The day before I finalized my previous post, I was attending our Institute’s annual public lecture: the Bernoulli Lecture, named after Johann Bernoulli, who worked for a while at the University of Groningen.

This year’s lecture, Can Machines Do Mathematics?, was organized with Studium Generale Groningen and given by Johan Commelin, mathematician at the University of Groningen, director of the Mathlib initiative and co-organizer of the recent workshop on Mechanization and Mathematical Research at the Lorentz Center (it ended with a public symposium which is well worth watching as well).

This was a very timely topic, spanning across the boundaries of our departments and possibly more, with finally a young, not to mention excellent, speaker. I genuinely think he did an incredible job, and I hope that you have one hour, so that you can stop reading here, watch the lecture and then come back to read the rest of this post. Please do, the rest will contain spoilers.

What I loved most was that the lecture ended with a call to action, inviting the mathematical community to avoid being passive or succumbing to despair, and instead rise in a dialogue to set our values and shape our future. This could not resonate more with me. Johan also announced a manifesto on this call to action, born from the discussion that mathematicians, philosophers and historians had at the Lorentz Center. The overall gist is already present in one of the last slides of the lecture, which I quote here:

Five focus areas and recommendations:

1. Values — root technology in our epistemic and aesthetic values
2. Practice — engage with how our relationship to mathematics will change
3. Teaching — emphasize posing problems, communicating ideas, evaluating arguments
4. Technology — build open-source, academically governed infrastructure
5. Ethics — develop a living statement of ethical principles

If you have read any of my previous posts on these topics, you can already see how these recommendations align perfectly with my own views. I was reeeeally curious to read the extended elaboration and, finally, this morning, the manifesto appeared on the arxiv: Shaping the Future of Mathematics in the Age of AI by Johan Commelin, Mateja Jamnik, Rodrigo Ochigame, Lenny Taelman, Akshay Venkatesh.

It is just four pages. So, again, stop reading, and take ten minutes to go read the manifesto!

It sets the stage by defining what the mathematical community is in a broad sense, and as promised, clarifies the meaning of the five points above. I really like that it acknowledges the many different views and opinions that exist within our community and is first and foremost a call to dialogue and to action.

And it is exactly due to this variety of views that the start should be a “deliberate examination of our values in all their complexity”, with an open attitude aimed at shaping our shared future and with the urgency imposed by the “scale and speed of recent developments”. Importantly, this examination should “remain firmly rooted in our own epistemic and aesthetic values”. Here, ’epistemic’, if you are unfamiliar with the term, refers to what guides our pursuit of knowledge.

The second point is a look at the mathematical practice itself. Here the authors raise an important question: “what will qualify as a mathematical discovery” in the future? And, crucially, this ultimately leads to asking “how can we nurture the next generation of mathematicians when the very nature of our practice is in flux?”

That last question landed close to home. After the lecture, many of our students and PhD candidates were sad or afraid about their future. While this is very natural, we should not forget the deeply social nature of mathematics (at least if you agree with my arguments) and that, as such, it is a practice that is deeply rooted in our communities. So we are still agents of change, and we can steer it. Setting the pace of change aside for a moment, we also need to remember that our discipline has been changing for centuries, and that we have been able to adapt to it. We can do it again, and we should do it together. These are precisely the kinds of questions the manifesto’s third point sets out to address, and hopefully it will lead to answers that will offer much-needed peace of mind.

The discussion then moves to how teaching should change. I honestly think that the explosion of AI only accelerated a change that was already needed. I fully agree that we should “emphasize a broader range of skills, including posing problems, communicating ideas, and critiquing purported logical arguments”. I also think that we should encourage students to engage with the history and philosophy of mathematics.

It is long past time we left our ivory towers. With a growing divide between academia and society, which started well before the current AI hype, we now need more than ever to connect with the world. To do that we must reconnect with our history and our humanity, and we must reach out and share the magic that permeates what we do instead of keeping it with ourselves. This requires learning to talk to non-mathematicians, adjusting our language and our message to our interlocutors, and learning to listen to them as well. These skills are useful to us as well as our students, and it is about time we started teaching them.

Of course, a discussion on mechanization cannot be complete without mentioning technology. Here there is a crucial emphasis on openness and transparency. Whatever we do, we should promote the circulation of ideas and reduce the divide. We have been pioneers of this, with arXiv. Let’s keep pioneering and foster open knowledge and open technology! Building such an infrastructure is also a good way to “maintain intellectual independence”.

And last but not least, all of this should be done within a strong ethical framework. Here the authors call for a “living statement of ethical principles to guide academic mathematicians and mathematical institutions in their interactions with AI systems and developers”. Apparently a group of people from the workshop is already working on a first draft. I am really looking forward to reading it!

The manifesto closes with its own call to action. Fittingly, it reads ELDeR: Engage, Learn, Discuss, Reflect. We are not alone; it is time to stand together.