In which the author harangues mathematicians for failing to "live deliberately" when contemplating a mechanical future

The first half of an essay, so far untitled, to appear soon in the AMS Bulletin special issue on automation and mathematics

The call for a special issue, provisionally entitled “Will Machines Change Mathematics?,” was launched in response to last fall’s successful Fields Medal Symposium in Toronto. I am a contributor as well as an editor of this special issue. While waiting to complete new material for this newsletter I have decided to share my contribution here, rather than to post it on the arXiv, where several of the articles have already appeared. The article is too long for Substack’s email limit, so it will appear in two parts.

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We're just a biological speculation
Sittin' here, vibratin'
And we don't know what we're vibratin' about (Parliament Funkadelic, 1972)

                                                       Entering the woods

One-line summary: The Venkatesh essay provides a unique opportunity to reflect on our values.

Toward the end of his talk at the 2022 Fields Medal Symposium that gave rise to this volume of BAMS, Jeremy Avigad found a helpful way to look at our current assignment in Thoreau's Walden.  Thoreau

went to the woods because [he]  wished to live deliberately, to front only the essential facts of life…

What would it mean for mathematics to "live deliberately" in the face of the existential challenge posed by artificial intelligence?  At a minimum, living deliberately¹ entails knowing where we, as a community, want to go; and this in turn entails knowing what mathematics is for, "what we're vibratin' about."

A necessary first step is to free ourselves from the grip of misleading metaphors.  The "intelligence" part of AI, for example, can only be a metaphor, a particularly tenacious one, since there is no agreed definition that embodies everything that may go by the name of "intelligence."   The consequence is that discussion tends to be dominated by the definitions promoted most aggressively — not coincidentally those preferred by the industry that stands to profit from turning "intelligence" into something monetizable.  But "community" is also a metaphor:  it's more accurate to describe mathematics as a landscape of overlapping villages.   Geometers, logicians, analysts, and applied mathematicians differ in the potential reasons they may welcome AI, or be curious, or be indifferent, or feel threatened.  Certainly we don't all expect to be vibratin' about the same thing.  

A second step, one that is apparently particularly challenging for mathematicians, is to recognize that other disciplines have much to teach us.  The Columbia library catalogue lists 4628 books with "Automation" in the title.  Even accounting for duplication this is a substantial literature that reflects a vast variety of approaches to the topic, some of which are likely to be directly relevant to mathematics even though the books are written from the standpoint of history, economics, political science, sociology, even the humanities. 

When mathematicians talk about automated proof verification, automated theorem proving, or the other challenges or opportunities to be expected from information technology, there is no acknowledgment of this vast literature.² The discussion instead tends to be limited to  the sort of comments that economist Robert J. Shiller cites in what he calls the (viral) narrative "Automation and Artificial Intelligence Replace Almost All Jobs"³ — in other words, the pop culture version of technological determinism, which is also pervasive in Silicon Valley.  I have no objection to quoting pop culture; George Clinton, for example, quoted above, is as astute an observer of the contemporary condition as any academic.  But living "deliberately" should entail seeking out and engaging with any coherent thoughts that may shed light on the most urgent questions about our future.

Venkatesh's essay marks a break:  it is the first explicit call for mathematicians to examine the implications of AI for what we value most in the practice of our profession and to take action to defend our acknowledged values.  Among other benefits, it provides an opportunity to agree on what these values are — Venkatesh helpfully points out that these values are constructed and indicates a few "mechanisms" that contribute to this construction — and to frame definitions of intelligence suited to our own priorities.  Like most of the other authors represented in this collection, I intended my text to be a response to Venkatesh's call.  I will not, however, respond directly to the explicit questions that inform his essay.  Instead, my text is a lengthy detour through some ways of thinking that are not often practiced in connection with mathematics but that might help us understand our shared values. 

                                    Crisis of vocation, crisis of institutions

One-line summary: The kind of reflection described here is extremely common in the literary disciplines and social sciences, and is related to a perceived legitimacy crisis of the institutions in which we work.

Both insofar as the mind has clear and distinct ideas, and insofar as it has confused ideas, it strives, for an indefinite duration, to persevere in its being and it is conscious of this striving [conatus] it has.⁴ (Spinoza, Ethics, Part III, Proposition 9)

You might think that a profession's very first steps, upon pledging to live deliberately, would be to clarify the nature of the "being" in which it "strives" — its Spinozian conatus —  to persevere; and, having done that, to reflect on the material conditions without which its striving is futile.  For the profession of mathematics, the conatus is largely concerned with the values highlighted in Venkatesh's essay.  It is exceedingly rare, however, for mathematicians to undertake such a reflection.  The most recent such reflection that comes to mind is the exchange in response to the 1993 Jaffe-Quinn article on "Theoretical Mathematics."⁵  I'm sure one reason these texts — especially Thurston's article "On Proof and Progress in Mathematics" — are so widely cited is that such wide-ranging discussions among mathematicians are so infrequent.

As for the material conditions, they are primarily provided by the research university⁶, an institution that only relatively recently (compared to the history of mathematics) adopted its present form, and whose fragile balance of conflicting priorities has been evident for my entire career.  I know of no serious attempt on the part of mathematicians to come to grips with the deep instability of the model of higher education that provides our livelihood.  To grasp the scale of the endlessly postponed crisis we need to leave the departmental silo and cross the campus to meet our colleagues in the humanities and social sciences.  It's only a slight exaggeration to say that for years they have been speaking of nothing else than this vocational crisis. 

The crisis in higher education as a whole is reflected in contradictory metaphors. Is higher education itself an industry, that "shapes who our future leaders and builders will be"? Or is it a public good? Is it primarily the means by which its self-reproducing elites (like the one described at greater length below) reproduce themselves? Or is it the primary locus of the debate in which the contours of this and other elites are questioned?

Material conditions for disciplinary self-reproduction are already in crisis at institutions like mine. Here is how Columbia administrators responded recently to faculty questions about the decision to reduce graduate admissions, particularly in humanities and social sciences (my emphasis):

…admissions numbers set decades ago were in many cases not reflective of the current realities of our disciplines.

The median figure for that hope [to pursue an academic career] was 89%; and yet the median figure for the actual academic placements for those same departments is 58%. What do we tell our students about their unattainable futures? Our pedagogical dependence on our PhD students cannot take precedence over their desires and hopes for their own professional lives. … all peer graduate schools have already rethought or are currently rethinking the number of PhDs produced, as well as their responsibilities to those students.

Mathematicians don't need to be reminded of our departments' "pedagogical dependence" on graduate students.  Discussions of their "unattainable futures," when they do occur, mainly take the form of reminders that mathematics PhDs have many options, often lucrative, outside the academy, more in any case than those in many other fields.  I have never seen their "desires and hopes" placed at the center of such a discussion — nothing like this:

In society as we know it, it is a rare privilege to be offered the conviction that a meaningful vocation awaits you.  A meaningful vocation doesn't await everyone.  And most people know it.⁷

Venkatesh acknowledges that if automation does provoke a crisis in mathematics — which I think is likely, given the balance of forces — it will be in the first place a crisis of meaning.  This understanding is rare in mathematics but it is commonplace for scholars in the humanities⁸.   The above quotation is taken from one of two new books, by colleagues in Columbia's English and Architecture departments, that I read last fall.  Both books, unsurprisingly, are exercises in disciplinary deliberate living.    Reinhold Martin's Knowledge Worlds treats the material basis for the university in the broadest possible terms, as

…a world of gates, screens, departments, papers, reports, and other media,

and emphasizes how each of these factors, and the architectural structure of the campus, is an expression of the values of the social forces behind the university's creation.⁹  Bruce Robbins's Criticism and Politics deals with his discipline's material existence more directly.

Always eager to slash public budgets and make education a matter of private investment in consumer preference, public decision-makers might decide they can dispense with such an esoteric and unprofitable branch of knowledge production.

The fact that market relations have tightened their grip on both [journalism and literary criticism] since the eighteenth century… provok[e] both journalists and academics to recall or invent a more independent past … [But] [t]here was no earlier autonomy.  The critics/journalists depended on the market.  They were never truly autonomous.¹⁰

I quote these texts because the disciplines housed on the other side of campus routinely practice self-examination on a scale that mathematicians only encounter when faced with extraordinary challenges.  If we learn from our colleagues in the humanities and social sciences we may be better prepared to face the challenge of automation.   It's true that literary scholars have for quite some time been made aware of the material conditions of their discipline by its very subject matter.  Here, for example, is the illiterate publisher Dauriat, in Balzac's Lost Illusions, published around 1840:

…from this day forward … If anybody comes here with manuscripts … ask him whether it is poetry or prose; and if he says poetry, show him the door at once. Verses mean reverses in the booktrade.

Disciplinary values remain central when our literary colleagues address the implications of the new technology, even when they are not talking about crises.  The document announcing an upcoming conference on AI-Technology as Interactional Human Culture. lists the following "Research Questions," among others, as topics for the conference:

• •    How do social values and cultural traditions, among them beliefs about machines, commercial interests, technological affordances and notions of language frame the development of AI technology?

• •    How do traditions of writing, established language norms, the dominance of English and people’s beliefs about language shape the programming of speech-enabled AI or translation technologies?

• •    And, finally, what do we learn from all this with regards to the question what constitutes democratic, culturally-sensitive and human-centred AI?

All of these questions, suitably adapted, are relevant to mathematics as well.   Is the debate on AI within mathematics concerned with "Language, Data Practice, and Social Struggle" — the conference's subtitle?  If not, why not?

Venkatesh's analysis shares an attention to value with the above list of "Research Questions," and with the torrent of articles that have appeared in practically every venue,¹¹ especially since the release of ChatGPT about the implications of AI for one discipline or another.  The only visible reaction of mathematicians, apart from the Fields Medal Symposium and this special issue, is in the conference on Machine Assisted Proofs that was taking place at IPAM as I wrote these lines, and at the upcoming workshop, organized by the National Academies of Sciences, Engineering, and Medicine, on "AI to Assist Mathematical Reasoning."  The word value appears nowhere in the Overview of the IPAM program; nor is there any mention of values, traditions, or cultural sensitivity in the blurb for the National Academies program.   Both programs seem designed to show mathematicians vibratin' in unison with the tech industry.  Moreover, they sidestep the very real possibility that obtaining the massive datasets needed to train AI systems to formalize or generate proofs may involve copyright infringement, a concern already raised by class action lawsuits against AI corporations by artists.¹²  A failure to live deliberately could hardly be more blatant.

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The second half of the essay will appear later. I prepared the first half for publication here while virtually attending this conference on history of mathematics, a sharp reminder of the richness of historical engagement of mathematicians with philosophical speculation — deliberate engagement — in the effort to define the value and purpose of mathematics. While individual mathematicians defended their respective positions forcefully at the time, what I find most precious in this history is that the questions remained unsettled and the controversies remained lively. As an example, readers are encouraged to contemplate this slide from Nicolas Michel’s talk, which was largely devoted to the 19th century Danish geometer H. G. Zeuthen’s thoughts about the relation between intuition and symbolic computation:

The depressing attempts to close off the debate by reducing the content of mathematics to computer code not only fail to respect this history but lend legitimacy to claims that machine learning will “solve mathematics,” a claim that only makes sense if the conatus of mathematics is settled once and for all in alignment with the priorities of the tech industry.

1. I intend to try the reader's patience by squeezing as much as I can out of Thoreau's expression; it's really the most apt description I know of what the profession needs at this turning point (if it is one), as much for the character of the person who wrote it as for what it says.

2. The bibliography in Marcus du Sautoy's The Creativity Code, which "wonder[s], with the onslaught of new developments in AI, if the job of mathematician will still be available to humans in decades to come," includes exactly one item with "automation" in the title.

3. Robert J. Shiller, Narrative Economics, Chapter 14.  One contributor to this volume once wondered in public whether I might be a Luddite — a character of the narrative treated in Shiller's Chapter 13.

4. Translation from Stanford Encyclopedia of Philosophy. 

5. Bulletin of the AMS, 29 (1993) 1-13. 

6. At some point the population of mathematicians beyond academia will have to be included in this deliberate living.  The National Security Agency, for example, home to "the largest collection of mathematicians and linguists in the country and possibly the world," according to James Bamford's Body of Secrets, published in 2001, is entirely absent from this reflection.

7. Bruce Robbins, Criticism and Politics, Stanford University Press (2022), p. 40.

8. The Chronicle of Higher Education has just published a collection of essays, its second in three years, on the "collapse of the discipline" of literary studies.

9. Reinhold Martin, Knowledge Worlds, Columbia University Press (2021), p. 7.  On pp 62-3 of this book, for example, I learned something that every American mathematician deserves to know — that chalk and blackboard made their first appearance at an American campus at the US Military Academy at West Point, as an introduction of French methods of training future officers in the basics of geometry.    

10. Ibid. p. 82, p. 90.

11. Among others, in the MIT Technology Review, The New Yorker, Times Higher Education, The Chronicle of Higher Education, … 

12. S. Escalante-Mattei, "Artists Are Suing Artificial Intelligence Companies and the Lawsuit Could Upend Legal Precedents Around Art,"  Art in America, May 5, 2023.