You may recognize the title as an altered quote from Georg Cantor, the father of modern mathematics. From a mathematical perspective, posing a question is valued more than solving it because a great question can create entirely new fields of study. While a solution closes a specific chapter, a profound question can alter the trajectory of mathematics for centuries.
- Good questions reveal deep, unsuspected connections between seemingly unrelated mathematical areas. The “good question” that mathematicians Marc Lackenby and András Juhász at the University of Oxford posed to the AI was: “Is there a mathematical relationship between the hyperbolic geometry of a knot and its algebraic invariants?” The result was a new mathematical concept: the slope conjecture.
- Solving a radically new problem requires mathematicians to invent entirely new frameworks and techniques. For example, Fermat’s Last Theorem posed a simple question, but the 350-year quest to solve it forced the development of modern algebraic number theory and the theory of elliptic curves.
- A single unresolved problem can dictate the research focus of thousands of mathematicians for decades. “More than 2,200 years ago, the Greek geometer Apollonius of Perga posed a deceptively simple question: How many circles can touch three given circles, each at exactly one point? The puzzle would resist solution for nearly two millennia before mathematicians finally proved the answer — eight.” [1]
Let’s investigate the following questions concerning mathematicians’ and AI’s ability to uncover mathematical truths.
- Do mathematicians ask the right questions?
- Are mathematicians specialists?
- Does AI specialize?
- How is AI limited?
- Is the future humans versus AI or humans using AI?
Do mathematicians ask the right questions?
Before we can address the question above, we need to clarify the extent to which mathematicians possess broad, cross‑disciplinary mathematical expertise versus deep, highly specialized knowledge confined to their particular field. Once this is established, we must ask whether the questions a mathematician formulates will limit their search for solutions to the boundaries of their specialty, or whether those questions can open pathways to insights, methods, and results that originate in entirely different areas of mathematics.
Alon Amit addresses whether mathematicians are well-versed in all areas of mathematics or are limited to their field of study.
“No, it is no longer possible, and has been impossible for about 100 years. Hilbert and Poincaré are considered to be the last mathematicians to have mastered so much of the mathematics of their day that they could create and build in most fields.
Mathematics has grown in both breadth and depth to such an extent that producing original research in more than a few areas is extremely difficult, and writing papers cutting through “all” modern fields of math is truly impossible.
As an example, Fields medalist Tim Gowers … In 2010, while attending the International Congress of Mathematics, Gowers wrote:
…I decided on two prodigies: Marianna Csörnyei and Jacob Lurie. Marianna I have known quite well for many years, and she works in areas that I can be expected to understand reasonably well (which is not quite the same as saying that I do understand them reasonably well, but in fact I do usually follow quite a bit of her talks). Jacob Lurie is the opposite: I had never met him, or even seen him, and had absolutely no chance of understanding anything he would say, so I was going for the sole purpose of gawping.” [2]
Despite their being possibly being limited, mathematicians expand their field by discovering new abstract concepts, proving deep logical conjectures, and solving ancient puzzles. They build on past work to create modern frameworks. These new tools help advance physics, computer science, and data security.
Mathematicians often stay within their specific field because of specialization. Specialization in mathematics occurs because the total volume of human mathematical knowledge has grown too vast for any single mind to master and because of the practical demands of academic and scientific careers.
How much human mathematical knowledge currently exists? Over 4.6 million mathematical publications are tracked globally by mathematical indexing databases, such as zbMATH and MathSciNet. An estimated 100,000 to 125,000 new items are added to these databases each year. No one person can read all the papers or learn all the theories in a lifetime. [3][4]
The majority of these mathematical research papers are produced primarily by academic researchers and university professors, with the highest volume of output originating from institutions in the United States and China. Professors and graduate students focus on narrow thesis topics to produce original results because breaking down complex areas into smaller subfields (e.g., algebra, topology, or geometry) enables them to conduct deep research. Why? An academic’s professional success is largely judged by the number and quality of publications (“publish or perish”), since high publication rates help universities boost their global standing and prestige.
Etienne Toussaint notes that ‘…academic careers reward specialization. Tenure committees want to see “a clear and focused research agenda.” Grant agencies fund experts in specific areas. Conference organizers invite scholars known for their particular domain expertise. Publishers seek authors who can speak authoritatively on specific topics to defined audiences.’ [5]
There is nothing inherently limiting about conducting research entirely within one’s own mathematical specialty; doing so deepens and advances knowledge in that field. The difficulty arises when progress stalls. At that point, a researcher unfamiliar with neighboring or distant areas of mathematics may not know which external concepts, tools, or frameworks could help. They may be unsure what questions to pose, which connections to explore, or even which other subfields might contain the missing insight needed to move the work forward.
Are mathematicians specialist?
Mathematicians don’t merely stay within their field of expertise; they also test its boundaries. Many of the most powerful ideas in mathematics emerge when someone dares to carry a concept into unfamiliar territory—where the questions are different, the language is new, and the math must evolve to keep up. AI tools connect mathematical silos by spotting hidden structural patterns, translating continuous numeric data into discrete symbolic logic, and pairing large language models with formal proof assistants. They link distinct sub-disciplines such as graph theory, algebraic geometry, and knot theory.
NovaPrism notes that in school, “math is taught in strict, isolated silos—algebra one year, geometry the next. But in professional research, staying in your lane guarantees you’ll never solve the hardest problems. Modern mathematics resembles a massive, interconnected web. When a problem remains unsolved for decades or centuries in one specific area, the breakthrough almost always comes from translating the issue into the language of another domain.” [6]
For example, Grigori Perelman solved the 100-year-old Poincaré conjecture in topology using differential geometry and analysis. He used Ricci flow, a geometric version of the heat equation—to smooth out shapes and prove a fundamental truth about 3D spaces.
A researcher strictly confined to their area of expertise would likely never have found the proof.
Recent advances show that artificial intelligence systems can independently devise surprising and sophisticated strategies for tackling long‑standing mathematical challenges. The following examples show how AI models discovered structural connections, higher-dimensional mappings, and counterexamples that challenged previous mathematical consensus:
- The Jacobian Conjecture Solution: A massive 87-year-old math conundrum was solved by Anthropic’s Claude Fable 5, which found an incredibly compact counterexample AI’s solution to 87-year-old riddle takes mathematicians by surprise.
- The Erdős Unit Distance Problem: An OpenAI model disproved a central geometry conjecture by translating the problem to a higher-dimensional lattice of points, a completely unexpected structural step An OpenAI model has disproved a central conjecture in discrete geometry.
- The First Proof Initiative: Recaps a special challenge where AI models tackled unstudied research-level questions, finishing an equivalent of graduate school. The AI Revolution in Math Has Arrived | Quanta Magazine.
These developments have sparked debate about whether AI’s success in solving difficult math problems is a rare exception or a sign of a broader emerging capability.
Does AI specialize?
AI doesn’t specialize. Because so much mathematical knowledge is now available electronically—across the internet, digital libraries, and indexing databases—AI systems can draw from an enormous amount of information. This breadth gives them remarkable range, but it also comes with limitations: they can access vast material, yet they do not possess the depth, judgment, or domain‑specific intuition of a true specialist.
Shizhe Liang notes: “While present-day AI tools may still struggle with high-level reasoning and the creation of genuinely novel mathematical ideas, this article has shown that they excel at tasks such as suggesting formal proof steps, identifying patterns, generating plausible conjectures, and constructing mathematical objects — especially when integrated with other problem-solving tools. These strengths underscore that AI’s role as a complementary resource that can enhance our intellectual pursuits.” [7]
As noted by [Fields Medalist Timothy] Gowers, ‘large language models such as ChatGPT have an “encyclopaedic knowledge of mathematics”. Moreover, they can follow huge numbers of speculative lines of enquiry, even those unlikely to lead anywhere, without human time constraints. The latter seems to be what provided the key to success here. In hindsight, it seems an expert given a small number of hints would be likely to be able to reach the same proof. As Gowers notes: Many of the ideas needed for the proof were present in the literature already, and for such ideas either no hint is needed, since the expert is aware of that piece of literature, or a highly generic “look it up” hint would be enough.“’[8]
AI today sits in a fascinating tension: it is fundamentally limited—bound by the data it has seen, the rules it follows, and the fact that it doesn’t truly understand the world—yet it often feels capable of almost anything. AI can write essays, solve equations, generate images, analyze patterns, and hold conversations that seem uncannily human. This creates an illusion of boundless capability, when in reality every output is the result of careful constraints, statistical reasoning, and engineered guardrails rather than genuine comprehension or autonomy. The paradox is that the more convincingly AI performs, the easier it is to forget that behind the curtain is a system that imitates intelligence rather than possesses it.
How is AI limited?
“The question of whether machines can think is about as relevant as the question of whether submarines can swim.” ~ Edsger W. Dijkstra
Dijkstra used the submarine as a metaphor to critique how people frame questions about artificial intelligence. A submarine is designed to move efficiently through water, but we don’t call that “swimming” — because swimming is a biological, anthropomorphic term. Similarly, a computer is designed to process information and solve problems, but “thinking” is a human‑centric concept. Asking whether a computer “thinks” imposes an irrelevant standard on a machine that operates differently.
In his article, John Pavlus examines the growing uncertainty within the scientific community about what it means for modern AI systems to “reason.” Large reasoning models have recently demonstrated striking capabilities: they solve difficult mathematical problems, earn medals at the International Mathematical Olympiad, and even assist professional mathematicians such as Terence Tao in rediscovering or refining proofs. These successes have led some researchers to claim that genuine reasoning has emerged in these systems. Yet other studies reveal that the models often rely on shallow shortcuts, fail under simple variations, and produce step‑by‑step explanations that appear disconnected from the internal processes that generated the final answer. [9]
Tom Zahavy adds his insight into whether AI can “reason.”
“I explore the fundamental nature of scientific invention, highlighting the critical gap between the ability of humans to create computational systems from physical intuition and current artificial intelligence capabilities. While modern Generative AI excels at pattern recognition (induction) and logical proofs (deduction), we argue it fundamentally lacks the capacity for “abduction“—the intuitive leap required to generate novel explanatory hypotheses.” [10]
The prevailing consensus in many articles is that AI tools help connect different areas of math by spotting patterns across fields that typically don’t interact. They can translate ideas from one branch into forms that make sense in another and search vast collections of research for shared themes. In doing so, they reveal hidden links among areas such as algebra, geometry, and knot theory that might otherwise go unnoticed.
Is the future humans versus AI or humans using AI?
“AI will not replace humans, but those who use AI will replace those who don’t.” –Ginni Rometty, Former CEO of IBM
For decades, every advancement has sparked the same uncomfortable question: Will this replace human workers? A few key tools that fundamentally changed how we do mathematics include algebra (a shift from concrete arithmetic to abstract symbolic reasoning), book printing (mathematics books for the dissemination and expansion of human knowledge), Napier’s logarithms (simplifying multiplication), calculus (modeling motion), slide rules (enabling quick computation), calculators (changed accounting), spreadsheets (transformed financial analysis), and computers (enabling simulation). Each new tool was exciting because it made mathematics easier. Many of the tools sped up mathematical calculations, but a user still had to understand what was being done. [11]
“They [people using AI productively] got good at asking the right questions. When AI can produce a decent analysis in minutes, the thing that makes your analysis valuable isn’t the analysis itself. It’s the judgment that went into deciding what to analyze, what question to actually answer, and what the person asking it really needed to hear. Anyone can now get a chart. Anyone can get a summary. The hard part — the part that still requires a human with real context — is knowing whether that chart is answering the right question, whether that summary is missing something important, and whether the recommendation makes sense for this specific business at this specific moment. That’s not a technical skill. It’s a judgment skill. And it compounds over time in a way that tool proficiency never fully does. That’s not a technical skill. It’s a judgment skill. And it compounds over time in a way that tool proficiency never fully does.” [12]
Karen Davidson notes that someone ‘still has to determine which questions are worth pursuing. Someone still has to supply the initial insight that launches an entirely new line of inquiry. AI hasn’t assumed that role, at least not yet. “Our goal is not to replace the mathematician,” he says. “But to augment their abilities. Human plus AI should be better than just AI.”’ [13]
Conclusion
Mathematics has always advanced through a delicate interplay between human insight and the tools that extend human capability. This dynamic is now entering a new era. Mathematicians remain essential because they possess the judgment, intuition, and creative spark required to pose the kinds of questions that open new worlds. As noted, “a profound question can alter the trajectory of mathematics for centuries,” and even today breakthroughs emerged because humans asked questions that AI could not have invented on its own.
At the same time, AI is beginning to act as a powerful amplifier of human mathematical thought. It can scan vast bodies of literature, explore countless speculative paths, and uncover structural patterns across fields that rarely interact. As Timothy Gowers observes, AI models possess an “encyclopaedic knowledge of mathematics” and can pursue lines of inquiry far beyond human time constraints. Yet this breadth does not replace the depth of human expertise; AI still lacks the abductive reasoning, intuition, and conceptual creativity that define true mathematical invention. As Tom Zahavy emphasizes, AI excels at induction and deduction but “fundamentally lacks the capacity for abduction.”
The future, then, is not a contest between humans and machines. It is a partnership. AI expands the search space; humans decide where to search. AI proposes patterns; humans determine which patterns matter. AI can solve or disprove long‑standing conjectures, but humans still choose the questions worth asking and interpret the meaning of the answers.
The message is clear: mathematicians who learn to wield AI thoughtfully will shape the next era of discovery, just as those who mastered algebra, calculus, computers, and every transformative tool before them. As Ginni Rometty’s quote reminds us, “AI will not replace humans, but those who use AI will replace those who don’t.” The mathematician’s role is evolving—not diminishing—and the art of asking the right question remains the most human, most irreplaceable part of the enterprise.
References
[1] 
Leung, Barry. “New Mathematics Breathes Life Into Geometry’s Most Ancient Mysteries!” Math Games, July 12, 2026. https://medium.com/math-games/new-mathematics-breathes-life-into-geometrys-most-ancient-mysteries-d113fc0136e3.
[2] Amit, Alon. “Are mathematicians typically well-versed in all areas of mathematics?” Quora, Accessed July 31, 2026. https://qr.ae/pFwSAlon.
[3] AMS. “MathSciNet by the Numbers.” American Mathematical Society, August 1, 2026. https://mathscinet.ams.org/mathscinet/help/byTheNumbers.html.
[4] “zbMATH Open – Search Smart.” Search Smart, Accessed August 1, 2026. https://www.searchsmart.org/results/zbmath.
[5] Toussaint, Etienne. “Sharpening Your Research Niche.” The Tenure Track, November 1, 2025. https://www.thetenuretrack.com/p/sharpening-your-research-niche.
[6] NovaPrism. “Are mathematical researchers restricted to their specific fields, losing the chance to make discoveries in other areas?” Quora, May 15, 2026. https://qr.ae/pFwSUi.
[7] Liang, Shizhe, Wei Zhang and Tianyang Zhong. “Mathematics and Machine Creativity: A Survey on Bridging Mathematics with AI.” arXiv. December 21, 2024. https://arxiv.org/html/2412.16543v1.
[8] Lee, Melissa. “An AI Solution to an 80-Year-Old Problem Has Shocked Mathematicians.” The Conversation, May 26, 2026. https://theconversation.com/an-ai-solution-to-an-80-year-old-problem-has-shocked-mathematicians-283686.
[9] Pavlus, John. “Is AI Reasoning Right for the Wrong Reasons?” Quanta Magazine, July 31, 2026. https://www.quantamagazine.org/is-ai-reasoning-right-for-the-wrong-reasons-20260731/.
[10] Zahavy, Tom. “LLMs Can’t Jump (PDF).” Tom Zahavy, June 27, 2026. https://www.tomzahavy.com/projects/llms-cant-jump.
[11] Mathematical Mysteries. “What Is a Mathematician?” June 1, 2026. https://mathematicalmysteries.org/2026/06/01/what-is-a-mathematician/.
[12] Uttam, Analyst. “Everyone Is Learning AI. The Smartest People Are Learning Something Else.” Data Science Collective, June 24, 2026. https://medium.com/data-science-collective/everyone-is-learning-ai-the-smartest-people-are-learning-something-else-0ee1d82c9878.
[13] Davidson, Karen and Mark Evans. “How AI Is Redefining Research.” Texas AI, July 21, 2026. https://ai.utexas.edu/news/0006/how-ai-redefining-research.
Additional Reading
Johnson, Tom. “Asking Questions Is More Important than Finding Answers – Why?” I’d Rather Be Writing, April 27, 2012. https://idratherbewriting.com/2012/04/27/asking-questions-is-more-important-than-finding-answers-why/.
Asking questions seems to drive creativity. It cultivates an open mind. The questions we ask lead us to new knowledge. Questions drive us to answers we never thought to consider until we asked the question. Of course one cannot simply ask question after question, without giving any thought to answers. Such a method seems insincere in the question-asking. But rather questions lead naturally to a consideration of answers, which lead to more questions, which lead to more answers, which lead to more questions. The two move back and forth, like a lumberjack’s saw at an old oak tree, sawing through the rings with each back and forth motion until you reach the core. Not all the questions may be worth exploring, but for every dozen, there’s a golden one that causes us to wonder. The question moves us into an idea or answer we hadn’t yet explored. The golden question cuts through several rings at once. It takes a bit of meandering until we find the question, but once found, it holds us with wonder.
Kakaes, Konstantin. “The AI Revolution in Math Has Arrived.” Quanta Magazine, April 13, 2026. https://www.quantamagazine.org/the-ai-revolution-in-math-has-arrived-20260413/.
Sensory Edge. “Why Questions Are More Important Than Answers.” SensoryEdge, July 15, 2019. https://blog.sensoryedge.com/why-questions-are-more-important-than-answers/.
Questions are important because they help us create a framework for understanding and discovery, whereas answers are (at most) temporary fixes to our problems. Both questions and answers must be updated over time as things change. Formulating good questions is the best tool for learning and applies across all fields, and questions can do a lot of things, such as cause someone to rethink their position or to look more closely at a problem. Good questions are clear where bad questions are unclear. Bad questions are misleading in style and content, and require learners to guess what instructors are thinking. Also, a question might be poorly timed. A good question is timed well and is succinct and clear. It is open-ended and encourages further thinking and creativity. Make sure to carefully revise your questions until it is saying exactly what you want it to say and provoking the correct response, which is further thought and learning rather than spitting out the “right” answer that the teacher wants to hear. Hopefully your question will provoke further questions and learning in the students’ minds.
Sircar, Anisha. “AI Solved A Mathematical Problem That Had Stumped The World’s Best Minds For Decades.” Forbes, April 17, 2026. https://www.forbes.com/sites/anishasircar/2026/04/17/ai-solved-a-mathematical-problem-that-had-stumped-the-worlds-best-minds-for-decades/.
Skuse, Benjamin. “AI in Mathematics Is Forcing Big Questions.” What it Means to Be a Mathematician When AI Does the Math. IEEE Spectrum, June 25, 2026. https://spectrum.ieee.org/ai-in-mathematics.
Tsang, Ivan. “When AI Can Solve the Problems, What Should Learning Mathematics Become?” Academic Ink, March 12, 2026. https://medium.com/academic-ink/when-ai-can-solve-the-problems-what-should-learning-mathematics-become-c6ea48b04b8d.
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