Conjunction, Inference and the Limits of Analysis

Kraemer, Eric Russert

doi:10.1111/j.1467-9973.1986.tb00843.x

“…Another way to see that extricability claims can be problematic is to consider the question of what would be left if one subtracted someone is thirsty from I'm thirsty (Yablo 2012); or the question of what would be left if one subtracted the tomato is red from the tomato is scarlet (Searle & Körner 1959, Woods 1967, Kraemer 1986, Yablo 2012)? It's not clear that there are well-defined answers to be given-unless, of course, one is prepared to say 'nothing'.…”

Section: Trivialist Infinitarianismmentioning

confidence: 99%

Nominalism, Trivialism, Logicism

Rayo¹

2014

Philosophia Mathematica

0

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This paper is an effort to extract some of the main theses in the philosophy of mathematics from my book, The Construction of Logical Space. I show that there are important limits to the availability of nominalistic paraphrase-functions for mathematical languages, and suggest a way around the problem by developing a method for specifying nominalistic contents without corresponding nominalistic paraphrases.Although much of the material in this paper is drawn from the book-and from an earlier paper (Rayo 2008)-I hope the present discussion will earn its keep by motivating the ideas in a new way, and by suggesting further applications. NominalismMathematical Nominalism is the view that there are no mathematical objets. A standard problem for nominalists is that it is not obvious that they can explain what the point of a mathematical assertion would be. For it is natural to think that mathematical sentences like 'the number of the dinosaurs is zero' or '1 + 1 = 2' can only be true if mathematical objects exist. But if this is right, the nominalist is committed to the view that such sentences are untrue. And if the sentences are untrue, it not immediately obvious why they would be worth asserting. * For their many helpful comments, I am indebted to Vann McGee, Kevin Richardson, Bernhard Salow and two anonymous referees for Philosophia Mathematica. I would also like to thank audiences at Smith College, the Università Vita-Salute San Raffaele, and MIT's Logic, Langauge, Metaphysics and Mind Reading Group. Most of all, I would like to thank Steve Yablo. 1A nominalist could try to address the problem by suggesting nominalistic paraphrases for mathematical sentences. She might claim, for example, that when one asserts 'the number of the dinosaurs is zero' one is best understood as making the (nominalistically kosher) claim that there are no dinosaurs, and that when one asserts '1 + 1 = 2' one is best understood as making the (nominalistically kosher) claim that any individual and any other individual will, taken together, make two individuals.1 Such a strategy faces two main challenges. The first is to explain why mathematical assertions are to be understood non-standardly. One way for our nominalist to address this challenge is by claiming that mathematical assertions are set forth 'in a spirit of makebelieve' (Yablo 2001). She might argue, in particular, that when one makes a mathematical assertion one is, in effect, claiming that the asserted sentence is true in a fiction, and more specifically a fiction according to which: (a) all non-mathematical matters are as in reality, but (b) mathematical objects exist with their standard properties. This proposal leads to the welcome result that fictionalist assertions of mathematical sentences can convey information about the real world. For instance, one can use a fictionalist assertion of 'the number of the dinosaurs is zero' to convey the information that there are no dinosaurs, since the only way for 'the number of the dinosaurs is zero' to be true in a fiction whereby mat...

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“…Suppose A is Lord of the Rings and B states the existence of hobbits. One looks in vain for an R that is suitably independent of B and such that A is equivalent to B &R. Try to subtract The tomato is red from The tomato is scarlet (Searle and Körner [1959], Woods [1967], Kraemer [1986]). Try to subtract Someone is thirsty from I'm thirsty, or I swam from I swam three laps.…”

Section: Nominalistic Contentmentioning

confidence: 99%

Explanation, Extrapolation, and Existence

Yablo¹

2012

Mind

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Colyvan [2010] raises two problems for "easy road" nominalism about mathematical objects. The first is that a theory's mathematical commitments may run too deep to permit the extraction of nominalistic content. Taking the math out is, or could be, like taking the hobbits out of Lord of the Rings. I agree with the "could be," but not (or not yet) the "is." A notion of logical subtraction is developed that supports the possibility, questioned by Colyvan, of bracketing a theory's mathematical aspects to obtain, as remainder, what it says "mathematics aside." The other problem concerns explanation. Several grades of mathematical involvement in physical explanation are distinguished, by analogy with Quine's three grades of modal involvement. The first two grades plausibly obtain, but they do not require mathematical objects. The third grade is likelier to require mathematical objects. But it is not clear from Colyvan's example that the third grade really obtains.

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Nominalism, Trivialism, Logicism

Rayo

¹

2014

Philosophia Mathematica

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This paper is an effort to extract some of the main theses in the philosophy of mathematics from my book, The Construction of Logical Space. I show that there are important limits to the availability of nominalistic paraphrase-functions for mathematical languages, and suggest a way around the problem by developing a method for specifying nominalistic contents without corresponding nominalistic paraphrases.Although much of the material in this paper is drawn from the book-and from an earlier paper (Rayo 2008)-I hope the present discussion will earn its keep by motivating the ideas in a new way, and by suggesting further applications. NominalismMathematical Nominalism is the view that there are no mathematical objets. A standard problem for nominalists is that it is not obvious that they can explain what the point of a mathematical assertion would be. For it is natural to think that mathematical sentences like 'the number of the dinosaurs is zero' or '1 + 1 = 2' can only be true if mathematical objects exist. But if this is right, the nominalist is committed to the view that such sentences are untrue. And if the sentences are untrue, it not immediately obvious why they would be worth asserting. * For their many helpful comments, I am indebted to Vann McGee, Kevin Richardson, Bernhard Salow and two anonymous referees for Philosophia Mathematica. I would also like to thank audiences at Smith College, the Università Vita-Salute San Raffaele, and MIT's Logic, Langauge, Metaphysics and Mind Reading Group. Most of all, I would like to thank Steve Yablo. 1A nominalist could try to address the problem by suggesting nominalistic paraphrases for mathematical sentences. She might claim, for example, that when one asserts 'the number of the dinosaurs is zero' one is best understood as making the (nominalistically kosher) claim that there are no dinosaurs, and that when one asserts '1 + 1 = 2' one is best understood as making the (nominalistically kosher) claim that any individual and any other individual will, taken together, make two individuals.1 Such a strategy faces two main challenges. The first is to explain why mathematical assertions are to be understood non-standardly. One way for our nominalist to address this challenge is by claiming that mathematical assertions are set forth 'in a spirit of makebelieve' (Yablo 2001). She might argue, in particular, that when one makes a mathematical assertion one is, in effect, claiming that the asserted sentence is true in a fiction, and more specifically a fiction according to which: (a) all non-mathematical matters are as in reality, but (b) mathematical objects exist with their standard properties. This proposal leads to the welcome result that fictionalist assertions of mathematical sentences can convey information about the real world. For instance, one can use a fictionalist assertion of 'the number of the dinosaurs is zero' to convey the information that there are no dinosaurs, since the only way for 'the number of the dinosaurs is zero' to be true in a fiction whereby mat...

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Conjunction, Inference and the Limits of Analysis

Cited by 3 publications

References 3 publications

Nominalism, Trivialism, Logicism

Nominalism, Trivialism, Logicism

Explanation, Extrapolation, and Existence

Nominalism, Trivialism, Logicism

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