Dancy Cartwright: Particularism in the philosophy of science

This paper aims to explore the space of possible particularistic approaches to Philosophy of Science by examining the differences and similarities between Jonathan Dancy’s moral particularism—as expressed in both his earlier writings (e.g., Moral Reasons, 1993), and, more explicitly defended in his book Ethics without Principles (2004)—and Nancy Cartwright’s particularism in the philosophy of science, as defended in her early collection of essays, How the Laws of Physics Lie (1983), and her later book, The Dappled World: A Study of the Boundaries of Science (1999). I shall argue that Dancy’s particularism is more radical, but also more plausible, than Cartwright’s, concluding that we have good reason to embrace a scientific particularism that is far closer to Dancy’s ethical particularism than any view defended by Nancy Cartwright, or any other philosopher from the ‘Stanford school’ of scientific theory.     

Deformed virasoro algebras andw-algebras from elliptic algebras

Deformedw _n -algebras in the classical case (i.e., Poisson structures) and in the quantum case (exchange algebras) are constructed from the elliptic algebras . In memory of M. Saveliev When I think about Misha, and this has been happening rather frequently these recent months, it is his enthusiasm and his communicative energy that first come to mind. Something very important to Misha was friendship. We scientists have the priviledge to possess friends, real friends, among our colleagues, and Misha was a friend. Something else was fundamental for Misha, his family, i.e., his parents, his wife, and his daughter. As Professor Manin said, summarizing Misha’s behavior very simply and elegantly, Misha was “a good son, a good husband, a good father.” Misha’s parents died about three years ago, and I remember how he was sad when I met him in Georgia in September 1996 and he informed me about his father’s and mother’s deaths a few months before. After that, he concentrated all his forces to protect his wife and daughter, the two Svetlanas, as he used to say! He was so proud of his gifted daughter, a good musician, a good painter, and a brilliant student. I wish to present them my most respectful thoughts. Misha was a great expert in integrable systems. He was deeply interested in developing models, which he did so nicely in the framework of Toda theories. I learned much from him about Toda theories and also continuous Lie algebras. Misha was highly convinced of the importance of symmetries and was especially interested in the use of infinite symmetries, the W-algebras among them. The landscape of continuous symmetries, and particularly of infinite-dimensional symmetries, is becoming richer and richer these years. I dare to think that Misha would have been interested in the modest seminar I now present. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 311–322, May, 2000.     

Fermats ?ad?quare? – und kein Ende?

In his papers on the determination of maxima and minima and on the calculation of tangents Pierre Fermat uses two different Latin verbs, ?quare and ad?quare, which do not differ semantically but are used by him obviously in different meanings. While ?quabitur is used unambiguously in the sense of “is equal” the meaning of ad?quabitur is disputed by the experts since Tannery’s French translation (Œuvres complètes de Fermat, Vol. III, 1896). Herbert Breger (Arch. Hist. Exact Sci. 46, 193–219, (1994), p. 197 f), for instance, holds the view that Fermat used the word ad?quare in the sense of “to put equal” and adds: In a mathematical context, the only difference between “?quare” and “ad?quare” (if there is any) seems to be that the latter gives more stress on the fact that the equality is achieved. In contrast to this Michael Mahoney holds the thesis that ad?quare describes a counterfactual equality (Mahoney, M.S.: Fermat, Pierre de. In: Dictionary of Scientific Biography, vol. IV (1971), p. 569) or a pseudo-equality (Mahoney, M.S.: The Mathematical Career of Pierre de Fermat (1601–1665), (1973), p. 164), whatever that may mean. This viewpoint has been taken up again recently by Enrico Giusti (Ann. Fac. Sci. Toulouse, Math. (6), 18 fascicule spécial, 59–85 (2009)) in order to bring arguments to bear against Breger. In contrast to these (and other) authors, I show that Fermat makes a subtle logical distinction between the words ?quare and ad?quare. The same distinction is made by Nicolas Bourbaki introducing his ?théorie égalitaire?. Notwithstanding: both verbs stand for a ?relation d’égalité?. On this premiss, I describe—using six selected examples—that Fermat’s “method” may be justified right down to the last detail, even from the view of today’s mathematical knowledge.     

‘Seeing is believing’ and ‘believing is seeing’

The principal concern of my paper is a distinction between two ways of appreciating works of art, characterised here in terms of the phrases ‘seeing is believing’ and ‘believing is seeing’. I examine this distinction in the light of an epistemological requirement at times at least grounded in what David Davies, in his Art as Performance, refers to as the ‘common sense theory of art appreciation’ in order to assess exactly what aspect of the philosophical approach generally known as aesthetic empiricism his account commits him to reject. I argue that the ‘experiential requirement’, if only conceived in a slightly broader way than is usual, might very well have an important role to play not only in the appropriate appreciation of works that do not demonstrate the need for such a requirement (primarily works of late modern and conceptual art), but also in the ontological account Davies himself favours.     

Onσ-finite invariant measures for Markov processes

Sufficient conditions are given for the existence ofσ-finite invariant measure for conservative and ergodic Markov processes. This paper is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem uuder the supervision of Professor S. R. Foguel. The author wishes to thank him for his helpful advice and kind encouragement.     

Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper

The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values of the q-logarithm function, with natural choice of~q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main tools are Euler-type integrals and hypergeometric series. 2000 Mathematics Subject Classification Primary—11Y60; Secondary—11J72, 33C20, 33D15 The work of the second author is supported by an Alexander von Humboldt research fellowship Dedication: To Leonhard Euler on his 300th birthday.     

A Warranted-assertability Defense of a Moorean Response to Skepticism

According to a Moorean response to skepticism, the standards for knowledge are invariantly comparatively low, and we can know across contexts all that we ordinarily take ourselves to know. It is incumbent upon the Moorean to defend his position by explaining how, in contexts in which S seems to lack knowledge, S can nevertheless have knowledge. The explanation proposed here relies on a warranted-assertability maneuver: Because we are warranted in asserting that S doesn’t know that p, it can seem that S does in fact lack that piece of knowledge. Moreover, this warranted-assertability maneuver is unique and better than similar maneuvers because it makes use of H. P. Grice’s general conversational rule of Quantity—“Do not make your contribution more informative than is required”—in explaining why we are warranted in asserting that S doesn’t know that p.

Kvanvig on Pointless Truths and the Cognitive Ideal

Jonathan Kvanvig has recently attempted to reconcile the problem of (apparently) pointless truths with the claim that the value of truth is unrestricted—that truth is always and everywhere valuable. In this paper, I critically evaluate Kvanvig’s argument and show it to be defective at a crucial juncture. I propose my own alternative strategy for generating Kvanvig’s result—an alternative that parts ways with Kvanvig’s own conception of the cognitively ideal.     

The life and work of André Cholesky

Cholesky’s method for solving a system of linear equations with a symmetric positive definite matrix is well known. In this paper, I will give an account of the life of Cholesky, analyze an unknown and unpublished paper of him where he explains his method, and review his other scientific works. I dedicated this work to John (Jack) Todd with esteem and respect at the occasion of his 95th anniversary.     

Bar-On on Self-Knowledge and Expression

I critically discuss the account of self-knowledge presented in Dorit Bar-On’s Speaking My Mind (OUP 2004), focusing on Bar-On’s understanding of what makes our capacity for self-knowledge puzzling and on her ‘neo-expressivist’ solution to the puzzle. I argue that there is an important aspect of the problem of self-knowledge that Bar-On’s account does not sufficiently address. A satisfying account of self-knowledge must explain not merely how we are able to make accurate avowals about our own present mental states, but how we can reasonably regard ourselves as entitled to claim self-knowledge. Addressing this aspect of the problem of self-knowledge requires confronting questions about the metaphysical nature of mental states, questions that Bar-On’s approach seeks to avoid.     

The Pepsi Putnam Challenge

Opening a copy of The Mathematical Intelligencer you may ask yourself uneasily, “What is this anyway—a mathematical journal, or what?” Or you may ask, “Where am I?” Or even “Who am I?” This sense of disorientation is at its most acute when you open to Cohn Adam’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be hannless.     

Problems for Broome’s Cognitivist Account of Instrumental Reasoning

In this paper, I examine an account of instrumental reasoning recently put forth by John Broome. His key suggestion is that anyone who engages in reasoning about his intentions also believes that he will do what he intends to do and that combined with a belief about necessary means this creates rational pressure towards believing that one will take the necessary means. I argue that Broome’s model has three significant problems; his key premise is false—the sincere expression of an intention does not entail the belief that one will successfully execute that intention; his account yields a model of instrumental reasoning that is uncomfortably reflective; he seems unable to explain the rational pressure towards taking necessary means that arises directly from having an end and an instrumental belief. All three problems, I argue, are a consequence of Broome’s inadequate position on what it is to intend to do something.     

German history experienced: My studies, my teachers

Conclusion  I wanted to describe my meetings with personalities who have influenced my Me and show how different from today our world half a century ago actually was. I also wanted with gratitude to bring to life the memory of people who were not only important scientists but also—Menschen. This article is based on a talk given at the University of Stuttgart, October 22, 1998. The author and the Editor thank George Seligman for his advice in preparing the present version. Much of the material appeared also in “Menschen, mein Studium, meine Lehrer” inMitteilungen der DMV 4 (1999), 43-52.     

The Incoherence Objection in Moral Theory

J.J.C. Smart famously complained that rule utilitarianism is incoherent, and that rule utilitarians are guilty of “rule worship”. Much has been said about whether Smart’s complaint is justified, but I will assume for the sake of argument that Smart was on to something. Instead, I have three other goals. First, I want to show that Smart’s complaint is a specific instance of a more general objection to a moral theory—what I will call the Incoherence Objection. Second, I want to illustrate how the Incoherence Objection can apply both to consequentialist and, surprisingly, some nonconsequentialist theories. Finally, I want to demonstrate at least one way nonconsequentialist theories that make use of rules, principles, and the like can dodge the Incoherence Objection.     

Expressing and Describing Experiences. A Case of Showing Versus Saying

Experiences are interpreted as conscious mental occurrences that are of phenomenal character. There is already a kind of (weak) intentionality involved with this phenomenal interpretation. A stricter conception of experiences distinguishes between purely phenomenal experiences and intentional experiences in a narrow sense. Wittgenstein’s account of psychological (experiential) verbs is taken over: Usually, expressing mental states verbally is not describing them. According to this, “I believe” can be seen as an expression of one’s own belief, but not as an expression of a belief about one’s belief. Hence, the utterance “I believe it is raining” shows that I believe that it is raining, although it is not said by these words that I believe that it is raining. Thinking thoughts such as “I believe it is raining, but it is not raining” (a variant of Moore’s paradox) is an absurdity between what is already said by silently uttering “It is not raining” and what is shown by silently uttering “I believe it is raining.” The paper agrees with a main result of Wittgenstein’s considerations of Moore’s paradox, namely the view that logical structure, deducibility, and consistency cannot be reduced solely to propositions—besides a logic of propositions, there is, for example, a logic of assertions and of imperatives, respectively.     

Irreflexive Banach spaces are imperfect

It is shown that irreflexive Banach spaces are imperfect: thus the “perfect” spaces are exactly the reflexive Banach spaces and “mixed” spaces do not exist. This paper is part of the author’s M.Sc. dissertation, performed under guidance of Dr. D. Amir at the Technion — Israel Institute of Technology. The author wishes to thank Dr. Amir for his help.     

An Answer to S. Simons’ Question on the Maximal Monotonicity of the Sum of a Maximal Monotone Linear Operator and a Normal Cone Operator

The question whether or not the sum of two maximal monotone operators is maximal monotone under Rockafellar’s constraint qualification—that is, whether or not “the sum theorem” is true—is the most famous open problem in Monotone Operator Theory. In his 2008 monograph “From Hahn-Banach to Monotonicity”, Stephen Simons asked whether or not the sum theorem holds for the special case of a maximal monotone linear operator and a normal cone operator of a closed convex set provided that the interior of the set makes a nonempty intersection with the domain of the linear operator. In this note, we provide an affirmative answer to Simons’ question. In fact, we show that the sum theorem is true for a maximal monotone linear relation and a normal cone operator. The proof relies on Rockafellar’s formula for the Fenchel conjugate of the sum as well as some results featuring the Fitzpatrick function.      

Is There a Conceptual Connection Between Intentionality and Consciousness?

It is argued that Searle’s argument for the thesis that there is a conceptual connection between intentionality and consciousness suffers from serious ambiguities. Searle’s argument is reconstructed as consisting of three premises that contain the notions of aspectual shape and irreducible subjectivity. Having identified two different readings of ‘aspectual shape’ and ‘irreducibly subjective,’ I conclude that each version of Searle’s argument incorporates at least one false premise. This paper is a descendant of a paper presented at the Wittgenstein Symposium at Kirchberg am Wechsel in August 2004. I would like to thank the reviewer of this journal for his/her helpful comments on a previous draft.     

Squared polynomial extrapolation methods with cycling: an application to the positron emission tomography problem

Roland and Varadhan (Appl. Numer. Math., 55:215–226, 2005) presented a new idea called “squaring” to improve the convergence of Lemaréchal’s scheme for solving nonlinear fixed-point problems. Varadhan and Roland (Squared extrapolation methods: A new class of simple and efficient numerical schemes for accelerating the convergence of the EM algorithm, Department of Biostatistics Working Paper. Johns Hopkins University, , 2004) noted that Lemaréchal’s scheme can be viewed as a member of the class of polynomial extrapolation methods with cycling that uses two fixed-point iterations per cycle. Here we combine these two ideas, cycled extrapolation and squaring, and construct a new class of methods, called squared polynomial methods (SQUAREM), for accelerating the convergence of fixed-point iterations. Our main goal is to evaluate whether the squaring device is effective in improving the rate of convergence of cycled extrapolation methods that use more than two fixed-point iterations per cycle. We study the behavior of the new schemes on an image reconstruction problem for positron emission tomography (PET) using simulated data. Our numerical experiments show the effectiveness of first- and higher-order squared polynomial extrapolation methods in accelerating image reconstruction, and also their relative superiority compared to the classical, “unsquared” vector polynomial methods.     

The Pfaff/Cauchy derivative identities and Hurwitz type extensions

More than 200 years ago, Pfaff found two generalizations of Leibniz’s rule for the nth derivative of a product of two functions. Thirty years later Cauchy found two similar identities, one equivalent to one of Pfaff’s and the other new. We give simple proofs of these little-known identities and some further history. We also give applications to Abel-Rothe type binomial identities, Lagrange’s series, and Laguerre and Jacobi polynomials. Most importantly, we give extensions that are related to the Pfaff/Cauchy theorems as Hurwitz’s generalized binomial theorems are to the Abel-Rothe identities. We apply these extensions to Laguerre and Jacobi polynomials as well. Dedicated to Dick Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—05A19; Secondary—33C45