A value-based approach to mathematics

[This is a guest post by Ivan Corwin. This blog post was initially written in a different file format and converted using AI. — T.]

Mathematics and those mathematicians who guide its development have been central to societal advancement since antiquity. AI, like many tools that were invented by mathematicians (e.g., the abacus, slide rule, calculator, and computer), holds the potential to continue to advance this positive trajectory. Indeed, in each epoch of history where a new mathematical tool becomes available, the role and responsibility of mathematicians to society has grown. At the same time, it comes with hazards and challenges that if improperly handled will impede rather than accelerate progress and impact on the world.

Most of the value that flows into the world from mathematics and the work of mathematicians will not be replaced by AI, but could be magnified. In applied fields outside of mathematics, what if AI could help bring tools, ideas and modes of thought that mathematicians continue to develop to bear on real world problems and in areas where it was not anticipated. This could lead to new interactions between those fields and mathematicians and help draw mathematics closer to its immense applications. In mathematical research, AI systems have shown promise in connecting and sharing ideas between fields and in using those connections to solve some concretely posed problems (e.g. the unit distance conjecture). It would be a great outcome if AI helps to reduce the siloing of mathematical inquiry and knowledge and allow mathematicians to ask and pursue bigger questions and grapple with more complex systems.

Improper usage of AI offloads thinking, understanding and learning, thus mortgaging future mathematical development. Students and researchers may not experience the growth that comes from failing and struggling with a problem, and may become risk-averse and engage more with the AI than each other. We have already seen a burst in the number of arXiv postings and problems being solved by AI. If AI is used to indiscriminately solve open problems without mathematicians being deeply involved, it may not produce much societal value from the process or even the product. AI companies have built their tools using the freely shared work and expertise of generations of mathematicians, yet seem to have very little concern for this possibility. They are concerned with their valuation and do so at the cost of trampling on the very community norms that led to the free flow of ideas on which they are built.

I advocate for a value-based approach to navigating these waters. Moreover, based on this approach, the mathematical community should clearly articulate and communicate its value to society and demonstrate how that value will only increase with the mass adoption of AI. In my framing, there are four loci where the work of mathematicians produces value. This includes the value from cutting edge research, but is also grounded in the broader influence of mathematical thinking on society as well as the value of training and mentoring students at all levels in the traditions and modes of thought of mathematics.

Society. History is replete with examples of specific mathematical ideas and methods that have driven progress in society and revolutionized science and knowledge. I am a probabilist and though probability theory is a late-comer into the canon of mathematics (early 20th century), it has been a clear success story in terms of impact. The central limit theorem and extreme value distributions figure centrally in classical statistics. Stochastic processes like Brownian motion fuel the analysis behind the world of finance, and model real diffusive systems across science. Random matrices and spin-glasses provide a starting point to understand how long to train LLMs and which algorithms work best. Understanding of why such probabilistic AI systems work and how and why they fail will require study of novel probabilistic models as well as new vision.

In all of these examples, problem solving came at the last stage of maturation of a topic, and much of the creativity and innovation came earlier in identifying new areas of interest, developing intuition and predictions, testing that and then formulating precise questions. In this way, though it is often not presented this way, mathematics is quite similar to other sciences.

It is not just mathematical models and results that impact society, but rather the very essence of mathematics — its modes of thought and language slowly seep deep into the way humanity approaches the world. Toddlers learn to count and young students learn number systems, both basic forms of abstraction that map different systems onto the same internal understanding. Debaters, writers, lawyers and all of us learn to craft and structure arguments based on ideas of rigor and logic. Probabilistic ideas like conditioning, expected value, sampling and independence have become a powerful and widely spoken language to deal with uncertainty and chance.

Mathematics is not just of practical importance. Mathematicians ensure that it is an ever flowing source of creativity, wonder, beauty and playful joy. Children exercise their minds musing over puzzles, patterns and games that are all inherently mathematical. Fractals, chaos and the process of emergent behavior and phase transitions captures the imagination of the many beyond the world of mathematics.

The advances, applications and implications of mathematics seldom come in the form of a concrete problem to solve or conjecture to resolve. Rather, they represent the true form of mathematics as it has been for thousands of years. Math is the pursuit of new knowledge and understanding, both in the formal systems in which it is constructed but also in contact with the real world which it serves. That process is non-linear and unpredictable. Sometimes it is driven by societal need, and sometimes other factors such as curiosity, beauty and structure drive mathematical advances forward long before an application becomes apparent. For example, probability theory is built on measure theory, quantum mechanics on Hilbert space theory, and cryptography on number theory. The theory, developed with relatively little concern for its application, ended up being the key to advance more practical and applied understanding.

Students. Here I focus on students who, at the college or graduate level have the opportunity to receive rigorous mathematical training from mathematicians. Beyond the specific topics that a student learns and their importance in other fields (e.g. an engineer must learn calculus and linear algebra to understand a mechanical system), mathematical course-work and training teaches students many other lessons. These include how to learn and work within a technical field, how to operate in an abstract or logical system, how to grapple with complex systems, how to work in an open-ended area where it is not always clear what questions should be asked or answers given, how to communicate complex ideas and develop precise language as well as illustrative language to share understanding, and how to fail over and over again while learning something important from that process.

Mathematical training and education is a vehicle for these deeper lessons and for the maturity that comes with them. When math majors and PhDs are hired, it is in part for what they know, but also in large part for their training and preparedness to take on new and increasingly complex and daunting projects at large scale. This, along with the abstract and technical capacity gained through learning mathematics, explains why math students who do not pursue academia so often land great jobs and are often behind transformative innovation.

Community. Here I refer to the community of mathematicians (though much could also be said of the role of mathematicians within other communities including key roles they play in their colleges and universities), often divided into smaller subfields. Perhaps due to a lack of big money or the general nature of many mathematicians, the communities in which mathematicians operate tend to be supportive, collaborative and hold themselves to high ethical standards. The currency within many of these communities tends to be less about results (i.e., problem solving) and more about understanding. When colleagues meet, they often share the questions they are thinking about, the ideas behind their recent progress and new or unexpected phenomena they have discovered. Those who broaden or create new fields through introducing techniques, objects of study or questions are often held in higher regard than those who prove definitive results that end discussion. Slow and deliberate math is valued, in part because of the quality it brings but also because it allows the community to digest and share in the understanding, and for students to become a part of the process. Students are protected and valued, whether or not they plan to stay in academia. Unlike other fields of science and perhaps due to the limited need for resources, mathematical communities tend to avoid hegemonies though they do develop ways to promote their members, raise up their students and support new directions of inquiry. Perhaps this is also in recognition of the fact that new ideas and understanding often arise in unpredictable ways; big ideas do not materialize in a vacuum but rather nucleate on the collective efforts of many smaller results or calculations.

Individuals. Mathematicians derive considerable personal joy and fulfillment from their own process and work, and this explains in large part why people enter mathematics and eventually devote their lives to it. Math has many different facets and appeals to different types of thinkers. It can look like a puzzle or game, a challenge or competition; it can be abstract, purely logical and elegant or beautiful; it can be informative and applicable, connected to other fields and the real world. Some find gravity and gratification in being part of the long conversation (in Barry Mazur’s words) which has been unfolding over thousands of years and will continue far beyond today. Many mathematicians are attracted to the focus on deep and careful thinking, and the often shared notion that each mathematician should understand the basis of what they do down to the very last detail. In that spirit, mathematics is a domain in which one must not just understand why something works, but also understand the boundary of why — how it can be broken and why other approaches fail. Finally, each individual mathematician may derive personal value from being part of a community, from teaching and training students and from the impact of mathematics on society.

Call to action. The mathematics community should understand, articulate and communicate its value to the broader society and to people who control funding for mathematics. I have provided some of my thoughts on this above, but everyone will have their own version. As a community, we should ground our discussions and decisions about mentoring and resource/reward allocation schemes in pragmatic terms based on the long-term value they produce. These discussions should include younger members such as undergraduate and graduate students. In the past they have been generally sheltered from such considerations that were left to those involved in stewardship of the professor; but this moment calls for their input and the future trajectory of mathematics relies on their decisions.

We should develop new infrastructure focused on understanding, articulating and communicating the value of mathematics. I believe mathematics will be well-served by creating venues (e.g. journals and conferences) for mathematicians to articulate to the broader public the value of their mathematics and the broader value of their field. This could be done in conjunction with experts who use mathematical ideas in other domains of science and beyond, or with involvement of historians who can help recreate the problems that drove the development of new math in generations past. Other efforts like the new journals of Essential Number Theory (and soon also Essential Analysis), Mathematical Discourse, and Galileo (and of course Quanta) should be supported and people who contribute meaningfully to broader understanding should be given more credit and respect by the community. Research institutes (like SLMath, for which I co-chair the scientific advisory committee) can play a key role here in allocating resources in support of the value of mathematics.

We must also recenter our focus on training and all of the positive outcomes from the process of doing mathematics. Teaching, at all levels, should be seen as an opportunity to bring great value to students, not as a matter of service to the university. Great teaching and thoughtful development of new courses that help share mathematics more broadly should be encouraged and rewarded.

Conclusion. I have struggled with this blog post for the last two weeks, writing and rewriting it as my thinking has evolved through discussions with family, colleagues, students and self-reflection. Incidentally, this period overlapped with the Jewish period between Rosh Hashanah and Yom Kippur which is called the ten days of T’shuvah. This is a period of deep personal and communal reflection and the word T’shuvah literally means “return” or “turning back”. While I do not think we should turn away from AI and the potential it brings, I do believe that we need to return to and revisit the value of mathematics and ensure that value is preserved and enhanced going forward.

I could have asked AI to write this blog or even help write it and edit it, but then I would not have taken the time to formulate and solidify my thoughts, to share, discuss and then reformulate and rewrite. I would not have thought through the arguments that I ultimately abandoned and did not write here, and I would not have seen this as a call to action for myself. I would have missed out on the learning opportunities from my conversations and email correspondences. The value of math, like so many other forms of knowledge, also comes from the process, not just the product. AI should be used in so far as it enhances the value derived from both.