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Why mathematics is set to be revolutionized by AI

๐ŸŒˆ Abstract

The article discusses how machine learning and artificial intelligence (AI) are transforming the field of mathematics, particularly in the area of generating mathematical conjectures. It highlights how AI can help identify patterns and form conjectures, but also notes the limitations of AI in fully understanding the broader context and importance of these conjectures.

๐Ÿ™‹ Q&A

[01] Giving birth to a conjecture

1. What is a mathematical conjecture, and how does it differ from an educated guess?

  • A mathematical conjecture is a proposition that is suspected to be true but requires definitive proof, unlike an educated guess.
  • Formulating mathematical conjectures requires a combination of genius, intuition, and experience.
  • Even mathematicians can struggle to explain their own discovery process for conjectures.

2. How have researchers applied machine learning to mathematical data, and what have they discovered?

  • Researchers at the London Institute for Mathematical Sciences have applied machine learning to mathematical data, including discovering that simple AI classifiers can predict the rank of an elliptic curve, which is a crucial step towards solving one of the Millennium Problems.
  • This was unexpected, as the high-stakes arena of the Millennium Problems was thought to be beyond the reach of AI.

[02] AI's impact on mathematical discovery

1. What are some examples of how AI has made inroads in mathematical discovery?

  • A computer program called the Ramanujan Machine produced new formulas for fundamental constants like ฯ€ and e by exhaustively searching through families of continued fractions.
  • Researchers at Google DeepMind trained a neural network on data for different knots and discovered an unexpected relationship between their algebraic and geometric structures.

2. How has AI been able to make a difference in areas of mathematics where human creativity was thought to be essential?

  • In mathematics, a single counterexample can falsify a conjecture, and AI can help identify such counterexamples more efficiently.
  • Mathematical data, such as primes, knots, and integer sequences, are abundant and can be used to train AI systems.
  • AI can help spot patterns and form conjectures, but not all conjectures are equally valuable in advancing mathematical understanding.

3. What are the limitations of AI in the context of mathematical discovery?

  • AI may struggle to identify the most important conjectures that can lead to significant advancements in mathematics, as this requires an intuition for how the field will evolve, which remains out of AI's reach for now.
  • The imagination and intuition of human mathematicians will still be required to make sense of the output of AI tools and to determine the broader significance of the conjectures.

[03] The role of conjectures in mathematics

1. What is the importance of publishing more conjectures in mainstream mathematics journals?

  • Some of the most significant problems and conjectures in mathematics have shaped the course of the field, and publishing more conjectures can speed up research by pointing researchers in the right direction.
  • Journal articles about conjectures, backed by data or heuristic arguments, can accelerate the discovery process.

2. How does the role of AI in mathematical discovery compare to its role in other fields, such as materials science?

  • In materials science, AI can predict a large number of potential new crystal structures, but determining their stability, synthesizability, and practical applications still requires the expertise of human researchers.
  • Similarly, in mathematics, AI will act as a catalyst for human ingenuity, rather than a substitute for it, as the broader context and significance of the conjectures will still require the intuition and imagination of human mathematicians.
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