Currently, AI is profoundly impacting the field of mathematics, forcing mathematicians to rethink "what mathematics is and what it means to do mathematics." The author of this article is a topologist and a quantum computing researcher - AI has shown him opportunities. He believes that truly revolutionary mathematical ideas are born from the unknown and uncertainty, and the vitality of mathematics will not disappear due to AI. On the contrary, in the mathematical universe expanded by AI and quantum computing, those immature and constantly growing "growth-type mathematics" may usher in a new prosperity.
Translation | Snowing
On August 8, 1900, Hilbert delivered that famous opening speech at the International Congress of Mathematicians (ICM) in Paris, where he posed the question: "Is there anyone among us who would not be eager to lift the veil that shrouds the future?" Now, as artificial intelligence (AI) is transforming and shaping the future of mathematics, I am more eager than ever to know: what lies ahead for me and for future mathematicians.
The Chinese characters for "crisis" consist of "危" (danger) and "机" (opportunity). While discussions surrounding the "危" (danger) posed by AI to mathematics are intense and ongoing, I will focus on the "机" (opportunity), or the potential for mathematics to flourish and reach new heights. I am optimistic that after a period of pain and uncertainty, mathematics will adapt and evolve, as it always has, towards a brighter future - because mathematics has always been humanity's most resilient and enduring intellectual experience.
What is mathematics?
As a mathematics professor, I have never actually spent time pondering what "mathematics" is, because I have always been practicing it. Most mathematicians do not need to define mathematics: mathematics is what they do. As a topologist, I often have to explain what I do to others on planes or at dinner tables. When people hear the word topology, the most common reaction is to associate it with topography. Even at my wedding, someone enthusiastically and correctly explained that topology involves a technique for taking off a shirt without unbuttoning it. However, the most amusing anecdote comes from my high school English teacher: he immediately concluded that I was studying the best subject in the world, because just like zoology, my field is the study of "top" things. To me, mathematics is primarily about quantifying the world through numbers, geometrizing it through shapes, and thus abstracting and exploring it, which gives rise to branches of mathematics such as number theory, analysis, and geometry.
In this discussion, I would like to categorize mathematics into three interrelated and overlapping types: artistic, organic, and utility or applied. These types are not mutually exclusive, but rather dynamically evolving, so the same subject can belong to different types at different stages: in the early stages, it may be more inclined towards natural growth, while a mature discipline may be more artistic. Both artistic and organic mathematics are driven purely by curiosity, with the goal of exploring internal structures, rather than serving other disciplines or real-world applications. The main difference between artistic and organic mathematics lies in their level of maturity: artistic mathematics has been thoroughly researched, with many textbooks that provide consistent definitions of the basic objects being studied, leaving no ambiguity at the foundational level; whereas organic mathematics is still in the development stage, with definitions that are often tentative, and structural results that clearly exhibit immature characteristics. Examples of artistic mathematics include plane geometry and elementary number theory; examples of organic mathematics include quantum topology and mathematical quantum field theory.
As a quantum computing researcher, I have always been very concerned about what AI can achieve in mathematics and natural sciences. I recently supervised a paper on a famous open problem in topology, namely the stable Andrews-Curtis conjecture. It remains a significant research question to this day, without any revolutionary results. Overall, the current generation of large language models performs poorly in topology, and the reason is unclear - this may simply be a matter of time and training. However, I am more interested in whether there are some fundamental reasons behind this.
Topology is a relatively new field in mathematics, belonging to a part of general geometry. It studies the overall properties of space, such as the number of "holes" in a graph. For decades, the foundation of this discipline has been unstable, but this has not hindered its remarkable progress. Just as Newton did not need to define limits in strict ε–δ language to spark a scientific revolution, Poincaré made epoch-making contributions to this field without giving a clear definition of topological space. One of his conjectures, the three-dimensional Poincaré conjecture (3DPC), was the first Millennium Prize Problem to be solved, and the book "Perfect Rigor" by M. Gessen provides an in-depth analysis of the mathematician G. Perelman, who accomplished this feat. The author makes a thought-provoking comment in the book: how important is Perelman to the entire edifice of mathematics? I quote as follows:
In the world of top mathematicians, the intellectual elite are those who open up new horizons by proposing questions that no one else has thought of. Slightly below them are those who come up with methods to answer these questions. They often join the elite early in their careers - for example, just a few years after receiving their PhD, they first prove other people's theorems, and then begin to propose their own theorems. Finally, there are a handful of extremely rare individuals who complete the final steps of the proof. These mathematicians, who are meticulous, patient, and persevering, ultimately pave the way that others have dreamed of and outlined. In our story, Poincaré and Thurston belong to the first category, Hamilton belongs to the second category, and Perelman is the one who completes the entire work.
At this milestone in mathematics: Poincaré posed the initial question, while Thurston incorporated it into the grand framework of all three-dimensional manifold geometrization — where topological structure is completely determined by geometric structure, just as Poincaré and others had accomplished in the two-dimensional case. Hamilton then invented the Ricci flow, a tool that could lead to geometric structure, thereby solving the three-dimensional Poincaré conjecture and Thurston's entire geometrization conjecture. However, he became stuck for many years, until Perelman brought this program to a grand conclusion.
After Deep Blue and AlphaGo, I am convinced that human intelligence is not as extraordinary as I once wished to believe. However, we can still pose a hypothetical question: if all knowledge of topology and geometry had been fed into AI before Thurston's breakthrough, how long would it have taken for the AI to propose Thurston's Geometrization Conjecture and ultimately solve it? It is still unclear to me when and how machines will reach that level. Before that ultimate singularity arrives - which may take longer than the lifespan of our universe - humans still have a vast amount of mathematics to explore.
I have no doubt that AI will be extremely helpful for various types of mathematics, but at this stage, AI seems to be most suitable for artistic and practical mathematics, that is, fields with abundant literature and a high degree of formalization. However, for fields with a large amount of uncertainty, its assistance is much smaller. This may be because the investment and training are still insufficient, but I would rather believe that it is at least partially a fundamental problem faced by the current generation of large language models: they are limited by existing knowledge and lack the ability to truly internalize a subject as a "skill" to freely create.
AI is making knowledge almost effortlessly accessible, and can devour libraries one after another. We have seen similar moments in history, such as the advent of printing making books easily accessible. The world did change as a result, but churches and universities continued to endure. The impact of AI on intellectual labor is not much different from the impact of the Industrial Revolution on physical labor. The question is: what is the mathematics that truly belongs to humans? What is the difference between humans and machines?
I can envision many possibilities: future mathematicians may be like Indy 500 drivers, each with their own vehicle, chasing the fastest finish time; but airplanes are obviously faster than cars, and airplanes are not equivalent to birds. A mathematician who relies on AI for research may not be the kind of mathematician I aspire to be, but it is entirely possible that this will become the choice of many people in the future. And it will certainly be exciting to see how far we can go. So, what is left for me? Firstly, to continue the racing analogy, we still have our own Olympic Games.
Mathematics is not a speed competition, nor is it similar to a sport. At its core, mathematics enables us to understand the world around us - whether natural or man-made - through its inherent logic, and to invent or discover universal patterns and truths related to this world. The vitality of mathematics lies in genuine human experience and in the fact that the results obtained are meaningful - indeed, they can self-consistently explain things. What else could it be? After all, it is not a product fabricated out of thin air, even if it sometimes appears to be so.
The power of AI is driving transformation, but it will not change everything. AI relies on classical computers and is therefore constrained by classical complexity theory and classical logic. Classical computing itself has limitations and stability issues when it comes to numerical calculations. While AI can absorb the entire internet's knowledge, this does not guarantee that it will produce revolutionary ideas. Having too much education is not always a good thing, as a famous example in topology illustrates.
In 1981, when Michael Freedman announced his proof of the four-dimensional Poincaré conjecture, he was met with serious opposition from a world authority. However, this expert failed to recognize Freedman's ingenuity. Freedman was not familiar with the standard mathematical approach of using function approximation to determine homeomorphism; instead, he envisioned an approximation through a series of "relations" in his mind. He would stare at a piece of paper for hours, without writing down a single mathematical symbol. Truly revolutionary ideas, by definition, are unimaginable beforehand. This was the case with the solution to the four-dimensional Poincaré conjecture, and it was also the case with the development of superconductivity theory - electrons should have repelled each other, but superconductivity theory required them to form pairs and attract each other.
As a researcher who has spent decades studying quantum computing, I have been asked many times: what is the use of quantum computers? While they certainly have enormous potential to benefit society, to me, their most important application lies in their role as a scientific instrument, similar to microscopes or telescopes. Microscopes allow us to observe the microscopic world, telescopes allow us to see distant worlds, and quantum computers can enable us to truly "see" and "experience" the vibrant quantum world - which will become a brand new human experience. This also leads to a mathematical development direction that transcends AI and is somewhat fanciful: quantum mathematics. As American philosopher Hilary Putnam once asked: is logic empirical? Fundamental physics tells us that the underlying nature of our world is quantum. If we have powerful quantum computers in the future, and form a richer, more direct experience of the quantum world, will our logic and reasoning methods also change? I don't know, but I tend to think the answer is yes: in quantum physics, we will no longer just use real numbers and probabilities for calculations, but rather use wave functions to count, and amplitudes to argue. This may usher in a whole new iteration of mathematics.
As a mathematician, I wholeheartedly welcome and embrace the assistance of a tool that possesses nearly infinite knowledge and can be programmed in English. At the same time, I also see the danger of using AI irresponsibly and contaminating mathematics - claiming major breakthroughs without careful and rigorous verification. Verifying Perelman's work took a long time, and the process was not without drama. Those irresponsible mathematical assertions remind me of a true story that occurred in China in the 1980s: after Chen Jingrun proved the "1+2" form of the Goldbach conjecture and became a national hero, some amateur math enthusiasts claimed to have proved the "1+1" form of the Goldbach conjecture, and they came to the Chinese Academy of Sciences' Mathematics Institute with their "proofs" to ask experts for evaluation. Many real mathematicians even had to leave work through the back door because they were powerless to deal with the endless harassment. Later, someone even posted a notice saying: if you can find one error in my proof, I'll give you x yuan; if you can find two errors, I'll give you y yuan... In mathematics, a proof is invalid as long as it contains one error, one inaccuracy, or is simply
There is no doubt that mathematics and its education are standing at a historic crossroads. In a sense, mathematics has been caught off guard, and we have yet to find an appropriate way to respond. Perhaps other fields can learn from the plight of mathematics and start discussing the impact of AI as soon as possible.
I believe that in this ever-expanding mathematical universe, growth-oriented mathematics will not vanish like tears in rain, but will instead thrive and multiply like hydra in the Great Lakes.
Disclaimer: This article was written without the assistance of any AI tools. Its imperfections are an inherent part of the human experience.
This article is authorized and translated from Zhenghan Wang, The Future of Mathematics, original article: https://www.linkedin.com/pulse/future-mathematics-zhenghan-wang-y30le/.
The original title of this article was "The Future of Mathematics", and the current title was added by the editor.
本文原题为《数学的未来》，现标题为编者所加。