The Impact of AI and Formalization on Mathematics: A City Planning Analogy by Terence Tao

In a recent post, renowned mathematician Terence Tao (@tao@mathstodon.xyz) drew an insightful analogy between the impact of automobiles on urban planning and the influence of AI and formalization on mathematics. This comparison highlights how new technologies can reshape existing infrastructures and practices, often with both positive and negative consequences.

The Analogy: Cars and Urban Planning

Before cars, city streets were narrow and designed for pedestrians, horses, and carriages. When automobiles arrived, they brought unprecedented speed and power but also congestion and safety issues. Over time, cities adapted by building new roads, railways, and freeways to accommodate these vehicles, leading to significant changes in urban landscapes and lifestyles.

However, the initial focus on making cars faster and more efficient did not solve all problems. Urban sprawl, environmental degradation, and traffic congestion became major concerns. It was only through thoughtful urban planning and the development of social and legal rules that cities managed to balance the benefits of automobiles with the needs of pedestrians.

The Mathematics Infrastructure

In this analogy, the narrow pre-automotive roads represent the existing infrastructure of mathematics:

• Papers and Journals: These are the primary means of disseminating mathematical knowledge.
• Conferences and Workshops: Platforms for mathematicians to meet, discuss, and collaborate.
• Citations: A way to acknowledge and build upon previous work.
• Advisor-Student Relationships: The mentorship model that guides new researchers.
• Job Postings and Funding Opportunities: The economic and career pathways in the field.

These elements are designed primarily for human use, with limited computer assistance. They form a social network that supports the mathematical community's activities and progress.

The Arrival of AI and Formalization

AI and formalization tools (like proof assistants) are like automobiles in this analogy. They offer significant advantages:

• Speed and Efficiency: AI can process large datasets and perform complex calculations much faster than humans.
• Formal Verification: Proof assistants can rigorously verify the correctness of mathematical proofs, reducing errors.

However, these tools also introduce challenges:

• Congestion and Overload: The rapid generation of results and data can overwhelm traditional publication and review processes.
• Accessibility Issues: Not all mathematicians are equally adept at using AI or formalization tools, potentially creating a divide within the community.

Adapting to the New Reality

Just as cities had to build new infrastructure and establish rules for traffic management, the mathematical community needs to adapt to the integration of AI and formalization:

• New Publishing Models: Journals and conferences may need to evolve to handle the volume and complexity of AI-generated results.
• Formal Verification Standards: Establishing standards for using proof assistants can ensure that new results are reliable and reproducible.
• Training and Education: Providing training in AI and formalization tools can help bridge the skill gap within the community.
• Ethical Considerations: Addressing ethical issues, such as data privacy and algorithmic bias, is crucial to maintaining trust in mathematical research.

Conclusion

Terence Tao's analogy underscores that while AI and formalization offer exciting opportunities for mathematics, they also require careful planning and adaptation. By learning from the lessons of urban development, the mathematical community can leverage these tools effectively while preserving the integrity and inclusivity of its practices.