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What Numbers are Real?

Published online by Cambridge University Press:  28 February 2022

Kenneth L. Manders
Kenneth L. Manders*
Affiliation:
University of Pittsburgh
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Extract

Perhaps one of the chief items of pride of mathematical philosophy in the last century and a half is the insight that mathematics is the science of formal structures; as opposed to the traditional view, that ‘the proper and exclusive subject matter of mathematics is…quantity.’ Closely associated with this insight is the distinction between pure mathematics, the beneficiary of the freedom conferred by the new status, and applied mathematics (in the philosopher's rather than the mathematician's sense of the word), which has been sent into philosophical limbo, supposedly under the care of philosophy of empirical science.

These two insights allow us to avoid many embarrassments. Our postulated freedom to adore gods other than Quantity allows us to break the seemingly endless circle of pointless debate, whether or not negative and complex numbers are legitimate quantities; our assumption that mathematical truth is independent of physical truth relieves us of the worry, whether Euclidean or non-Euclidean geometries are legitimate.

Type
Part VII. Mathematics
Information
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1986 , Issue 2: Volume Two: Symposia and Invited Papers 1986 , 1986 , pp. 253 - 269
Copyright
Copyright © 1987 by the Philosophy of Science Association

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Footnotes

1

I would like to thank H. Bos and W. Sieg for helpful conversations and for help on sources. I gratefully acknowledge financial support by NSF Grant SES 85-11229, and use of document production facilities at Carnegie-Mellon University. However, views expressed are exclusively the author's responsibility.

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