This article provides a detailed analysis of the transfer of a key cluster of ideas from mathematical logic to computing. We demonstrate the impact of certain of Turing’s logico-philosophical concepts from the mid-1930s on the emergence of the modern electronic computer—and so, in consequence, Turing’s impact on the direction of modern philosophy, via the computational turn. We explain why both Turing and von Neumann saw the problem of developing the electronic computer as a problem in logic, and we describe their (...) joint journey from logic to electronic computation. While much has been written about Turing’s and von Neumann’s individual contributions to the development of the computer, this article investigates less well-known terrain: their interactions and mutual influences. Along the way we argue against ‘logic skeptics’ and ‘Turing skeptics’, who claim that neither logic nor Turing played any significant role in the creation of the modern computer. (shrink)

The topic of this article is Hilbert’s axiom of solvability, that is, his conviction of the solvability of every mathematical problem by means of a finite number of operations. The question of solvability is commonly identified with the decision problem. Given this identification, there is not the slightest doubt that Hilbert’s conviction was falsified by Gödel’s proof and by the negative results for the decision problem. On the other hand, Gödel’s theorems do offer a solution, albeit a negative one, in (...) the form of an impossibility proof. In this sense, Hilbert’s optimism may still be justified. Here I argue that Gödel’s theorems opened the door to proof theory and to the remarkably successful development of generalized as well as relativized realizations of Hilbert’s program. Thus, the fall of absolute certainty came hand in hand with the rise of partially secure and reliable foundations of mathematical knowledge. Not all was lost and much was gained. (shrink)

I develop a theory of counterfactuals about relative computability, i.e. counterfactuals such as 'If the validity problem were algorithmically decidable, then the halting problem would also be algorithmically decidable,' which is true, and 'If the validity problem were algorithmically decidable, then arithmetical truth would also be algorithmically decidable,' which is false. These counterfactuals are counterpossibles, i.e. they have metaphysically impossible antecedents. They thus pose a challenge to the orthodoxy about counterfactuals, which would treat them as uniformly true. What’s more, I (...) argue that these counterpossibles don’t just appear in the periphery of relative computability theory but instead they play an ineliminable role in the development of the theory. Finally, I present and discuss a model theory for these counterfactuals that is a straightforward extension of the familiar comparative similarity models. (shrink)

No chance of seeing her for another fortnight and it is 11 days since I saw her. Went solitary walk felt miserable but to some extent staved it off by reflecting on |$\langle$|Continuum Problem|$\rangle$|1The occasion for this review article on the life and accomplishments of Frank Ramsey is the publication in the last eight years of three important books: a biography of Frank Ramsey by his sister, Margaret Paul, a book by Steven Methven on aspects of Ramsey’s philosophy, and the (...) recent biography of Ramsey written by Cheryl Misak. In a companion piece [Mancosu, 2020] I discuss Ramsey’s contacts with logicians in continental Europe and publish previously unpublished letters from Ramsey to Abraham Fraenkel and to Heinrich Behmann.1. INTRODUCTIONFrank Plumpton Ramsey (1903–1930) will be well known to the readers of this journal. His contributions to philosophy, mathematics, and economics have left a lasting legacy. In mathematical logic and philosophy of mathematics, Ramsey’s solution to the paradoxes by means of a simple theory of types [1926a] and Ramsey’s theorem in combinatorics (both finite and infinite [1930]) are among Ramsey’s most celebrated contributions. Ramsey is a key presence in accounts of Wittgenstein’s, Russell’s, and Moore’s lives and work. It is well known that it was Ramsey who translated the Tractatus into English and wrote what is still considered a very insightful review [1923] of the Tractatus. Wittgenstein mentioned the importance of his conversations with Ramsey in 1929 in Cambridge as having influenced his transition to the views presented in the Philosophical Investigations. Ramsey also contributed, in 1924, to the second edition of Principia Mathematica (see [Linsky, 2011]). While Ramsey’s friendship with Moore does not have an analogous concrete precipitate as the work on the Tractatus or Principia Mathematica, it is evident from the correspondence and other documents that Ramsey’s work in philosophy (truth, propositions, universals, etc.) influenced Moore. Of course, the direction of influence goes also the other way around. Wittgenstein, Russell, and Moore, among others, were essential in shaping Ramsey’s approach to philosophy. (shrink)