sketched in this scenario. Readers are invited to propose alternatives! There may not even be clear-cut transitions; a computer’s internal clock synchonizes its operations but doesn’t necessarily aggregate sequences of operations into turning points in a narrative. In what follows, however, we will assume that we have reinterpreted Kantor’s challenge as an invitation to tell a story about how Klara 3.0 came up gradually but spontaneously and autonomously with a Conjecture about the geometry of 3-dimensional manifolds.

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                                                  Cheating not allowed

However, we don’t want the Klara 3.0 we have placed in front of the chalkboard to be genuinely autonomous, or even an Artificial General Intelligence; otherwise Panel 1 may just inspire them to prepare to audition for American Idol rather than to focus on posing a novel question, much less a question whose solution will one day be worth one million dollars. Something in Klara 3.0’s circuits, like the blinders on a carriage horse, must instead predispose them, upon contemplating the chalkboard, to do something, without the intervention of a conscious subject (human or otherwise), that is likely to lead to the generation of the kind of data relevant to the eventual formulation of the Poincaré Conjecture. Panel 2 is a picture of that something, but ChatGPT’s version looks too unstructured. Too many centuries of different kinds of mathematics are crowding that hypothetical chalkboard to be meaningfully rearranged into anything that bears on the geometry of 3-dimensional manifolds, or of any kind of manifold.

Many of my colleagues who subscribe to AG’s grim predictions seem to imagine א(4) as a kind of oracle. At the beginning the human mathematician approaches א(4) —our Klara 3.0 — with a specific question. For example, we might ask: “What algebraic invariant characterizes the sphere among all 3-dimensional manifolds?” This resets the machine at the start of Panel 3, and after a humanly incomprehensible number of petaflops (at less than 200 kilowatts apiece) it outputs the Poincaré Conjecture, preferably with proof included. This already suffices to make humans feel that searching for proofs on their own is futile. Soon enough, however, the oracle realizes that its superior capacity makes the step involving the human superfluous; it decides to ignore the human questions and generates its own, along with their answers. Visitors who arrive with questions — including the question of what Klara 3.0 has learned while playing with themselves — are sent on their way, invited to shop instead at a different subsidiary of the oracle’s parent company.

If AG’s challenge is of that nature, then it is situated in the process that takes place in the last two panels, while Kantor’s challenge is to find a plausible process for automatically filling in Panel 2. But maybe AG has something else in mind, beyond the familiar argument that:

• They said an AI could never learn Go

• But AlphaZero learned Go

• Go is a game

• Mathematics is also a game

• Therefore א(0) will learn mathematics.

I’m particularly dubious about the fourth line, independently of the plausibility of the analogy as a whole. But I think that it’s incumbent on anyone who wants to argue that the analogy is valid — that includes you, AG! — to provide a plausible mechanism for the last line.

Meanwhile, I’m going to return to Kantor’s challenge and try to imagine what happens in Panel 2.

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                                                                       Two days later

…it is easy to set aside considerations on actions, activities and agency and to only focus on the conditions that need to be met to attribute knowledge to an epistemic subject, without describing what a subject needs to do to acquire this knowledge in the first place.

(Yacin Hamami, Agency in Mathematical Practice)

As promised, I have been trying to imagine how to fill in Panel 2, but after two days I have made no progress. In the meantime the European Research Council (ERC) had invited me to participate in a survey on

Use and impact of AI in the scientific process.

To the question

By 2030 how could the use of AI further develop the scientific process in general and in your specific field? In your view, what will be the three key applications or advancements? 2000 character(s) maximum

I replied

In my field I expect there will regularly be highly publicized announcements of important developments by research groups sponsored by leading tech corporations or their subsidiaries. Upon closer examination these announcements will prove to be exaggerated.

Because of the overwhelming domination of the news cycle by tech giants, whose priorities have little to do with those of mathematicians, I do not expect any significant developments in pure mathematics based on AI by 2030.

This means specifically that I do not share Christian Szegedy’s assessment that, by 2029, some AI system will be able to provide completely autonomous proofs of “10% of problems” from a list of “100 open human conjectures.”

My guesses about how AI might be used in the sciences in 2030 are indicated in the box at the top of this post. Note that I was not asked to speculate on what the situation might be in 2050, and I do not do so now. Nor did I suggest that the ERC might want to take up Kantor’s challenge as part of its question about “conducting autonomously a scientific process end-to-end.” Does a “scientific process” begin after the hypothesis (in this case the Poincaré Conjecture) has been formulated, or is the formulation of the hypothesis the starting point of the process?

In spite of Thurston’s well-documented interest in and use of computers⁵ there’s no tension in my appeal to his ideas. Recall what he wrote:

…the important things in mathematics … are somehow alive; they live in people’s brains, they live in communities of people talking to each other.

This means, and I agree, that whatever Klara 3.0 puts in Panel 2 will not be important unless Klara 3.0 is part of a community of people. Of course, what we consider important is liable to change, perhaps under pressure from institutions like the ERC. I have absolutely no objection to using computers to “visualize and explore abstract mathematical objects,” nor to carry out calculations beyond what is possible on a human time-scale, as long as “people’s brains” and “communities of people” are directing the computers rather than the other way around.

This still leaves Kantor’s challenge open. As I mentioned in Footnote 3, I have made a substantial concession by granting that Wilton’s arguments provide a plausible scenario for filling in Panels 3 and 4. For one thing, neither Seifert nor van Kampen was born when Poincaré was writing his Analysis Situs, and they were certainly aware of the Poincaré Conjecture when they proved their theorem about presentations of fundamental groups. To invoke their work one would have to suppose, counterfactually, that Poincaré found a reason to formulate their theorem before he reached Panel 1. For that matter, the kinds of algorithms invoked in Wilton’s “rigorous heuristics” presumably didn’t exist at the turn of the 20th century, so one would again have to find a plausible motivation for Poincaré (or Klara 3.0) to have invented them.

Nevertheless, I’m willing to grant Panels 3 and 4 for the sake of argument. Still, I’m stuck on Panel 2. What, looking at the totality of mathematics up to the year 1900, would induce Klara 3.0 — fully autonomous but wearing mathematical blinders — to ask a question like: “What algebraic invariant characterizes the sphere among all 3-dimensional manifolds?” I have now run out of space, so I will have nothing more to say about this challenge; but readers are encouraged to make suggestions.

1. This was intentionally written in such a way as to confuse the reader. So I should remind you that Kantor does not believe a machine can duplicate Poincaré’s insight, whereas AG suggests, or worries, that it is inevitable. Wilton apparently believes the question is open. I am sympathetic to Kantor’s position but feel that more than rhetoric is required to relieve AG’s anxieties.

2. It’s ChatGPT who guessed Klara 3.0’s pronouns, without being prompted.

3. This is a rather substantial concession, for historical reasons, among others; see the discussion in the final section.

4. I accept Wilton’s implicit claim that this does not lead to combinatorial explosion.

5. I first heard Thurston speak in a colloquium talk at Harvard, probably in the 1980s, in which he explained his ideas about the topology of parallel computing.