The outer limits of reason. What science, mathematics, and logic cannot tell us (Q2844487)

In a book aimed at the educated reader, which avoids as much as possible formulas, the author presents the present state of knowledge on the provably impossible to know. Starting with the mere paradoxical, such as liar's paradox, Zeno's paradoxes, Russell's paradox, Yablo's paradox, Berry's paradox, the Monty Hall paradox, the author moves on to infinite sets, Cantor's diagonal argument, the Zermelo-Fraenkel axioms, the Banach-Tarski paradox, and then to the heart of the limitations of reason, computability, and decidability. These include NP-complete problems (such as the traveling salesman problem, the partition problem, the subset sum problem, the halting problem), the \(n\)-body problem, various limitations arising from quantum mechanics (superposition, Heisenberg's uncertainty principle, the Kochen-Specker theorem, Schrödinger's cat, Bell's theorem, the Conway-Kochen free will theorem), from relativity theory (length contraction, time dilation, the relativity of simultaneity), as well as several ``mathematical obstructions'', in historical order: the incommensurability of the side and the diagonal of a square, the unsolvability by ruler and compass of three of the classical Greek construction problems, the undecidability of the tiling problem, Tarski's theorem on the undefinability of truth in arithmetic, Gödel first incompleteness theorem (with mention of the unprovability of Goodstein's theorem in PA), Parikh's theorem of 1971, Löb's theorem, Gödel's second incompleteness theorem, Gentzen's theorem (that the consistency of ZFC implies the consistency of PA). The meaning and impact of these limitations as well as of Wigner's ``unreasonable effectiveness'' are discussed from two philosophical points of view, nominalism and Platonism, and deeper philosophical issues are addressed, including Popper's and Kuhn's views on the philosophy of science, as well as the anthropic principle.