Yemima Ben-Menahem

This book provides a survey of the major issues in the philosophy of mathematics, such as ontological questions regarding the nature of mathematical objects, epistemic questions about the acquisition of mathematical knowledge, and the intriguing riddle of the applicability of mathematics to the physical world. Some of these issues go back to the nascent years of mathematics itself, others are just beginning to draw the attention of scholars. In addressing these questions, some of the papers in this volume wrestle with them directly, while others use the writings of philosophers such as Hume and Wittgenstein to approach their problems by way of interpretation and critique. The contributors include prominent philosophers of science and mathematics as well as promising younger scholars. The volume seeks to share the concerns of philosophers of mathematics with a wider audience and will be of interest to historians, mathematicians and philosophers alike.

According to Mark Steiner, Wittgenstein’s intense work in the philosophy of mathematics during the early 1930s brought about a distinct turning point in his philosophy. The crux of this transition, Steiner contends, is that Wittgenstein came to see mathematical truths as originating in empirical regularities that in the course of time have been hardened into rules. This interpretation, which construes Wittgenstein’s later philosophy of mathematics as more realist than his earlier philosophy, challenges another influential interpretation which reads Wittgenstein as moving in the opposite direction, from a more realist toward a less realist position. Both of these readings, I argue here, tend to overlook the crucial role of conventionalism in shaping Wittgenstein’s later philosophy. I show that during the transition period, Wittgenstein was first strongly attracted to conventionalism, but then, upon discovery of the rule following paradox, came to realize its weaknesses. Did this realization mark the end of the liaison with conventionalism and a wholehearted acceptance of the empirical regularities account? Illustrating Wittgenstein’s ongoing ambivalence toward conventionalism and pointing to his novel understanding of this position, I answer this question in the negative.

The ArgumentEinstein's concept of causality as analyzed in this paper is a thick concept comprised of: (a) regularity; (b) locality; (c) symmetry considerations leading to conservation laws; (d) mutuality of causal interaction. The main theses are: 1. Since (b)–(d) are not elements of Hume's concept of causality, Einstein's concept, the concept embedded in the theory of relativity, is manifestly non–Humean. 2. On a Humean conception, Newtonian mechanics is a paradigmatically causal theory. Einstein, however, regarded this theory as causally deficient, for it fails to comply with both (b) and (d). Special relativity was (partly) motivated by the wish to correct the first of these failures; general relativity the second. 3. Ironically, general relativity, based on the thick concept of causality, opens the way for a conventionalist understanding of that concept. 4. With regard to human freedom, Einstein professed to be a Spinozist. However, he suggested a version of soft determinism, not found in Spinoza.

This book subjects the traditional concept of law of nature to critical examination. There are two kinds of reasons that invite this reexamination, one deriving from philosophical concerns over the traditional concept, the other motivated by theoretical and practical changes in science. One of the philosophical worries is that the idiom of law of nature, especially when combined with the notion of laws 'governing' individual events and processes, is no longer as intelligible as it used to be in the theistic context in which the formulation of laws became central to science. The traditional concept is also challenged in various ways by contemporary scientific theories such as quantum mechanics, chaos theory and the general theory of relativity. It is no longer clear that there are any universal laws, laws do not always guarantee predictability, and the border between physical and mathematical considerations is constantly shifting. The most difficult challenge, perhaps, is to come up with a scientific explanation of the origin of laws. Wrestling with these intriguing problems, the papers in this volume broaden both our understanding of the natural order and our desiderata of scientific explanation.

Poincaré’s conventionalism has thoroughly transformed both the philosophy of science and the philosophy of mathematics. Not only proponents of conventionalism, such as the logical positivists, were influenced by Poincaré, but also outspoken critics of conventionalism, such as Quine and Putnam, were inspired by his daring position. Indeed, during the twentieth century, most philosophers of mathematics and of science engaged in dialogue with conventionalism. As is often the case with such complex clusters of ideas, there is no consensus about the meaning of conventionalism in general, and Poincaré’s original version of it, in particular. Nonetheless, notions such as the under-determination, empirical equivalence, implicit definition, holism and conceptual relativity, all of which can be linked to Poincare's writings have become central to philosophy. This essay explores the impact of some of these notions on twentieth century philosophy of science. In addition to inspiration based on Poincare's actual views, it emphasizes directions based on misreading and unjustified appropriations of Poincaré.

Quine adduced two arguments that point to the obstacles facing an objective understanding of the language-world relation: one is based on the underdetermination of theory by observation, the other pointing to the inscrutability of reference (also referred to as ontological relativity). Together, these arguments generate the indeterminacy of translation—a thesis of ongoing concern not only to Quine, but also to many of his interpreters, among them Putnam and Davidson. This paper begins with an examination of these arguments, and proceeds to monitor the changes they underwent as Quine, Putnam, and Davidson were struggling with their justification and import. It is shown that Quine wavered on the underdetermination of theory whereas Putnam vacillated on ontological relativity. Davidson, in turn, despite his numerous expressions of deference to Quine and his philosophy of language, actually deviated quite significantly from Quine’s conception of meaning. The differences between their positions notwithstanding, Quine, Putnam, and Davidson shared a pragmatist rejection of scepticism and an avowal of commonsense realism. These commitments, I will argue, underlie their responses to the worrisome indeterminacies of the language-world relation. © 2021 Elsevier B.V., All rights reserved.

Poincaré’s conventionalism has thoroughly transformed both the philosophy of science and the philosophy of mathematics. In the former it gave rise to new insights into the complexities of scientific method, in the latter to a new account of the nature of (so-called) necessary truth. Not only proponents of conventionalism, such as the logical positivists, were influenced by Poincaré, but also outspoken critics of conventionalism, such as Quine, Putnam, and (as I will argue) Wittgenstein, were deeply inspired by conventionalist ideas. Indeed, during the twentieth century, most philosophers of science and mathematics engaged in dialogue with conventionalism. As is often the case with complex ideas, there is no consensus about the meaning of conventionalism in general and Poincaré’s original version of it in particular. Nonetheless, notions such as underdetermination (of theory), empirical equivalence (of incompatible theories), implicit definition, holism, and conceptual relativity, all of which can be linked to Poincaré’s writings, have become central to philosophy. In tracing the flow of Poincaré’s ideas through the philosophical space of the twentieth century, this article emphasizes not only the stimulus provided by his actual views but also the effects of misreading and unjustified appropriation.

The ArgumentIdealizations and approximations are an indispensable tool for the scientist. This paper argues that idealizations and approximations are equally indispensable for the philosopher of science. In particular, it is shown that the deductive model of scientific theories is an idealization in precisely the same sense that frictionless motion is an idealization in mechanics. By its very nature, an idealization cannot be criticized as not being absolutely true to the facts, for it need not be. Thus, the usual type of criticism levelled against the deductive model is shown to be irrelevant. The main step in the argument consists in using a logic of approximation to clarify the notions of approximate implication and approximate entailment, which are usually rejected as hopelessly vague. More generally, the conception underlying the paper is that the desiderata of science should be formulated in such a way that complying with them could be a matter of degree, and that pragmatic considerations could be taken into account. This conception is motivated by the wish to introduce some quantitative concepts in the philosophy of science, thereby bringing the philosopher's perspective closer to that of the scientist.

According to the current epistemic interpretation of quantum probabilities, the quantum correlations manifesting nonlocality can be derived from purely probabilistic and information-theoretic constraints. As such, they do not constitute a spacetime phenomenon and cannot lead to conflict between QM and any spatial-temporal constraints. This paper compares recent epistemic interpretations with earlier probabilistic interpretations, noting their merits as well as the difficulties they encounter. In particular, the implications of the recent PBR theorem are examined. While generally seen as undermining the epistemic interpretation, I argue that the PBR theorem actually suggests an epistemic position more radical than previously ones. This review of probabilistic interpretations enables reassessment of the epistemic theorist's attempt to disentangle nonlocality from spacetime considerations.

When Sadi Carnot carried out the pioneering work on heat engines which led to the second law of thermodynamics, he contemplated an ideal heat engine, one that was completely reversible. Carnot's use of idealization was particularly successful ‐ while the ideal engine cannot actually be constructed, the conclusions he derived for the ideal engine hold a fortiori for actual heat engines. For example, the greater the temperature difference between the two heat reservoirs, the higher the engine's efficiency. But this is not always the case. What holds true in the ideal limit may be false in reality. Nevertheless, the ideal case, simple and tractable, can be expected to shed light on actual cases, the precise treatment of which is impossible or impractical. Idealizations abound in science: ideal gases, closed systems, perfectly rational agents, and evolutionarily stable strategies. Indeed, thinkers as diverse in their outlook as Edmund Husserl and Albert Einstein have pointed to idealizations as the hallmark of modern science.

Critique involves reflection, specifically self-reflection, and as such it is inherently linked with philosophy. Critique calls for change, awareness, liberation from false conceptions, and reshaping of spheres of action and belief. Consequently it is closely linked with the moral and the political. Critique aspires to enhance truth, beauty, and justice and is thus an integral part of science, art, and social action. The present volume tackles issues of critique through a selection of papers originally presented at the workshop on “Models of Critique in the Sciences, Society, and the Arts,” held in Jerusalem and Tel Aviv during May 1996. As the full program indicates, this selection provides only a partial picture of the topics discussed at the workshop. A major consideration has been the publication policy of Science in Context, with its emphasis on science, which is reflected here. We wish to express our gratitude to all the participants for their presentations and contributions to a stimulating exchange of ideas.