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

Quanta Magazine
31 Aug 202306:34

TLDRIn the realm of mathematics, the traditional view of axiom-based proofs is being challenged by the advent of AI. Mathematician Andrew Granville discusses the philosophical and practical implications of computer-assisted proofs, questioning the role of mathematicians as AI systems like Lean begin to verify and potentially generate proofs. The conversation delves into the historical foundations of proof and the evolving nature of mathematical verification, hinting at a future where the line between human and machine in mathematical discovery may blur.

Takeaways

  • 📚 The traditional view of mathematics is based on a solid foundation of axioms and deductive reasoning, but this ideal is not entirely accurate.
  • 🤖 The role of AI in mathematics is evolving, with machines potentially outperforming humans in guessing and proving mathematical steps.
  • 🧐 The philosophical question arises: what do we value in proofs, and how does AI alter our understanding and expectations of mathematical proof?
  • 📈 Andrew Granville, an expert in analytic number theory, discusses the intersection of mathematics and computation, including his work on Fermat's Last Theorem and L functions.
  • 🎨 Granville's interest in popularizing mathematics led to a collaboration with his writer sister on a graphic novel, which attracted the attention of philosopher Michael Hallett.
  • 🔍 The debate on what constitutes a proof involves the concept of 'primitives' or axioms, which are the self-evident truths that underpin all mathematical arguments.
  • 📚 Historically, mathematical truths were verified by consulting published papers in libraries; today, AI systems like Lean store and verify proofs within their programs.
  • 🤓 Lean acts as a rigorous and persistent colleague, challenging mathematicians to clarify and simplify their proofs, as illustrated by Peter Scholze's experience.
  • 🚀 The potential for computers to not only assist but also to lead in the creation of proofs is an emerging and exciting development in the field of mathematics.
  • 🤔 There is concern about the future role of mathematicians if computers can handle the details of proofs, which could change the nature of mathematical training and practice.
  • 🔮 The impact of computer-generated proofs on the profession and the essence of doing mathematics remains uncertain, with the potential for significant changes in the coming decades.

Q & A

  • What is the common undergraduate fantasy about the nature of mathematics according to the script?

    -The undergraduate fantasy is that mathematics is built solely on a bedrock of axioms through deductive argument, creating a towering and incontrovertible edifice of brilliant mathematics.

  • What is the philosophical question posed by the use of A.I. in mathematics?

    -The philosophical question is at what point does the machine do a better job of helping guess the next step in a proof, and what happens when mathematicians start inputting their ideas or proofs into a machine.

  • What are the 'massive questions' that are being raised about proofs and AI according to Andrew Granville?

    -The massive questions include what we want from proofs, what we have historically needed from proofs, what we believe when something's proved, and how AI changes our understanding and expectations of proofs.

  • What is Andrew Granville's field of study?

    -Andrew Granville works in analytic number theory, focusing on ideas of L functions and multiplicative functions.

  • What is the significance of the graphic novel developed by Andrew Granville and his sister?

    -The graphic novel was developed to explore the philosophy of mathematics and to portray the way mathematics is done, which caught the interest of philosopher of mathematics Michael Hallett.

  • What does Aristotle define as the foundation of a proof?

    -Aristotle defines the foundation of a proof as resting on what he called 'primitives' or things already known to be true, which eventually leads to axioms that cannot be defined any further.

  • How does the traditional method of verifying mathematical truths differ from the AI approach?

    -Traditionally, one would go to a library to check a book for the exact details of a proof. In contrast, AI stores information within a program, allowing for proofs to be input and verified by the program based on axioms.

  • What role does the AI program Lean play in the verification of mathematical proofs?

    -Lean acts as a rigorous and persistent colleague, asking questions and demanding explanations until it understands the proof. It helps in identifying any unclear or incorrect parts of a proof.

  • Can you provide an example of how AI has been used to verify a proof, as mentioned in the script?

    -Peter Scholze used Lean to verify a difficult proof he was not completely sure of. Lean's persistent questioning helped Scholze identify and clarify the parts of the proof that he was initially uncomfortable with.

  • What are the implications of AI-generated proofs for the future of mathematics and mathematicians?

    -AI-generated proofs could change the nature of mathematics by potentially leading mathematicians to rely more on machines for proof details, which may affect the way they are trained and what they value in their profession.

  • What is the potential future impact of computer-generated proofs on the role of mathematicians?

    -The potential impact is that mathematicians may become more like physicists, proposing ideas and relying on computers to verify them, which could change the focus of their training and the nature of their work.

Outlines

00:00

📚 The Myth of Axiomatic Mathematics and AI's Role

Andrew Granville, a renowned mathematician in analytic number theory, discusses the common misconception that mathematics is built solely on a foundation of axioms and deductive reasoning. He points out that this ideal is not only unattainable but also misleading. Granville explores the intersection of artificial intelligence and mathematics, questioning how AI can be used to predict and verify mathematical proofs. He delves into the philosophical implications of AI's involvement in mathematics, touching upon historical perspectives on proof and axioms, as well as the potential shift in mathematicians' roles and the nature of mathematical proof verification in the future.

05:03

🤖 The Future of Mathematics: Human and Machine Collaboration

This paragraph contemplates the future impact of AI on the field of mathematics. It raises concerns about the value and identity of mathematicians as AI becomes more capable of handling complex proofs. The potential for AI to not only suggest but also generate proofs is examined, posing the question of what the role of mathematicians will be if machines can verify and even produce proofs. The paragraph speculates on a future where mathematicians might become more like physicists, relying on computers to verify their work, and ponders the profound changes this could bring to the practice and perception of mathematics.

Mindmap

Keywords

💡Axioms

Axioms are fundamental starting points or assumptions in mathematics that are accepted as true without proof. They serve as the foundation upon which mathematical theories are built. In the context of the video, the speaker discusses the traditional view of mathematics as a structure built on these axioms through deductive reasoning. The script mentions that 'we have this bedrock of axioms, and everything is built solely on these axioms,' highlighting the importance of axioms in the classical understanding of mathematical proof.

💡Deductive Argument

Deductive argument is a logical process where conclusions are drawn from premises that are assumed to be true. In the realm of mathematics, deductive arguments are used to establish theorems based on axioms. The video script alludes to the 'deductive argument' as the traditional method by which mathematical knowledge is believed to be built, creating a 'towering edifice of brilliant mathematics' that is 'solidly established in an incontrovertible way.'

💡A.I.

A.I., or Artificial Intelligence, refers to the simulation of human intelligence in machines that are programmed to think and act like humans. In the video, the speaker mentions working with A.I. to explore the philosophical and practical implications of machines assisting in the creation of mathematical proofs. The script notes, 'If you start always inputting your ideas, or proofs into a machine, at what point does the machine do a better job of helping guess the next step?' This reflects the ongoing discourse on the role of A.I. in advancing mathematical knowledge.

💡Proofs

In mathematics, a proof is a logical argument that establishes the truth of a statement. The script delves into the evolving nature of proofs with the advent of A.I., questioning what we historically needed from proofs and how A.I. might change that. The speaker ponders, 'what is it we want from proofs?' and 'how does AI change that?', indicating that the traditional concept of a proof is being challenged by the capabilities of artificial intelligence.

💡Analytic Number Theory

Analytic number theory is a branch of mathematics that uses methods of analysis to solve problems in number theory. The speaker, Andrew Granville, mentions working in this field and having previously worked on Fermat's Last Theorem. The script states, 'I work in analytic number theory,' and 'When I was young, I worked on Fermat's Last Theorem before it was proved,' showcasing the speaker's expertise and the application of analytic methods in advancing mathematical theorems.

💡Graphic Novel

A graphic novel is a form of literature that combines art and narrative in a book format. The speaker discusses the idea of creating a graphic novel in mathematics to make the subject more accessible and engaging. The script mentions, 'We worked together to develop it and eventually wrote it,' referring to the collaborative effort to produce a graphic novel that could introduce mathematical concepts in a novel way.

💡Philosophy of Mathematics

The philosophy of mathematics is a branch of philosophy that explores the foundations, methods, and implications of mathematics. The script introduces Michael Hallett, a philosopher of mathematics, who is interested in the portrayal of mathematics in the speaker's graphic novel. The video discusses the philosophical implications of A.I. in mathematics, such as the debate over what it means to have something proved and the nature of axioms.

💡Aristotle

Aristotle was an ancient Greek philosopher whose works have influenced many areas of knowledge, including the philosophy of mathematics. The script references Aristotle's view on proof, stating, 'Aristotle said that to prove something is true, to establish it's true, your argument should rest on what he called primitives, things you already knew to be true.' This reflects the historical underpinnings of the concept of proof in mathematics.

💡Lean

Lean is a proof assistant, a software tool that helps mathematicians verify the correctness of mathematical proofs. The speaker discusses using Lean to formalize and verify proofs, comparing it to an 'obnoxious colleague who just won't let go.' The script describes Lean's role in the process of proof verification, where it 'acts like an obnoxious colleague... asking the most questions' and helping the mathematician refine their proof.

💡Peter Scholze

Peter Scholze is a renowned mathematician known for his work in algebraic geometry. The script mentions an example involving Scholze, where he used Lean to verify a difficult proof. The video states, 'This is particularly famous example by Peter Scholze. He had a very, very difficult proof that he wasn't 100% sure of.' This illustrates the practical application of proof assistants like Lean in the work of contemporary mathematicians.

💡Computer-Generated Proofs

Computer-generated proofs are proofs that are created or verified by computers, often using proof assistants or A.I. The video script raises questions about the future of mathematics as computers become more capable of generating proofs. The speaker contemplates, 'will machines change mathematics?' and discusses the potential for computers to 'lead you in a proof, not just follow you or make suggestions, but prove things itself.' This reflects the ongoing discussion about the role of computers in the creation and verification of mathematical knowledge.

Highlights

The traditional view of mathematics as a deductive system based on axioms is challenged by the reality of the field.

A.I.'s role in mathematics raises questions about the nature of proofs and the future of mathematicians.

Andrew Granville discusses the philosophical and practical implications of A.I. in analytic number theory.

The debate on the definition of 'proof' and its evolution with the advent of computer-assisted proofs.

Aristotle's concept of primitives and axioms in the context of mathematical proofs.

The historical method of verifying mathematical truths through published papers and libraries.

How A.I. programs like Lean store and verify mathematical proofs within their systems.

The analogy of Lean as an 'obnoxious colleague' that challenges and refines a mathematician's proof.

Peter Scholze's experience using Lean to verify and refine a complex proof.

The potential for A.I. to not only suggest but also generate new proofs in mathematics.

The concern that reliance on A.I. might diminish the value and understanding of profound mathematical proofs.

The possibility of mathematicians becoming more like physicists, relying on A.I. for proof verification.

The uncertainty of the role and purpose of doing mathematics in the future with A.I. advancements.

The impact of computer-generated proofs on the training and skill set of mathematicians.

The philosophical questions arising from the interplay between human mathematicians and A.I. assistants.

The potential shift in the mathematician's role from proof creator to idea generator.

The exploration of the graphic novel medium as a means to convey mathematical philosophy and ideas.

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