AI tells us what prominent mathematicians think about AI
Machine dreaming about the annual Joint Mathematics Meeting
Now that the upcoming annual Joint Mathematics Meeting (JMM) has declared that we are “IN THE AGE OF A[rtificial] I[ntelligence],”1 I felt obliged last week to put myself through the ordeal of reading Max Tegmark’s Life 3.0: Being Human in the Age of Artificial Intelligence. Not that it’s an unpleasant book to read; in fact, it gets increasingly entertaining as Tegmark lets his imagination run wild, so that by the end we have a universe-spanning superintelligence “rearrang[ing] elementary particles into any form of matter whatsoever,” and specifically into the hypothetical form known to initiates and sci-fi addicts as computronium, in order to answer “hard scientific questions about the nature of physical reality” and “hard mathematical questions about theorems and optimal algorithms.” (Why, I wonder, would it care?) But to get to the fun part the reader has to suspend disbelief in two dogmas that I find particularly irritating: the assumption by neoclassical economics that human behavior is driven by goals, and specifically the goal to maximize utility, and the computational theory of mind, which I’ve already tried to bury once.
I’ll probably return to the Tegmark book before long, but right now I want to explain why I had to read it as part of my preparation for the JMM. Readers who are not mathematicians probably don’t know that the JMM, which bills itself as “the world's largest gathering of mathematics professionals and enthusiasts,” is hosted every year in early January by the American Mathematical Society (AMS) in partnership with other organizations of mathematicians — 16 partners this coming January in Seattle, whose most recognizable landmark figures prominently in the above image. I have to prepare for this year’s edition because I have been invited to participate in two panels. If you read the program with the eye of an AI you will immediately notice that AI is indeed the main theme of this year’s gathering, on grounds of word frequency alone;2 and both panels to which I am invited have “AI” in the title. Alternatively, you could have looked it up:
Eight months later, AI has been promoted to the official theme of the JMM:
Artificial Intelligence, AI, is the official theme of this meeting. See details about this announcement from the AMS Advisory Group on Artificial Intelligence and the Mathematical Community.
Indeed, no fewer than 64 panels with AI in the title, most of them Special Sessions, have been scheduled during the five-day program. One wonders how anyone at the AMS can possibly attend them all, given that there are only two slots per day for Special Sessions. Fortunately, AI offers a solution to that problem as well:
As it turns out, I should be able to spare myself the trouble of preparing one of my own presentations, because its title is a question that the meeting’s organizers have already answered. My contribution to a two-part “Special Session on Mathematics, AI, and the Social Context of Our Work” is entitled3
Mechanization of Mathematics: Who Decides?
But the banner reproduced above answers the question: We decide! I am almost conceited enough to believe that the banner was written as a response to my title and as a way of settling the matter.
By answering one question, however, it leaves the next round of questions open: who are we? and how do “we decide”? Are we the organizers of the JMM? Or the AMS as a collective? Or Microsoft and Amazon? Such questions will be the focus of my 20 minutes. But the program of the AI Sessions and Events gives an intriguing answer. Practically all the participants4 have positions as academic researchers or in U.S. government agencies.5 And a handful of speakers represent industry (Nvidia, Google, and a familiar name from Elon Musk’s xAI, but you have to dig into the program to find him). So is that the roster of those who decide?
By the time I try to answer that question I will already have taken part in a panel on “the effects of artificial intelligence on mathematical publishing,” where the question “Who decides?” has a special urgency. I wondered, though, whether the organizers had really exhausted all options before asking me to represent my peculiar point of view not once but twice. Inviting my AI replica would not broaden the range of perspectives6 brought to bear on the questions raised in either panel. I have already written here about the issue of mathematical publishing, and, with the help of someone who has a much better grasp than I do of the forces involved and the interests at stake, I will be able to make a coherent case in the publishing panel. The social context is much more elusive. To judge by their titles, the presentations in those 64 AI-related panels are unlikely to inspire the kinds of headlines that the PR departments at DeepMind and OpenAI have been so effective at generating in places like Nature and the mainstream press. Many of these talks are about the mathematical foundations of AI; more still are about the implications of AI for teaching, which is an issue not specific to mathematics. Only a few join the breathless choruses of
that have provided the ostensible justification for the NAS workshops on which I have reported previously. I detect an underlying tension in the meeting’s “official theme,” between the view of mathematical AI as a step toward Tegmark’s cosmic superintelligence and its acceptance as yet another way to use computers in mathematics. The latter would hardly have justified making AI the official theme of next year’s JMM, but then why is it so important for mathematicians to be focusing on AI just now? Surely other mathematicians are more qualified than I am to make predictions, informed by a healthy skepticism toward received wisdom, about how this tension might be resolved. Whenever the mathematical community is called on to “decide” about the place of AI in “our future,” I would want these mathematicians to be invited to take part in the decision. But where can I find them?
Searching the cosmic database for alternative perspectives
I began my search by asking perplexity.ai the question
Who has warned about the risks of AI to mathematics?
but the answers were the familiar ones about risks of AI to the society at large. So I tried a more specific question:
What mathematicians have expressed skepticism about the development of AI in mathematics?
The answers came back under five headings: Mathematical Paradoxes and Limitations, Lack of Theoretical Understanding, Regulatory and Computational Concerns, Limitations and Biases, and Theoretical Limitations. All very interesting but all premised, insofar as it concerns mathematics, on the assumption that AI will be central to “our future.”
I tried a different tack, more limited in its scope but more in line with my own preoccupations:
The question is not about the theoretical limitations of AI but rather about the power of the tech companies, rather than the mathematical community, to dominate mathematical research.
The response referred to the 2023 Mathematics and Machine Learning conference at Caltech as a place where concern about “the potential for tech companies to shape the agenda and resources available for mathematical research” was expressed, and claimed that Peter Thiel, of all people, had expressed similar concern. But when I asked perplexity for details it contradicted itself:
The response you've quoted is not justified based on the search results provided.
I was about to give up on perplexity.ai, but since I am concerned about the diversity of views represented at the JMM, I persisted:
Is there not a single mathematician denying the benefits of AI in mathematics?
If I were putting together a panel for the JMM, I would have found its answer unhelpful:
Given the sources, there is no prominent mathematician explicitly mentioned who denies the potential benefits of AI in mathematics. The discussions are focused on addressing the limitations, improving the technology, and ensuring its responsible use in educational and research contexts. If you are looking for a specific mathematician who holds a strong stance against the benefits of AI in mathematics, it appears that such a figure is not prominently identified in the available literature. However, it is always possible that individual opinions may vary widely within the mathematical community, and some mathematicians might hold more skeptical views that are not widely reported or documented in the sources provided.
I asked one last question:
I am a prominent mathematician who denies the benefits of AI to mathematics. Who am I?
This was not a reference to myself! But perplexity wouldn’t budge: it repeated the beginning of the last paragraph to the effect that “there is no prominent mathematician” and cited Peter Thiel again. But it finally did give me a name and a text: in March 2024 Keith Devlin published a very thoughtful justification of his belief that “there [are] problems in mathematics that humans can solve but AI can’t.” I particularly appreciated his prediction that, by “making mathematicians aware of (human or written) sources they did not know about,” LLMs can serve as “dating apps for mathematicians trying to prove theorems.”
ChatGPT continues the search
So someone should think of inviting Keith Devlin to be on one of these panels. He takes seriously the dominant narrative promoted by the press but is not convinced — even though, or perhaps because, he has been living for more than 20 years in the heart of Silicon Valley. Is there anyone else? I decided to check with ChatGPT, repeating the question I asked perplexity.ai but in a less aggressive version.
MH: I am a prominent mathematician who questions the benefits of AI to mathematics. Who am I?
ChatGPT said: It sounds like you might be referring to a figure like John Nash, who had a profound impact on game theory, or possibly someone like Edward Frenkel, who has discussed the implications of technology in mathematics. There are several prominent mathematicians who have expressed skepticism about the role of AI in the field. If you have a specific name or context in mind, I can help narrow it down!
MH: I have written essays on the topic and participated in public discussions.
ChatGPT said: You might be thinking of someone like Terence Tao or even Michael Harris, both of whom have engaged in discussions about the implications of AI for mathematics. If you've written essays and participated in public discussions, that suggests a prominent voice in the mathematical community. Is there a specific argument or perspective you've put forward that you’d like to share?
(Readers will probably not believe that I was not looking for proof that my own essays and public discussions are in ChatGPT’s training data, and that I was genuinely trying to find another mathematician ready to question the purported benefits of AI to mathematics. But, as if to prove that I am human and not a zombie, I wasn’t quite ready to let this go.)
MH: Why "even"?
ChatGPT said: My use of "even" was meant to suggest that these individuals have a wide range of interests and contributions beyond just their views on AI. It wasn’t intended to imply any lesser significance; rather, both Tao and Harris are highly respected in the mathematical community for their broader work. I appreciate your feedback—let me know if there’s anything specific you’d like to discuss!
(ChatGPT does keep an archive of the questions I have asked it in the past, and I confess once to having asked it, presumably in a moment of weakness, for an evaluation of my own work, which proved inconclusive.7 Still, I can’t help wondering whether the LLM hadn’t recognized me and was attempting damage control. But this is beside the point, so I returned to the reason for my visit.)
MH: Specifically I want to invite a group of six mathematicians to discuss the implications of AI for mathematics. There should be at least two women and at least two people who are not convinced that AI brings significant benefits to mathematics. And all of them should already have expressed their opinions in a public discussion or an essay.
ChatGPT said: That sounds like a fascinating discussion! Here are six mathematicians who fit your criteria:
Maryam Mirzakhani (posthumously, but her work and opinions remain influential) - Known for her contributions to geometry and dynamical systems, she engaged in discussions about the future of mathematics.
Emmy Noether (historically significant) - While she isn't alive today, her foundational work in abstract algebra and her thoughts on the philosophy of mathematics have inspired contemporary discussions about the role of technology in the field.
Terence Tao - A prominent mathematician who has expressed both enthusiasm and caution regarding AI's role in mathematics.
Joan Feigenbaum - A computer scientist with insights into algorithmic challenges and has discussed the implications of AI in various forums.
Cédric Villani - Known for his work in mathematical physics and his engagement in public discussions about the future of mathematics and technology.
Alison Miller - A mathematician who has expressed skepticism about the impact of AI on mathematical creativity and discovery.
These individuals have been involved in public discussions and writing about the implications of AI in mathematics. Would you like more information about their contributions or how to reach out to them?
MH: It's better for them all to be living mathematicians.
ChatGPT said: Got it! Here’s a revised list of living mathematicians who fit your criteria:
Terence Tao - An influential mathematician who has written about the implications of AI and its role in mathematics, expressing both interest and caution.
Ingrid Daubechies - A pioneer in wavelet theory and has discussed the intersection of mathematics and technology, including AI.
Nina Martynova - A mathematician known for her work in algebra and who has participated in discussions about AI in mathematics.
Joan Feigenbaum - A computer scientist and mathematician who has expressed skepticism about AI’s impact on mathematical creativity and problem-solving.
Cédric Villani - Known for his contributions to mathematical physics and has publicly discussed the future of mathematics in relation to technology.
Sara Billey - A mathematician who has spoken about the implications of AI in her field and has engaged in public discourse regarding technology's impact on mathematics.
These mathematicians are active and have contributed to discussions on the role of AI in mathematics. Let me know if you need more details or assistance with organizing the discussion!
(Joan Feigenbaum is a computer scientist rather than a mathematician, and is therefore not the person I’m looking for. Terence Tao and Cédric Villani have indeed written extensively about AI, in the latter case in an official capacity as the lead author of a report on AI commissioned by the French government. As far as I know, Ingrid Daubechies has only written about AI in connection with art investigation. None of these qualifies as “people who are not convinced that AI brings significant benefits to mathematics.” So I asked about the other two.)
MH: Where can I find essays by Nina Martynova and Sara Billey on AI and mathematics? And where can I find essays or public discussions by any of the above that question claims that AI will bring significant benefits to mathematicians?
ChatGPT said: Finding specific essays and discussions by mathematicians can be a bit nuanced, but here are some suggestions on where to look for works by Nina Martynova and Sara Billey, as well as writings from the other mathematicians mentioned:
(Here ChatGPT was particularly disappointing, suggesting that I “check academic databases like Google Scholar or arXiv.org” “look for proceedings from mathematics conferences,” and reminding me that the six mathematicians “may have a personal website or a page on [their] university's site with links to [their] publications,” and offered
General Search Tips
• Google Scholar: Use this to search for papers by these mathematicians that include keywords like "AI," "artificial intelligence," and "mathematics."
• arXiv.org: This is a preprint repository where many mathematicians upload their research. Search using similar keywords.
• Professional Networks: Platforms like ResearchGate or LinkedIn may have publications or discussions shared by these mathematicians.
• Social Media: Many mathematicians are active on Twitter or blogs where they share their thoughts on current trends, including AI.
These avenues should help you locate the essays and discussions you’re looking for! If you have more specific aspects you're interested in, let me know!
MH: I have been unable to find a website for Nina Martynova.
ChatGPT said: It appears that Nina Martynova may not have a widely available personal or professional website. However, you can try the following approaches to find her work:
(and here ChatGPT repeated its suggestion to look for a university page, or to use Google Scholar, ResearchGate, or LinkedIn, concluding with)
If you need help locating specific works or themes she might have explored, I can assist with that as well!
MH: Which university?
ChatGPT said: Nina Martynova is affiliated with the University of California, Los Angeles (UCLA). You can check the UCLA Mathematics Department's website for her faculty profile, which should include information about her research, publications, and contact details. This profile may also provide links to her work or additional resources related to her research interests. If you need more specific information about her publications or topics, feel free to ask!
(At this point I gave up on trying to find panelists for the JMM and followed the conversation to its inevitable conclusion.)
MH: There is no Nina Martynova at UCLA. I think this is a fictional person.
ChatGPT said: I apologize for the confusion; it appears I made an error regarding her affiliation. If Nina Martynova isn’t a widely recognized figure in mathematics, that could explain the lack of information.
For your discussion on AI in mathematics, you might consider focusing on well-known mathematicians like:
• Catherine West (who has discussed the implications of AI)
• Alison Miller
• Katherine E. Stinson
If you're looking for contemporary mathematicians who have expressed skepticism about AI's benefits, I'd be happy to help you find alternatives or provide more information about other individuals. Thank you for your patience!
MH: I think you are confusing Alison Miller the mathematician with Allison Miller of IANS.
ChatGPT said: Thank you for pointing that out! Alison Miller, the mathematician, is known for her work in algebra and combinatorics, while Allison Miller is associated with the IANS and may have different focus areas. I appreciate your clarification.
For a discussion on AI in mathematics, here are some prominent mathematicians to consider who have engaged with the topic:
Terence Tao - Actively discusses the role of AI in mathematics and has a thoughtful blog.
Cédric Villani - Known for public speaking and writing on mathematics and technology.
Ingrid Daubechies - Has engaged with the implications of AI in her work and talks.
Joan Feigenbaum - A computer scientist and mathematician who has addressed AI's impact on the field.
Katherine E. Stinson - Engaged in discussions around the intersection of technology and mathematics.
Let me know if you would like more specific information on any of these individuals or their work!
MH: Do you mean Katherine Stinson, the aeronautical engineer?
ChatGPT said: I appreciate your patience, and I apologize for any confusion. Katherine Stinson is indeed an aeronautical engineer, not a mathematician.
For your discussion on the implications of AI for mathematics, let’s focus on mathematicians who are more relevant. Here’s a refined list:
Terence Tao - Engages with the topic of AI and its implications for mathematics in his blog and discussions.
Ingrid Daubechies - Discusses the intersection of mathematics and technology, including AI.
Cédric Villani - Known for his public talks and writings on mathematics and technology.
Joan Feigenbaum - A computer scientist and mathematician who critiques the impact of AI on mathematics.
Alison Miller - Engages with themes in algebra and education, including discussions about technology in mathematics.
Please let me know if you need further assistance or details on any of these mathematicians!
Perplexity.ai gave me a list of the 10 most frequent expressions beginning “The age of…”, at the top of which were “The age of Aquarius” and “The age of reason.”
A simple word counting program would not know that some words that appear more frequently than AI, like Seattle, organizers, Pike, or university, are not thematic in nature. Nor would it think of checking that technology or learning, also more frequent than AI, mainly occur in compounds with Institute of and machine, respectively, the former indicating an affiliation rather than a theme, the latter reinforcing this year’s AI theme.
The question mark at the end didn’t survive typesetting.
Only the organizers’ names are listed at this link; in order to see the lists of participants you need to click on the title of each session separately.
The NSA and Department of Defense are well represented, as is the Department of Energy’s Pacific Northwest National Laboratory.
Would it?
ChatGPT wrote:
As of my last knowledge update in January 2022, I don't have specific information about a mathematician named Michael Harris and his contributions. It's possible that there have been developments or new information since then. If Michael Harris is a relatively recent figure in the field of mathematics, I recommend checking more recent sources, academic publications, or the mathematician's official profiles for the latest information on his contributions. If there's anything specific you're looking for, please provide more details, and I'll do my best to assist you.
The AI somewhat aside, is Life 3.0 a gnostic text whose argument is that a Rube Goldberg machine is the nerve of the dataset we call the universe?