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In this work Dummett discusses, section by section, Frege's masterpiece The Foundations of Arithmetic and Frege's treatment of real numbers in the second volume ... |
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In arithmetic, if only because many of its methods and concepts originated in India, it has been the tradition to reason less strictly than in geometry, ... |
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A paper in this journal by Fraser MacBride, ‘Can Ante Rem Structuralism Solve the Access Problem?’, raises important issues concerning the epistemological goals and burdens of contemporary philosophy of mathematics, and perhaps philosophy of science and other disciplines as well. I use a response to MacBride's paper as a framework for developing a broadly holistic framework for these issues, and I attempt to steer a middle course between reductive foundationalism and extreme naturalistic quietism. For this purpose the notion of entitlement (...) |
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Charles Parsons; X*—Mathematical Intuition, Proceedings of the Aristotelian Society, Volume 80, Issue 1, 1 June 1980, Pages 145–168, https://doi.org/10.1093/ari. |
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I consider different versions of a structuralist view of mathematical objects, according to which characteristic mathematical objects have no more of a 'nature' than is given by the basic relations of a structure in which they reside. My own version of such a view is non-eliminative in the sense that it does not lead to a programme for eliminating reference to mathematical objects. I reply to criticisms of non-eliminative structuralism recently advanced by Keränen and Hellman. In replying to the former, (...) |
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This work is the long awaited sequel to the author’s classic Frege: Philosophy of Language. But it is not exactly what the author originally planned. He tells us that when he resumed work on the book in the summer of 1989, after a long interruption, he decided to start afresh. The resulting work followed a different plan from the original drafts. The reader does not know what was lost by their abandonment, but clearly much was gained: The present work may (...) |
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Charles Parsons has argued that we have the ability to apprehend, or "intuit", certain kinds of abstract objects; that among the objects we can intuit are some which form a model for arithmetic; and that our knowledge that the axioms of arithmetic are true in this model involves our intuition of these objects. I find a problem with Parson's claim that we know this model is infinite through intuition. Unless this problem can be resolved. I question whether our knowledge that (...) |
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One kind of structuralism holds that mathematics is about structures, conceived as a type of abstract entity. Another denies that it is about any distinctively mathematical entities at all—even abstract structures; rather it gives purely general information about what holds of any collection of entities conforming to the axioms of the theory. Of these, pure structuralism is most plausibly taken to enjoy significant advantages over platonism. But in what appears to be its most plausible—modalised—version, even restricted to elementary arithmetic, it (...) |
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How we see and how we visualize: why the scientific account differs from our experience. |
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Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic (...) |
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6. Seeing With the Mind’s Eye 1: The Puzzle of Mental Imagery .................................................6-1 6.1 What is the puzzle about mental imagery?..............................................................................6-1 6.2 Content, form and substance of representations ......................................................................6-6 6.3 What is responsible for the pattern of results obtained in imagery studies?.................................6-8.. |
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