DIGITAL LIFE

Mathematics and AI

The creator of ChatGPT announced on Tuesday that one of its AI models had solved a century-old mathematical equation in less than four days.

That announcement was overshadowed by public statements from Australian mathematician Tristan Buckmaster, who questioned the similarities between his own work—predating OpenAI's project—and that of the AI ​​model.

Artificial intelligence "offers the potential to strengthen and accelerate the study and understanding of mathematics," renowned researchers wrote in an open letter on Friday.

"The mathematics profession will have to adapt to these changes in various ways," they explained.

For mathematicians, it is "human decisions controlling this technology" that will determine, "to a large extent," whether it proves beneficial to the field or has a destructive effect.

The 25 medalists therefore call on leading AI companies to "urgently address" these questions.

Since last year, several of these companies have reported breakthroughs in the world of mathematics, but the solution to the Navier-Stokes equation, announced on Tuesday, is the most notable.

It is one of the seven "Millennium Prize Problems" selected in 2000 by the research-supporting Clay Mathematics Institute, each carrying a $1 million prize.

Over the past few months, the mathematical capabilities of large language models (LLMs) have improved dramatically, to the point where they can solve major unsolved problems in many fields of mathematics. However, the push by AI companies to solve mathematical problems as a benchmark is detrimental to the field of mathematics and to the mathematical community. The goals of AI companies and those of the mathematical community are severely misaligned. We view these as part of broader alignment issues affecting other scientific and creative professions, as well as society as a whole.

Mathematical research focuses on understanding the basic structures of shapes, numbers, and natural phenomena. Over the course of generations, it has built up a vast body of sophisticated ideas, methods, abstractions, and other tools for understanding the mathematical landscape. In turn, modern technologies and sciences are based on mathematical tools.

Famous problems have often served as landmarks and beacons against which one can gauge a deeper understanding of this field. Solving one of these problems has been a sure sign of new insights and interesting methods, which would then be studied by a community of mathematicians through a long and arduous process of talks, discussions, and simplifications. At the end of this process, one will ideally find a textbook presentation of the results suitable for any graduate or even undergraduate student to study. Some of these mathematical ideas continue their journey even further, becoming—decades or centuries later—tools that are understood and used by the general public.

The mathematical community functions, in many ways, as a miniature version of humanity. It consists of individuals who use a wide variety of different approaches, united by core values. The most precious resources of our profession are students and ideas, and we nurture them with great care. We feel responsible for helping them reach their full potential, until they can stand on their own in the mathematical world. For students, we often suggest problems with the primary goal of developing skills that will position them well for advances in research and beyond. We disseminate our ideas through talks, private discussions, and carefully crafted write-ups, connecting them to the previous ideas of others. These processes invariably take time and are based on human interaction.

In recent months, AI’s success in solving major mathematical problems has made headlines even outside mathematical circles. But problem-solving is only a tool and a means to an end—the primary goal of conceptual understanding and insight. Forgetting this in the world of AI may cause the tool to work against that primary goal. Indeed, the mass production of “true/false” statements at an ever-faster pace could destroy fertile ground instead of breathing life into new ideas.

Often these solutions are announced in a hurry, leaving no time for a proper write-up, the identification of new methods and ideas, and the citation of relevant prior work by others. As in all creative professions, this raises serious questions regarding attribution and plagiarism. Moreover, without the willing mathematicians who must oversee their development and integration into the mathematical canon, AI-conceived ideas would never fully come to life, and the crucial human chain of transmission among mathematicians would be lost.

We are witnessing a general threat to intellectual work, with a misalignment between the outcome of AI use and its original purpose. In many fields and activities, years of training have traditionally served not only to produce a final answer or product, but also to develop understanding and the ability to formulate new questions and ideas. However, building on a vast body of previous human work, AI systems are becoming increasingly capable of producing the results of such work directly, and these goals are no longer aligned. The challenges the mathematical community now faces are similar to those facing other scientific and creative professions, and point to challenges that all of humanity may face: how to ensure that, as AI changes the way work is done, we do not lose sight of what that work was meant to achieve in the first place.

AI offers the potential to enhance and accelerate genuine mathematical study and understanding. The field of mathematics will need to adapt to these changes in several ways. However, whether these changes ultimately benefit the field or have a destructive effect will largely depend on the decisions made by the people in control of this new technology.

Mathematicians rebel against AI...Here is the statement, signed by Terry Tao among many other math notables, most of you probably have read it by now.  I do not accept the most cynical interpretations of this proclamation.  Some of you for instance may recall that I made and indeed stressed a similar point in the last chapter of my recent “generative book” on marginalism.  In some near future, perhaps fewer economists will carry around marginalist insights and modes of thought in their heads, since you can just get the right answer by pressing the proverbial button on the AI.

I find this future disturbing, and not altogether pleasant for me personally, given how much personal status I have wrapped up in particular modes of economic thought.  Yet I also know the Bastiat distinction between the seen and the unseen, and I expect the benefits to economic science from AI will be enormous, even if current practitioners cannot foresee most of those benefits today.

I do very much differ with at least one part of the mathematicians’ proclamation.  They write: “…whether these changes ultimately benefit the field or have a destructive effect will in large part be determined by the decisions of the humans in control of this new technology.”  There is no actual argument for that proposition, and I would sooner expect that the main “action variable” is how well the mathematicians adapt to the new reality.  For instance there is nothing stopping the mathematics community from awarding status, pay, and promotions to people who “fill in the important blanks in math understanding,” even if an AI already has proven or disproven the underlying theorems.  If that kind of work is so important, we still can do it and reward it professionally.  In the meantime, I expect the funding for mathematics, and the interest in the topic, to rise considerably, at least in the medium term.  All of a sudden, math matters much more than it used to, all the more so if P vs. NP happens to go the wrong way, or if the distribution of the primes turns out to be a little too predictable.

The mathematicians may not in every way enjoy being the subordinates or handmaidens of the AIs, but that is a change in status they simply will have to get used to, just as I realize AIs someday will end up as better column and blog writers than I am.  I do not look to the companies — which I fully expect to “act like companies” — to somehow manage, moderate, or assuage that pending trend.  It really is up to me to parlay my current intellectual portfolio into new, more AI-compatible intellectual and yes also marketing approaches.  I’ve been given plenty of “legs up” along the way already, as is true for the Fields Medal winners as well, and it is up to me to figure out how to contribute in the future.

Might someone not invent/discover/prompt a way to use AIs to produce, articulate, and teach “more mathematical understanding” along the way?  I get that solving famous dramatic math problems is the current commercial priority of the major AI companies.  But as the AI space grows, these other paths hardly seem unlikely to me, and in fact the human mathematicians are the ones who can do the most to lead the way along those dimensions.

In this regard the current manifestation of complaints seems oddly early.  “I didn’t like the first week or two of your intellectual revolution” is an accurate, and perhaps better reframed way of putting it.  At which point perhaps a bit of patience is needed before anything else?  These days, we all have more mathematical resources at our disposal, and so a bit of celebration is in order as well.

mundophone