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Are there any essays on real numbers (in general?).

Specifically I want to learn more about:

  • The history of (the system of) numbers;

  • their philosophical significance through history;

  • any good essays on their use in physics and the problems of modeling a 'physical' line.

Cheers.

I left this vague as google only supplied Dedekind theory of numbers which was quite interesting but not really what I was hoping for.

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5 Answers 5

You might try consulting The World of Mathematics, edited by James R. Newman. This is a four-volume compendium of articles on various topics in mathematics. It was published in 1956 so is not exactly cutting-edge, but then again, neither is our understanding of the construction of the real numbers. It contains an essay by Dedekind himself, which is just part of a section of articles on the number concept.

There is also a book simply called Number by Tobias Dantzig that is a classic history of the number system.

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I'm not sure what you're looking for but try these books:

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The following books should cover your interests. Not all are strictly about real numbers, but are concerned with modeling the "physical continuum" or providing different foundations for analysis. A good starting point might be the entry Continuity and Infinitesimals in the Stanford Encyclopedia of Philosophy by J. L. Bell, who has lots of related articles on his homepage.

John L. Bell, The Continuous and the Infinitesimal in Mathematics and Philosophy

Philip Ehrlich, Real Numbers, Generalizations of the Reals, and Theories of Continua

Oliver, Deiser, Reelle Zahlen: Das klassische Kontinuum und die natürlichen Folgen (German)

Richard Dedekind, Essays on the Theory of Numbers

Hermann Weyl, The Continuum: A Critical Examination of the Foundation of Analysis

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A nice introduction to the history of mathematics is Morris Kline's Mathematics: the loss of certainty. He talks about the difficulties in characterising the real numbers. Also of interest might be one of the appendices to Imre Lakatos' Proofs and refutations where he discusses the history of the concept of continuity, and the difficulties it caused.

More philosophically, there is a paper by Douglas Gasking called (I think) Mathematics and the world from the Australasian Journal of Philosophy and a reply by Hector Neri-Castañeda in the same journal called Arithmetic and reality. These papers are not easy to find, however. The basic question at stake in these papers is whether, if we had developed different methods for measuring the world, whether our number system would have been different.

Finally there is a little book by Donald Gillies calle Frege, Dedekind and Peano on the foundations of arithmetic but this is more about integers than the reals.

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A fresh perspective on the real numbers is provided by Edward Nelson's approach called internal set theory. Here the background set theory is enriched through the introduction of a new unary predicate "standard". The result is that one can now identify a new type of real number which behaves like an infinitesimal, as well as real numbers that behave like infinite numbers. Nelson's article

Nelson, Edward (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society 83(6):1165–1198

was recently reprinted, also in the Bulletin of the American Mathematical Society; see http://www.ams.org/mathscinet-getitem?mr=2823017

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