Over the years I read different versions of how the point in geometry (and subsequently in maths) came to be defined as an abstract, dimensionless entity.
I read that it was Architas who influenced Euclid, but also that it was a mistake made by translators of Plato, rendering 'the point hasn't any parts' with the wrong "the point has no part".
Can you give a detailed account of how the debate evolved and who was actually responsible for the definitive decision?
See Sir Thomas Heath's edition of The Thirteen Books of Euclid's Elements, Volume 1, Book I, Definitions, page 155-on.
"A point is that which has no part."
Pre-Euclidean definitions.
It would appear that this was not the definition given in earlier textbooks; for Aristotle (Topics VI. 4,141 b 20), in speaking of "the definitions" of point, line, and surface, says that they all define the prior by means of the posterior, a point as an extremity of a line, a line of a surface, and a surface of a solid. The first definition of a point of which we hear is that given by the Pythagoreans (cf. Proclus, p. 95, 21); who defined it as a "monad having position" or "with position added". It is frequently used by Aristotle, either in this exact form (De anima I. 4, 409 a 6) or its equivalent: e.g. in Metaph. 1016 b 24 he says that that which is indivisible every way in respect of magnitude and qua magnitude but has not position is a monad, while that which is similarly indivisible and has position is a point.
Aristotle's conception of a point as that which is indivisible and has position is further illustrated by such observations as that a point is not a body (De caelo II. 13, 296 a 17) and has no weight (ibid. III. 1, 299 a 30); again, we can make no distinction between'a point and the place where it is (Physics IV. I, 209 a II).
Euclid's definition itself is of course practically the same as that which Aristotle's frequent allusions show to have been then current, except that it omits to say that the point must have position. [...] The definition has been over and over again criticised because it is purely negative.
Simplicius (quoted by an-Nairizi) says that "a point is the beginning of magnitudes and that from which they grow; it is also the only thing which, having position, is not divisible." He, like Aristotle, adds that it is by its motion that a point can generate a magnitude: the particular magnitude can only be "of one dimension," viz. a line, since the point does not "spread itself". Simplicius further observes that Euclid defined a point negatively because it was arrived at by detaching surface from body, line from surface, and finally point from line. "Since then body has three dimensions it follows that a point [arrived at after successively eliminating all three dimensions] has none if the dimensions, and has no part."
On Archytas of Tarentum, see :
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.
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.
We can see also Archytas as Mathematician.
We can see also Lucio Russo, The Definitions of Fundamental Geometric Entities Contained in Book I of Euclid's Elements, AHES (1998), for the interesting thesis that Euclid's definitions has been interpolated in late antiquity into Euclid’s treatise and their source is the Definitiones by Heron of Alexandria (c.10 AD – c.70 AD).
