Uses of construction in problems and theorems in Euclid’s Elements I–VI

Sidoli, Nathan

doi:10.1007/s00407-018-0212-4

Cited by 11 publications

(5 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Many have pointed to this as evidence that Euclid's constructions do not correspond to actual ruler and compasses, since no actual compasses are collapsible (Sidoli, 2018b, 422, Acerbi, 2010b. This conclusion clashes with the operationalist reading of geometry.…”

Section: The "Collapsible" Compassmentioning

confidence: 99%

Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry

Blåsjö

2021

Found Sci

View full text Add to dashboard Cite

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid (implicit diagrammatic reasoning, superposition) can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised.

show abstract

Section: The "Collapsible" Compassmentioning

confidence: 99%

Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry

Blåsjö

2021

Found Sci

View full text Add to dashboard Cite

show abstract

“…This means essentially Postulates 1-3 and 5 of Book I, with some minor accommodations for informal assumptions that Euclid does not articulate explicitly. See, e.g., [Sidoli 2018], where, in particular, Euclid's postulates (419-424) and problems (426-431) are both interpreted as functions introducing certain geometrical entities.…”

Section: Analysis As Justification Of Meansmentioning

confidence: 99%

Rigorous Purposes of Analysis in Greek Geometry

Blåsjö¹

2021

philosophiascientiae

View full text Add to dashboard Cite

Dans la géométrie grecque, l'analyse est traditionnellement vue comme une technique heuristique. Cependant, de nombreuses analyses dans les traités mathématiques anciens sont difficiles à expliquer en ces termes. Cet article vise à montrer que l'analyse géométrique grecque peut aussi servir des enjeux mathématiques déterminés, qui, sans son apport, rendraient la synthèse incomplète. D'abord, lorsque la solution d'un problème est précédée par une analyse, celle-ci démontre rigoureusement qu'il n'y a pas d'autres solutions au problème que celles offertes dans la synthèse. Ensuite, chaque fois qu'on est en présence d'une hypothèse de construction qui dépasse l'utilisation de la règle et du compas, l'analyse démontre que le problème n'est pas seulement résolvable à partir de cette assomption, mais qu'il est en fait rigoureusement équivalent à elle.

show abstract

“…Euclid seems never to question the universal replicability of his constructions, and many scholars have advanced hypotheses on the role of constructive schemata in the process of universal generalization of the theorems. The most refined and subtle attempts towards a constructivist reading of the Elements have been made by Ferreirós (2016, 130), Sidoli (2018b, 434), and Blåsjö (2022).…”

mentioning

confidence: 99%

“… 40 The thesis was argued in Sidoli’s important paper (Sidoli 2018b, 422–23), which also offers in footnote 45 a subtle argument on how to transform the proof of Elements I, 46 in order to make P4 unnecessary in it, even though the procedure cannot be generalized (the footnote also claims, I think wrongly, that Elements I, 14 and 15 do not need P4). For a fuller treatment of the notion of givenness, see also Sidoli 2018a.…”

mentioning

confidence: 99%

Euclid’s Fourth Postulate: Its authenticity and significance for the foundations of Greek mathematics

De Risi

2022

Sci Context

View full text Add to dashboard Cite

Argument The Fourth Postulate of Euclid’s Elements states that all right angles are equal. This principle has always been considered problematic in the deductive economy of the treatise, and even the ancient interpreters were confused about its mathematical role and its epistemological status. The present essay reconsiders the ancient testimonies on the Fourth Postulate, showing that there is no certain evidence for its authenticity, nor for its spuriousness. The paper also considers modern mathematical interpretations of this postulate, pointing out various anachronisms. It further discusses the validity of the ancient proof by superposition of the Fourth Postulate. Finally, the article proposes an interpretation of the history of the concept of angle in Greek geometry between Euclid and Apollonius, and puts forward a conjecture on the interpolation of the Fourth Postulate in the Hellenistic age. The essay contributes to a general reassessment of the axiomatic foundations of ancient mathematics.

show abstract

Uses of construction in problems and theorems in Euclid’s Elements I–VI

Cited by 11 publications

References 27 publications

Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry

Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry

Rigorous Purposes of Analysis in Greek Geometry

Euclid’s Fourth Postulate: Its authenticity and significance for the foundations of Greek mathematics

Contact Info

Product

Resources

About