Figure 1: Knot Theory,Trefoil Knot. Source: Towards Data Science
Mathematics, Intuition and Artificial Inteligence
In December 2021, a paper was published in Nature, showing how machine-learning tools can be used to guide the intuition of mathematicians. The paper particularly outlines a framework for how artificial intelligence and mathematicians can work together for advancing the field of pure mathematics and make powerful contributions to open problems. Two fields are particularly the focus of the paper: 1. the algebraic and geometric structure of knots and the combinatorial invariance conjecture for symmetric groups. The paper is quite interesting and it raises a lot of issues. One is concerned with the very field of mathematics and the concept of intuition. A recent rational solution found to a famously 40-year long difficult mathematical problem called the Cursed Curve or the equation:
\[y^4 + 5x^4 − 6x^2y^2 + 6x^3z + 26x^2yz + 10xy^2z − 10y^3z − 32x^2z^2\\ − 40xyz^2 + 24y^2z^2 + 32xz^3 − 16yz^3 = 0\]
was proven using a closed-source computer system (magma). In this talk at Collège de France on November 15, 2021, Professor and Mathematician Kevin Buzzard uses this example to discuss the concept of proof within the field of Philosophy of Mathematical Practice and the role of computer-based solutions. How can one call proof something that is closed and difficult to verify by the community at large? How to take stock of the issue of proof itself as a scientific practice and its validity, especifically in a field where this is so essential such as mathematics. Would there be fundamental differences between human proof and computer proof? Can we even call a computer proof a proof?
In fact, the role of computational power, data and programs is becoming important for all fields. It is definitely shaping how we are currently developing knowledge in the 21st Century. It is an important and fast shift. Even in a field like demography, for instance, increased computer power has definitely enabled to advance so many topics and opened other research avenues. I often feel that we are more and more becoming data scientists and programming has become more essential for our everyday tasks. Use of different sources of data as well as big data have also inflood demographic reasearch.
However, as much as I value and consider this an important advancement, I cannot stop but wonder how will this shift shape the way we do, think and interpret research. In an area where we don´t have “cursed curves”, but “live curves”, how can we find a balance where knowledge is traceable? Who is held accountable for producing knowledge that often guides policy and informs important human phenomena? So I join in the concerns of mathematicians who wonder how can machine learning tools guide something so unique such as intuition or whether making proofs aided by AI can be considered an actual proof. Isn’t mathematics a fundamentally human activity?
In order to have a brief but provocative discussion on this, we have to resort, as usual, to philosophy, and start asking some fundamental questions.
What is Mathematics?
I always recall a teacher I had in elementary school when I asked in a lesson of arithmetics what was mathematics. I guess that is when the problem started - rewinding to my 8-year-old self. Those “who-gave-those-oranges-to-whom-in-a-basket-of-apples-somewhere” questions were always difficult for me as I started building scenarios in my mind, different patterns and started wandering away from the right answer. In one opportunity, I asked the teacher what was mathematics, as he wrote in an exam that the answer was wrong and then made a note “this is not art class, this is mathematics”. His answer was that mathematics was counting. Of course, he was not entirely wrong, but let’s say that indeed mathematics is much more than just counting and could as well just attend the art class, as it is about endless creativity and abstractness. Curiously enough, despite being one of the most abstract things in the world, people feel that they have an opinion of what is mathematics in a much stronger way than they do about other things. The closeness and the somewhat objectivity of numbers or the fact that numbers make up a great part of a common day - like paying a bill, setting up dates and hours and many others give a proximity to one of the aspects of mathematics.
I of course do not intend to make you all dive deep into the rabbit hole as this question is complex and a research topic on itself. However, it is important to start with the platonic and non-platonic accounts of mathematics.
Let´s then consider that mathematics is If you want to go deep, read this entry at the Stanford Encyclopedia of Philosophy to have more details and references.
With this, we also need to ask what is intuition?
What is Intuition for Mathematicians?
A second important, and related question concerns the role of intuition for mathematicians. Intuition is a tricky concept and that is a topic on its own - maybe for another post : ). But for mathematicians it has a more specific concept - which is key. To approach the topic I will recover the very notion of proof and how mathematicians go about making proofs, so that we can dive into the topic in a more “practical way”.
In one another part of my website I mention Kevin Buzzard on Proof and this talk at Collège de France on November 15, 2021, Professor and Mathematician Kevin Buzzard prompts the topic of “what is proof?”, within the field of Philosophy of Mathematical Practice. This sets the tone for what actually is a mathematical proof. There are other problems that he mentions to take stock of the issue of proof itself as a scientific practice and its validity, especifically in a field where this is so essential such as mathematics. Another interesting topic he raises are the differences between human proof and computer proof. So how do mathematicians use their intuition?
Is Mathematics still Mathematics when not “Human”?
Now this all raises the question of is Mathematics still Mathematics when not solely “human”. Are we “humanizing” AI or is it mathematizing us?
–VDL