Hard Proofs and Good Reasons (2410.18994v1)

Abstract: Practicing mathematicians often assume that mathematical claims, when they are true, have good reasons to be true. Such a state of affairs is "unreasonable", in Wigner's sense, because basic results in computational complexity suggest that there are a large number of theorems that have only exponentially-long proofs, and such proofs can not serve as good reasons for the truths of what they establish. Either mathematicians are adept at encountering only the reasonable truths, or what mathematicians take to be good reasons do not always lead to equivalently good proofs. Both resolutions raise new problems: either, how it is that we come to care about the reasonable truths before we have any inkling of how they might be proved, or why there should be good reasons, beyond those of deductive proof, for the truth of mathematical statements. Taking this dilemma seriously provides a new way to make sense of the unstable ontologies found in contemporary mathematics, and new ways to understand how non-human, but intelligent, systems might found new mathematics on inhuman "alien" lemmas.

• The paper's main contribution is its analysis of the tension between vast proof complexity and the quest for cognitively satisfying mathematical reasons.
• It uses examples like the Goldbach Conjecture and principles such as No-Coincidence to highlight the gap between computational proofs and heuristic reasoning.
• The study discusses implications for human cognition and AI-assisted verification, challenging traditional views on mathematical ontology.

Understanding the Interrelation of Hard Proofs and Good Reasons in Mathematics

The paper, "Hard Proofs and Good Reasons," authored by Simon DeDeo, addresses a significant and often understated issue in contemporary mathematics: the dichotomy between the adequacy of reasons for mathematical truths and the complexity of their proofs. This work explores the phenomena observed by mathematicians, where true mathematical statements are often believed to have good reasons, despite formidable results from computational complexity suggesting many theorems possess only exponentially-long proofs. Such proofs generally fail to provide the cognitive satisfaction expected as good reasons for their corresponding truths.

The primary argument presented is that there exists a tension between the intuitive sense that mathematical claims are true for good reasons and the stark reality of proof complexity where numerous theorems entail proofs that are infeasible in practical terms. This leads to a thought-provoking question: Are mathematicians naturally approaching those truths that can be efficiently proven, or do they operate under a belief system where good reasons diverge from efficient proofs?

Key Insights and Contributions

The paper highlights several noteworthy contributions:

1. The "No-Coincidence" and "No-Miracles" Principles: DeDeo connects the empirical practices of mathematicians, which involve heuristics like Timothy Gowers's No-Coincidence Principle. This principle suggests that if an apparent mathematical coincidence occurs, a reason must exist, akin to a probabilistic or philosophical expectation. These principles underline the mathematician's belief in the sufficiency of reasons without necessarily relying on formal proofs at the preliminary stages.
2. Examples Examined: DeDeo examines examples, such as the Goldbach Conjecture, where substantial computational verification supports the conjecture's truth, yet a comprehensive proof eludes us. These examples illuminate how mathematical reasoning often relies on probabilistic or statistical laws rather than immediate deductive proofs.
3. The Gap between Proofs and Reasons: The theoretical foundation discusses proof complexity, which shows numerous tautologies necessitate long proofs, raising the question of how these align with the notion of good reasons. This gap proposes a further exploration of whether good reasons are indeed equivalent to efficient proofs, a central theme throughout the paper.
4. Theoretical Implications: The paper highlights significant theoretical implications regarding the adaptability of human reasoning to a domain where true statements with manageable proofs are sparse. It contemplates how artificial intelligence might intervene by accepting the validity of truths that, although infeasible for human cognition, hold under computationally intensive methods.
5. Impacts on Mathematical Ontology: The exploration of how modern mathematicians navigate between conjectural and formal proofs suggests transformations in the ontology of mathematics. DeDeo posits that the evolving landscape of mathematical exploration, where notions of truths and proofs coexist in new dynamics, influences the philosophical understanding of mathematical truths.

Future Directions and Speculative Insights

The paper opens avenues for future research that could probe deeper into how mathematicians can efficiently navigate mathematical landscapes dominated by seemingly insurmountable proof complexities. Moreover, it suggests an exploration into the role AI might play in discovering new mathematical fields based on complex proofs verified through automated systems.

While the paper refrains from sensationalizing its findings, it does cleverly situate its claims in the broader context of mathematical practice and theory, proposing a framework where future mathematicians may rely on computational tools for proving statements that defy conventional reasoning.

In conclusion, "Hard Proofs and Good Reasons" invites the academic community to critically analyze the philosophical and practical implications of proof complexity. It suggests a shift from traditional human-centric methods of mathematical reasoning toward a paradigmatic integration with artificial computational assistance, redefining the ontological and epistemic boundary of what constitutes a mathematical truth.