At the root of all these advances is computation, and in particular, the pioneering work of Kurt Gödel and Alan Turing in the 1930s.
This work and its legacy is the focus of the volume under review.
Such a view seems to have been part of Brouwer's belief that mathematical thought is essentially unformalizable.
One might instead maintain the need for higher-order logic to formalize adequately the concepts and inferences of branches of mathematics that implicate the infinite, such as real analysis.
Extending the case made in his 2006 article on the same subject, Shapiro articulates a view of Friedrich Waismann that our pre-theoretical mathematical concepts are generally not determined in all possible ways, so that there are typically many ways to sharpen concepts further, rather than just one "correct" way (as the Platonist would have it).
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He illustrates the "open texture" of mathematical concepts by drawing on Lakatos' famous dialogue on the concept of polyhedra, in which a series of proposed definitions of polyhedra are confronted with confounding cases that suggest a series of revisions to these proposals.A reader of this volume will acquire a broad acquaintance with the history of the theory of computation in the twentieth century, and with ways in which this theory will continue to develop in the twenty-first century.At the heart of the twentieth-century revolution of computation are Gödel's incompleteness theorems of 1931, which assert the existence of arithmetic sentences that are true in the standard natural numbers but unprovable in any formalized axiomatic theory of those natural numbers.Kripke holds that even if his thesis is only understood as a reduction of Church's thesis to Hilbert's thesis, he has amplified the Church-Turing thesis in a substantive way.Stewart Shapiro's article makes the case, in contrast to Kripke's, that the Church-Turing thesis cannot be proved.The question is whether these sharpenings were somehow "tacit" in our original pre-theoretic concept, or whether these revisions are instead replacing the original concept with something new.The advocate of open texture holds that the original concept was not precise enough to fix any particular revision as being right.The Gödel completeness theorem for first-order logic entails that P is a first-order consequence of a first-order theory T if and only if P can be deduced from T by first-order logic.Taking P as the conclusion of the given computation/deduction and T as the premises of this deduction, the completeness theorem yields that there is a first-order deduction of P from T.It would be no exaggeration to say that computation changed the world in the twentieth century.Implicated in nearly all contemporary technology, today's computers empower so-called "intelligent systems" to negotiate the world of human reasoners, even driving cars.