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- Mary Leng (2005). Platonism and Anti-Platonism: Why Worry? International Studies in the Philosophy of Science 19 (1):65 – 84.This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti-Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the 'no miracles' argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific 'realists' should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism.Reading list | Discuss | Edit | Categorize |My bibliography
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A response is given here to Benacerraf's 1973 argument that mathematical platonism is incompatible with a naturalistic epistemology. Unlike almost all previous platonist responses to Benacerraf, the response given here is positive rather than negative; that is, rather than trying to find a problem with Benacerraf's argument, I accept his challenge and meet it head on by constructing an epistemology of abstract (i.e., aspatial and atemporal) mathematical objects. Thus, I show that spatio-temporal creatures like ourselves can attain knowledge about mathematical objects by simply explaininghow they can do this. My argument is based upon the adoption of a particular version of platonism — full-blooded platonism — which asserts that any mathematical object which possiblycould exist actuallydoes exist.
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| | Other links: jstor.org | Scholar | At my library | More options ... According to standard mathematical platonism, mathematical entities (numbers, sets, etc.) are abstract entities. As such, they lack causal powers and spatio-temporal location. Platonists owe us an account of how we acquire knowledge of this inaccessible mathematical realm. Some recent versions of mathematical platonism postulate a plenitude of mathematical entities, and Mark Balaguer has argued that, given the existence of such a plenitude, the attainment of mathematical knowledge is rendered non-problematic. I assess his epistemology for such a profligate platonism and find it unsatisfactory because it lacks an adequate semantics, in particular, an adequate account of reference.
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| | Scholar | At my library | More options ... Mark Balaguer's Platonism and Anti-Platonism in Mathematics presents an intriguing new brand of platonism, which he calls plenitudinous platonism, or more colourfully, full-blooded platonism. In this paper, I argue that Balaguer's attempts to characterise full-blooded platonism fail. They are either too strong, with untoward consequences we all reject, or too weak, not providing a distinctive brand of platonism strong enough to do the work Balaguer requires of it.
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| | Other links: philmat.oxfordjournals.org | Scholar | At my library | More options ... Platonism in the philosophy of mathematics is the doctrine that there are mathematical objects such as numbers. John Burgess and Gideon Rosen have argued that that there is no good epistemological argument against platonism. They propose a dilemma, claiming that epistemological arguments against platonism either rely on a dubious epistemology, or resemble a dubious sceptical argument concerning perceptual knowledge. Against Burgess and Rosen, I show that an epistemological anti-platonist argument proposed by Hartry Field avoids both horns of their dilemma.
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| | Other links: analysis.oxfordjournals.org dx.doi.org | Scholar | At my library | More options ... Platonism about mathematics (or mathematical platonism as I will mostly call it) is typically defined as the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. If true, mathematical platonism would be of great philosophical significance: it would be a counterexample to common physicalist views, and it would put great pressure on the epistemology of mathematics. The view would also be of significance for mathematical practice.
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| | Scholar | At my library | More options ... This is an extended, critical review of Mark Balaguer's book *Platonism and Anti-Platonism in Mathematics* (New York: Oxford University Press, 1998). After describing his theory ("full-blooded Platonism"), we raise two criticisms. The first concerns the fact that Balaguer's theory offers no way to uniquely identify the denotations of the terms appearing in mathematical theories. The second concerns the fact that Balaguer overlooks the possibility that the fact, that Platonism and anti-Platonism agree on numerous points but differ only on whether mathematical objects exist, can be explained if both views turn out to be two different interpretations of the same formal theory.
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| | Other links: philmat.oxfordjournals.org dx.doi.org | Scholar | At my library | More options ... Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. In this survey article, the view is clarified and distinguished from some related views, and arguments for and against the view are discussed.
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| | Scholar | At my library | More options ... On the face of it, platonism seems very far removed from the scientific world view that dominates our age. Nevertheless many philosophers and mathematicians believe that modern mathematics requires some form of platonism. The defense of mathematical platonism that is both most direct and has been most influential in the analytic tradition in philosophy derives from the German logician-philosopher Gottlob Frege (1848-1925).2 I will therefore refer to it as Frege’s argument. This argument is part of the background of any contemporary discussion of mathematical platonism.
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| | Scholar | At my library | More options ... In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good argument for or against platonism, but that we could never have such an argument and, indeed, that there is no fact of the matter as to whether platonism is correct.
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| | Scholar | At my library | More options ... This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti?Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the ?no miracles? argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific ?realists? should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism.
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| Discuss | Edit | Categorize | My bibliography | Export citation | Other links: tandfonline.com dx.doi.org | Scholar | At my library | More options ...
| | Other links: tandfonline.com dx.doi.org | Scholar | At my library | More options ... Discussion of Mary Leng, Platonism and anti-platonism: Why worry?
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