I argue on the basis of an example, Fourier theory applied to the problem of vibration, that Field's program for nominalizing science is unlikely to succeed generally, since no nominalistic variant will provide us with the kind of physical insight into the phenomena that the standard theory supplies. Consideration of the same example also shows, I argue, that some of the motivation for mathematical fictionalism, particularly the alleged problem of cognitive access, is more apparent than real.

Scientific Realism and Antirealism Debates about scientific realism concern the extent to which we are entitled to hope or believe that science will tell us what the world is really like. Realists tend to be optimistic; antirealists do not. To a first approximation, scientific realism is the view that well-confirmed scientific theories are approximately true; … Continue reading Scientific Realism and Antirealism →.

SummaryBenacerraf challenges us to account for the reliability of our mathematical beliefs given that there appear to be no natural connections between mathematical believers and mathematical ontology. In this paper I try to do two things. I argue that the interactionist view underlying this challenge renders inexplicable not only the reliability of our mathematical beliefs, construed either platonistically or naturalistically , but also the reliability of most of our beliefs in physics. I attempt to counter Benacerraf's challenge by sketching an (...) alternative conception of reliability explanations which renders explicable the reliability of our beliefs in physics and in mathematics but in which mathematical and formal considerations themselves play a central role. My main thesis is that abstract objects do not strike us, but that this is irrelevant to the reliability of our mathematical and physical beliefs. (shrink)

In this paper I argue three things: (1) that the interactionist view underlying Benacerraf's (1973) challenge to mathematical beliefs renders inexplicable the reliability of most of our beliefs in physics; (2) that examples from mathematical physics suggest that we should view reliability differently; and (3) that abstract mathematical considerations are indispensable to explanations of the reliability of our beliefs.

In this paper I discuss lessons that metaphysicians might learn from Duhem. Given Duhem’s well known antipathy to metaphysics, you will likely think that this is a fairly inauspicious beginning with a predictable ending: i.e., physics is one thing, metaphysics another, and never the twain shall meet. If you will bear with me, however, I hope to persuade you differently. On the contrary, I will argue, Duhem was both a common sense and metaphysical realist, his nuanced views about the relationship (...) between physics and metaphysics are poorly understood, and properly understood contain important lessons for contemporary metaphysics. I’ll spend most of the paper rehabilitating Duhem and finish with some lessons for metaphysicians. (shrink)

It is commonly presupposed that all instances of the deflationary reference schema ‘F’ applies to x if and only if x is ‘are correct. This paper argues, mainly on the basis of concrete example, that we have little reason to be confident about this presupposition: our tendency to believe the instances is based on local successes that may not be globally extendible. There is a problem of semantic projection, Ii argue, and standard accounts that would resolve or dissolve the problem (...) are problematic. (shrink)

According to externalism, reference is a relation between uses of an expression and features of the environment. Moreover, the reference relation is normative , and the referential relata of our expressions are explanatory of successful language use. This paper largely agrees with the broad conception underlying externalism: it is what people do with words that makes them have the references they have, and the world constrains what people can successfully do with words. However, the paper strongly disagrees with the details (...) . A centrally important feature of what people do with words is how they use them in inferential contexts. When due attention is given to the reference-determining role played by inferential properties of expressions, I argue, we arrive at a more satisfactory account of semantic norms and explanations. Much of the argument is based on a detailed look at the language of chemical classification used in the late 19th century. (shrink)

Book Information Knowledge, Cause, and Abstract Objects: Causal Objections to Platonism. Knowledge, Cause, and Abstract Objects: Causal Objections to Platonism Colin Cheyne , Dordrecht: Kluwer Academic Publishers , 2001 , xvi + 236 , £55 ( cloth ) By Colin Cheyne. Dordrecht: Kluwer Academic Publishers. Pp. xvi + 236. £55.

Most science requires applied mathematics. This truism underlies the Quine-Putnam indispensability argument: we cannot be mathematical nominalists without rejecting whole swaths of good science that are seamlessly linked with mathematics. One style of response (e.g. Field’s program) accepts the challenge head-on and attempts to show how to do science without mathematics. There is some consensus that the response fails because the nominalistic apparatus deployed either is not extendible to all of mathematical physics or is merely a deft reconstrual equivalent to (...) standard mathematics. A second style of response (suggested, e.g., by Balaguer and Maddy) denies that indispensability entails realism: when we mathematize a physical problem we treat its physical content as if it were the mathematical representation; provided the two are sufficiently similar, we can use the mathematics to draw conclusions about the physics; even if we cannot represent physical facts without mathematical tools, as-if-fictionalism is reasonable. In this paper I argue that uses of mathematics in science reach deeper than is appreciated by this second response and, indeed, in the more general literature. More specifically, our confidence that we can use the mathematics to draw conclusions about the physics itself depends on mathematics. If the mathematical premises we employ in concluding that a certain application is trustworthy are false, we may lack a justification for supposing that the application will reliably lead us from correct input to correct output. For example, solutions to many physical problems require the determination of a function satisfying a differential equation. Sometimes (e.g., if the differential equation is linear) the existence of a solution for initial value problems can be established directly; where direct methods fail, the existence of a solution must be established indirectly, generally by constructing a sequence of functions that converges to a limit function that satisfies the initial value problem. Moreover, the solution often cannot be evaluated by analytic methods, and scientists must rely on finite element numerical methods to approximate the solution. Mathematical analysis of errors provides further useful information governing the choice of approximation method and of the step size and number of elements needed for the approximation to reach a desired precision. Mathematical physicists rely on the background mathematical theories (e.g., theory of differential equations) presupposed in proving the existence of the solutions and approximating them. It is difficult to see how they could do this while adding the fictionalist disclaimer, “But, you know, I don’t believe any of the mathematics I’m using”. It is difficult to see how a fictionalist pursuing the second strategy can account for the soundness of mathematical reasoning in mathematical physics and elsewhere in the sciences. The paper will fill out this argument by appeal to examples and attempt to make clear (a) how mathematics is indispensable to understanding – and thus underwriting our confidence in – applications that would otherwise be shaky approximations and idealizations and (b) how this role is difficult to square with fictionalism. (shrink)

Most science requires applied mathematics. This truism underlies the Quine-Putnam indispensability argument: we cannot be mathematical nominalists without rejecting whole swaths of good science that are seamlessly linked with mathematics. One style of response accepts the challenge head-on and attempts to show how to do science without mathematics. There is some consensus that the response fails because the nominalistic apparatus deployed either is not extendible to all of mathematical physics or is merely a deft reconstrual equivalent to standard mathematics. A (...) second style of response denies that indispensability entails realism: when we mathematize a physical problem we treat its physical content as if it were the mathematical representation; provided the two are sufficiently similar, we can use the mathematics to draw conclusions about the physics; even if we cannot represent physical facts without mathematical tools, as-if-fictionalism is reasonable. In this paper I argue that uses of mathematics in science reach deeper than is appreciated by this second response and, indeed, in the more general literature. More specifically, our confidence that we can use the mathematics to draw conclusions about the physics itself depends on mathematics. If the mathematical premises we employ in concluding that a certain application is trustworthy are false, we may lack a justification for supposing that the application will reliably lead us from correct input to correct output. For example, solutions to many physical problems require the determination of a function satisfying a differential equation. Sometimes the existence of a solution for initial value problems can be established directly; where direct methods fail, the existence of a solution must be established indirectly, generally by constructing a sequence of functions that converges to a limit function that satisfies the initial value problem. Moreover, the solution often cannot be evaluated by analytic methods, and scientists must rely on finite element numerical methods to approximate the solution. Mathematical analysis of errors provides further useful information governing the choice of approximation method and of the step size and number of elements needed for the approximation to reach a desired precision. Mathematical physicists rely on the background mathematical theories presupposed in proving the existence of the solutions and approximating them. It is difficult to see how they could do this while adding the fictionalist disclaimer, “But, you know, I don’t believe any of the mathematics I’m using”. It is difficult to see how a fictionalist pursuing the second strategy can account for the soundness of mathematical reasoning in mathematical physics and elsewhere in the sciences. The paper will fill out this argument by appeal to examples and attempt to make clear how mathematics is indispensable to understanding – and thus underwriting our confidence in – applications that would otherwise be shaky approximations and idealizations and how this role is difficult to square with fictionalism. (shrink)

Russell’s views about the proper logical and epistemological treatment of names conspired to lead him to set aside considerations that support the claim that names are not definite descriptions. Though he appreciated those considerations, he famously argued that ordinary names are truncated definite descriptions. Nevertheless, his appreciation of the distinctive semantic behavior of ordinary names combined with his view that acquaintance comes in degrees led him to attempt to secure a semantically privileged status for ordinary names: only special kinds of (...) descriptions can go proxy for ordinary names “used as names”. The paper attempts to tell this story, filling in gaps where Russell doesn’t provide sufficient elaboration, and to draw some general conclusions about acquaintance-based approaches to names and singular thoughts. (shrink)

History and Philosophy of Logic, Volume 0, Issue 0, Page 1-2, Ahead of Print.

In this book physicist Roland Omnès addresses some big questions in philosophy of mathematics. Anyone who reflects on the history and practice of mathematics and the sciences, especially physics, will naturally be struck by some remarkable coincidences. First, often newly developed mathematics was not well understood. But its successful applications and its agreement with intuitive representations of reality promoted confidence in its correctness even absent clear foundations . Later, this confidence is vindicated when a proper setting for the concepts and (...) techniques is discovered . Second, often mathematical concepts designed for one purpose later turn out to have pervasive applications that could not have been imagined by the original practitioners. Third, many of the most important results obtained in physics since the late nineteenth century were driven by the search for precise, comprehensive, consistent theoretical frameworks: the sequence special relativity, general relativity, relativistic quantum mechanics, string theory can be seen as one that increases comprehensiveness by consistent unification. The fundamental theoretical work has little to do with empirical investigation and a lot to do with mathematical and conceptual investigation of invariances and symmetries. Fourth, mathematical principles guarantee existence principles needed by physics . Such coincidences naturally invite questions: Why is confidence in the consistency of a successful piece of mathematics so often vindicated? Why does mathematics turn out to be so comprehensive and fruitful in unexpected …. (shrink)