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Http Dx Doi Org 10 1080 00048408812343401, Jun 2, 2006
The British Journal for the Philosophy of Science, 1991
Philosophers have recently expressed interest in accounting for the usefulness of mathematics to ... more 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.
Philosophical Topics, 1989
The British Journal of Sociology, 1990
An academic directory and search engine.
Australasian Journal of Philosophy, 1988
Philosophy and Phenomenological Research, 1988
Frege invented a theory of quantification in order to provide a means to test the validity of pro... more Frege invented a theory of quantification in order to provide a means to test the validity of proofs in mathematics. A striking innovation in this theory was to construe quantifiers as second-level concepts which applied to first-level concepts. A few years later he introduced his theory of arithmetic in Die Grundlagen der Arithmetik. In the Grundlagen, Frege launched an extensive critique of the views of others concerning arithmetic. After finding these views unsatisfactory, he concluded that "the content of a statement of number is an assertion about a concept" ([i], ?46). Although he presented his view as an improvement on earlier attempts, the failure of other views served mainly as independent support for an already motivated theory. This central thesis of the Grundlagen follows from the earlier work on logic, for attributions of cardinality are special cases of Frege's theory of quantification. According to this result, attributions of cardinality apply higher-level "cardinality concepts" to lower-level concepts. However, Frege did not conclude that the cardinal numbers are kinds of quantifiers. In the Grundlagen he proceeded to argue that the cardinal numbers are objects, not concepts. He finally defined the numbers as certain objects, the extensions of those concepts. The close connection between quantification and cardinality has not received the attention it deserves. This is not surprising, for it is in apparent conflict with the thesis that numbers are objects. Frege's ontology encompassed two ontological categories, function and object.' These ontological categories were mutually exclusive; no function was an object. And concepts were a type of function, namely those which took a single argument, yielding a truth-value. So no concept could be an object. Frege introduced quantifiers as certain second-level concepts, with no
Http Dx Doi Org 10 1080 00048408812343401, Jun 2, 2006
The British Journal for the Philosophy of Science, 1991
Philosophers have recently expressed interest in accounting for the usefulness of mathematics to ... more 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.
Philosophical Topics, 1989
The British Journal of Sociology, 1990
An academic directory and search engine.
Australasian Journal of Philosophy, 1988
Philosophy and Phenomenological Research, 1988
Frege invented a theory of quantification in order to provide a means to test the validity of pro... more Frege invented a theory of quantification in order to provide a means to test the validity of proofs in mathematics. A striking innovation in this theory was to construe quantifiers as second-level concepts which applied to first-level concepts. A few years later he introduced his theory of arithmetic in Die Grundlagen der Arithmetik. In the Grundlagen, Frege launched an extensive critique of the views of others concerning arithmetic. After finding these views unsatisfactory, he concluded that "the content of a statement of number is an assertion about a concept" ([i], ?46). Although he presented his view as an improvement on earlier attempts, the failure of other views served mainly as independent support for an already motivated theory. This central thesis of the Grundlagen follows from the earlier work on logic, for attributions of cardinality are special cases of Frege's theory of quantification. According to this result, attributions of cardinality apply higher-level "cardinality concepts" to lower-level concepts. However, Frege did not conclude that the cardinal numbers are kinds of quantifiers. In the Grundlagen he proceeded to argue that the cardinal numbers are objects, not concepts. He finally defined the numbers as certain objects, the extensions of those concepts. The close connection between quantification and cardinality has not received the attention it deserves. This is not surprising, for it is in apparent conflict with the thesis that numbers are objects. Frege's ontology encompassed two ontological categories, function and object.' These ontological categories were mutually exclusive; no function was an object. And concepts were a type of function, namely those which took a single argument, yielding a truth-value. So no concept could be an object. Frege introduced quantifiers as certain second-level concepts, with no