Understanding vs. Reasoning
Mathematics and the Undead, Part 2: A philological adventure
This is the continuation of Mathematics and the Undead, the previous entry in this newsletter, which was terminated abruptly by Substack’s email limit. The original plan was to conclude the entry with an invitation to mathematicians to engage in what is conventionally called “critical thinking” when faced with a variety of challenges from the tech industry, the most insidious of which is a challenge to the right of self-definition. That invitation remains as the conclusion of this entry; it is preceded by an extended exercise in philology, which might be considered critical thinking’s zeroth degree.
I call philology an “adventure” because, if conducted with full awareness, it should lead to the realization that the language we use speaks us rather than the other way around. Our words, much older than we are, began to shape us before we could have imagined thinking about them in a critical spirit. What words, after all, would have allowed us that freedom? It is therefore a form of courage to continue to use words — like “understand” — after we realize that we can’t explain their meaning. This paradox is the sense of Nietzsche’s famous dictum:
We have to cease to think if we refuse to do it in the prison-house of language [in
dem sprachlichen Zwange, literally “in the constraints of language”]; for we cannot reach further than the doubt which asks whether the limit we see is really a limit.
Some colleagues see the only escape from this paradox in the mechanization of reasoning, and formalization as the necessary tool to effectuate this escape — for the robots in our future, if not for us.
Mathematicians explore Silicon Valley’s “private kingdom”
…There is the dream and the will to found a private kingdom… what may be attained by industrial or commercial success is still the nearest approach to medieval lordship possible to modern man… (Joseph Schumpeter, The Theory of Economic Development, 1911 )
In our last episode we saw the academic-industrial AI complex throw down the gauntlet to mathematics by organizing a workshop whose
goal … is to find out “when machines can surpass human experts in different mathematical domains?”
and by inviting fellow travelers to
learn more about how AI is disrupting mathematics.
We wouldn’t expect a private kingdom to invite us to a respectful dialogue of equals. So why would self-respecting academics trade in our professional autonomy for the chance to speak at a corporate-sponsored forum? Could it be to immunize ourselves against the presumptuousness of those Silicon Valley billionaires and their aspiration to medieval lordship? “In poison there is physic,” said the Earl of Northumberland in Henry IV, Part II (Act I, Scene 1). Shakespeare would have sympathized:
As, to prevent our maladies unseen,
We sicken to shun sickness when we purge, (Sonnet 118)
But if there were a dialogue, what would be the common language? Even if we are using the same words, why would we and the corporations understand them in the same way?
Understanding vs Reasoning
The results of philosophy are the discovery of some piece of plain nonsense and the bumps that the understanding has got by running up against the limits of language. They — these bumps — make us see the value of that discovery. [Die Ergebnisse der Philosophie sind die Entdeckung irgend eines schlichten Unsinns und Beulen, die sich der Verstand beim Anrennen an die Grenze der Sprache geholt hat. Sie, die Beulen, lassen uns den Wert jener Entdeckung erkennen.] (Wittgenstein, Philosophical Investigations, 119)
The words “understand” and “discuss” topped the frequency chart1 I compiled from remarks of participants in the Fields Medal Symposium reproduced in the previous newsletter. The most common retort to the Central Dogma's assertion that the purpose of mathematics is to create formal (or formally verifiable) proofs is the claim that (to quote Thurston, author of the most eloquent exposition of this claim):
what [mathematicians] really want… is understanding.
It may just be a coincidence that zombies, who were not invited to the Symposium, are characterized by their structural incapacity to "understand." If you try to define “understanding” unambiguously you will probably bump up against the paradox that you have set yourself the task of making the word comprehensible to zombies. Rodrigo Ochigame put his finger on this dilemma in his comments on the Toronto event:
I think the kinds of understanding and insight that matter to mathematicians are quite different from those offered by existing techniques of "interpretable/explainable ML.”
Indeed, there is this from the overview of the 1st MATH-AI:
one defining property of advanced intelligence – reasoning – requires a much deeper understanding of the data beyond the perceptual level; it requires extraction of higher-level symbolic patterns or rules. Unfortunately, deep neural networks have not yet demonstrated the ability to succeed in reasoning.
You just read the word “reasoning” twice, and it's no accident: it tops the frequency chart I compiled from the overviews announcing a few recent and upcoming conferences2 with MATH and AI in the title. My method was completely ad hoc and unreliable but it strongly suggests, in comparison with the frequency chart for the Fields Medal Symposium participants, that “reasoning” is the engineer’s alternative to “understanding,” the mechanizable part (see note 9 of the previous newsletter).
The distinction Ochigame identified is not innocent. When the engineers refer to "understanding" in their conference overviews it is as a means to another objective; for the colleagues in Toronto understanding was the objective. Not one of these multi-billion dollar corporations is trying to design an AI for which understanding is an end in itself. Human mathematicians, at least for Kantian ethics, are ends in ourselves. For a corporate person, an AI is a means to an end. Big difference!
My “Little Data” seem very much to suggest that the opposition understanding/reasoning traces the fault line between mathematicians and those who promise or threaten to disrupt mathematics. It also conveniently parallels Ian Hacking’s opposition cartesian proof/leibnizian proof that we have encountered repeatedly, most recently in Ochigame’s presentation in Toronto. But the result is suspiciously neat, and sweeping conclusions are hardly warranted by such a small sample. Besides, although in my normal life I use the word “understanding” with reckless abandon, I can hardly claim to know what it means. Nor can you; otherwise you would already have invented artificial general intelligence (AGI). As St. Augustine wrote in reference to the question What is time, in his Confessions:
If no one asks me, I know; if I want to explain it to a questioner, I do not know.
So I decided to open my 1300-page (in double-columns) Dictionary of Untranslatables3 (henceforward: the Dictionary) to the entry for understanding on p. 1184, written by Denis Thouard, author of several books on hermeneutics and philology. To my great astonishment, the Dictionary’s entry informed me that the opposition understanding/reasoning has been central to the history of western philosophy practically from the very beginning.
Formalize that, if you dare!
The passage that grabbed my attention is just above the red box in the scanned page above:
As a translation of the Latin intellectus, it inherits a long conceptual history that contrasts it, as an act of intuition (Gr. nous [νους]), with rational discursive acts (Gr. dianoia [διάνοια], and it is defined in contradistinction to reason (Lat. ratio).4 (Emphasis added)
It’s tempting to think that “understanding” (intellectus, nous) is the secret shibboleth of mathematicians, just as “reasoning” is that of computer scientists and the industry that employs them. The Dictionary’s entry tells us that noein, the verb from which nous is derived, “gives substance to the link between perception and thought… in the suddenness, the immediacy of a perception”; whereas dianoia “moves by way of hypotheses and demonstrations.” So that even if we think we are talking to one another there’s no reason to believe that meaning is preserved in the passage from emitter to receiver.
The passage continues, however, with a warning:
But these words, being defined in terms of one another, exchange characteristics several times; the more prestigious one becomes ordinary, the ordinary one is reevaluated.
If every word in the mathematical vocabulary were as protean — and as Untranslatable — as these,5 we would be tempted to "cease to think." What you see reproduced above is just the first of seven equally dense Dictionary columns. But in fact, real live mathematicians are leibnizian as well as cartesian, and “hypotheses and demonstrations” are inseparable from our working lives. I have to assume that real live computer scientists also live for their eureka! moments, their flashes of sudden understanding. Nevertheless, it’s intuitively satisfying to imagine that the spontaneous tendency to write about understanding rather than reasoning, or vice versa, must reflect a preference deeply ingrained in the self-images of the respective professions.
At the limit, then, one understands how computer scientists can imagine robots taking their jobs, and how they can equally imagine robots taking our jobs. But if our job as mathematicians is primarily to understand, to experience thinking on the model of “the suddenness, the immediacy of a perception,” how can we meaningfully imagine robots taking our place? Does it make more sense to replace us in understanding than it does in reading, or singing, or dancing, or eating, or breathing?
The Dictionary helps us locate the source of this muddle in Descartes’ use of the word entendement, the French translation of intellectus, that blurs its distinction from raison (i.e., ratio). Reading that "this tendency… becomes cemented in Anglo-Saxon6 philosophy" — the dominant school in philosophy of mathematics — we understand why it is so difficult to recover the distinction and to take understanding seriously as an end in itself. The distinction then gets blurred in a different way when it becomes Kant’s “devaluation of the understanding (Verstand) in favor of reason (Vernunft),” and the post-Kantian German idealists introduce more permutations.
Interestingly, one philosopher escapes this blurring:
Spinoza’s example shows how his intellectus … reconciles mathematical discursiveness and the intuition of the mind…
Thouard illustrates this with a reference, too long to reproduce here, to the four modes of knowledge in Spinoza’s Tractatus de emendatione intellectus. You can expect to see more of Spinoza in these pages in the coming months.
Metaphors and their limits
There are more tangible reasons for the “devaluation of the understanding” in rhetoric. Metaphors are available for reasoning as an activity that develops in time. Reasoning, on the model of a formal proof, is a step-by-step process. The problem for designers of automated theorem proving software is to generate the leap from one step to the next. Anyone with an experienced team and an ample budget can design a robot that keeps taking steps, one after another, until its battery runs out. The challenge is to induce it to go somewhere, preferably somewhere new, rather than to walk around in circles or into a wall.
The challenge to build a reasoning machine is a real one, but at least there are metaphors, like those in the preceding paragraph, that allow us to picture the desired outcome. The theory of digital reasoning, the archetype of formalization, can be implemented on machines entirely in the language of symbols — a language consisting of two symbols largely suffices. But it’s hardly an accident that modern programming grew into a flourishing academic discipline, an industry worth trillions of dollars, and a dominant force in contemporary civilization, only after Turing introduced the visually compelling metaphor of an impossible machine making marks, one after another, on an infinite tape.7
What, now, does it look like to understand something? Mathematicians have as a common heritage a vivid picture for that: Poincaré putting his foot on the step of the omnibus, realizing that “that the transformations [he] had used to define the Fuchsian functions were identical with those of non-Euclidean geometry.” But this is a picture of Poincaré, not of understanding. No one expects the engineers will have to equip their AGI with feet and bus tickets in order to mimic mathematical creativity. But the engineers’ metaphors for intelligence (intellectus, a proxy for understanding) are mechanical by nature, as in this excerpt from AI pioneer John McCarthy’s answer to the question WHAT IS ARTIFICIAL INTELLIGENCE:
Intelligence is the computational part of the ability to achieve goals in the world.… Intelligence involves mechanisms, and AI research has discovered how to make computers carry out some of them and not others
If mathematics accepts the engineering framework, then understanding will be used purely instrumentally, as in McCarthy’s influential text. Tim Gowers has been working for years, probably more than anyone, to design mechanical systems for doing mathematics in ways that mimic human understanding. Gowers writes: “it may be hard to give a precise definition of what understanding is.” In his recent manifesto variants of the word understanding occur far more frequently than reasoning, but are well behind words like proof, problem, and theorem. The metaphors his text offers for understanding in mathematics look like its goals and its output, symbols on a page, hardly metaphors at all.
Taking the engineers’ goals seriously suggests an explanation of the prevalence of reasoning over understanding in their discourse that is more banal than a failure to generate adequate metaphors for the latter. Reasoning, after all, can be automated, sold, monetized. In contrast, there is no profit to be made in understanding, a subjective experience that can only be stimulated but not sold as such.
It’s a safe bet that a paradigm shift for mathematics that has been outsourced to the tech industry will be marked by an obsessive attention to the bottom line. When contemplating participation in conferences sponsored by the tech industry, it would be naive to trust speakers who are also employees of these massive corporations, whatever their standards of personal integrity, to call their employer’s business strategy into question, purely to serve the hypothetical and Untranslatable priorities of the mathematical profession. Such behavior risks termination of the speaker’s contract; at the least it discourages investment, thereby depriving the vampire of the blood it “must continually extract… to remain undead.”
But what’s in it for the tech industry? What’s the payoff? Automated software verification is and always has been a major priority. And this is yet another hint of why understanding is dispensable for the engineers. We can place human activities along a spectrum according to how important understanding is to their realization. As long as the software works as advertised no one cares whether or not it was verified by a zombie. The less my thermostat (or my refrigerator, or my browser, or my electric meter) understands of my preferences, the happier I am. The engineers seem to believe that mathematics belongs on the zombie end of the spectrum, along with chess and go. Mathematicians feel otherwise.
There’s also the assumption that automating mathematics is a giant step in the direction of AGI: the subtitle of the first MATH-AI Workshop was
The Role of Mathematical Reasoning in General Artificial Intelligence
Beyond the walls of university labs AGI is the mythical motor of the fourth (or maybe the fifth) industrial revolution. And this, in turn, means someone thinks there’s money to be made if mathematics can be automated.
Looking at Silicon Valley’s balance sheets, we may ask: why would anyone need to make so much money? The answer (naturally) is: that’s all zombie corporations know how to do. For the rest of us, living in a world governed by the logic of the undead, it can be a survival strategy.
Finally, automating mathematics, like chess or go, offers the opportunity for triumphant press releases, which the industry’s AI labs cook up with all the freshness and spontaneity of a branding campaign.
Critical thinking
Whatever understanding may be, critical thinking is the recommended means by which it can be attained.8 It is also one of the skills the liberal arts education promises to transmit,9 and for this reason it’s an indispensable aspect of the academic’s professional style. Mathematicians don’t stress this very often but that may be because critical thinking is the essence of mathematical practice — arguably more essential than proving.
Ideally, critical thinking should be applied not only to our disciplinary domain but also to the material and intellectual conditions that make our discipline possible. This kind of thinking is incompatible with corporate culture, as Mario Savio reminded his Berkeley audience in 1964:
I don’t really expect AI researchers, nor their admirers among mathematicians, to turn their backs on the ethos of 2010’s Silicon Valley —
…those who understood our cultural moment saw that selling out—corporate positions, partnerships, sponsors—would become our generation’s premier aspiration, the best way to get paid. (Anna Weiner, Uncanny Valley, p. 34)
if the alternative is that of 60s Berkeley
There comes a time when the operation of the machine becomes so odious, makes you so sick at heart, that you can't take part, you can't even passively take part. And you've got to put your bodies on the gears, and upon the wheels, upon the levers, upon all the apparatus. (Mario Savio again)
Nor do I really expect readers to forego support from sponsors of conferences like those listed above. But I do hope to move some readers to ask whether associating with such sponsors may indirectly help them to promote an agenda with whose implications they may not agree. This would be an example of what Jeremy Avigad, quoting Thoreau, means by “living deliberately.”
Once again I want to thank Avigad for not only finding the right formulation of the attitude needed to address the challenges mechanization poses to mathematics, but also for finding the right author to quote. (I suspect Thoreau didn’t really expect to end slavery and war by refusing to pay his taxes.) What kind of “civil disobedience” can resist the hegemonic narrative promoted by Silicon Valley’s zombie corporations? Recovering an alternative narrative is a start:
These matters can recover their original urgency for us only if they are retold within the unity of a single great collective story; only if, in however disguised and symbolic a form, they are seen as sharing a single fundamental theme — … the collective struggle to wrest a realm of Freedom from a realm of Necessity.10
Human understanding is the “realm of Freedom” that mathematicians wrest from the Necessity of what Wittgenstein called the “hardness of the logical must.” This is how I understand Cantor’s claim that “the essence of mathematics lies in its freedom.” Mathematicians have no reason to accept as necessary the disruption of this understanding, the devaluation of the mathematical experience, the curtailing of our autonomy, the neglect of our aspiration to freedom, by a powerful industry bent on setting the agenda.
Silicon Reckoner has just published two essays in quick succession and will be taking a break until the end of December. Future essays will be devoted to a report on David Bessis’ book Mathematica and to the staging of the mechanization of mathematics as spectacle, in the spirit of the Situationist International.
…after terms like “mathematics,” “computer,” and AI, which referred directly to the theme of the symposium.
Edited by Barbara Cassin, translation [sic!] edited by the high-powered team of Emily Apter, Jacques Lezra, and Michael Wood. It should hardly come as a surprise that Apter has written a penetrating essay entitled “The Prison-House of Translation” which, among other gems, provides a source for the transformation of Nietszche’s “constraint of language” to the metaphorical “prison-house,” and in which I found Wittgenstein’s quotation, copied above, about “bumps” suffered by the understanding.
A philologist friend tells me that the pair عقل/فهم is the corresponding opposition in classical Arabic philosophy, and was already present in the Qur’an. Like their ancient Greek and Latin counterparts, the contents of these words are fluid, but they remain current in modern Arabic, with roughly the same meaning.
The Dictionary’s entry for TRUTH fills more than 40 columns. Mathematicians have confided to me that they suspect that the colleagues in departments that speak in Untranslatables don’t bother to listen to each other; and I suspect we hold this suspicion is in common with computer scientists. But of course those colleagues do listen, very very carefully.
This is the term in the French original, where it refers to philosophy in English, not to the authors of Beowulf.
You can actually observe one in action on YouTube.
For the purposes of this essay, of course. In other circumstances and in other cultures prayer or meditation may be more effective.
My Columbia colleague Bruce Robbins writes, in connection with the use of the expression by Walter Jackson Bate:
The phrase “critical thinking” has come to sound portentously empty, as if inviting the wrong people to celebrate its emptiness as a kind of authoritative neutrality. University administrators can’t seem to do without it.
But he goes on to say that
Criticism for [Bate] is a kind of permanent opposition. It assumes that something is fundamentally wrong with the culture of the present and that its task is to make that wrongness known.
This is the task that Silicon Valley, as the self-appointed remodeler of the “culture of the present,” cannot undertake; but it is just right for academics who choose to “live deliberately.”
Frederic Jameson, The Political Unconscious. Jameson is referring to Marxism, which is irrelevant here. The Prison-House of Language is the title of one of Jameson’s first books, and the book in turn contributed to making the expression into a cliché — and maybe DALL-E can turn it into a meme.
Searle's Chinese Room gedanken experiment casts some measure of doubt on attributing "Understanding" to programmed entities.