Critique of Obscure Reason: Artificial Intelligence in the Perception of Mathematicians

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Mathematicians at HSE University believe that there is no need to fear losing jobs because of the widespread use of AI, while at the same time they warn against uncritical acceptance of works and projects prepared with its help. AI, however, can be a useful tool in research, creating models and processing large volumes of information.

The HSE Faculty of Mathematics held a research seminar entitled ‘How Artificial Intelligence Changes the Nature of Mathematical Discovery.’ The meeting was attended by practising mathematicians, students, as well as high school students and teachers from leading Moscow mathematical schools. In her opening remarks at the seminar, the Dean of the Faculty, Alexandra Skripchenko, stressed the importance of discussing both the opportunities and challenges of using AI in mathematics. She noted that some scholars believe AI has made mathematicians, as its authors, the most sought-after specialists in the world, while others fear that artificial intelligence may even leave mathematicians unemployed.

She then gave the floor to Andrei Okounkov, winner of the Fields Medal (one of the most prestigious mathematical awards in the world) and professor at HSE University and Columbia University (USA). He presented a talk entitled ‘Critique of Obscure Reason’ and invited the audience to reflect on the process of mathematical cognition and the role played in it by computational machines and words. ‘Mathematics is unthinkable without precision and honesty. I am not an expert, nor even a very active user of AI and large language models, but I do have something to say about them,’ the professor remarked.

Andrei Okounkov

Andrei Okounkov recalled Immanuel Kant’s Critique of Pure Reason, in which the German philosopher writes that mathematics provides the most brilliant example of pure reason, which successfully expands its own space spontaneously. This, Andrei Okounkov noted, is also a play on words, something large language models (LLMs) often engage in. He reminded the audience that Immanuel Kant was also interested in the natural sciences, even writing a treatise on the Solar System in which he claimed that all its planets were inhabited, and that the further one moved from the Sun, the higher the moral level of their inhabitants.

Such reflections would hardly be possible today, since modern science speaks the language of mathematics. The first to articulate this idea was the great Italian scientist Galileo Galilei: ‘The book of nature is written in the language of mathematics, and its characters are circles, triangles, and other geometrical figures.’ Yet each new chapter of this book requires a new mathematics, one that does not follow from the previous but, on the contrary, provides its very foundation.

Concepts and the terms invested in them expand over time. Mathematics does not contemplate things a priori; its essence lies in the crystallisation of new structures, concepts, and phenomena. One only needs to think about the development of the meaning of the word ‘space’ over the past 150 years. According to modern ideas, our space changes dynamically, oscillating both classically and quantum-mechanically. Yet there is something more fundamental even than space.

Discussing artificial intelligence from a scientific perspective, Andrei Okounkov argued that neural networks in the brain and in machines are structurally and functionally similar. Human and machine cognition of mathematics cannot be stopped, and there is nothing traumatic in this. People once feared that computers would displace mathematics, but numerical and symbolic methods instead gave mathematicians strength and confidence. ‘I have used them all my life and I strongly advise everyone not to forget the old algorithms, both exact and those close to classical mathematics,’ the scholar recommended.

In his view, some large language models resemble postgraduate students: they often produce pleasant surprises, but at times they merely repeat clever words superficially without delving into the true meaning of the subject. It is therefore essential to analyse in detail any work produced with the help of AI.

For instance, while working with Kazhdan–Lusztig numbers, an area of mathematics clearly advancing thanks to machine learning, the speaker asked ChatGPT and other models to produce a table of these numbers. Yet errors appeared already at the most elementary stage—a stage with which humans would have had no difficulty.

Andrei Okounkov noted that some students had stopped learning independently, choosing instead to rely on AI. While a few may benefit from such a strategy, for many it will lead to a dead end. In mathematics, it is essential to distinguish right answers from wrong ones, and critical thinking plays a central role when the results produced by language models are often unreliable. ‘AI has not yet completed its doctorate, and we should not forget that ordinary intuition has always played, and continues to play, an important role in mathematics,’ remarked the professor.

The human brain is capable of recognising with precision that certain phenomena are structurally similar—something AI cannot always achieve. Even in the age of rapid AI development, it remains immensely valuable to have colleagues and friends who can unexpectedly spot analogies between constructions from entirely different areas of mathematics. ‘It would be wonderful if AI had such hints at hand,’ the scholar said.

Mathematicians are accustomed to many conjectures proving false, and even when they come close to the truth, much has to be worked out along the way. Most mathematicians are both inclined and able to critically analyse bold hypotheses of all kinds. ‘A single AI can cook up far more porridge than all the scholars can digest,’ Prof. Andrei Okounkov lamented. In such a situation, he concluded, mathematical intuition and critical thinking remain the key ingredients of successful research—with or without AI—and the importance of high-quality mathematical education cannot be overstated.

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The second part of the seminar was devoted to the presentation of real outcomes of applying AI in fundamental mathematics, delivered by early-career researchers from the HSE Laboratory for Complex Networks, Hypergraphs, and their Applications. HSE mathematics master’s student Sergei Usanov presented a paper entitled ‘A New Tool of Science: How to Use AI in Mathematics,’ prepared under the supervision of Fedor Pavutnitskiy (St Petersburg State University) and Prof. Vasily Gorbounov (HSE Faculty of Mathematics).

He spoke about the use of machine learning in knot theory. Knots are closed curves in space, and invariants are functions that help identify equivalent knots which may look very different visually. The researcher also explained chord diagrams and the reduction of a number of knot-theory-related problems to graphs, before introducing the audience to the basic ideas of machine learning.

Machine learning works with an object that must be trained, for example, to predict the price of a flat based on a set of features. The ideal function may vary widely, since in reality objects are discrete. A model is a family of functions that must be calibrated and approximated to reality. However, if a prediction is 100% accurate, this usually signals a deficiency in the model.

To train a model, one must select a large number of objects, a target (dependent variable), and search for a function. In knot theory, this target is the number of edges in an intersection graph. The function is a polynomial with a given number of chords, which can be calculated recursively with considerable time and effort, or alternatively by attempting to express the graphs numerically and calculate the number of subgraphs.

Using machine learning, the team was able to develop a formula that produced sufficiently accurate predictions and made it possible to identify a pattern through linear regression. This result was confirmed experimentally.

‘AI can be very useful for noticing the dependency you need, but it is not easy—sometimes it is hard to find the right model, or not always possible to represent objects numerically,’ Sergei Usanov concluded.

Artem Malko

HSE mathematics master’s student Artem Malko delivered a presentation entitled ‘Large Language Models—A New Tool of Science,’ prepared under the supervision of Fedor Pavutnitskiy. His talk provided a detailed overview of existing models currently applied in mathematics.

The early-career researcher reminded the audience that the creators of LLMs follow different approaches—ranging from open natural language to fully formal proofs, as well as highly specialised models in which flexibility is traded for reliability and controllability.

He explained that transformer-based architectures, such as those behind ChatGPT, are easy to use but difficult to verify, and not sufficiently efficient when solving narrowly defined tasks. 

Other models, meanwhile, involve machine verification of proofs in formal syntax, offering higher guarantees of correctness and reproducibility, but at the cost of significant overheads for formalisation and a steeper entry threshold.

He also referred to a recently published article in Nature, in which the authors presented a supervised learning architecture that proposed new options for machine learning, and where a connection was identified between algebraic and hyperbolic components.

When preparing a paper on knot classification, the authors trained a transformer-based model to predict knot types, where knots were encoded as words in groups, and algebraic data were used in deep models.

On long group words, the best models achieved over 95% accuracy. They also applied reinforcement learning to test the Andrews–Curtis conjecture, but this proved challenging due to the vast action space, sparse rewards, and enormous number of codes involved.

‘AI is not going to take our jobs, but it can be used in areas of mathematics that we do not know well enough. It is also a method for processing large volumes of information,’ Artem Malko concluded.