The Joy of Why – Terence Tao on Good Mathematics

Introduction

Host: Steve Strogatz
Guest: Terence Tao, UCLA,
- Winner of Fields Medal, Breakthrough Prize, MacArthur Fellowship
Topic: Revisiting "What is Good Mathematics?"
Original Essay by Tao (2007): Sought to define good mathematics (rigor, elegance, real-world utility)
Question: Do we need to rethink what good mathematics is now, 15 years later?

Essence of Good Mathematics:
- Rigor, aesthetic beauty, problem-solving, technical power, real-world applications
- Often, multiple criteria converge to define the same "good" mathematics
- Eugene Wigner's essay: Unreasonable effectiveness of mathematics in physical sciences
Historical Perspective:
- Example: Riemannian geometry for general relativity
- Mathematics that seem elegant often provide deep insights
- Platonic approach: Objective "good" mathematics exists, different paths converge to it

Von Neumann's Comment:
- Mathematical ideas originate in empirical sources but evolve to take on a creative, aesthetic life
- Danger of abstract inbreeding: Subject may degenerate if too removed from empirical roots.
- Von Neumann: Empirical grounding is crucial to prevent this.
Practical Illustrations:
- Bourbaki era: Abstraction in algebra and topology
- Pendulum swing: Current focus on interconnections and applications

Philosophical Perspective:
- Mathematics makes fundamental, non-obvious connections (e.g., Descartes' Cartesian coordinates)
- Parable of the Elephant: Different perspectives reveal a cohesive whole
- Implication: Mathematics feels "real" when connections are made and the subject matures

Advising Students:
- Tao's motivation: Misconceptions about what is valuable in mathematics
- Broadened view: Importance of multiple perspectives in good mathematics
- Example: Collaboration and learning from others in the field
Breadth of Interest:
- Example: Work in fluid dynamics
- Concept: Hedghogs (deep domain experts) vs Foxes (connective generalists)

Computer-Assisted & AI-Generated Proofs:
- Increasing acceptance of computer-assisted proofs
- Forthcoming challenges with AI-generated proofs
- Shift from secretive preprints to public sharing (e.g., arXiv)
- Greater collaboration; interdisciplinary and large-scale projects
Potential for Amateur Involvement with AI:
- Similar to citizen science in astronomy
- Technology (e.g., proof assistants) enabling new collaboration modes and amateur contributions

Tao's optimistic view: Technological tools empower mathematicians, enhancing collaboration and discovery
Internet and accessibility: Facilitating global and interdisciplinary collaboration

Tao utilizes online resources (e.g., Wikipedia) to start learning about new subjects
Emphasis on connections: More connections lead to better mathematical discoveries

Interdisciplinary program collaboration led to significant improvements in MRI speed and efficiency
Algorithm Impact: Software for faster and more accurate MRI imaging is now implemented in machines

Value for society: Applications, intellectual interest, insights into phenomena
Importance of resource allocation: Highlight and publicize valuable advances
Public Accessibility: Tao’s blog and articles aimed at making math understandable