Mathematical Practices Can Be Metaphysically Laden
01 Jan 2020-pp 1-26
About: The article was published on 2020-01-01. It has received 6 citations till now.
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01 Jan 2014
TL;DR: The experience of being compelled by proof, the sense that something must be true, that a result is certain, generates the philosophy of mathematics and also creates the illusion that mathematics is certain this paper.
Abstract: Mathematics plays an inordinate role in the work of many of famous Western philosophers, from the time of Plato, through Husserl and Wittgenstein, and even to the present. Why? This paper points to the experience of learning or making mathematics, with an emphasis on proof. It distinguishes two sources of the perennial impact of mathematics on philosophy. They are classified as Ancient and Enlightenment. Plato is emblematic of the former, and Kant of the latter. The Ancient fascination arises from the sense that mathematics explores something ‘out there’. This is illustrated by recent discussions by distinguished contemporary mathematicians. The Enlightenment strand often uses Kant's argot: ‘absolute necessity’, ‘apodictic certainty’ and ‘a priori’ judgement or knowledge. The experience of being compelled by proof, the sense that something must be true, that a result is certain, generates the philosophy. It also creates the illusion that mathematics is certain. Kant's leading question, ‘How is pure mathematics possible?’, is easily misunderstood because the modern distinction between pure and applied is an artefact of the 19th century. As Russell put it, the issue is to explain ‘the apparent power of anticipating facts about things of which we have no experience’. More generally the question is, how is it that pure mathematics is so rich in applications? Some six types of application are distinguished, each of which engenders its own philosophical problems which are descendants of the Enlightenment, and which differ from those descended from the Ancient strand. Keywords: philosophy of mathematics, Plato, Kant, necessity, a priori, application.
37 citations
Journal Article•
TL;DR: In this article, it was shown that if ZFC is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory S4.2.
Abstract: A set theoretical assertion psi is forceable or possible, written lozenge psi, if psi holds in some forcing extension, and necessary, written square psi, if psi holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory S4.2.
7 citations
TL;DR: In this paper, the authors explore how intellectual humility manifests in mathematical practices, and employ accounts of this virtue as developed by virtue epistemologists in three case studies of mathematical activity.
Abstract: In this paper I explore how intellectual humility manifests in mathematical practices. To do this I employ accounts of this virtue as developed by virtue epistemologists in three case studies of mathematical activity. As a contribution to a Topical Collection on virtue theory of mathematical practices this paper explores in how far existing virtue-theoretic frameworks can be applied to a philosophical analysis of mathematical practices. I argue that the individual accounts of intellectual humility are successful at tracking some manifestations of this virtue in mathematical practices and fail to track others. There are two upshots to this. First, the accounts of the intellectual virtues provided by virtue epistemologists are insightful for the development of a virtue theory of mathematical practices but require adjustments in some cases. Second, the case studies reveal dimensions of intellectual humility virtue epistemologists have thus far overlooked in their theoretical reflections.
3 citations
Cites background from "Mathematical Practices Can Be Metap..."
...See (Rittberg, 2020) for an analysis of the interplay between philosophy and set-theoretic activity. everything is “inside” and we cannot make sense of the “outside” of the universe inside the theory ZFC itself, except in a metamathematical approach....
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...See (Rittberg, 2020) for more on monism as the orthodoxy of contemporary set theory....
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References
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TL;DR: In this paper, what is the continuoustime problem of the Cantor's Continuum Problem and how to solve it is discussed. But it is not a continuous problem, it is continuous.
Abstract: (1947). What is Cantor's Continuum Problem? The American Mathematical Monthly: Vol. 54, No. 9, pp. 515-525.
473 citations
Book•
01 Jan 1997
Abstract: Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem-realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.
324 citations