Rethinking geometrical exactness

@article{Panza2011RethinkingGE,
  title={Rethinking geometrical exactness},
  author={Marco Panza},
  journal={Fuel and Energy Abstracts},
  year={2011}
}
  • M. Panza
  • Published 1 February 2011
  • Philosophy
  • Fuel and Energy Abstracts
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Axiomatizing Changing Conceptions of the Geometric Continuum I: Euclid-Hilbert
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Constructibility and Geometry
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It is argued that geometry is essentially characterized by hypothetical and potential constructions: geometrical objects are not effectively constructed, but they are only constructible.
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References

SHOWING 1-10 OF 49 REFERENCES
Shifting the foundations: Descartes's transformation of ancient geometry
The twofold role of diagrams in Euclid’s plane geometry
TLDR
The purpose is to reformulate the thesis that many of Euclid’s geometric arguments are diagram-based in a quite general way, by describing what he takes to be the twofold role that diagrams play in Euclid's plane geometry (EPG).
The Algebra of Geometric Impossibility: Descartes and Montucla on the Impossibility of the Duplication of the Cube and the Trisection of the Angle
Today we credit Pierre Wantzel with the first proof (1837) of the impossibility of doubling a cube and trisecting an arbitrary angle by ruler and compass. However two centuries earlier Descartes had
Descartes and the cylindrical helix
Descartes and Mathematics
Descartes is one of the few geniuses in history who was able to bequeath epoch-making contributions to philosophy and to mathematics at the same time. In addition, his contributions to these areas
The Concept of Existence and the Role of Constructions in Euclid's Elements
Abstract This paper examines the widely accepted contention that geometrical constructions serve in Greek mathematics as proofs of the existence of the constructed figures. In particular, I consider
The Works of Archimedes
THIS is a companion volume to Dr. T. L. Heath's valuable edition of the “Treatise on Conic Sections” by Apollonius of Perga, and the same patience, learning and skill which have turned the latter
Mathematical Thought and Its Objects: Sources and Copyright Acknowledgments
In Mathematical Thought and Its Objects, Charles Parsons examines the notion of object, with the aim of navigating between nominalism, which denies that distinctively mathematical objects exist, and
Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century
TLDR
Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period with a sophisticated picture of the subtle dependencies between technical development and philosophical reflection.
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