Arithmetic, set theory, reduction and explanation

George Boolos. The iterative conception of set. The journal of philosophy, vol. 68 (1971), pp. 215–231. - Dana Scott. Axiomatizing set theory. Axiomatic set theory, edited by Thomas J. Jech, Proceedings of symposia in pure mathematics, vol. 13 part 2, American Mathematical Society, Providence1974, pp. 207–214. - W. N. Reinhardt. Remarks on reflection principles, large cardinals, and elementary embeddings. Axiomatic set theory, edited by Thomas J. Jech, Proceedings of symposia in pure mathematics, vol. 13 part 2, American Mathematical Society, Providence1974, pp. 189–205. - W. N. Reinhardt. Set existence principles of Shoenfield, Ackermann, and Powell. Fundament a mathematicae, vol. 84 (1974), pp. 5–34. - Hao Wang. Large sets. Logic, foundations of mathematics, and computahility theory. Part one of the proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada–1975, edited by Robert E. Butts and Jaakko Hintikka, The University of Western Ontario series in philosophy of science, vol. 9, D. Reidel Publishing Company, Dordrecht and Boston1977, pp. 309–333. - Charles Parsons. What is the iterative conception of set?Logic, foundations of mathematics, and computahility theory. Part one of the proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada–1975, edited by Robert E. Butts and Jaakko Hintikka, The University of Western Ontario series in philosophy of science, vol. 9, D. Reidel Publishing Company, Dordrecht and Boston1977, pp. 335–367. - Volume 50 Issue 2

Literally speaking, "Philosophy of biology" is a rather old expression. William Whewell coined it in 1840, at the very time he introduced the expression "philosophy of science". Whewell was fond of creating neologisms, like Auguste Comte, his French counterpart in the field of the philosophical reflection about science. Historians of science know that a few years earlier, in 1834, Whewell had generated a small scandal when he proposed the word "scientist" as a general term by which "the students of the knowledge of the material world" could describe themselves, and distinguish themselves from artists. The term "philosopher", Whewell argued, was too wide. A new generic term, more or less equivalent to the French term "savant", was needed in order to prevent the disintegration of science that seemed to flow from its specialization in modern times.1 When Whewell first introduced the terms "philosophy of science" and "philosophy of biology" in his 1840 Philosophy of the inductive sciences, the latter term was merely a special branch of "philosophy of science". The expression "philosophy of science" itself had two justifications: firstly, this phrase expressed the idea that "science" remained cognitively coherent enough to justify a critical enquiry into its methodological unity and its foundation; secondly, the phrase "philosophy of science" was required in order to distinguish a properly "philosophical" enquiry from a "historical" approach to science. Although Whewell's 1840 Philosophy of the inductive sciences 2 had approximately the same chapter structure as his 1837 History of the Inductive Sciences 3 (that is, a series of chapters successively devoted to the concept of science in general, and although there was considerable overlap between the contents of the two books, then to particular sciences), its theoretical purpose was different. Clearly, Whewell was not willing to confuse the genres of history and philosophy as Auguste Comte had done in his 1830 Cours de philosophie positive.4 Note also that the word "biology" was still extremely rare in English when Whewell used it in 1840. In fact, Whewell's 1837 History of the Inductive Sciences does not make use of the word "biology": Whewell successively examines "botany", "zoology", "physiology" and "comparative antomy" as special branches of "analytico-classificatory science", then discusses palaeontology as a special case of the "palaeo-etiological sciences". Three years later, in his Philosophy of the Inductive Sciences, Whewell does use "biology" as a generic term for all the sciences dealing with life. The various sciences which were separately examined in the previous book are now collectively considered. Furthermore, the main philosophical problem raised by biology is its dual nature: biology is both nomological and a historical science. Modern philosophers of biology are generally unaware of the story of the origins of the expression "philosophy of biology", but Whewell's dual theoretical nature of biology is still a major concern for modern "philosophy of biology".

A major controversy in contemporary philosophy of science concerns the possibility and desirability of its ‘naturalization.’ In this paper I review the philosophical controversy concerning naturalism, and investigate the role it might play in the science classroom. I argue that science students can benefit from explicit study of this controversy, and from explicit consideration of the extent to which philosophy of science can be studied naturalistically. More specifically, I suggest that such consideration can enhance students' understanding of the nature of ‘natural’ science, of the nature and importance of philosophy of science, and of the relationship between the two — and that these benefits accrue to science education whichever philosophical view concerning naturalization proves to be correct. My hope is that the paper demonstrates the benefits to be gained from explicit consideration in the science classroom of an important issue in the philosophy of science.

In this article, we examine the anglo-saxon archaeological interpretation of the philosophy of science throughout the last century. With reference to an idea developed by Jane Kelley and Marsha Hanen, we define the former interpretation as the “narrow perspective” of the philosophy of science. This approach has focused exclusively on the analytical tradition of the philosophy of science; the foremost example is the New Archaeology interest in logical positivism. In the second section, we examine the consequences of this narrow perspective and we conclude that it is necessary to develop a “wide perspective” of the philosophy of science in archaeology.

In the second half of the 20th century, the crisis of positivism led to the rise of new directions in the philosophy of science. In the same period many economists expressed an increasing dissatisfaction with the so far dominant economic paradigms. The aim of this paper is to point out the relation between the main trends in the philosophy of science and the debate on the nature and the method of economics. The paper will review the main thesis of logical empiricism, proposed by Hutchison as research guidelines in the field of economics, and the transition from verificationism to Popper's falsicationism. It will then examine how the empirical-deductive method of positivism was interpreted by Friedman and Samuelson, who many consider the main exponents of neo-classic economics. With Kuhn and Lakatos, the emphasis of the «new» philosophy of sciences shifts from what scientists «should do» to what they actually «do» in the research practice. Another important evolution occurs with the works of Feyerabend and Rorty and the emergence of «post-modernist» trends, which deny any a kind of epistemology and maintain that a theory is acceptable only inasmuch it is endorsed by the scientific community. The relation between these radical changes in the philosophy of sciences and the debate on the methods and status of economics is complex. Heterodoxes economic theories, wishing to address the crisis of mainstream economics and the «economists' uneasiness», do not imply a refusal of positivism, and «post-modernist» approach had a limited appeal among economists. The paper presents the contributions of McCloskey and Rosenberg, two economists who particularly developed the arguments against positivist epistemology applied to economics, but it observes that, if interesting, such ideas do not provide a solid enough basis to overcome neoclassical orthodoxy. The conclusions briefly examine a still evolving and promising research approach, the «critical realism» recently promoted by the «Cambridge Group», which links criticism to the empiricaldeductive method with the proposal of alternative economic theories.

The Interface Between Statistics and the Philosophy of Science

Philosophy of mathematics is arguably one of the oldest branches of philosophy, and one that bears significant connections with core philosophical areas, particularly metaphysics, epistemology, and (more recently) the philosophy of science. This entry focuses on contemporary developments, which have yielded novel approaches (such as new forms of Platonism and nominalism, structuralism, neo-Fregeanism, empiricism, and naturalism) as well as several new issues (such as the significance of the application of mathematics, the role of visualization in mathematical reasoning, particular attention to mathematical practice and to the nature of mathematical explanation). Excellent work has also been done on particular philosophical issues that arise in the context of specific branches of mathematics, such as algebra, analysis, and geometry, as well as particular mathematical theories, such as set theory and category theory. Due to limitations of space, this work goes beyond the scope of the present entry.

If we are to solve the central problems in the philosophy of science, Paul Churchland argues, we must draw heavily on the resources of the emerging sciences of the mind/brain. In "A Neurocomputational Perspective, "Churchland illustrates the fertility of the concepts and data drawn from the study of the brain and from the study of artificial networks that model the brain. These concepts bring an unexpected coherence to scattered issues in the philosophy of science, new solutions to old philosophical problems, and new possibilities of science itself. "A Neurocomputational Perspective "collects the most influential of Churchland's recent essays in the philosophy of mind, and introduces a number of ground breaking new essays in the philosophy of science. The essays in part 1 form a systematic treatment of the major problems in the philosophy of mind, as seen from the perspective of the philosophy of science and also from the perspective of computational neuroscience and connectionist AI. The nature of mental representations, of sensory qualia, of psychological explanation and of human self understanding all receive significant new treatment, and Churchland reaffirms his well known position that our "folk psychology" must eventually be reduced, perhaps to a mature cognitive neurobiology, or else be displaced by some new framework that does cohere with emerging theories of brain function. In part 2, Churchland provides a coherent and unified alternative conception, in neurocomputational terms of the many issues central to the philosophy of science. These include the nature of theories, the theory-ladenness of perception, the nature of conceptual unification and theoretical simplicity, thenature of paradigms, the kinematics and dynamics of conceptual change, the nature of abductive inference, and the nature of explanatory understanding. Collectively these present a new conception of science. Paul M. Churchland is Professor of Philosophy and member of the Cognitive Science Faculty at the University of California at San Diego. A Bradford Book.

How was Poincare’s conventionalism connected to his relationism? How, in other words, is it the case that the basic principles of geometry and mechanics are, ultimately, freely chosen conventions and that, at the same time, science reveals to us the structure of the world? This lengthy study aims to address these questions by setting Poincare’s philosophy within its historical context and by examining in detail Poincare’s developing views about the status and role of conventions in science and the status and role of relations in science. It is argued that Poincare was a neither a rampant conventionalist (as Edouard Le Roy was arguing) nor a pure structuralist (as Bertrand Russell was urging). But he was both a conventionalist and a relationist, aiming to delineate a position which allowed room for both freely, but not arbitrarily, chosen constitutive principles of science and the acquisition of objective, though, relational knowledge of the natural world.

Empirical investigations use empirical methods, data, and evidence. This banal observation appears to favour empiricism, especially in philosophy of science, though no rationalist ever denied their importance. Natural sciences often provide what appear to be, and are taken by scientists as, realist, causal explanations of natural phenomena, often in terms of forces or entities we do not perceive with our normal, unaided human senses. Empiricism has never been congenial to realism about such scientific posits. Bas van Fraassen’s “Constructive Empiricism” purports that realist interpretations of any “unobservables” mentioned by a scientific theory in principle always transcend whatever can be justified by that theory’s empirical adequacy, and that “explanations” are merely pragmatic, insofar as they are context‑specific to the presuppositions of whomever poses the question an explanation is to answer. Here I argue that “Constructive Empiricism” rests upon a series of flawed presumptions about natural science and about epistemology. I draw upon two main resources. One resource is the constraints upon specifically c ognitive reference to particulars, first identified by Kant (and later by Evans). The second is William Harper’s (2011) brilliant re‑analysis and defense of Newton’s Prin c ipia , which shows that, and how, Newton justified his realism about gravitational force. One surprise is that Kant’s semantics of singular cognitive reference (examined in §3) directly and strongly supports Newton’s Rule 4 of scientific method (§4), which strongly supports his realism about gravi‑ tational force (summarized in §2). A further surprise is that Hegel first recognized that this semantics of singular cognitive reference directly and strongly supports Newton’s meth odological Rule 4 of experimental philosophy in ways which support Newton’s realism about gravitational force, and about distance forces generally. The textual and exegetical issues these attributions require I examine elsewhere. Here I make these important findings available to philosophers and historians of science. (*) Invited paper

Axiomatic set theory is almost universally accepted as the basic theory which provides the foundations of mathematics, and in which the whole of present day mathematics can be developed. As such, it is the most natural framework for Mathematical Knowledge Management. However, in order to be used for this task it is necessary to overcome serious gaps that exist between the "official" formulations of set theory (as given e.g. by formal set theory ZF) and actual mathematical practice. In this work we present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity.

I shall attempt to demonstrate here that the general categories of hermeneutics can be applied just as well to the natural sciences as to the humanities, and I shall explore the implications of this demonstration for both types of disciplines. In doing so, it will be necessary for me to raise some unfashionable issues about the objectivity and rationality of critical interpretations, and, as a result, I may appear to be cast in the role of advocatus diaboli. The value of lowering the critical barriers that have been set up between the human and natural sciences, however, will outweigh any incidental paradoxes that may crop up en route.

Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century.

Rudolf Carnap. Inductive logic and inductive intuition. The problem of inductive logic, Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, volume 2, edited by Imre Lakatos, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1968, pp. 258–267. - M. Bunge, J. W. N. Watkins, Y. Bar-Hillel, K. R. Popper, J. Hintikka, R. Carnap. Discussion. The problem of inductive logic, Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, volume 2, edited by Imre Lakatos, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1968, pp. 268–314. - Volume 40 Issue 3

“Map Theory et Antifondation”