Algebraic number theory and the complex plane

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

I review Nagel’s model of theory reduction, with its projection of a collapse of two discourses into one, and then Jeremy Butterfield’s critique of it. He argues that in physics, definitional extension is both too strong and two weak to conform to Nagel’s strict criteria, and so blocks the collapse. I then argue that the same holds true for attempts at theory reduction in mathematics, by tracking the expositions of alleged reductions in a logic textbook by H. B. Enderton. I also review the proofs of a few of Fermat’s conjectures by means of the alliance of number theory with complex analysis, and show that these truly ampliative and explanatory proofs open up the study of the rational numbers to the study of algebraic number fields, an extension of number theory that is once too strong and too weak to look like Nagelian theory reduction, which is precisely why it turns out to be so fruitful.

Original languageEnglish (US)
Title of host publicationStudies in Applied Philosophy, Epistemology and Rational Ethics
PublisherSpringer International Publishing
Pages59-80
Number of pages22
DOIs
StatePublished - Jan 1 2016

Publication series

NameStudies in Applied Philosophy, Epistemology and Rational Ethics
Volume30
ISSN (Print)2192-6255
ISSN (Electronic)2192-6263

Fingerprint

Number Theory
Algebra
Alliances
Discourse
Exposition
Thomas Nagel
Mathematics
Logic
Fermat
Physics
Textbooks

All Science Journal Classification (ASJC) codes

  • Philosophy

Cite this

Grosholz, E. R. (2016). Algebraic number theory and the complex plane. In Studies in Applied Philosophy, Epistemology and Rational Ethics (pp. 59-80). (Studies in Applied Philosophy, Epistemology and Rational Ethics; Vol. 30). Springer International Publishing. https://doi.org/10.1007/978-3-319-46690-3_4
Grosholz, Emily Rolfe. / Algebraic number theory and the complex plane. Studies in Applied Philosophy, Epistemology and Rational Ethics. Springer International Publishing, 2016. pp. 59-80 (Studies in Applied Philosophy, Epistemology and Rational Ethics).
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Grosholz, ER 2016, Algebraic number theory and the complex plane. in Studies in Applied Philosophy, Epistemology and Rational Ethics. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol. 30, Springer International Publishing, pp. 59-80. https://doi.org/10.1007/978-3-319-46690-3_4

Algebraic number theory and the complex plane. / Grosholz, Emily Rolfe.

Studies in Applied Philosophy, Epistemology and Rational Ethics. Springer International Publishing, 2016. p. 59-80 (Studies in Applied Philosophy, Epistemology and Rational Ethics; Vol. 30).

Research output: Chapter in Book/Report/Conference proceedingChapter

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Grosholz ER. Algebraic number theory and the complex plane. In Studies in Applied Philosophy, Epistemology and Rational Ethics. Springer International Publishing. 2016. p. 59-80. (Studies in Applied Philosophy, Epistemology and Rational Ethics). https://doi.org/10.1007/978-3-319-46690-3_4