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Did Plato or anyone else discuss/define the geometric point? I know Pythagoras discussed the math point.

I read that Euclid's definition (with no part) is a mistake in translation from the original text that said (without any parts= indivisible) is that true?

Can you report the original text(s)?

  • "I read that Euclid's definition (with no part) is a mistake in translation"; can you give us the source of this information, please ? – Mauro ALLEGRANZA Sep 16 '18 at 13:37
  • 2
    "Infinitesimal: how a dangerous mathematical theory shaped the modern world", by Amir R. Alexander, 2014. My local library had this as an Ebook. – Gordon Sep 16 '18 at 16:51
  • @MauroALLEGRANZA Euclid's "definitions" are likely late interpolations, possibly borrowed from Heron, see Russo The Definitions of Fundamental Geometric Entities Contained in Book I of Euclid's Elements. – Conifold Sep 16 '18 at 21:36
  • "interpolation" does not mean "mistake in translation". Translation to Greek from what ? – Mauro ALLEGRANZA Sep 17 '18 at 13:30
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There are no Pythagoras' texts extant.

We have the ancient Pythagorean tradition; a source is Proclus's Commentary on Euclid's Elements, Book I :

the Pythagoreans define the point as a unit that has position.

A notable ancient Pythagorean was Archytas of Tarentum; see :

First, we must take into account that :

No list of Archytas’ works has come down to us from antiquity, so that we don’t know how many books he wrote. In the face of the large mass of spurious works, it is disappointing that only a few fragments of genuine works have survived.

See the English transaltion of Archytas’ Fragments.

Note: see Diels–Kranz numbering :

47Bn means the n-th fragment of Archytas : "B" stays for Ipsissima Verba: Literally translated to "exact words", and sometimes also termed "fragments", these are items containing exact words of the author in the form of quotations in later works.

47Am means the m-th text regarding Archytas : "A" stays for Testimonia: These are accounts of the authors' life and doctrines. Testimonia include commentaries on the works of the pre-Socratics and accounts of their lives and of their philosophical views.


The relevant source is 47A22, that is Arist.,Meta VIII. 2 (1043a14–26), where Aristotle comments about Archythas theory of definitions :

The same is true of the kind of definitions which Archytas used to accept; for they are definitions of the combined matter and form. E.g., what is "windlessness?" Stillness in a large extent of air; for the air is the matter, and the stillness is the actuality and substance.What is a calm? Levelness of sea. The sea is the material substrate, and the levelness is the actuality or form.

This reference points to Top, I,18 (108b25) :

Likewise, also, in the case of objects widely divergent, the examination of likeness is useful for purposes of definition, e.g. the sameness of a calm at sea, and windlessness in the air (each being a form of rest), and of a point on a line and the unit in number-each being a starting point. If, then, we render as the genus what is common to all the cases, we shall get the credit of defining not inappropriately. Definition-mongers too nearly always render them in this way: they declare the unit to be the startingpoint of number, and the point the startingpoint of a line.

See Huffman, page 499-500 :

There is reason to believe that Aristotle and Alexander may preserve even more definitions by Archytas, in passages where Archytas is not mentioned by name. Scholars have not commonly connected Aristotle’s discussions of Archytas’ definitions of windlessness and calm-on-the-ocean in the Metaphysics with his further discussion of the definitions of these terms in two passages of the Topics.

According to Aristotle, this is just what “those who give definitions” in fact do, for they say “the unit is the starting point of number and the point is the starting point of a line” Are these Archytas’ definitions of unit and point?

That these should be Archytas’ definitions of point and unit contradicts a common assumption (e.g. Heath 1925: I. 155) that the Pythagoreans defined the point as “a unit having position”. Of course, even if we accepted that this was the common early Pythagorean definition of the point, it is perfectly possible that Archytas, here, as elsewhere, represents a further development in that tradition. It is far from clear, however, that the definition of the point as “a unit having position” should be ascribed to the Pythagoreans. It is the definition that Aristotle himself adopts, and he uses it in contexts that suggest that it is connected to the early Academy; nowhere does he ascribe this definition to the Pythagoreans, however.

There is one important piece of external evidence that provides at least partial support for assigning the definitions of the point as “the starting point of the line” and the unit as “the starting point of number” to Archytas. It is precisely these definitions that appear in Nicomachus’ *Introduction to Arithmeticé (ii. 7; see Heath 1921: I. 69). Nicomachus’ treatise is the most important extant treatise on arithmetic in the Pythagorean tradition, and that treatise features a prominent quotation from Archytas’ Harmonics, so that it would not be surprising, if Nicomachus also adopted Archytas’ definitions of the point and unit, although he does not explicitly assign them to any particular thinker.

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