Philosophia Mathematica Advance Access originally published online on April 1, 2008
Philosophia Mathematica 2008 16(2):256-264; doi:10.1093/philmat/nkn003
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Copyright © The Author 2008. Published by Oxford University Press.
Book Review |
NATHANIEL MILLER. Euclid and his Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry. CSLI Studies in the Theory and Applications of Diagrams
* Department of Philosophy, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A.
Correspondence: jmumma@andrew.cmu.edu
Nathaniel Miller. Euclid and his Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry. CSLI Studies in the Theory and Applications of Diagrams. Stanford, Calif.: CSLI Publications, 2007. ISBN 978-1-57586-508-9 (pbk); 978-1-57586-507-2 (hbk). Pp. viii + 119.
| The first 150 words of the full text of this article appear below. |
It is commonplace to view the rigor of the mathematics in Euclid's Elements in the way an experienced teacher views the work of an earnest beginner: respectable relative to an early stage of development, but ultimately flawed. Given the close connection in content between Euclid's Elements and high-school geometry classes, this is understandable. Euclid, it seems, never realized what everyone who moves beyond elementary geometry into more advanced mathematics is now customarily taught: a fully rigorous proof cannot rely on geometric intuition. In his arguments he seems to call illicitly upon our understanding of how objects like triangles and circles behave rather than grounding everything rigorously in axioms.
Though widespread, the attitude is in a historical sense puzzling. For over two millenia, mathematicians of all levels studied the arguments in Elements and found nothing substantial missing. The book, on the contrary, represented the limit of mathematical explicitness. It served as