Mathematics After AI

Last year, mathematicians from more than a dozen universities met in the city of Leiden to discuss how artificial intelligence will affect the future of their field. The fruit of that meeting is “The Leiden Declaration on Artificial Intelligence and Mathematics,” a few dozen paragraphs of carefully measured prose concerning the threats posed by this new technology and how best to address them. At the time of writing, it has accrued over 3,500 signatures. 

One of the themes of the declaration is that the AI frenzy has created a dangerous culture of oversimplification. Its drafters have therefore endeavored to steer the narrow way between hype and hysteria. I tried manfully to read the declaration in this spirit, but as I reviewed the list of “potential threats,” I could not help but think that, were some Screwtape assigned by the infernal high command to bring down the mathematical community, he could hardly have dreamed up a more perfect instrument to effect his designs than AI. 

Take the first two threats on the list: AI is remarkably good at producing fluent mathematical nonsense and remarkably bad at attributing the work it cites to the correct human sources. Feed enough papers with “plausible but unreliable arguments” and unverified citations into the existing peer-review system, and it will quickly break down or otherwise be forced to abandon its traditional standards of rigor. These papers, once published, will also form part of the training material for the next generation of large language models. Only a few small errors compounding over time might be enough to push mathematics into a state of grave disorder. 

AI is also undermining the autonomy of mathematical research. In a technofetishist culture like our own, new technology is often “incentivized for its own sake.” Mathematicians will soon find themselves in thrall to the companies that train, host, and license the relevant models, companies that have so far shown a complete disregard for the norms of publication and a cavalier tendency to hype up results for the purposes of marketing, all to the detriment of “proper evaluation.” 

Indeed, research agendas are already being adjusted to prioritize areas of mathematics that are suited to automation. This can be seen in the excitement surrounding AI’s ability to tackle the so-called Erdős problems, a loose collection of hundreds of mathematical questions posed by Hungarian mathematician Paul Erdős. The media have been quick to report on any successes by large language models in solving the problems, all the while neglecting to mention that Erdős himself cautioned that there was no way of distinguishing between the “marshmallows”—his word for “tasty” but ephemeral mathematical curiosities—and the “acorns” from which a “mighty oak can develop.”

The Leiden declaration summarizes all of this admirably. But it omits a separate threat presented by AI concerning what we could call the “sapiential value” of mathematics. This phrase is taken from the title of a lecture delivered by Ennio De Giorgi at the University of Lecce in 1994. De Giorgi is best known today for his groundbreaking work on the regularity of solutions to elliptic equations. He was also a devout Catholic who would often take himself away to pray to God when stuck on a particularly recalcitrant theorem. He gave the most succinct definition of this sapiential value during an interview around the time of the Lecce address, in the context of a discussion about the “unreasonable effectiveness of mathematics in the natural sciences,” an old problem first posed by Hungarian-American theoretical physicist Eugene Wigner: 

For me the most suggestive indicator is in the Book of Proverbs, one of the most ancient books of the Bible, which at a certain point says that wisdom (which is wider than mathematics) was with God when He created the world and that this wisdom is to be found by men who search for it and adore it. Mathematics is one of the most significant manifestations of the love of wisdom . . . the world is made of things both visible and invisible and mathematics is the unique science with the capacity to pass from the observations of visible things to the imagination of things invisible.

In other words, there is a reason why the infernal high command might want to sow chaos among the mathematicians: They are the custodians of a tradition of human wisdom, one that, to borrow from John Henry Newman, has been the source of a continuous “enlargement of mind.” And, as with the other branches of wisdom, mathematics depends on and contributes to the cultivation of what De Giorgi calls the “sapiential virtues,” such as humility, hope, and conviviality. The last had a particular significance for De Giorgi, suggesting “shared knowledge, friendship, the search for mutual understanding”—all essential to mathematical research—while also recalling both the banquet of wisdom in Proverbs and that other great wisdom-poem, Dante’s Convivio. 

In the same lecture in Lecce, De Giorgi cautioned that the sapiential value of mathematics disappears when it is treated as a “little machine where one pushes the problem and immediately gets the solution.” Though he did all of his own work with pencil and paper, he was not averse to technology and saw in the computer an exciting source of new and strange mathematics. He also believed, however, that its effects would be fatal if it were ever treated as a substitute for human imagination. In the words of another outstanding Catholic mathematician of the last century, Marston Morse, if all of mathematics were duplicable by a machine, we would do well to “turn to another and truer art.” 

There may be a way of framing these ideas that would be acceptable to the Leiden signatories, irrespective of their religious beliefs. One potential model is the Universal Declaration of Human Rights, a document written in the secular language of international law, which De Giorgi nonetheless counted among the “highest expressions of the human mind.” Indeed, he believed there was a deep harmony between the practice of mathematics and the reality of human rights. As he put it, mathematics flourishes where men are allowed to “freely search for wisdom through their own personal conviction,” and this in turn depends on an unconditional respect for the “dignity of man.” 

But De Giorgi also believed that the love of wisdom leads beyond the boundaries of this world. He would no doubt have agreed with Aquinas that the branches of wisdom converge finally ad immortalitatis regnum (in the kingdom of the immortal). He said as much shortly before his own death with the sort of holy matter-of-factness that would no doubt shock the average research mathematician working today: 

The idea of the resurrection—that life does not end in the brief arc of years which we have here, that even our loved ones who have passed still live in some way—is one of the fundamental elements of my life and even of my research . . . I see this as a journey through which, until the end, one must love knowledge completely, expecting that this love will continue in another form, even after death.

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