When Computers Write Proofs, What's the Point of Mathematicians?
TLDRIn this transcript, mathematician Andrew Granville discusses the evolving role of mathematicians in the age of AI. He challenges the traditional view of mathematics as a deductive science based on axioms, noting the impact of AI on proof generation and verification. Granville explores the philosophical questions raised by computer-assisted proofs, such as the nature of 'proof' itself and the future of mathematical training. He also highlights the potential of AI to revolutionize mathematics, posing the question of what value mathematicians will add if machines can handle the intricacies of proof.
Takeaways
- 📚 The traditional view of mathematics as a solid and incontrovertible field based on axioms is a fantasy that doesn't reflect reality.
- 🤖 The integration of AI into mathematics raises questions about the role of mathematicians and the nature of proofs in the future.
- 🧠 Andrew Granville, a mathematician in analytic number theory, discusses the philosophical and practical implications of AI in mathematics.
- 🎨 Granville's interest in popularizing mathematics led him to collaborate with his writer sister on a graphic novel to explore mathematical philosophy.
- 📖 Philosopher Michael Hallett's reading of the novel sparked a debate on the meaning of proof and the nature of axioms in mathematics.
- 🔬 Aristotle's definition of proof as resting on known primitives or axioms is contrasted with the evolving role of AI in establishing mathematical truths.
- 📚 The traditional method of verifying mathematical truths through published papers and library resources is being challenged by AI's capacity to store and verify proofs.
- 🤖 Programs like Lean are being used to formalize proofs, acting as a relentless colleague that demands clarity and correctness in mathematical arguments.
- 🔍 The use of AI in proofs, as illustrated by Peter Scholze's experience, can help mathematicians identify and correct uncertainties in their work.
- 🔮 The potential for computers to not only assist but also to lead in mathematical proofs represents a significant shift in the field of mathematics.
- 🧐 The reliance on machines for proofs could diminish the value of traditional mathematical training and the role of profound proofs in the profession.
- 🚀 The future of mathematics may see a transformation in how mathematicians work and think, with computers taking on more of the burden of proof verification.
Q & A
What is the traditional conception of mathematics in terms of its foundation?
-The traditional conception of mathematics is that it is built on a bedrock of axioms, with everything established through deductive argument, creating a towering edifice of solid and incontrovertible mathematics.
How does Andrew Granville describe the current state of the traditional mathematical axiomatic system?
-Andrew Granville describes the traditional axiomatic system as a fantasy that is almost impossible to live up to, and not even close to the truth.
What role does Andrew Granville see for A.I. in the future of mathematics?
-Andrew Granville sees A.I. playing a significant role in guessing the next steps in proofs and potentially doing a better job at it than humans, which raises questions about the nature and purpose of proofs.
What is Andrew Granville's area of expertise in mathematics?
-Andrew Granville works in analytic number theory and has been involved in areas such as Fermat's Last Theorem and ideas of L functions and multiplicative functions.
What was the unique project Andrew Granville collaborated on with his sister?
-Andrew Granville collaborated with his sister, who is a writer, to develop and write a graphic novel in mathematics.
What debate does Andrew Granville highlight regarding the meaning of 'proof' in mathematics?
-The debate highlighted by Andrew Granville is about what we mean when we say something is proved, what we historically needed from proofs, and how AI might change our understanding and reliance on proofs.
According to Aristotle, what should an argument rest on to prove something is true?
-According to Aristotle, an argument should rest on what he called primitives or axioms—things already known to be true—to establish the truth of something.
How does the traditional method of verifying mathematical truths compare to the use of AI in proofs?
-Traditionally, verifying mathematical truths involved going to a library and checking books. In contrast, AI, such as the Lean program, stores proven information within its program and can verify proofs based on axioms.
What does Andrew Granville suggest about the future role of mathematicians if AI takes over detailed proof verification?
-Andrew Granville suggests that mathematicians might become more like physicists, relying on AI to verify proofs and potentially focusing less on the detailed process of proof verification themselves.
What example does Andrew Granville provide to illustrate the interaction between a mathematician and a proof-checking AI like Lean?
-Andrew Granville cites the example of Peter Scholze, who used Lean to verify a difficult proof. Lean acted like an obnoxious colleague, asking questions and forcing Scholze to clarify his proof, ultimately helping him realize the simplicity of his argument.
What concerns does Andrew Granville express about the potential impact of computer-generated proofs on the profession of mathematics?
-Andrew Granville expresses concerns that if machines can handle most of the proof details, it may diminish the value of traditional mathematical training and the role of mathematicians, possibly leading to a change in how mathematics is practiced and valued.
Outlines
📚 The Myth of Axiomatic Mathematics and AI's Role
Andrew Granville discusses the common misconception that mathematics is strictly built upon a foundation of axioms through deductive reasoning, creating an unassailable structure. He points out that this idealized view is not accurate and that even top mathematicians question it. Granville's work with A.I. explores how machines can assist in the proof process, raising philosophical questions about the nature of proof and what we believe when something is proven. He also shares his experience with analytic number theory, his interest in computational and algorithmic questions, and his venture into creating a graphic novel to popularize mathematics. The discussion includes the historical perspective on proof, referencing Aristotle's view on establishing truth through 'primitives' or axioms, and the modern approach of using programs like Lean to verify proofs. The narrative touches on the potential impact of A.I. on the future of mathematics, suggesting a shift in the role of mathematicians and the way proofs are generated and understood.
🤖 The Future of Mathematics: Human and Machine Collaboration
This paragraph delves into the implications of computer-generated proofs and the evolving role of mathematicians. It raises concerns about the value of human proof work in a future where machines could potentially handle the details. The paragraph suggests a potential shift in the identity of mathematicians, comparing them to physicists who might rely on computers to verify their theories. The author speculates on the unpredictable nature of mathematics in the coming decades, as computers become more capable and the boundaries of their capabilities become less clear. The summary captures the essence of the debate on the future of proof generation, the potential for A.I. to lead in proofs, and the philosophical and practical questions this raises for the mathematical community.
Mindmap
Keywords
💡Axioms
💡Deductive Argument
💡Analytic Number Theory
💡Fermat's Last Theorem
💡L Functions
💡Graphic Novel
💡Philosophy of Mathematics
💡Aristotle's Primitives
💡Artificial Intelligence (AI)
💡Proof Verification
💡Computer-Generated Proofs
Highlights
The traditional conception of mathematics as a deductive system based on axioms is challenged by the reality of mathematical practice.
A.I.'s role in mathematics is evolving, with machines potentially outperforming humans in guessing the next steps in proofs.
The philosophical and practical questions around what constitutes a proof and the role of A.I. in verification are being re-evaluated.
Andrew Granville's work in analytic number theory and his interest in the intersection of mathematics and computation.
The idea of a graphic novel in mathematics, developed in collaboration with Granville's writer sister, to explore mathematical philosophy.
Michael Hallett's interest in the portrayal of mathematical proof in the graphic novel and the debate on the meaning of proof.
Aristotle's definition of proof resting on primitives or axioms, and the challenge of identifying these foundational truths in mathematics.
The traditional method of verifying mathematical truths through published papers and library resources.
A.I. programs like Lean, which store and verify proofs based on axioms, acting as a rigorous and persistent colleague.
Peter Scholze's experience using Lean to verify a complex proof, highlighting the utility of A.I. in identifying gaps in understanding.
The potential for A.I. to lead in mathematical proofs, rather than just follow or suggest, represents a new frontier in the field.
The infancy of computer-generated proofs and the optimism for their future in contributing to significant mathematical advancements.
The existential questions raised for mathematicians as A.I. begins to handle more details of proofs, affecting the value and training in the profession.
The possibility of mathematicians becoming more like physicists, relying on A.I. for proof verification and focusing on higher-level conceptual work.
The uncertainty and excitement surrounding the future of mathematics and the role of human mathematicians in an A.I.-assisted landscape.
The emergence of computer-generated proofs as a new possibility that may redefine the limits of what computers can achieve in mathematics.