Hilbert's Program Then and Now

Hilbert's program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to "dispose of the foundational questions in mathematics once and for all, "Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, "finitary" means, one should give proofs of the consistency of these axiomatic systems. Although Godel's incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and meta-theory, both at the time and since. The article discusses the historical background and development of Hilbert's program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.Comment: 43 page

Kriesel and Wittgenstein

Georg Kreisel (15 September 1923 - 1 March 2015) was a formidable mathematical logician during a formative period when the subject was becoming a sophisticated field at the crossing of mathematics and logic. Both with his technical sophistication for his time and his dialectical engagement with mandates, aspirations and goals, he inspired wide-ranging investigation in the metamathematics of constructivity, proof theory and generalized recursion theory. Kreisel's mathematics and interactions with colleagues and students have been memorably described in Kreiseliana ([Odifreddi, 1996]). At a different level of interpersonal conceptual interaction, Kreisel during his life time had extended engagement with two celebrated logicians, the mathematical Kurt Gödel and the philosophical Ludwig Wittgenstein. About Gödel, with modern mathematical logic palpably emanating from his work, Kreisel has reflected and written over a wide mathematical landscape. About Wittgenstein on the other hand, with an early personal connection established Kreisel would return as if with an anxiety of influence to their ways of thinking about logic and mathematics, ever in a sort of dialectic interplay. In what follows we draw this out through his published essays—and one letter—both to elicit aspects of influence in his own terms and to set out a picture of Kreisel's evolving thinking about logic and mathematics in comparative relief.Accepted manuscrip

Cry "Good for history, Cambridge and Saint George"? Essay Review of Mary Jo Nye (Ed.); The Cambridge History of Science, Vol. 5. The Modern Physical and Mathematical Sciences, (Cambridge: Cambridge University Press, 2003)

FIRST PARAGRAPH This volume is the third thus far published of The Cambridge history of science, planned in eight parts over the last decade by Cambridge University Press. Noting the incompleteness of George Sarton’s heroic solo endeavour on a comparably magisterial scale (Sarton, 1953–1959), Cambridge general editors David Lindberg and Ronald Numbers adopted a more pragmatic multiple author approach in devising this new series. They devote the four latter volumes to that fertile wonder ‘modern science’, its modernity construed chronologically as the post-1800 era. While Volume 6 encompasses the biological and earth sciences ( Bowler & Pickstone, forthcoming), Volume 7 deals with the social sciences ( Porter & Ross, 2003), and Volume 8 examines the sciences in national and international setting ( Livingstone & Numbers, forthcoming). Lindberg and Numbers thus circumscribe the territory of Volume 5 to be the history of physics, chemistry, astronomy and mathematics in the Euro-American world. Although this might seem a fairly conventional—even conservative—subject clustering, few historians would have felt undaunted by the heterogeneity of such material, the narrowness of the brief and the long two-century period of coverage. This volume must therefore be judged with sensitivity to the difficulties of leading thirty-seven scholars in diverse specialisms to produce a coherent product, and the sheer impracticability of Sarton’s near-Shakespearean ambitions for unitary drama. Useful comparisons can thus be made with recent works that offer a multi-perspectival view over comparably broad terrain: John Krige and Dominic Pestre’s stimulating and uncomplacent Science in the twentieth century (1997), and the more radically inclusive bibliographical essays in Arne Hessenbruch (Ed.), The reader’s guide to the history of science (Hessenbruch, 2000)

Philosophy and the practice of Bayesian statistics

A substantial school in the philosophy of science identifies Bayesian inference with inductive inference and even rationality as such, and seems to be strengthened by the rise and practical success of Bayesian statistics. We argue that the most successful forms of Bayesian statistics do not actually support that particular philosophy but rather accord much better with sophisticated forms of hypothetico-deductivism. We examine the actual role played by prior distributions in Bayesian models, and the crucial aspects of model checking and model revision, which fall outside the scope of Bayesian confirmation theory. We draw on the literature on the consistency of Bayesian updating and also on our experience of applied work in social science. Clarity about these matters should benefit not just philosophy of science, but also statistical practice. At best, the inductivist view has encouraged researchers to fit and compare models without checking them; at worst, theorists have actively discouraged practitioners from performing model checking because it does not fit into their framework.Comment: 36 pages, 5 figures. v2: Fixed typo in caption of figure 1. v3: Further typo fixes. v4: Revised in response to referee

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Logical Form, the First Person, and Naturalism about Psychology: The Case Against Physicalist Imperialism

Physicalistic theories of psychology are a classic case of scientific imperialism: the explanatory capacity of physics, both with respect to its methods and to its domain, is taken to extend beyond the traditional realm of physics, and into that of psychology. I argue in this paper that this particular imperialistic venture has failed. Contemporary psychology uses methods not modelled on those of physics, embracing first-personal methodology where physics is strictly impersonal. I make the case that whether or not scientific imperialism is in general harmful, in this instance naturalists who reject first philosophy should give up physicalist imperialism. Using only general principles from the philosophy of logic plus accepted physicalist criteria of identity, I show that first-personal psychology embodies a minor but fruitful increase in expressive strength compared to impersonal psychology: the ability to distinguish descriptively indiscriminable posits