Proof, Discovery, and AI: Reimagining Mathematics and Beyond with Terence Tao
In his insightful speech at the International Mathematical Olympiad ("IMO"), Professor Terence Tao articulated the transformative potential of AI in mathematics. Two key ideas stood out to me: First, AI's ability to unearth hidden connections within vast datasets, acting as a tireless explorer to identify patterns and correlations that might elude human perception. Second, its computational prowess in formal proof, ensuring rigor and precision in verifying complex mathematical arguments, thus expanding the scope of human inquiry.
My Personal Reflections on AI's Capabilities and Limitation
Tao's insights underscore the immense potential of AI in mathematics (or in many other areas), particularly in discovering new connections and conducting formal proofs. AI's ability to traverse massive datasets and recognize patterns allows it to uncover associations that might be overlooked by humans. In the Knot Theory and Machine Learning example, Tao discussed how machine learning was applied to connect two seemingly disparate areas of knot theory: combinatorial invariants and geometric invariants. By training a neural network on a vast database of knots, researchers discovered a surprising correlation between the knot signature and certain geometric invariants. This unexpected connection led to researchers to come up with new conjectures and theorems, demonstrating AI's ability to help uncover hidden relationships within complex mathematical structures, or, AI can also help identify incorrect or misleading relationships, leading researchers to refine their hypotheses and reach more accurate conclusions.
AI's strength in formal proof is equally noteworthy. Tao cited the Four Color Theorem and the Kepler Conjecture as examples where AI's computational abilities have made significant contributions. The Four Color Theorem, which posits that no more than four colors are needed to color a map such that no adjacent regions share the same color, was confirmed with the aid of computer-assisted proofs. Similarly, the Kepler Conjecture, concerning the most efficient way to pack spheres in a container, benefited from AI-driven formal proofs to address its complex calculations and logical reasoning. AI's precise computational power ensures rigor in these proofs, helping to verify arguments that are often too intricate for manual verification. Additionally, AI can assist in large-scale case analysis and verification—feats beyond human capacity.
The Uncharted Territory: Navigating the Limitations of AI's Empirical Approach
From my perspective, while AI's capacity to process vast datasets and identify patterns is undeniably powerful, it is crucial to recognize the limitations of its empirical approach. AI's reliance on existing data can lead to a focus on "white swans"—the common or expected outcomes. This focus may inadvertently obscure "black swans"—rare and unpredictable events that lie beyond the confines of current knowledge. Such limitations could profoundly impact mathematical discovery and other fields where AI is applied.
Also, the current tools may not be as good as solving the problem alone. As Tao pointed out, "it (machine learning) doesn't directly solve the problem for you but it gives you all these really useful hints as to where the connections are and where to look at them, but you still need the human to actually make the connections."
Conclusion: AI as the Source of Inspiration and Verification Tool
Professor Terence Tao's presentation outlines a promising future for AI in mathematics. AI not only reveals hidden mathematical connections but also enhances the rigor of formal proofs, expanding the boundaries of human exploration. However, we must remain aware of AI's limitations, especially when addressing "black swan" that lie beyond existing data.
AI currently serves as both a source of inspiration and a powerful verification tool. It enables us to explore mathematical and other realms more efficiently, uncover new patterns, and validate complex proofs. While AI is not yet capable of independently solving major problems, it significantly supports research endeavors.
As AI technology evolves, we may witness the resolution of more complex challenges. Yet, no matter how advanced AI becomes, human creativity and insight will remain at the core of discovery. Let us keep an open mind, embrace the opportunities presented by AI, and collaborate with each other to chart new frontiers in exploration.
All opinions express the views of the author but not his employer. This article was created with the support of ChatGPT/Gemini.