Ideas and processes in mathematics: A course on history and philosophy of mathematics

This paper describes an attempt to develop a program for teaching history and philosophy of mathematics to inservice mathematics teachers. I argue briefly for the view that philosophical positions and epistemological accounts related to mathematics have a significant influence and a powerful impact on the way mathematics is taught. But since philosophy of mathematics without history of mathematics does not exist, both philosophy and history of mathematics are necessary components of programs for the training of preservice as well as inservice mathematics teachers.     

The architecture of modern mathematics: Essays in history and philosophy

It is a little understood fact that the system of formal logic presented in Wittgenstein’s Tractatusprovides the basis for an alternative general semantics for a predicate calculus that is consistent and coherent, essentially independent of the metaphysics of logical atomism, and philosophically illuminating in its own right. The purpose of this paper is threefold: to describe the general characteristics of a Tractarian-style semantics, to defend the Tractatus system against the charge of expressive incompleteness as levelled by Robert Fogelin, and to give a semantics for a formal language that is the Tractarian equivalent of a first-order predicate calculus. Of note in regard to the latter is the fact that a Tractatusstyle truth-definition makes no appeal to the technical trick of defining truth in terms of the satisfaction of predicates by infinite sequences of objects, yet is materially equivalent to the usual Tarski-style truth-definitions     

Duhem,the Arabs,and the history of cosmology

Duhem has generally been understood to have maintained that the major Greek astronomers were instrumentalists. This view has emerged mainly from a reading of his 1908 publication To Save the Phenomena. In it he sharply contrasted a sophisticated Greek interpretation of astronomical models (for Duhem this was that they were mathematical contrivances) with a naive insistence of the Arabs on their concrete reality. But in Le Système du monde, which began to appear in 1913, Duhem modified his views on Greek astronomy considerably; his more subtle understanding included the recognition that many Greeks subordinated mathematical astronomy to physical theory. But he could not completely repudiate his earlier views about Greek astronomy in part because his extreme nineteenth century prejudices led him to continue to insist on a clear-cut demarcation between Greek and Arabic astronomy. The inevitable result is a certain unevenness in the Système and some glaring inconsistencies.     

Modernizing the philosophy of mathematics

The distinction between analytic and synthetic propositions, and with that the distinction between a priori and a posteriori truth, is being abandoned in much of analytic philosophy and the philosophy of most of the sciences. These distinctions should also be abandoned in the philosophy of mathematics. In particular, we must recognize the strong empirical component in our mathematical knowledge. The traditional distinction between logic and mathematics, on the one hand, and the natural sciences, on the other, should be dropped. Abstract mathematical objects, like transcendental numbers or Hilbert spaces, are theoretical entities on a par with electromagnetic fields or quarks. Mathematical theories are not primarily logical deductions from axioms obtained by reflection on concepts but, rather, are constructions chosen to solve some collection of problems while fitting smoothly into the other theoretical commitments of the mathematician who formulates them. In other words, a mathematical theory is a scientific theory like any other, no more certain but also no more devoid of content.     

Some recent essays in the history of the philosophy of mathematics: A critical review

Research was partially supported by NSF grant DIR 9011694.     

"不可或缺性论证"与反实在论数学哲学

The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining abstract mathematical entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions. __________ Translated from Zhexue Yanjiu 哲学研究 (Philosophical Researches), 2006, (8): 74–83     

The pernicious influence of mathematics upon philosophy

We shall argue that the attempt carried out by certain philosophers in this century to parrot the language, the method, and the results of mathematics has harmed philosophy. Such an attempt results from a misunderstanding of both mathematics and philosophy, and has harmed both subjects.Portions of the present text have previously appeared inThe Review of Metaphysics 44 (1990), 259–271, are reprinted with permission.     

Einstein and Duhem

Pierre Duhem's often unrecognized influence on twentieth-century philosophy of science is illustrated by an analysis of his significant if also largely unrecognized influence on Albert Einstein. Einstein's first acquaintance with Duhem's La Théorie physique, son objet et sa structure around 1909 is strongly suggested by his close personal and professional relationship with Duhem's German translator, Friedrich Adler. The central role of a Duhemian holistic, underdeterminationist variety of conventionalism in Einstein's thought is examined at length, with special emphasis on Einstein's deployment of Duhemian arguments in his debates with neo-Kantian interpreters of relativity and in his critique of the empiricist doctrines of theory testing advanced by Schlick, Reichenbach, and Carnap. Most striking is Einstein's 1949 criticism of the verificationist conception of meaning from a holistic point of view, anticipating by two years the rather similar, but more famous criticism advanced independently by Quine in Two Dogmas of Empiricism.I wish to thank the Hebrew University of Jerusalem, which holds the copyright, for permission to quote from the unpublished letters of Einstein. Items in the Einstein Archive are cited by giving their number in the control index after the following format: EA nn-nnn. Similar formats are employed for citing other archival material. Thus AA refers to material in the Adler Archive at the Verein für Geschichte der Arbeiterbewegung, Vienna; and RC refers to material in the Rudolf Carnap collection at the Archive for Scientific Philosophy, Department of Special Collections, Hillman Library, University of Pittsburgh. The research for this paper was supported in part by a grant from the National Science Foundation, No. SES-8420140, as well as by grants from the Deutscher akademischer Austauschdienst, the American Philosophical Society, and the University of Kentucky Research Foundation.     

Phenomenological method and philosophy of history

Continental Philosophy Review - The purpose of this study is to examine the phenomenological method as it applies to the philosophy of history. This leads me to divide my inquiry into two parts. I...     

Maiocchi on Duhem,howard on Duhem and Einstein: Historiographical comments

These comments center on the methodological stance that Howard and Maiocchi recommend to us when we are doing history of philosophy. If Howard and Maiocchi are right, both Duhem and Einstein developed closely related versions of conventionalism and realism, and in both of their philosophies the conventionalist and realist moments were mutually compatible. Duhem's holism and, arguably, Einstein's as well, denies the need for across-the-board literalism, and both of them had important reasons for denying that convergence was required or even desirable for realism. Thus, for those who are caught up in the current disputes, serious consideration of the discrepancies between the standard current versions of realism and conventionalism and the positions that contextualist analyses reveal to have been advocated by Duhem and Einstein may uncover some of the tacit assumptions that impede the resolution or advancement of our disputes.     

Writing the history of Russian philosophy

This article addresses the writing of the history of Russian philosophy from the first of such works—Archimandrite Gavriil’s Russian Philosophy [Russkaja filosofija, 1840]—to philosophical histories/textbooks in the twenty-first century. In the majority of these histories, both past and present, we find a relentless insistence on the delineation of “characterizing traits” of Russian philosophy and appeals to “historiosophy,” where historiosophy is employed as being distinct from the historiographical method. In the 1990s and 2000s, the genre of the history of Russian philosophy has grown increasingly conservative with regards to content, with histories from this period demonstrating an almost exclusive Orthodox focus. This conservatism, in turn, has contributed to widespread contention in recent years over the status of these philosophical textbooks—disagreements that often lead to either (1) further appeals to “historiosophical” methods; or (2) denials of the domestic philosophical tradition altogether, where the response to the query “Is there philosophy in Russia?” is emphatically negative. This article argues that the contemporary disputes over the development and preservation of the Russian philosophical canon are in many ways part of a larger debate over the roles of Orthodoxy and the history of philosophy in post-Soviet philosophical thought.