Between 1937 and 1951 Wittgenstein had numerous philosophical conversations with his student and close friend, Rush Rhees. This article is composed of Rhees’s notes of twenty such conversations — namely, all those which have not yet been published — as well as some supplements from Rhees’s correspondence and miscellaneous notes. The principal value of the notes collected here is that they fill some interesting and important gaps in Wittgenstein ’s corpus. Thus, firstly, the notes touch on a wide range of (...) subjects, a number of which are only briefly addressed by Wittgenstein elsewhere, if at all. The subjects discussed include: explanation, ethics, anarchism, contradiction, psychoanalysis, colour, religion, concepts, classification, seeing-as, evolution, the relation between science and philosophy, and free will, amongst others. Secondly, the notes contain references to, and brief remarks about, philosophers of whom Wittgenstein otherwise says very little, if anything — such as Brentano, Heidegger, Aquinas, and Marx, amongst others. And thirdly, the notes provide us with valuable examples of Wittgenstein ’s use of some key ‘Wittgensteinian’ terms of art which are surprisingly rare in his written works, such as ‘surface-’ and ‘depth-grammar’, and ‘centres of variation’. (shrink)

The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This article analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humor is associated with risky forms of inference, which are essential (...) to creative mathematics. The components of the joke are explicated by argumentation schemes devised for application to topic-neutral reasoning. These in turn are classified under seven headings: retroduction, citation, intuition, meta-argument, closure, generalization, and definition. Finally, the wider significance of this account for the cognitive science of mathematics is discussed. (shrink)

Russell had two theories of definite descriptions: one for singular descriptions, another for plural descriptions. We chart its development, in which ‘On Denoting’ plays a part but not the part one might expect, before explaining why it eventually fails. We go on to consider many-valued functions, since they too bring in plural terms—terms such as ‘4’ or the descriptive ‘the inhabitants of London’ which, like plain plural descriptions, stand for more than one thing. Logicians need to take plural reference seriously (...) if only because mathematicians take many-valued functions seriously. We assess the objection (by Russell, Frege and others) that many-valued functions are illegitimate because the corresponding terms are ambiguous. We also assess the various methods proposed for getting rid of them. Finding the objection ill-founded and the methods ineffective, we introduce a logical framework that admits plural reference, and use it to answer some earlier questions and to raise some more. (shrink)

Since the publication of the Remarks on the Foundations of Mathematics, Wittgenstein’s interpreters have endeavored to reconcile his general constructivist/anti-realist attitude towards mathematics with his confessed anti-revisionary philosophy. In this article, the author revisits the issue and presents a solution. The basic idea consists in exploring the fact that the so-called “non-constructive results” could be interpreted so that they do not appear non-constructive at all. The author substantiates this solution by showing how the translation of mathematical results, given by the (...) possibility of translation between logics, can be seen as a tool for partially implementing the solution Wittgenstein had in mind. (shrink)

This article analyses in detail Wittgenstein’s ‘Cambridge period’ from his return to Cambridge in 1929 until his decease in 1951. Within the ‘Cambridge period’, scholars usually distinguish the ‘middle’ (1929–1936) and the ‘late’ (1936–1951) periods. The trigger point of Wittgenstein’s return to Cambridge and philosophy was his visit to Brouwer’s lecture on ‘Mathematics, Science, and Language’ in Vienna in March 1928. Dutch mathematician Brouwer influenced not only Wittgenstein’s ability to do philosophy again but also the development of some of his (...) ideas. Namely, for Brouwer, language was a natural development of the social history of human beings. With the help of his friends, F. P. Ramsey, J. M. Keynes, G. E. Moore, and B. Russell, in 1929 Wittgenstein returned to Cambridge. The author argues that this was the crucial turning point in his philosophy that led to the revision of some of his ideas. Wittgenstein returned from self-isolation in remote villages of Lower Austria to the intellectual academic environment, where he could discuss his ideas with intellectual interlocutors and receive their valuable remarks and comments. This article is organised in chronological order: it describes the very ‘early’ middle period of 1929–1935 involving the development of Wittgenstein’s ‘phenomenology’ and the origin of the ‘language-games’ concept in the Blue and Brown Books; 1935–1936 period of ‘romantic’ enthusiasm for Soviet Russia and a trip there; Wittgenstein’s life and work during the WWII-time; resign from teaching in Cambridge in 1947 to finally finish his Philosophical Investigations. The author suggests that the ‘romantic areal’ about the USSR was formed by Wittgenstein’s predilection for Russian poetry and literature of the nineteenth century, i.e., Tolstoy, Dostoevsky, and Pushkin. Analysing the development of Wittgenstein’s ideas on language in his Cambridge period, the author mentions the influence of N. Bakhtin and P. Sraffa. Also, the author touches upon the topic of the possibility of Marxism’s influence on Wittgenstein’s life and thought. (shrink)

I defend Putnam’s modal structuralist view of mathematics but reject his claims that Wittgenstein’s remarks on Dedekind, Cantor, and set theory are verificationist. Putnam’s “realistic realism” showcases the plasticity of our “fitting” words to the world. The applications of this—in philosophy of language, mind, logic, and philosophy of computation—are robust. I defend Wittgenstein’s nonextensionalist understanding of the real numbers, showing how it fits Putnam’s view. Nonextensionalism and extensionalism about the real numbers are mathematically, philosophically, and logically robust, but the two (...) perspectives are often confused with one another. I separate them, using Turing’s work as an example. (shrink)

ABSTRACT Historical structuralist views have been ontological. They either deny that there are any mathematical objects or they maintain that mathematical objects are structures or positions in them. Non-ontological structuralism offers no account of the nature of mathematical objects. My own structuralism has evolved from an early sui generis version to a non-ontological version that embraces Quine’s doctrine of ontological relativity. In this paper I further develop and explain this view.

The theoretical solid-state physicist Walter Kohn was awarded one-half of the 1998 Nobel Prize in Chemistry for his mid-1960s creation of an approach to the many-particle problem in quantum mechanics called density functional theory (DFT). In its exact form, DFT establishes that the total charge density of any system of electrons and nuclei provides all the information needed for a complete description of that system. This was a breakthrough for the study of atoms, molecules, gases, liquids, and solids. Before DFT, (...) it was thought that knowledge of the vastly more complicated many-electron wave function was essential for a complete description of such systems. Today, 50 years after its introduction, DFT (in one of its approximate forms) is the method of choice used by most scientists to calculate the physical properties of materials of all kinds. In this paper, I present a biographical essay of Kohn’s educational experiences and professional career up to and including the creation of DFT. My account begins with Kohn’s student years in Austria, England, and Canada during World War II and continues with his graduate and postgraduate training at Harvard University and Niels Bohr’s Institute for Theoretical Physics in Copenhagen. I then study the research choices he made during the first 10 years of his career (when he was a faculty member at the Carnegie Institute of Technology and a frequent visitor to the Bell Telephone Laboratories) in the context of the theoretical solid-state physics agenda of the late 1950s and early 1960s. Subsequent sections discuss his move to the University of California, San Diego, identify the research issue which led directly to DFT, and analyze the two foundational papers of the theory. The paper concludes with an explanation of how the chemists came to award “their” Nobel Prize to the physicist Kohn and a discussion of why he was unusually well suited to create the theory in the first place. (shrink)

In the years following Bertrand Russell's visit in China, fragments from his work on mathematical logic and the foundations of mathematics started to enter the Chinese intellectual world. While up...

We present a plural logic that is as expressively strong as it can be without sacrificing axiomatisability, axiomatise it, and use it to chart the expressive limits set by axiomatisability. To the standard apparatus of quantification using singular variables our object-language adds plural variables, a predicate expressing inclusion (is/are/is one of/are among), and a plural definite description operator. Axiomatisability demands that plural variables only occur free, but they have a surprisingly important role. Plural description is not eliminable in favour of (...) quantification; on the contrary, quantification is definable in terms of it. Predicates and functors (function signs) can take plural as well as singular terms as arguments, and both many-valued and single-valued functions are expressible. The system accommodates collective as well as distributive predicates, and the condition for a predicate to be distributive is definable within it; similarly for functors. An essential part of the project is to demonstrate the soundness and completeness of the calculus with respect to a semantics that does without set-theoretic domains and in which the use of settheoretic extensions of predicates and functors is replaced by the sui generis relations and functions for which the extensions were at best artificial surrogates. Our metalanguage is designed to solve the difficulties involved in talking plurally about individuals and about the semantic values of plural items. (shrink)

ArgumentIn the decades following World War II, the Cowles Commission for Research in Economics came to represent new technical standards that informed most advances in economic theory. The public emergence of this community was manifest at a conference held in June 1949 titledActivity Analysis of Production and Allocation. New ideas in optimization theory, linked to linear programming, developed from the conference's papers. The authors’ history of this event situates the Cowles Commission among the institutions of postwar science in-between National Laboratories (...) and the supreme discipline of Cold War academia, mathematics. Although the conference created the conditions under which economics, as a discipline, would transform itself, the participants themselves had little concern for the intellectual battles that had defined prewar university economics departments. The authors argue that the conference signaled the birth of a new intellectual culture in economic science based on shared scientific norms and techniques un-interrogated by conflicting notions of the meaning of either science or economics. (shrink)