# When Computers Write Proofs, What's the Point of Mathematicians?

TLDRAndrew Granville, a mathematician specializing in analytic number theory, discusses the evolving role of mathematicians in the era of AI-assisted proofs. He challenges the traditional view of mathematics as a deductive system based on axioms and raises questions about the future of proof verification with tools like Lean, an AI program that can check proofs for logical consistency. Granville also contemplates the philosophical implications of relying on machines for proofs, suggesting that mathematicians may become more like physicists, trusting in computer verification and potentially losing the deep understanding and training in proof that is currently valued.

### Takeaways

- π§ The traditional view of mathematics as a solid and incontrovertible tower of knowledge built on axioms is not accurate.
- π€ The role of mathematicians is evolving with the advent of AI, which can assist in guessing the next steps in proofs and even verify them.
- π The philosophical question of what constitutes a proof and the value of proofs in mathematics is being revisited due to AI's involvement.
- π Andrew Granville, an analytic number theorist, discusses the impact of AI on the field, including his personal experiences and the broader implications.
- π¨ Granville has also explored the intersection of mathematics and art, collaborating on a graphic novel to popularize mathematical concepts.
- π€ Philosopher Michael Hallett's insights on the nature of proof and the debate over what it means to have something proven are highlighted.
- π The historical method of proving mathematical truths by referencing established axioms and primitives is contrasted with the modern approach.
- π€ Programs like Lean are revolutionizing how mathematicians verify proofs by storing information and requiring detailed explanations for each step.
- π The process of inputting proofs into AI systems like Lean can lead to deeper understanding and refinement of mathematical arguments.
- π The potential for computers to not only assist but also to lead in the creation of proofs is an exciting and somewhat unsettling development.
- π§ The future of mathematics may see a shift in the role of mathematicians, possibly becoming more akin to physicists who rely on computers for verification.
- π€ The implications of computer-generated proofs on the training and value systems within the mathematical community are a cause for contemplation and concern.

### Q & A

### What is the 'undergraduate fantasy' in mathematics, according to the speaker?

-The 'undergraduate fantasy' is the idea that mathematics is built solely on a bedrock of axioms through deductive argument, forming a solidly established and incontrovertible edifice of brilliant mathematics.

### Why does the speaker believe this fantasy is not true?

-The speaker believes this fantasy is not true because it is almost impossible to live up to it. In reality, the process of proving and establishing mathematical truths is more complex and less certain than this idealized version.

### How does artificial intelligence (AI) relate to the process of proving mathematical theorems?

-AI can assist in proving mathematical theorems by storing proven information and verifying proofs through programs like Lean. These programs can act like meticulous colleagues who help ensure the logical correctness of proofs.

### What are the traditional methods of verifying mathematical proofs?

-Traditionally, mathematical proofs are verified by publishing papers in journals or books, which are then stored in libraries. Mathematicians check these publications to verify and build upon previous work.

### What is Lean, and how does it assist mathematicians?

-Lean is an AI program with a library of already proven theorems based on axioms. Mathematicians can input their proofs into Lean, which then verifies the logical steps and raises questions if there are uncertainties.

### What experience did Peter Scholze have with Lean?

-Peter Scholze used Lean to verify a very difficult proof he was unsure of. Lean asked many questions, particularly in areas where Scholze was uncomfortable, helping him confirm the proof's validity.

### What are some of the potential impacts of AI on the field of mathematics?

-AI could change the way mathematics is practiced by leading in proofs, not just following or making suggestions. This could shift the focus away from traditional proof verification, raising questions about the future role and training of mathematicians.

### How might the role of mathematicians change with the increased use of AI in proofs?

-Mathematicians might become more like physicists, focusing less on proofs and more on proposing ideas that AI can verify. This could change the way mathematical training and value are perceived.

### What philosophical questions arise from the use of AI in mathematical proofs?

-The use of AI in proofs raises questions about what we want from proofs, what we historically needed from them, what it means to prove something, and how AI changes these concepts.

### What concerns does the speaker have about the future of mathematics with AI?

-The speaker is concerned that reliance on AI for proof verification might diminish the value of human mathematicians' work, change the nature of mathematical training, and lead to an unclear future for the profession.

### Outlines

### π The Myth of Axiomatic Mathematics and AI's Role

The script begins by challenging the common undergraduate notion of mathematics being built on a solid foundation of axioms and deductive reasoning. Andrew Granville, an analytic number theorist, discusses the unrealistic nature of this fantasy and introduces the emerging intersection of AI with mathematics. He raises philosophical questions about the nature of proofs and the impact of AI on the field, highlighting the rapid advancements and the potential for AI to assist in conjecturing and verifying mathematical steps. Granville also shares his personal journey in mathematics, his work on Fermat's Last Theorem, and his interest in computational and algorithmic questions. The paragraph concludes with his collaboration with a philosopher of mathematics, Michael Hallett, to explore the evolving definition of 'proof' in the context of AI assistance.

### π€ The Future of Mathematics: Human and Machine Collaboration

This paragraph delves into the implications of AI-generated proofs and the existential questions they raise for mathematicians. It discusses the potential shift in the value system of the mathematical community, as the reliance on machines for proof details could diminish the importance of human proof generation skills. The paragraph speculates on the future of mathematics, suggesting that mathematicians might become more like physicists, relying on computational validation rather than deep understanding. It raises concerns about the loss of training in proof thinking and the identity crisis that could arise from such a paradigm shift. The discussion ends with a reflection on the unpredictable trajectory of mathematics, as AI continues to expand its capabilities, blurring the lines between human and machine contributions to the field.

### Mindmap

### Keywords

### π‘Axioms

### π‘Deductive Argument

### π‘A.I. in Mathematics

### π‘Proofs

### π‘Analytic Number Theory

### π‘Fermat's Last Theorem

### π‘L-functions and Multiplicative Functions

### π‘Graphic Novel in Mathematics

### π‘Philosophy of Mathematics

### π‘Aristotle's View on Proof

### π‘Lean (Proof Assistant)

### π‘Computer-Generated Proofs

### Highlights

Undergraduate mathematics is often taught as if built solely on a bedrock of axioms and deductive arguments, but this idealized conception is not true.

Top mathematicians are exploring philosophical questions about the role of AI in mathematics.

When inputting ideas or proofs into a machine, at what point does the machine do a better job of helping guess the next step?

AI's role in proofs is a rapidly developing field, raising questions about the future of mathematical proofs.

Andrew Granville discusses his work in analytic number theory and his interest in computational and algorithmic questions.

Granville, along with his sister, developed a graphic novel in mathematics to explore philosophical aspects of the field.

Philosopher Michael Hallett was interested in their portrayal of mathematical processes.

Granville reflects on Aristotle's view that proofs should be based on primitivesβself-evident truths that do not need justification.

Traditionally, mathematical proofs were verified by checking published works in libraries, but AI programs like Lean are now used for verification.

Lean acts like an inquisitive colleague, rigorously checking all steps of a proof and asking for clarification when needed.

Peter Scholze used Lean to verify a difficult proof, finding that it asked the most questions where he was unsure, thus helping validate his work.

The potential of AI in generating new proofs is still in its infancy, but there are promising developments.

The rise of computer-generated proofs poses questions about the future role and training of mathematicians.

Granville speculates that mathematicians might become more like physicists, relying on computers to verify their ideas.

The limits of what computers can achieve in mathematics are still unclear, presenting new possibilities for the future.

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