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Conclusion

Jordan Ellenberg, University of Wisconsin–Madison, provided concluding remarks to wrap up the 3-day workshop. He emphasized how multifarious the field of artificial intelligence (AI) to assist mathematical reasoning is, citing the presentations on topics ranging from reinforcement learning to generate counterexamples to conjectures, to formal language to codify mathematical reasoning in machine-processable forms. AI may even assist aspects of mathematics outside of proof such as conjecture forming. Advancement in this area will be a two-way street, he said, requiring communication between groups of researchers who speak and think differently.

Ellenberg pointed out that translation is an important theme in many senses. It is vital for communication between researchers. In addition, when one receives output from a machine, the information conveyed should be understandable, but even further, one aims to understand the workings of the language itself. He stressed that although this may feel new to mathematicians, in reality this has always been the experience of mathematicians. Mathematicians first encounter strange new phenomena; then the phenomena become familiar; and finally, they become legible, with true understanding of the underlying structures. He shared André Weil’s thoughts about the analogies between number fields and function fields: “These texts are the only source of knowledge about the languages in which they are written. In each column, we understand only fragments” (Weil 1960). Ellenberg underscored that translation between fields is an old idea.

Next, Ellenberg characterized AI as being able to support rapid and eccentric exploration. This exploration is considered rapid because of how quickly the field moves and eccentric because it works differently from how humans alone might operate. He emphasized that this eccentricity is what provides value.

Raising the theme of understanding that appeared throughout the workshop, Ellenberg pondered what it means to understand. He observed that “artificial intelligence” is now a common term, but “artificial understanding” is not—understanding is something associated with humans. And mathematics is primarily motivated by growing, enlarging, and deepening human understanding. The goal of formalizing mathematical reasoning is not to eliminate the need for informal reasoning but to aid understanding. Interpretability in machine learning (ML) is a related issue, and mathematics can be a testbed for interpretability, he said, in part because mathematics is such an established discipline that has consensus on what counts as insight. Having this consensus can aid the study of what interpretability in ML means.

Ellenberg cited the simile from Rebecca Willett, University of Chicago, that investing in ML without understanding mathematical foundations is like investing in healthcare without understanding biology. He considered it an apt statement precisely because societies do invest in healthcare without understanding biology—there is so much still not understood in biology—and the two areas work in an iterative interplay. Applied work will leap ahead while biology seeks to understand why certain treatments are effective, and foundational biology guides applied work in directions more likely to be productive. The two support one another, as AI research and mathematics do.

Delving into how invigorating collaboration can be, Ellenberg urged all pure mathematicians to collaborate with others whose goals are aligned but not completely the same. The charm of collaboration is that unique perspectives are brought together. He recalled a statement in the workshop overview by Moshe Vardi, Rice University, that if one is not authentically open to changing one’s mind when speaking with another person, it is not really a conversation. Ellenberg suggested that while this standard is a high bar, it can be aspirational for research communities and inspire better collaboration.

Ellenberg expressed his belief in historical incrementalism, imagining that the most likely future resembles the past. Machines have been assisting mathematics for a long time by recategorizing certain tasks as computation instead of mathematics, freeing humans to explore further. Everything being done right now is new, but in another sense it is not; “it is new in a way that rhymes with the past,” he said. Summarizing the key themes of the workshop—translation, exploration, and

understanding—Ellenberg shared that despite the barriers discussed, these three actions could allow AI to progress from assisting mathematics to collaborating with mathematics.

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