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

Quanta Magazine
31 Aug 202306:34

TLDRThe transcript explores the evolving nature of mathematical proof in the age of AI. It challenges the traditional view of mathematics as a solid structure built on axioms, highlighting the role of AI in verifying and potentially discovering proofs. The speaker, Andrew Granville, discusses his experiences with AI in mathematics, the philosophical questions it raises, and the impact on the identity and future of mathematicians. The conversation touches on the use of AI platforms like Lean for proof verification and the potential for machines to lead in mathematical discovery, raising concerns about the value and training of mathematicians in a future where computers handle proofs.

Takeaways

  • 📚 The common misconception about mathematics is that it is built purely on a foundation of axioms and deductive reasoning, creating an unassailable structure of knowledge.
  • 🤖 The integration of AI into mathematics, particularly through tools like Lean, is challenging traditional notions of proof and verification.
  • 🔍 Top mathematicians are exploring the philosophical implications of AI's role in mathematics, questioning what we historically needed from proofs and how AI alters that understanding.
  • 💡 AI can assist in the proof process by acting as a critical colleague, pushing for clarity and simplicity in explanations, as exemplified by Peter Scholze's experience.
  • 📈 The development of computer-generated proofs is still in its infancy, with most work not yet contributing significantly to the broader mathematical landscape.
  • 🌟 There is optimism that AI may lead to the generation of interesting new proofs, potentially revolutionizing the field of mathematics.
  • 🧠 The reliance on machines for proof verification raises concerns about the future training and value of mathematicians, as the profession may shift towards a more physicist-like approach.
  • 🌐 The increasing capabilities of computers in mathematics raise questions about the future role and nature of human involvement in the field.
  • 🤔 The debate over the meaning of 'proof' is highlighted by the discussion of axioms and primitives, questioning the self-evident nature of mathematical truths.
  • 📚 The traditional method of verifying mathematical truths involved accessing published papers and books, a process that is now being supplemented by AI technology.
  • 🌟 The potential for AI to change the landscape of mathematics is vast, with unknown limits to its capabilities, suggesting a transformative future for the discipline.

Q & A

  • What is the common misconception about the foundation of mathematics according to the transcript?

    -The common misconception is that mathematics is built solely on a bedrock of axioms through deductive argument, and that there exists a towering and incontrovertible structure of mathematical knowledge.

  • How does the use of AI in mathematics challenge traditional notions of proof?

    -AI challenges traditional notions by assisting in guessing the next step in proofs and potentially doing a better job than humans at times, which raises questions about what we have historically needed from proofs and how our belief in mathematical truth might change.

  • What is Andrew Granville's field of expertise?

    -Andrew Granville works in analytic number theory.

  • What was Andrew Granville's contribution to Fermat's Last Theorem?

    -He worked on Fermat's Last Theorem before it was proved.

  • What other interests does Andrew Granville have in mathematics besides his main field?

    -He is interested in computational and algorithmic questions, as well as popular writing and graphic novels in mathematics.

  • How does the philosopher of mathematics, Michael Hallett, view the foundation of mathematical proofs?

    -Michael Hallett is interested in the portrayal of mathematics being done, particularly the debate on what it means to prove something, and the concept of axioms as self-evident truths.

  • What is the role of AI in storing mathematical proofs?

    -AI stores information within programs like Lean, which contain a library of already proven theorems based on axioms, allowing mathematicians to input their proofs for verification.

  • How does the Lean program assist mathematicians?

    -Lean acts like an obnoxious colleague that asks persistent questions to ensure the proof is logically sound, helping mathematicians refine their arguments and catch potential errors.

  • What was Peter Scholze's experience with Lean?

    -Peter Scholze used Lean to verify a difficult proof he wasn't completely sure of, and Lean's persistent questioning helped him refine the areas he was uncertain about.

  • How might computer-generated proofs impact the future of mathematics?

    -Computer-generated proofs could lead to new possibilities and challenges the traditional value of profound proofs in the mathematical profession, potentially changing how mathematicians think about and approach proofs.

  • What concerns does the transcript raise about the role of mathematicians in the future with AI?

    -The transcript raises concerns about the potential devaluation of the mathematicians' training and the profundity of their work if AI can handle most of the proof details, leading to a shift in the nature of mathematical inquiry.

Outlines

00:00

📚 The Myth and Reality of Mathematical Axioms

This paragraph discusses the common misconception among students that mathematics is built upon a solid foundation of axioms and deductive reasoning, leading to an unassailable structure of knowledge. The speaker, Andrew Granville, an analytic number theorist, dispels this fantasy by sharing his experiences with AI in mathematics. He explores the implications of AI's role in conjecturing the next steps in proofs and the philosophical questions it raises about the nature and purpose of proofs. Granville also delves into the historical context of proof verification through libraries and the evolution of this process with AI, exemplified by the Lean program. The narrative includes a personal anecdote where AI challenged and refined a complex proof by Peter Scholze, highlighting the potential of AI in mathematics.

05:03

🤖 The Future of Proofs: AI's Impact on Mathematicians

The second paragraph ponders the future implications of AI in mathematics, particularly the potential for AI to generate new and interesting proofs. It raises concerns about the value and identity of mathematicians if AI can handle the intricate details of proofs. The speaker speculates on the possibility of mathematicians taking on a role similar to physicists, relying on computers to verify their work without deep contemplation. The paragraph reflects on the shifting landscape of mathematics due to computer-generated proofs and questions the long-term effects on the profession, emphasizing the uncertainty of the field's future as computers become increasingly capable.

Mindmap

Keywords

💡axioms

Axioms are fundamental principles or statements that are accepted as true without proof, serving as the basis for logical arguments and reasoning in mathematics. In the context of the video, axioms are contrasted with the reality of how mathematics is often developed, suggesting that the idealized view of mathematics as strictly axiom-based is not entirely accurate. The speaker mentions axioms in relation to the misconceptions students have about the nature of mathematical proofs and the role of axioms in building mathematical knowledge.

💡deductive argument

Deductive argument is a method of reasoning that starts with one or more premises and reaches a logically certain conclusion based on those premises. In mathematics, this often involves starting with axioms and using logical steps to derive new truths. The video highlights the student's misconception that mathematics is purely based on deductive arguments from axioms, whereas the reality is more complex and involves various methods of discovery and proof.

💡A.I. (Artificial Intelligence)

Artificial Intelligence refers to the development of computer systems that can perform tasks typically requiring human intelligence, such as learning, reasoning, and problem-solving. In the video, AI is discussed in relation to its potential to assist in the process of mathematical proof development, raising questions about the role of AI in the future of mathematics and how it might change the way mathematicians work and think about proofs.

💡proofs

In mathematics, a proof is a logical demonstration that a statement is true. Proofs are central to the advancement of mathematical knowledge, as they provide a rigorous basis for accepting new theorems. The video discusses the historical and philosophical aspects of proofs, the evolving nature of what constitutes a proof, and the impact of AI on the traditional process of proving mathematical statements.

💡analytic number theory

Analytic number theory is a branch of mathematics that uses techniques from analysis to solve problems in number theory, which is the study of integers and their properties. The speaker, Andrew Granville, works in this field and has been involved in significant mathematical endeavors, such as working on Fermat's Last Theorem before it was proved. His interest in this area exemplifies the application of mathematical concepts and the pursuit of deep understanding in number theory.

💡computational and algorithmic questions

Computational and algorithmic questions pertain to the study of how mathematical problems can be solved using computers and the development of algorithms that efficiently perform these computations. In the video, these questions are part of the broader exploration of the role of technology and AI in mathematics, highlighting the intersection of mathematical research with computational methods.

💡graphic novel

A graphic novel is a book or novel that uses sequential art, such as comics, to tell a story. In the context of the video, the speaker and his sister collaborated on a graphic novel in mathematics, aiming to make the subject more accessible and engaging through a different medium. This project reflects the speaker's commitment to popularizing mathematics and exploring unconventional methods of communicating complex ideas.

💡philosophy of mathematics

The philosophy of mathematics is a branch of philosophy that explores the foundations, methods, and implications of mathematics. It includes questions about the nature of mathematical truth, the role of axioms and proofs, and the process by which mathematical knowledge is acquired. In the video, the speaker's work with a philosopher of mathematics, Michael Hallett, delves into these philosophical aspects, particularly the meaning of proof in mathematics.

💡Lean

Lean is a powerful interactive theorem prover and programming language designed for mathematical reasoning and formalizing mathematical knowledge. It allows mathematicians to input their proofs and verify them within the system, ensuring logical correctness based on a set of axioms. In the video, Lean is presented as a tool that can assist mathematicians in the proof process, acting as an 'obnoxious colleague' that challenges and clarifies the proof steps.

💡computer-generated proofs

Computer-generated proofs refer to mathematical proofs that are created or verified with the assistance of computer programs, rather than solely by human mathematicians. These proofs open up new possibilities for the advancement of mathematics, as they may lead to the discovery of new proofs and methods that were previously unattainable. The video discusses the potential impact of computer-generated proofs on the future of mathematics, including the possible changes in the role of mathematicians and the nature of mathematical discovery.

Highlights

The common undergraduate misconception that mathematics is built solely on a bedrock of axioms through deductive argument.

The reality that the beautiful conception of mathematics as an incontrovertible tower of knowledge is not true.

Top mathematicians grappling with the philosophy of mathematics in the context of AI.

The role of AI in assisting with mathematical proofs and guessing the next steps.

The question of what we historically needed from proofs and how AI changes that.

Andrew Granville's work in analytic number theory, including early work on Fermat's Last Theorem.

Granville's interest in computational and algorithmic questions within mathematics.

The collaboration with Granville's sister to create a graphic novel in mathematics.

Philosopher Michael Hallett's interest in the portrayal of mathematics in the graphic novel.

The debate on the meaning of 'proof' and the role of axioms in mathematics.

Aristotle's view on proving something as true by resting on primitives or axioms.

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

The emergence of AI in storing and verifying mathematical proofs within programs like Lean.

Lean's function as an obnoxious colleague that challenges and verifies every step of a proof.

Peter Scholze's experience with Lean验证 his difficult proof.

The potential of AI to change the nature of mathematical proofs and the role of mathematicians.

The concern that reliance on machines for proofs may alter the value and training of mathematicians.

The uncertainty of the future of mathematics with the advent of computer-generated proofs.

The exploration of new possibilities with AI in generating interesting new mathematical proofs.

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