Nathaniel Miller. Euclid and his twentieth century rivals: Diagrams in the logic of euclidean geometry. Csli studies in the theory and applications of diagrams
Philosophia Mathematica 16 (2):256-264 (2007)
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 the paradigm for careful and exact reasoning. How it could enjoy this reputation for so long is mysterious if careful and exact reasoning demands that all inferences be grounded in a modern axiomatic theory in the way Hilbert did in his famous Foundations of Geometry. By these standards, Euclid's work is deeply flawed. The holes in his arguments are not minor and excusable, but massive and cryptic.With his book Euclid and His Twentieth Century Rivals, Nathaniel Miller makes substantial progress in clearing this mystery up. The book is an explication of FG , a formal system of proof developed by Miller which reconstructs Euclid's deductions as essentially diagrammatic. The holes in Euclid's arguments are taken to appear precisely at those steps which are unintelligible without an accompanying geometric diagram. Interpreting the reasoning in the Elements in terms of a modern axiomatization , …
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
References found in this work BETA
No references found.
Citations of this work BETA
A Formal System for Euclid's Elements.Jeremy Avigad, Edward Dean & John Mumma - 2009 - Review of Symbolic Logic 2 (4):700--768.
Marcus Giaquinto. Visual Thinking in Mathematics: An Epistemological Study. [REVIEW]Jeremy Avigad - 2008 - Philosophia Mathematica 17 (1):95-108.
Similar books and articles
Diagrams in the Theory of Differential Equations (Eighteenth to Nineteenth Centuries).Dominique Tournès - 2012 - Synthese 186 (1):257-288.
Review of J. Norman, After Euclid: Visual Reasoning and the Epistemology of Diagrams. [REVIEW]F. Janet - 2007 - Philosophia Mathematica 15 (1):116-121.
Peirce and the Logical Status of Diagrams.Sun-Joo Shin - 1994 - History and Philosophy of Logic 15 (1):45-68.
A Brief Proof of the Full Completeness of Shin's Venn Diagram Proof System.Nathaniel Miller - 2006 - Journal of Philosophical Logic 35 (3):289 - 291.
Added to index2009-01-28
Total downloads50 ( #105,823 of 2,178,148 )
Recent downloads (6 months)3 ( #112,524 of 2,178,148 )
How can I increase my downloads?