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- L. Luce (1991). Literalism and the Applicability of Arithmetic. British Journal for the Philosophy of Science 42 (4):469-489.Philosophers have recently expressed interest in accounting for the usefulness of mathematics to science. However, it is certainly not a new concern. Putnam and Quine have each worked out an argument for the existence of mathematical objects from the indispensability of mathematics to science. Were Quine or Putnam to disregard the applicability of mathematics to science, he would not have had as strong a case for platonism. But I think there must be ways of parsing mathematical sentences which account for applicability of mathematics and also do not require us to believe in entities we have no evidence for, other than through reading these sentences literally. We will explore a particular way to interpret sentences of arithmetic which promises to account for their applicability without bringing in metaphysics not also brought in by science. The investigation will be limited to the arithmetic of cardinal numbers. The general strategy is to argue for the analogy between arithmetic and science, rather than to argue for one case having a particular characteristic independently of the other.Reading list | Discuss | Edit | Categorize |My bibliography
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A number of people have recently argued for a structural approach to accounting for the applications of mathematics. Such an approach has been called “the mapping account”. According to this view, the applicability of mathematics is fully accounted for by appreciating the relevant structural similarities between the empirical system under study and the mathematics used in the investigation of that system. This account of applications requires the truth of applied mathematical assertions, but it does not require the existence of mathematical objects. In this paper, we discuss the shortcomings of this account, and show how these shortcomings can be overcome by a broader view of the application of mathematics: the inferential conception.
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| | Other links: onlinelibrary.wiley.com dx.doi.org | Scholar | At my library | More options ... One recent trend in the philosophy of mathematics has been to approach the central epistemological and metaphysical issues concerning mathematics from the perspective of the applications of mathematics to describing the world, especially within the context of empirical science. A second area of activity is where philosophy of mathematics intersects with foundational issues in mathematics, including debates over the choice of set-theoretic axioms, and over whether category theory, for example, may provide an alternative foundation for mathematics. My central claim is that these latter issues are of direct relevance to philosophical arguments connected to the applicability of mathematics. In particular, the possibility of there being distinct alternative foundations for mathematics blocks the standard argument from the indispensable role of mathematics in science to the existence of specific mathematical objects.
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| | Other links: jstor.org blackwell-synergy.com dx.doi.org | Scholar | At my library | More options ... We examine, from the partial structures perspective, two forms of applicability of mathematics: at the “bottom” level, the applicability of theoretical structures to the “appearances”, and at the “top” level, the applicability of mathematical to physical theories. We argue that, to accommodate these two forms of applicability, the partial structures approach needs to be extended to include a notion of “partial homomorphism”. As a case study, we present London's analysis of the superfluid behavior of liquid helium in terms of Bose‐Einstein statistics. This involved both the introduction of group theory at the top level, and some modeling at the “phenomenological” level, and thus provides a nice example of the relationships we are interested in. We conclude with a discussion of the “autonomy” of London's model.
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| | Other links: journals.uchicago.edu jstor.org dx.doi.org | Scholar | At my library | More options ... Kant’s theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant’s theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant’s theory of arithmetic can be picked up, improved by reconstruction and defended under a contemporary perspective: the theory of numbers as products of rule following construction presupposing successive synthesis in time and the theory of arithmetic equations, sentences or “formulas”—as Kant says—as synthetic a priori. In order to do so, two calculi in terms of modern mathematics are introduced which formalise Kant’s theory of addition as a form of synthetic operation.
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| | Other links: jstor.org | Scholar | At my library | More options ... This paper explores how to explain the applicability of classical mathematics to the physical world in a radically naturalistic and nominalistic philosophy of mathematics. The applicability claim is first formulated as an ordinary scientific assertion about natural regularity in a class of natural phenomena and then turned into a logical problem by some scientific simplification and abstraction. I argue that there are some genuine logical puzzles regarding applicability and no current philosophy of mathematics has resolved these puzzles. Then I introduce a plan for resolving the logical puzzles of applicability.
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| | Other links: philmat.oxfordjournals.org dx.doi.org | Scholar | At my library | More options ... In this paper I defend mathematical nominalism by arguing that any reasonable account of scientific theories and scientific practice must make explicit the empirical non-mathematical grounds on which the application of mathematics is based. Once this is done, references to mathematical entities may be eliminated or explained away in terms of underlying empirical conditions. I provide evidence for this conclusion by presenting a detailed study of the applicability of mathematics to measurement. This study shows that mathematical nominalism may be regarded as a methodological approach to applicability, illuminating the use of mathematics in science.
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| | Other links: philmat.oxfordjournals.org dx.doi.org | Scholar | At my library | More options ... This paper offers an explanation of how philosophy of science in the second half of the 20th century came to be so conspicuously silent on the problem of how to explain the applicability of mathematics. It examines the idea of the early logicists that the analyticity of mathematics accounts for its applicability, and how this idea was transformed during Carnap's efforts to establish a consistent and substantial philosophy of mathematics within the larger framework of Logical Empiricism. I argue that at the end point of this development, philosophical discussion of the applicability problem was terminated although important aspects of the logicists' original response to the applicability problem had had to be sacrificed along the way.
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| | Scholar | At my library | More options ... Discussions of the applicability of mathematics in the natural sciences have been flawed by failure to realize that there are multiple senses in which mathematics can be ‘applied’ and, correspondingly, multiple problems that stem from the applicability of mathematics. I discuss semantic, metaphysical, descriptive, and and epistemological problems of mathematical applicability, dwelling on Frege's contribution to the solution of the first two types. As for the remaining problems, I discuss the contributions of Hartry Field and Eugene Wigner. Finally, I argue that there are epistemological problems concerning the applicability of mathematics that nobody in the philosophical community has yet confronted, though the problems are well known to physicists.
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| | Other links: philmat.oxfordjournals.org dx.doi.org | Scholar | At my library | More options ... In this paper I argue for the view that structuralism offers the best perspective for an acceptable account of the applicability of mathematics in the empirical sciences. Structuralism, as I understand it, is the view that mathematics is not the science of a particular type of objects, but of structural properties of arbitrary domains of entities, regardless of whether they are actually existing, merely presupposed or only intentionally intended.
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| | Scholar | At my library | More options ... Much recent discussion in the philosophy of mathematics has concerned the indispensability argument—an argument which aims to establish the existence of abstract mathematical objects through appealing to the role that mathematics plays in empirical science. The indispensability argument is standardly attributed to W. V. Quine and Hilary Putnam. In this paper, I show that this attribution is mistaken. Quine's argument for the existence of abstract mathematical objects differs from the argument which many philosophers of mathematics ascribe to him. Contrary to appearances, Putnam did not argue for the existence of abstract mathematical objects at all. I close by suggesting that attention to Quine and Putnam's writings reveals some neglected arguments for platonism which may be superior to the indispensability argument.
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| | Other links: springerlink.com jstor.org | Scholar | At my library | More options ... Discussion of L. Luce, Literalism and the applicability of arithmetic
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