Euclid’s Common Notions and the Theory of Equivalence

Abstract

The "common notions" prefacing the Elements of Euclid are a very peculiar set of axioms, and their authenticity, as well as their actual role in the demonstrations, have been object of debate. In the first part of this essay, I offer a survey of the evidence for the authenticity of the common notions, and conclude that only three of them are likely to have been in place at the times of Euclid, whereas others were added in Late Antiquity. In the second part of the essay, I consider the meaning and uses of the common notions in Greek mathematics, and argue that they were originally conceived in order to axiomatize a theory of equivalence in geometry. I also claim that two interpolated common notions responded to different epistemic needs and regulated diagrammatic inferences. §1. IntroductionThe Elements of Euclid are introduced by three sets of principles: definitions, postulates and common notions. The number of common notions (κοιναὶ ἔννοιαι) varies in different Greek, Arabic and Latin manuscripts. Most of these manuscripts comprise either nine or ten of them. Ancient commentators inform us that still further principles had been suggested in late antiquity but never found their way into the text of the Elements. 1 Following the textual indications of these commentaries and other philological considerations, Johan Ludvig Heiberg produced a critical edition of the Elements containing five of these common notions: CN1: Things which are equal to the same thing are also equal to one another. CN2: If equals be added to equals, the wholes are equal. CN3: If equals be subtracted from equals, the remainders are equal. CN4: Things which coincide with one another are equal to one another. CN5: The whole is greater than the part.This list is accepted nowadays by most historians of mathematics, all of whom agree with Heiberg's excision of many other principles from the text. Nonetheless, a few authoritative Aknowlwdgments. I thank Gregg de Young, Eduardo Giovannini, Marco Panza, and an anonymous referee, for their insightful remarks. I am especially grateful to Mattia Mantovani, whose many suggestions and relentless requests of better evidence substantially improved the final version of this essay. 1 For a first appraisal of the number of common notions in various manuscripts and Euclidean editions, see my V. DE RISI, The Development of Euclidean Axiomatics. The systems of principles and the foundations of mathematics in editions of the Elements from Antiquity to the Eighteenth Century, "Archive for History of Exact Sciences", 70 (2016), pp. 591-676.

Euclid After Computer Proof-checking

“…We must add the required infrastructure, but it won't matter exactly how we do it, so we do not go further into the matter here. 10 If five betweenness axioms had been required instead of three, we would have just added five betweenness axioms.…”

“…The name of the lemma is taken from Euclid's Common Notion 5; but Euclid does not cite CN5, in I.1 or anywhere else, and De Risi[10] after careful study reaches the conclusion that the original Euclid had only the first three of the five common notions given in the Heath translation. It seems, however, that he should have had CN5, and should have cited it in I.1 18.…”

Poincaré on the value of reasoning machines

“…Historians debate whether Euclid's Elements originally meant to give strict logical deductions from axioms[8,21]. That is outside our topic 5.…”

Decoding Book II of the Elements

“…See, for example,(Mueller, 2006), or (De Risi, 2021). 13 Fig.15is taken from Van der Waerden's (Van der Waerden, 1961) and illustrates his interpretation of Euclid's II.14.…”