 
Summary: Why Philosophers Should Care About Computational Complexity
Scott Aaronson
Abstract
One might think that, once we know something is computable, how efficiently it can be com
puted 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 the
orythe field that studies the resources (such as time, space, and randomness) needed to solve
computational problemsleads 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.
Contents
1 Introduction 2
1.1 What This Essay Won't Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Complexity 101 5
3 The Relevance of Polynomial Time 6
3.1 The Entscheidungsproblem Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Evolvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Known Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
