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

Quanta Magazine
31 Aug 202306:34

TLDRIn the realm of mathematics, the traditional approach of building proofs on axioms is being challenged by AI's role in proof generation. Andrew Granville discusses the philosophical and practical implications of AI in mathematics, questioning the future of mathematicians as AI systems like Lean begin to verify and even assist in creating proofs. The summary delves into the philosophical debate of what constitutes a proof and the potential shift in mathematicians' roles, hinting at a future where the line between human and machine contributions to mathematics becomes blurred.

Takeaways

  • 📚 The traditional view of mathematics is that it's built 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 verifying the next steps in proofs.
  • 🔍 Mathematicians are questioning what we truly seek from proofs and how AI could alter our understanding and belief in what constitutes a proof.
  • 👨‍🎓 Andrew Granville, an analytic number theorist, discusses the intersection of mathematics, computation, and philosophy, 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 caught the attention of philosopher Michael Hallett.
  • 📖 Aristotle's concept of proof resting on 'primitives' or axioms is contrasted with the self-evident truths in mathematics, highlighting a philosophical debate on the nature of proof.
  • 📚 The historical method of verifying mathematical truths involved physical libraries and books, which is now being mirrored by AI programs storing and verifying proofs digitally.
  • 🤖 Programs like Lean act as rigorous and persistent colleagues, questioning every step of a proof and aiding mathematicians in refining their arguments.
  • 🧠 The example of Peter Scholze's proof illustrates how AI can help verify the validity of complex proofs and identify areas of uncertainty.
  • 🔮 The future of mathematics with AI is uncertain, with the potential for machines to lead in proof generation, which could redefine the role and training of mathematicians.
  • 🧐 The philosophical and professional implications of relying on AI for proofs are profound, possibly shifting the identity of mathematicians towards a more physics-like approach.

Q & A

  • What is the traditional conception of how mathematical knowledge is built?

    -The traditional conception is that mathematical knowledge is built on a bedrock of axioms through deductive argument, creating a solid and incontrovertible structure of brilliant mathematics.

  • What is the role of A.I. in the context of mathematicians' work as discussed in the transcript?

    -A.I. is being explored for its potential to assist in guessing the next steps in mathematical proofs and to handle the input of ideas or proofs, raising questions about the future interaction between mathematicians and machines.

  • What are some of the philosophical questions arising from the use of A.I. in mathematics?

    -The philosophical questions include what we want from proofs, what we have historically needed from proofs, and how A.I. changes our beliefs about what constitutes a proof.

  • Who is Andrew Granville and what are his areas of interest in mathematics?

    -Andrew Granville is a mathematician working in analytic number theory. He has worked on Fermat's Last Theorem and is currently interested in L functions and multiplicative functions, as well as computational and algorithmic questions.

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

    -The graphic novel was an attempt to popularize mathematics and develop philosophical ideas within it. It caught the interest of philosopher Michael Hallett, who engaged with the portrayal of mathematics being done in the novel.

  • What is the debate about the meaning of 'proof' in mathematics?

    -The debate revolves around what we mean by 'proof', how we establish something as true based on axioms or primitives, and the role of self-evident truths in mathematical arguments.

  • How does the traditional method of verifying mathematical truths differ from the use of A.I. in proofs?

    -Traditionally, verification involved consulting published papers in libraries. In contrast, A.I. systems like Lean store proven information within the program, allowing mathematicians to input and verify proofs directly through the system.

  • What is the function of the Lean program in the context of formal proofs?

    -Lean acts as a rigorous and persistent colleague, questioning every step of a proof until it is fully understood and verified according to its library of axioms and proven statements.

  • Can you provide an example of how A.I. has been used to verify a difficult proof?

    -Peter Scholze used Lean to verify a difficult proof he was unsure about. The areas where Lean asked the most questions corresponded to the parts of the proof Scholze was most uncomfortable with, providing a form of verification.

  • What are the implications of computer-generated proofs for the future of mathematics?

    -Computer-generated proofs could change the nature of mathematical work, potentially making mathematicians more like physicists who propose ideas and rely on computers to verify them. This raises questions about the value and training of mathematicians in the future.

  • What is the current state of computer-generated proofs and their impact on significant mathematical work?

    -While computer-generated proofs are in their infancy and have not yet contributed much to the bigger picture of mathematics, there is optimism that new ideas may lead to the generation of interesting new proofs.

Outlines

00:00

📚 The Myth of Axiomatic Mathematics and AI's Role

Andrew Granville, an expert in analytic number theory, discusses the common misconception that mathematics is a solid structure built on axioms through deductive reasoning. He points out that this idealized view is not the reality and that even top mathematicians are questioning this philosophy. Granville shares his experience working with AI in mathematics, pondering when machines might outperform humans in conjecturing the next steps in proofs. He raises significant questions about the nature and purpose of proofs in the era of AI, and how AI might alter our fundamental beliefs about mathematical proof verification.

05:03

🤖 The Future of Mathematics: Human or Machine?

The script delves into the implications of AI-generated proofs and the potential shift in the role of mathematicians. It suggests that as AI becomes more capable of handling detailed proofs, the traditional value we place on the profundity of human proofs may diminish. This could lead to a change in mathematicians' training and identity, possibly making them more akin to physicists who rely on computational validation. The narrative speculates on the future of mathematics, suggesting that as computers take on more tasks, the limits of their capabilities in this field remain unclear, presenting both exciting possibilities and existential challenges for the profession.

Mindmap

Keywords

💡Axioms

Axioms are the foundational statements or principles that are accepted as true without proof, serving as the basis for logical reasoning and argumentation in mathematics. In the video, the concept of axioms is challenged by the idea that even the most established mathematical structures may not be as solid as traditionally believed, and the role of axioms in the context of computer-assisted proofs is discussed.

💡Deductive Argument

Deductive argument is a method of reasoning that moves from one or more statements assumed to be true to reach a logically certain conclusion. The script discusses the classical view of mathematics as a deductive structure built upon axioms and how this view is being questioned in the era of AI and computer-generated proofs.

💡A.I. in Mathematics

A.I. in Mathematics refers to the use of artificial intelligence to assist in mathematical proofs and problem-solving. The video script explores the impact of A.I. on the traditional roles of mathematicians, particularly in the context of proof generation and verification.

💡Proofs

Proofs in mathematics are rigorous demonstrations that a statement is true based on previously established statements, such as theorems, axioms, or definitions. The script raises questions about the nature and purpose of proofs in light of advancements in A.I. and computational tools.

💡Analytic Number Theory

Analytic Number Theory is a branch of number theory that uses methods of analysis to solve problems related to the distribution of prime numbers and other aspects of number sequences. Andrew Granville, mentioned in the script, works in this field, and it serves as an example of the diverse areas of mathematics that can be influenced by computational methods.

💡Fermat's Last Theorem

Fermat's Last Theorem is a famous problem in number theory that states there are no three positive integers a, b, and c that can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two. The script mentions that the speaker worked on this theorem before it was proved, highlighting the historical context of mathematical discovery.

💡L-functions

L-functions are complex functions that play a crucial role in analytic number theory, particularly in the study of the distribution of prime numbers. The script mentions that Andrew Granville works with ideas of L-functions, indicating the ongoing research and development in number theory.

💡Graphic Novel in Mathematics

A graphic novel in mathematics is a creative way to present mathematical concepts and narratives in a visual and engaging format. The script discusses the speaker's collaboration with his sister, a writer, to develop a graphic novel that includes philosophical discussions about mathematics.

💡Philosophy of Mathematics

The Philosophy of Mathematics is the branch of philosophy that explores the foundations, methods, and implications of mathematics. The script references a philosopher of mathematics, Michael Hallett, who is interested in the portrayal of mathematical practice in the graphic novel, particularly the nature of proof and justification.

💡Lean

Lean is a proof assistant, a software tool that helps mathematicians verify the correctness of proofs by checking them against a formal system of axioms and rules. The script describes Lean as acting like an 'obnoxious colleague' that questions every step of a proof, aiding in the verification process.

💡Computer-Generated Proofs

Computer-generated proofs are proofs that are created or verified by computer programs, which can handle complex calculations and logical deductions that may be beyond human capacity. The script discusses the potential of these proofs to revolutionize mathematics and the role of mathematicians in the future.

Highlights

The traditional conception of mathematics as a deductive system based on axioms is challenged by modern realities.

AI's role in mathematics is evolving, prompting questions about the nature and purpose of proofs.

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

The debate over what constitutes a 'proof' in mathematics is reignited by advancements in AI.

Aristotelian principles of proof based on self-evident 'primitives' or axioms are questioned in the context of AI.

The historical method of verifying mathematical truths through published papers and libraries is being transformed by AI.

Programs like Lean are changing how mathematicians input, verify, and trust proofs.

Lean acts as a rigorous colleague, challenging and refining mathematicians' proofs.

Peter Scholze's experience with Lean highlights how AI can uncover uncertainties in complex proofs.

The potential for AI to generate new proofs, rather than just verify existing ones, is a new and exciting frontier.

The impact of AI on the profession of mathematicians raises existential questions about their role and training.

AI's ability to handle proof details may lead mathematicians to adopt a more empirical approach, similar to physicists.

The future of mathematics may see a shift in the value and training placed on proof generation and understanding.

Granville speculates on the changing landscape of mathematics in the next 30 to 40 years due to computer-generated proofs.

The philosophical implications of AI in mathematics are as profound as the practical, with the potential to redefine the field.

The graphic novel collaboration between Andrew Granville and his writer sister introduces a new way to explore mathematical philosophy.

Michael Hallett's interest in the graphic novel underscores the ongoing dialogue about the nature of mathematical proof.

The self-evidence of mathematical axioms is scrutinized in light of AI's ability to process and question foundational truths.

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