Why Philosophers Should Care About Computational Complexity (1108.1791v3)

Abstract: One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed case that one would be wrong. In particular, I argue that computational complexity theory -- the field that studies the resources (such as time, space, and randomness) needed to solve computational problems -- leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume's problem of induction, Goodman's grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing aspects of complexity theory itself that could benefit from philosophical analysis.

• The paper argues that computational complexity theory offers crucial insights into long-standing philosophical debates on the nature of knowledge, intelligence, and rationality.
• It emphasizes the philosophical significance of the polynomial vs. exponential distinction in computational resource constraints for understanding feasibility and limits.
• The essay explores how complexity theory sheds light on topics such as the Turing Test, logical omniscience and bounded rationality, quantum computing foundations, and economic models.

Philosophical Implications of Computational Complexity

The essay "Why Philosophers Should Care About Computational Complexity" by Scott Aaronson explores the intricate relationship between computational complexity theory and several philosophical debates. The author meticulously argues that computational complexity offers fresh insights into enduring philosophical questions, particularly through considering the resource constraints inherent to computational processes.

Key Themes and Philosophical Implications

1. Computational Complexity and Philosophical Inquiry: Aaronson postulates that computational complexity extends beyond mere practical application, touching upon the nature of mathematical knowledge, strong AI, and the problem of logical omniscience. Complexity theory, dealing with resources necessary to solve computational problems, unveils new perspectives on philosophical issues previously dominated by computability theory.
2. The Polynomial vs. Exponential Barrier: Aaronson emphasizes the significance of the distinction between polynomial and exponential growth rates in resource usage, both practically and on a conceptual basis. This distinction sheds light on philosophical discourse regarding feasibility and computability, offering clarity in discussions about the limits of human knowledge and machine intelligence.
3. Complexity and the Turing Test: In addressing whether machines can exhibit genuine intelligence, Aaronson argues that complexity theory provides a framework for understanding the computations necessary to simulate human thought. The computational resources required for a machine to plausibly pass the Turing Test underscore practical boundaries in AI, rather than mere computability.
4. Logical Omniscience and Bounded Rationality: Complexity theory challenges classical notions of logical omniscience by introducing resource constraints, which more accurately model human thought processes. This perspective aligns with bounded rationality in economics, which suggests that limitations on computational resources are essential to understanding realistic decision-making scenarios.
5. Quantum Computing and the Foundations of Physics: Quantum computing brings complexity theory into conversation with the philosophical foundations of quantum mechanics. The potential exponential speed-up provided by quantum algorithms enriches discussions around the Many-Worlds interpretation and challenges to local realism, highlighting complexity’s role in the reconciliation of physics and computing.
6. Complexity in Economic Reasoning: Classical economic models frequently assume rational agents with perfect information, yet complexity theory suggests that finding equilibria or optimal strategies can be computationally infeasible. This introduces a new dimension to economic theory, proposing that infeasibility in computation might explain deviations from classical rational behavior.

Future Directions and Open Questions

Aaronson identifies several avenues for continued exploration, particularly concerning the empirical and philosophical status of asymptotic complexity claims. Complexity theory’s conjectures, such as $\mathsf{P} \neq \mathsf{NP}$, remain unproven, yet they hold substantial theoretical weight. Philosophers and complexity theorists alike may benefit from addressing the implications of these conjectures on our understanding of mathematical creativity and the interpretation of quantum mechanics.

Overall, Aaronson's essay serves as a clarion call for deeper interdisciplinary collaboration, inviting philosophers to integrate computational complexity insights into their analysis of the mind, knowledge, and rationality. Complexity theory, with its rich theoretical landscape, offers a profound toolkit to address some of philosophy’s most persistent challenges.