The concept of given in Greek mathematics

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Abstract

This paper is a contribution to our understanding of the technical concept of given in Greek mathematical texts. By working through mathematical arguments by Menaechmus, Euclid, Apollonius, Heron and Ptolemy, I elucidate the meaning of given in various mathematical practices. I next show how the concept of given is related to the terms discussed by Marinus in his philosophical discussion of Euclid’s Data. I will argue that what is given does not simply exist, but can be unproblematically assumed or produced through some effective procedure. Arguments by givens are shown to be general claims about constructibility and computability. The claim that an object is given is related to our concept of an assignment—what is given is available in some uniquely determined, or determinable, way for future mathematical work.

Notes

Acknowledgements

The core ideas of this paper go back some years now to my dissertation, and I thank Alexander Jones and Jan Hogendijk for their comments on that work. I presented an overview of this argument at a conference of the SAW Project, under the direction of Karine Chemla. The discussion following this presentation helped me to clarify some of my thinking. During the time that I was a guest of the SAW Project in Paris, 2015, some of the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013) / ERC Grant Agreement No. 269804. Ken Saito read an earlier draft of this paper and made a number of valuable suggestions. This paper has benefited considerably from the extensive notes made by Karine Chemla and Matthieu Husson.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School for International Liberal StudiesWaseda UniversityTokyoJapan

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