We show that Dedekind, in his proof of the principle of definition by mathematical recursion, used implicitly both the concept of an inductive cone from an inductive system of sets and that of the inductive limit of an inductive system of sets. Moreover, we show that in Dedekind’s work on the foundations of mathematics one can also find specific occurrences of various profound mathematical ideas in the fields of universal algebra, category theory, the theory of primitive recursive mappings, and set (...) theory, which undoubtedly point towards the mathematics of twentieth and twenty-first centuries. (shrink)

In other works, I’ve proposed a solution to the semantic paradoxes which, at the technical level, basically relies on failure of contraction. I’ve also suggested that, at the philosophical level, contraction fails because of the instability of certain states of affairs. In this paper, I try to make good on that suggestion.

An interview with Charles McCarty by Piotr Urbańczyk concerning mathematical explanation.

Wilfred Sieg and Dirk Schlimm. Dedekind's Analysis of Number: Systems and Axioms.

Formalism in the philosophy of mathematics has taken a variety of forms and has been advocated for widely divergent reasons. In Sects. 1 and 2, I briefly introduce the major formalist doctrines of the late nineteenth and early twentieth centuries. These are what I call empirico-semantic formalism, game formalism and instrumental formalism. After describing these views, I note some basic points of similarity and difference between them. In the remainder of the paper, I turn my attention to Hilbert’s instrumental formalism. (...) My primary aim there will be to develop its formalist elements more fully. These are, in the main, its rejection of the axiom-centric focus of traditional model-construction approaches to consistency problems, its departure from the traditional understanding of the basic nature of proof and its distinctively descriptive or observational orientation with regard to the consistency problem for arithmetic. More specifically, I will highlight what I see as the salient points of connection between Hilbert’s formalist attitude and his finitist standard for the consistency proof for arithmetic. I will also note what I see as a significant tension between Hilbert’s observational approach to the consistency problem for arithmetic and his expressed hope that his solution of that problem would dispense with certain epistemological concerns regarding arithmetic once and for all. (shrink)

The invention of “artificial perspective” revealed the ideal character of Euclidean geometry already in the Renaissance Europe of the fifteenth century. To the extent to which it made painting a “science” relying on mathematical rules, it made mathematics an “art” independent of the “geometry of nature.” It was the artistic vision emerging from perspective drawing that paved the way for scientific abstraction. However, it was only in the nineteenth century that the discovery of non-Euclidean geometry compelled mathematics to ponder the (...) visual evidence of its principles and the reliability of its abstract concepts. At that time, it was the mathematical vision that first championed the rights of ideal forms to a higher level of abstraction and, therefore, oriented science and art towards new representational spaces. (shrink)

Alonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Gödel's general recursiveness, not his own λ (...) -definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of effective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the character of my answers is reflected by an alternative title for this paper, Why Church needed Gödel's recursiveness for his Thesis. In Section 5, I contrast Church's analysis with that of Alan Turing and explore, in the very last section, an analogy with Dedekind's investigation of continuity. (shrink)

Wright claims that his and Hale’s abstractionist neologicist project is primarily epistemological in aim. Its epistemological aims include establishing the possibility of a priori mathematical knowledge, and establishing the possibility of reference to abstract mathematical objects. But, as Wright acknowledges, there is a question of how neologicist epistemology applies to actual, ordinary mathematical beliefs. I take up this question, focusing on arithmetic. Following a suggestion of Hale and Wright, I consider the possibility that the neologicist account provides an idealised reconstruction (...) of how people acquire cognition of numbers or arithmetical knowledge. I conclude that it is unlikely that this account can be used to reconstruct even idealised answers to pressing questions about our actual epistemology. (shrink)

This paper explores strengths and limitations of both predicativism and nominalism, especially in connection with the problem of characterizing the continuum. Although the natural number structure can be recovered predicatively (despite appearances), no predicative system can characterize even the full predicative continuum which the classicist can recognize. It is shown, however, that the classical second-order theory of continua (third-order number theory) can be recovered nominalistically, by synthesizing mereology, plural quantification, and a modal-structured approach with essentially just the assumption that an (...) -sequence of concrete atoms be possible. Predicative flexible type theory may then be used to carry out virtually all of scientifically applicable mathematics in a natural way, still without ultimate need of the platonist ontology of classes and relations. (shrink)

Essay review of Daniel Garber, 1992, Descartes' Metaphysical Physics, University of Chicago Press, Chicago and London, xiv + 389 pp., and Michael Friedman,: 1992, Kant and the Exact Sciences, Harvard University Press, Cambridge, Mass., and London, xvii + 357 pp. These two books display the historical connection between science and philosophy in the writings of Descartes and Kant. They show the place of science in, or the scientific context of, these authors' central metaphysical doctrines, pertaining to substance and its properties, (...) the metaphysics of force and causal agency, the relation of geometry to matter and to space, and the structure of cognition. In so doing, they provide an image of philosophy as an intellectually and culturally engaged enterprise, an image according to which it makes little sense to make pronouncements about the foundations of knowledge or the possibilities for science without direct engagement with the living practice of scientific and other cognitive enterprises. However, in some areas they did not engage the context as fully as needed; their interpretations of particular arguments and indeed of whole philosophical programs suffered thereby. (shrink)

Dans l'Appendice au livre I de The World and the Individual (1898), le philosophe américain Josiah Royce développe, en se fondant sur Was sind und was sollen die Zahlen ? de Dedekind, une critique détaillée du livre de Bradley Appearance and Reality. Se concentrant sur le fameux § 66, Royce maintient que la théorie de Dedekind peut être vue comme l'accomplissement du mouvement de pensée inauguré par Fichte et Hegel : le 'Soi idéal ' est infini et l'arithmétique est la (...) théorie de sa structure formelle. Cet article analyse le contenu de ce texte, en le mettant en relation avec la pensée de Russell et de Lotze. (shrink)

In the introduction, following the formulation of the theses on the concept ‘philosophy of science’, on interdisciplinarity in modern science, and on foundational studies in science, and on the bases of their content, a thesis on the interdisciplinary approach to foundations of science is formulated. In accordance with the latter, together with the canonical approach to foundations of science, which consists in an elaboration of the foundations of mathematics, physics and other fundamental canonical sciences, also an interdisciplinary approach to foundations (...) of science is realised, one that consists in the elaboration of four foundational complex sciences containing the most significant productive factors for the elaboration of various interdisciplinary theories: Methodology of S-modelling, Logic of science, Definitics, a complex theory of definite structures, aggregates, processes and systems in reality and in knowledge, Indefinitics, a complex theory of indefiniteness in its various forms in reality and in knowledge. Taking into account the internal connection between the first two complex sciences enlisted above, it is expedient in the process of investigations of their foundations that they be unified in yet another complicated complex science—logistics. (shrink)

We expose some basic elements of a style of programming supported by functional languages like Haskell by relating them to a coherent set of notions and techniques from Curry’s work in combinatory logic and formal systems, and their algebraic and categorical interpretations. Our account takes the form of a commentary to a simple fragment of Haskell code attempting to isolate the conceptual sources of the linguistic abstractions involved.

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses ofDirichlet characters in presentations of Dirichlet’s proof in the nineteenth and early twentieth centuries, with an eye toward understanding some of the pragmatic pressures that shaped the evolution of modern mathematical method.

Both Frege's Grundgesetze, and Lagrange's treatises on analytical functions pursue a foundational purpose. Still, the former's program is not only crucially different from the latter's. It also depends on a different idea of what foundation of mathematics should be like . Despite this contrast, the notion of function plays similar roles in their respective programs. The purpose of my paper is emphasising this similarity. In doing it, I hope to contribute to a better understanding of Frege's logicism, especially in relation (...) to its crucial differences with a set-theoretic foundational perspective. This should also spread some light on a question arisen by J. Hintikka and G. Sandu in a widely discussed paper, namely whether Frege should or should not be credited with the notion of arbitrary function underlying our standard interpretation of second-order logic. In section 1, I account for Lagrange's notion of function. In section 2, I advance some remarks on connected historical matters. This will provide an appropriate framework for discussing the role played by the notion of function in Frege's Grundgesetze. Section 3 is devoted to this. Some concluding remarks will close the paper. (shrink)

This is the first part of the essay devoted to the story of logicism, in particular to its Fregean version. Reviewing the classical period of Fregean studies, we first point out some critical moments of Frege‘s argumentation in the Grundla­gen, in order to be able later to differentiate between its salvageable and defec­tive features. We work on the presumption that there are no easy, catego­rical an­swers to questions like “Is logicism dead?“: Wittgenstein’s cri­tique of the foundational program as well as (...) the remarkable neo-Fregean dis­cov­eries of Boolos and Wright have to be confronted with the effects which the logi­cistic idea actually had on logico­matemati­cal practice. But that is another story, a sequel to this es­say, the purpose of which is systematic rather than criti­cal. (shrink)