Math is from Eros, Computing is from Thanatos
Math is from Eros, Computing is from Thanatos
On serendipitous epiphanies, and on what dogs and mathematics have in common
A post with a similar title was published on the Mathematics without Apologies blog more than six years ago. Someone named Gershom B. objected at the time that “programming and computer science are _extremely_ playful disciplines.” I am sure he was right about that; moreover, I have reliable reports that formalizing mathematics is “really good fun” and that this accounts for much of its appeal to a growing community. As usual, the point I am making is addressed to the rhetoric surrounding the mechanization of mathematics rather than to the experience of practitioners; nevertheless, we can speculate whether the machines themselves will ever experience proof verification as “extremely playful.” Discussion welcome on the MWA blog.
It's time to let my readers know what kind of person I am. I will not do this by revealing my scores on a five-factor personality test, which is designed to size me up on the five scales of extroversion, agreeableness, openness, conscientiousness, and neuroticism, which I treated as a theoretical oddity until we all learned that Cambridge Analytica had notoriously weaponized personality factor analysis. It's probably more revealing to say that I'm the kind of person who cheats on a five-factor personality test.
Maybe "cheat" isn't the right word; it's more accurate to say that I game the personality test. This is not because I have some pathetic need for an algorithm to reassure me that I really am the kind of person I wish I were but fear I am not. Nor is it exactly because I want to reassure myself that I have figured out which answers produce which results, although that's part of the interest in taking the test in the first place; anyone with sufficient motivation can game the test, and anyone who is paying attention probably will be able to figure out their personal results just by reading the questions. What I really want to do is to reassure myself that I can continue to slip through any cage in which the consumers of such tests, or data miners of any stripe, hope to confine me.
By the way, am I the only person who has trouble answering the three secret security questions that are supposed to guarantee my access to the digital accounts on which my continued participation in civilization increasingly depends? I use the word "civilization" as in the blurb for the film The Social Dilemma:
In The Social Dilemma, key tech insiders reveal how social media is reprogramming civilization…
It's not only social media. Just a few months ago the CNRS in France introduced a new barrier to access to my email account in the form of a list of three secret security questions, which for some reason were in English. Unlike the civilized person of the future, I played no favorite sport; my grandfathers had both died before I could know them and I have no idea what jobs they may have held, if any, during the Depression; and I neither had nor wanted a pet, as a child nor at any time since. On the other hand, I had not one but a string of successive best friends. "In Gattaca every child will have a pet and exactly one best friend." So I find myself obliged to invent a fictional avatar, a zombie who has my name and an invented list of likes and dislikes and family experiences but with no actual human experiences. Then somehow I have to remember the details of this avatar's fictional life if I don't want to be locked out of my bank account or my health care or my place of work.
And while we're on the subject of the civilized personalities of the future, the kind designed to interface fluidly with AI counterparts — maybe they are the AI counterparts — I've known ever since I watched my first TED talk that I am temperamentally unsuited to giving one. On the other hand, I've only recently come to realize that I would be perfect to give talks designed to bum out my listeners, to leave them in a foul mood. Those who have been interfacing with me daily — and this probably includes my successive childhood friends — have been trying to tell me this for years. Is there a market for that sort of thing? Anyway, you are welcome to assign me a score on extroversion, agreeableness, and neuroticism on the basis of that revelation. But don't think it entitles you to make generalizations about mathematicians!
I started writing this text, on the first really fine day of spring, while I was sitting in Tompkins Square Park in Manhattan's Lower East Side. There was no rational reason for me to be in that park. (I point this out because a future post will unpackage the myth that mathematics is a rational activity.) I don't live nearby, I didn't expect to meet anyone I know there, I can think of no definite activity that I can do there that I can't do just as well in many other places. Serendipity was the one precious substance I could hope to maximize in Tompkins Square Park, and I felt I needed a dose of that in order to jump start this text. A serendipitous rainstorm cut short my stay in the park and put an end to the jazz concert that I had not expected to find and that I had put down my writing to watch.
My path from the bench where I was writing to the source of the sounds of jazz at the other end of the park passed in front of the dog run. The concept of "dog run" is not an obvious one; the expression is only traced back to 1904. But it is a rational concept, a practical compromise between the rational organization of the modern city and dogs' need, or maybe I should come clean and call it dogs' desire, to run. The Park has two separate dog runs, one for dogs that weigh less than 25 lbs and another one for bigger dogs. Serendipity struck when I stopped outside the small dog run.
The sign on the dog run says: humans admitted only if accompanied by dogs. You'd think such humans must belong to the category of dog-lovers. The population of dog-haters probably swelled in New York during the pandemic, when some of their neighbors visibly slacked off in cleaning up after their beloved pets. My Tompkins Square epiphany was that I belong to neither category. I already mentioned that I never wanted a pet, dog or otherwise. This aversion probably stems from my excessive rationalism, which is not a virtue habitually attributed to dogs. Nevertheless I approve of and even celebrate the idea of dogs; I maintain a sense of wonder at the prominent roles dogs play in the epic of human history, and I can take pleasure in observing them from a distance.
What stunned me that day at the park into an epiphany was my realization that most of the four-legged creatures in the dog run appeared to be running laps, counterclockwise along the fence. But one terrier, followed closely by a second impressionable little dog of indefinite pedigree, would break the symmetry at (very) roughly the halfway point, turn around 180 degrees after a moment of indecision, and then scamper back to the starting point, where they would rejoin their comrades on the circuit. If the dog society on view that day had been an automated system, such behavior would have to be a bug. But maybe it was a feature? It certainly made the spectacle more intriguing to me. Does that mean I was the bug?
AI's public image had suffered a setback a few months earlier when a dog-shaped robot officially named Digidog, but that the press inevitably nicknamed "Robodog," bombed in its New York début last February in a police operation in the Bronx. Media reports made it clear that the locals didn't feel like movie extras; as spectators they were at best mildly entertained, with some likening the viral videos of this and subsequent incidents to reruns of Black Mirror's Metalhead episode. New York Mayor de Blasio called the (literally) wiry creature "Creepy" and Jamaal Bowman, who represents the North Bronx and parts of adjacent Westchester County in Congress, tweeted "This is ridiculous, y'all," as the artificial canine's off-Broadway run was cut short, unleashing a chorus of ironic dog-themed press headlines. "Robot Police Dog Sent Back With Tail Between Its Legs," read one, but Digidog needs no tail and has none: the Robot Police Dog has no social needs — no need to signal its acquiescence to its place in the Robot Dog hierarchy — and its built-in circuitry is adequate for whatever material advantages a tail confers on an organic dog.
Don't think I'm hinting that virtual dogs, and even robot dogs, differ from the live dogs in the dog run because they cannot be programmed to simulate play convincingly. The first fish-tank emulating screen savers appeared as early as the 1980s, and bionic butterflies and swifts that swarm are already on the market. This text is not addressed to those who insist that human consciousness is a simulation or an epiphenomenon; for such thinkers, distinguishing this programmed coordination from naturally evolved flight and swarming behavior is not a priority. I bring this up because it helps me draw another kind of distinction. You don't want Digidog to change direction halfway to destination like the dogs in the dog run or to stop in the median strip to dig up the flower bed. Assuming you can program self-driving cars, you wouldn't want them to sniff each other's tailpipes when their paths cross. The image helps to frame the question: should bionic mathematics be built in the likeness of dogs or of cars? Is mathematics a tool wedded to a specific and unchanging function or is it, like a dog, a thing in itself, or even a being for itself?
In a few paragraphs I will tell you my answer to that question in the park.
That observing dogs could help me understand the point of mathematics1 was less an epiphany than a belated realization. Digidog is dead, a pet zombie, and it has to be that way because otherwise it would not be fit for purpose. With two additional heads, like Cerberus, Digidog would be a more recognizable symbol of thanatos, the name given to the drive of all living matter to the reinstatement of lifelessness that Freud postulated as parallel to but opposed to the life instinct of eros, which encompasses the conventionally erotic, and so much more, in the register of joy. Extra heads would just be a literary device, an allusion to Greek mythology, and like its absent tail would serve neither economic nor ergonomic purposes. But this raises the question: for what purpose are those dogs in the small dog run fit? Too urban for herding sheep or hunting pheasants or pulling sleds, too small for defense against marauders, the purpose can only be companionship.
Let's postpone indefinitely the obligatory paragraph about robot companions for the surplus elderly and remember that canine companionship is a cultural structure in its own right. When we point out, as we often do, that it was our remote ancestors that designed the dog — the usual word is domestication — we should always add that, in so doing, they redesigned humans as beings whose purpose involves living among dogs, if not always with dogs. Reread the Odyssey, Book 17, before you disagree. If and when technology designs a better canine companion, it will by the same token be imposing an upgrade on humanity. Will Dog 2.0 be more lovable than last millenium's model? Or will the propensity to love dogs be scratched from the new human's specs, and will the genetically engineered dog of the future be as inscrutable and implacable as a Digidog?
Maybe you don't care about dogs, but you like sports? As Novak Djokovic promises to surpass the records that made Roger Federer and Rafael Nadal two of the most popular as well as respected tennis players of all time, speculation has intensified that Djokovic is a robot — fanned in large part by his repeated insistence to the press that "I'm not a robot." Maybe he is, maybe not. The consensus seems to be that he lacks some of the relatable yet very different human qualities that have made each of his two great rivals so beloved — the last word is pervasive, not coincidentally, in reporting about the two stars.
What's clear is that the degree of superiority to human practitioners, including mathematicians, that AI promises, is of an altogether different order than any superiority Djokovic could ever claim over Federer and Nadal. A tennis analogue of fully automated theorem proving would look like this: 100% flawless serves, the court melts into a 3D hypersonic blur, the match over in a few milliseconds, the scores announced instantly so that the result is known but no one in the stands has the faintest idea what happened, every match indistinguishable to human eyes from every other match. If Djokovic's "robotic" style proves itself durably superior with respect to the primary metrics that define professional tennis's value system, will fans have to be reprogrammed to repudiate their erotic attachment to Federer and Nadal and to write "Djokovic" as the secret answer to the secret security question "who is your favorite sports figure"?
(I realize that the implication three paragraphs back, that the dog is not a machine, runs counter to an influential tradition in western philosophy and explicitly contradicts a claim by Descartes. Some people do say they love their cars, of course, but most of them don't sincerely believe nor do they wish to believe that their cars love them back.)
Freud presented his theory of the death drive in Beyond the Pleasure Principle:
We have fixed our attention not on the living matter, but on the forces active in it, and have been led to distinguish two kinds of instincts: those the purpose of which is to guide life towards death, and the others, the sexual instincts, which perpetually strive for, and bring about, the renewal of life.
He had been anticipated, at least in spirit, by John Adams, who wrote: "It is action, not rest, that constitutes our pleasure."2 Thanatos is well-illustrated by the post-publication malaise every writer knows; the writer's desperation to see the book finished is the pursuit of thanatos, the drive to "return to the peace of the inorganic world" as Freud put it. Mathematicians are not immune! Finishing a paper is at best a bittersweet triumph.
And here's where I mention another epiphany: that my longstanding complaint about popular books on mathematicians — that they devote so much attention to personalities and personal histories rather than focusing on the work and its conceptual content — was completely wrong-headed. Humans, or rather the eros in humans, Freud's life-instinct, craves stories about people, a variety of different kinds of people; the polymorphously perverse mathematician is not satisfied with that unique childhood best friend.
What point am I trying to make with these random vignettes and absurd speculations?
My contention in this series of texts is that mathematics in its practice and purpose looks more like the dogs in the park than like Digidog, and has more in common with Federer and Nadal than with a self-driving car. This makes mathematics hard to define, but so is eros. And so is art; I've argued elsewhere that when mathematicians attempt to justify what they do as an art form it's because they are too proper to say that it provides pleasure. This may change, of course; the mathematics of the future may be like the Great Operation in Zamyatin's dystopian novel We, a procedure designed to root out everything irrational in the human, literally and physically:
No more delirium, no absurd metaphors, no feelings-only, facts. … I cannot help smiling; a splinter has been taken out of my head, and I feel so light, so empty!
But in the meantime my primary goal is to substantiate this contention. I insist that mathematics viewed in this way is perfectly consistent with the rich and growing incorporation of techniques from computing that loom on the horizon. It is not consistent, however — and explaining why this is the case will be another goal of these essays — with what is said too often about the motivation for incorporating these techniques, not to mention the predictions of inevitable subordination to these techniques, which makes mathematics look like nothing so much as a dead mechanical dog.
The digital will never capture the contingent.
This sentence, uttered spontaneously, and to all appearances contingently, by Gayatri Chakravorty Spivak, on November 4, 2020, during the course we were teaching jointly with the title Mathematics and the Humanities, immediately registered in my mind as the germinal idea that had been missing in my attempt to put a name on what formalization of mathematics fails to capture. That week's class was entitled Can Mathematics Be Done by Machine and Kevin Buzzard, founder of the Xena Project, whom readers will remember from an earlier entry, was our guest. After Kevin and I had both presented our points of view, it was my fellow teacher's turn to respond, and she did so by coining a remarkable series of epigrams. The one at just preceding this paragraph is the most perfect, and we will return to it repeatedly, but I could generate a 3000-word text to explicate any one of them. I am not yet ready to do that but I do want to quote another one:
To be haunted is as far away as possible from being in control. You cannot anticipate that you will be haunted. It is not periodical.
I might add that it is serendipitous. I did not anticipate that in my pursuit of serendipity I would be haunted by a vision of running dogs, but that is exactly what happened during the days following my visit to Tompkins Square Park.
Thanatos doesn't haunt. But it really is a drive. The need to be in control is a drive. You are driven to finish writing that paper, to know how the story ends, how the conjecture is proved. Then you will be in control of at least one more thing, but you will also be sad, face to face with your finitude. Triste est omne animale post demonstrationem. Leibniz's calculus ratiocinator is thanatos, Hilbert's program to read off the provability of a mathematicial proposition from its sheer form is thanatos, any program that promises to crystallize imagination in amber is thanatos.
And — stop press! — Juliette Kennedy has just informed me (on September 30, 2021) that Kurt Gödel, of all people, felt the same way:3
Remark (Philosophy): Is the difference between living and lifeless perhaps that its laws of action don’t take the form of “mechanical” rules, that is, don’t allow themselves to be “formalized”? That is then a higher level of complication. Those whose intuitions are opposed to vitalism claim then simply: everything living is dead.
A formal proof is thanatos precisely because it cannot be inserted in a discourse. A few years ago I wrote this in response to the recurrent obsession on the part of some computer scientists with formalizing Wiles's proof of Fermat's Last Theorem:
…a proof like the one Wiles published is not meant to be treated as a self-contained artifact. On the contrary, Wiles’ proof is the point of departure for an open-ended dialogue that is too elusive and alive to be limited by foundational constraints that are alien to the subject matter.
If the open-ended dialogue that began with Wiles's proof is alive, it follows that the drive to entomb it in a self-contained artifact, formalized or not, is the death drive. This is consistent with the erroneous vision of mathematics in Dialectics of Enlightenment by Adorno and Horkheimer:
enlightenment ... equates thought with mathematics. Thought is reified as an autonomous automatic process aping the machine it has itself produced so that it can finally be replaced by the machine... mathematics made thought into a thing, a tool, to use its own terms.
Not coincidentally, this perspective is compatible with the neoliberal identification of value with value for the market.
In the words of Matthew Handelman, for the Frankfurt school, in particular the authors of Dialectic of Enlightenment,
if philosophy became the formulae of mathematics, then, the critical theorists believed, thought would be impotent to resist and, therefore, complicit in the authoritarianism engulfing Europe.
Fortunately, actually existing mathematics demonstrates the limitations of the formalist perspective that the critical theorists made the mistake of taking literally. So it is possible to practice mathematics and not be complicit in authoritarianism, not to mention human extinction.
Update: I hadn’t realized that Tompkins Square Park occupies a special place in a parallel universe that revolves around dogs, but it turns out that it does. It was not foreordained that I would experience an epiphany in the park, but since I did, it’s not surprising that my epiphany also revolved around dogs.
See the subtitle of the previous entry.
On Machiavelli, Works, Vol V, p. 40.
This quotation can be found in Kennedy’s recent book Gödel, Tarski And the Lure of Natural Language: Logical Entanglement, Formalism Freeness.