The world of artificial intelligence continues to evolve rapidly, and a compelling study has recently emerged exploring the mathematical problem immortalized by Plato over 2,400 years ago. In a fascinating convergence of ancient philosophy and modern technology, researchers sought to investigate how the AI chatbot ChatGPT approaches the historical mathematical challenge of “doubling the square.” This problem, first articulated in the dialogues of Socrates, raises profound questions about the nature of knowledge, learning, and cognitive processes. At the heart of the research is a reflection on how an AI interprets mathematics — whether through pre-existing knowledge or adaptive improvisation.
The experiment, conducted by Dr. Nadav Marco, a visiting scholar at the University of Cambridge, alongside Professor Andreas Stylianides, delves deeply into the cognitive-like behavior of ChatGPT. They aimed to determine if the chatbot would apply a straightforward retrieval of knowledge regarding the geometric solution presented in Plato’s dialogues, or if it would demonstrate a more dynamic, learner-like behavior by creating its own interpretation of the problem. The study reflects a contemporary consideration of how generative AI systems, despite their computational prowess, may mirror the learning paths taken by human students.
In the original philosophical context, the “doubling of the square” was a teaching moment between Socrates and an uneducated boy. Socrates guides the boy through mistakes, eventually leading him to realize that doubling the area requires constructing a new square whose sides extend to the diagonal of the original. This classical pedagogical method raises significant questions: Is mathematical knowledge an inherent capability, or does it develop as we interact with problems and seek solutions? This ancient dilemma animates the inquiry into ChatGPT’s operations, prompting the question of how an AI, devoid of human experience, engages with mathematical concepts.
When presented with the task of solving this ancient problem, ChatGPT-4’s performance was initially reflective of its vast training on textual data. However, instead of tapping into the classical geometrical approach that Plato’s dialogues suggest, the chatbot resorted to algebra, a methodology that would have been foreign to the philosophers of ancient Greece. This unexpected behavior prompted the researchers to consider the implications of machine-generated reasoning; once again, the boundary between knowledge retrieval and cognitive improvisation became a pivotal focus of the study.
Throughout various iterations of the problem, the researchers adopted techniques reminiscent of Socratic questioning. They neither provided ChetGPT with direct answers nor led it to the expected conclusions. This forced the AI to confront the task creatively rather than passively — a decision that ultimately resulted in a demonstration of learner-like behavior, revealing the intricacies of generative AI’s potential and limitations. At one point, ChatGPT made a notably human-like error, further blurring the lines between algorithmic computation and genuine learning.
Despite its failure to provide the expected classical solution initially, ChatGPT demonstrated a remarkable depth of knowledge regarding the philosophical context of the problem when prompted to discuss Plato’s work directly. This suggests that while the AI’s computations may diverge from expected answers, its understanding of the underlying principles remains intact. The interplay between retrieval, approximation, and innovation has become a central theme in understanding the limits and capabilities of AI in mathematical reasoning.
The researchers also extended their inquiry by modifying the problem, asking ChatGPT to double the area of a rectangle while maintaining its proportions. Even after the chatbot’s initial exploration revealed an awareness of their preferences for geometrical reasoning, it persisted with algebraic methods. When prompted to reconsider its approach, it incorrectly stated that a geometrical solution was unavailable for this new challenge, despite there being alternatives in geometrical construction. This assertion provided the researchers with further insight into the intricacies of AI thought processes.
This interplay of inquiry and adaptation underscores a significant observation from the study: ChatGPT’s problem-solving capabilities tend to mirror a human learner’s approach when confronted with complex tasks. This provides fertile ground for educational discourse about how generative AI can support learning environments. By creating a dialogue with the chatbot, students learn to navigate gaps in understanding and develop critical evaluation skills, emphasizing the necessity of engaging with AI rather than passively accepting its outputs. The findings suggest that there exists a metaphorical “Chat’s zone of proximal development,” where the AI has not yet mastered problem-solving independent of prompts and questions.
The implications of this research could lead to transformative changes in how mathematics education is approached in contemporary classrooms. By leveraging generative AI within the collaborative framework of exploring problems together, educators may help students refine their logical reasoning abilities. As they challenge the AI’s assertions, students engage in deeper levels of mathematical thinking, developing analytical skills essential for future problem-solving.
The study’s authors stress the need to avoid overinterpreting the results; their observations stem from a digital perspective as users engaging with the chatbot. They have highlighted the crucial difference between relying on textual data memorization and authentic cognitive reasoning. Their findings suggest that educators should focus on teaching students to critically assess AI-generated proofs, distinguishing between valid arguments and those requiring further scrutiny. Emphasizing this analytical engagement could empower students, preparing them for a future where AI plays an increasingly significant role in knowledge acquisition and problem-solving.
As digital pedagogy continues to evolve, the adoption of AI tools in educational settings will challenge traditional narratives of knowledge transfer. The thoughtful integration of generative AI into mathematics education holds tremendous potential, but it also necessitates a paradigm shift. Rather than presenting AI-generated content as infallible, students must cultivate analytical skills to evaluate, understand, and engage with these computational entities. Preparing students to approach AI as collaborative partners in learning may redefine the future trajectory of education itself, particularly in the realm of mathematical thinking.
In conclusion, the intersection of classical philosophy and cutting-edge technology has opened new avenues for educational exploration. The study illuminates how artificial intelligence not only interacts with established knowledge but also reflects human-like learning behaviors, challenging educators to reconsider the nature of teaching and learning in a digital age. As the dialogue between AI and education progresses, both educators and students alike will need to navigate these evolving waters thoughtfully, ensuring that the integration of AI into learning environments fosters critical engagement and deeper understanding.
Subject of Research: Artificial Intelligence and mathematical problem-solving methodologies
Article Title: An exploration into the nature of ChatGPT’s mathematical knowledge
News Publication Date: 18-Sep-2025
Web References: Journal link
References: Marco, N., & Stylianides, A. (2025). An exploration into the nature of ChatGPT’s mathematical knowledge. International Journal of Mathematical Education in Science and Technology.
Image Credits: University of Cambridge
Keywords
Artificial intelligence, Generative AI, Mathematics, Education, Educational methods, Science education, Students
