The Future of Mathematical Proof: AI-Generated Coq Programs and the Riemann Hypothesis

In the realm of mathematics, the verification of complex conjectures like the Riemann Hypothesis has long been a monumental challenge. Imagine a scenario where the first rigorous proof of such a conjecture isn't just lengthy and intricate, but also generated by an AI as a Coq program. This article explores the implications of such an event, delving into the potential impact on the mathematical and computational communities.

The Challenge of Verification

Let us assume that a proof for the Riemann Hypothesis or another major conjecture has been discovered, but it is not straightforward to verify. The proof, written in Coq, is so complex and extensive that no human or team of humans can follow it entirely. However, by this point, we would have advanced proof-reading engines and 'reader’s digest' algorithms capable of transforming technical proofs into more accessible formats. Nonetheless, the initial reaction would likely involve a mix of confidence and sorrow.

People would recognize that such a proof, if verified, would solidify the nature of the Riemann Hypothesis, but the deep complexity and lack of human interpretability might take away some of the 'glory' of understanding the problem fully. This could challenge our perspective on the role of humans in reasoning and computation.

The Role of AI in Mathematical Proofs

Consider the scenario where an AI generates a Coq program as a proof for a significant theorem. If the proof is verifiable, no one would mind the AI's contribution, but if the proof is unreadable and no one can understand it, it raises serious questions. People might demand a ZFC-based check to ensure the consistency and reliability of the proof. The community's response would be particularly significant if the AI proves a controversial result, such as P NP.

The fear is that such an event could undermine confidence in the community's belief in proof-checking, the consistency of Coq's logic, or the trust in ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice). To maximize the fun, one might even use zero-knowledge proofs to demonstrate the existence of a proof without revealing its content, thereby first challenging cryptography before undermining trust in proof-checking methods.

Emerging Trends in Mathematical Proofs

As computer proof-assistants evolve into proof-discoverers, the landscape of mathematical proofs is expected to shift. Initially, these tools are likely to generate proofs for smaller lemmas and less complex results. As tools improve, they will be able to tackle larger, more significant theorems. However, even for these massive results, the goal remains to make the proofs understandable and interpretable.

Proofs generated by automated tools, like Coq programs, might appear daunting at first glance. However, these programs are not unreadable; they are just extensive and require advanced tools for interpretation. The blueprint and dependency graph of a Lean proof, for instance, provides a framework for understanding and breaking down these proofs. This trend suggests that the efforts to produce human-comprehensible proofs are not in vain; they will merely need to broaden to include the analysis of computer-generated proofs.

Conclusion

The advent of AI-generated proofs, particularly in Coq, challenges our entrenched notions of mathematical proof and the role of humans in mathematical discovery. While the initial reactions might be mixed, this is an exciting opportunity for the mathematical and computational communities to adapt and evolve. The future is not about abandoning our reasoning power but about expanding our capacity to understand and interpret the vast proofs generated by advanced computational tools.