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  1. A point is that of which there is no part.
  2. And a line is a length without breadth.$^1$

If above definition on point, expresses on point as to be indivisible length, as seems to be expressed in the question, Who originated the concept of making the point dimensionless?, or if it expresses point as to be nothing, is there any source which gives data on the reasoning to call point as either of them? If there are any reasons, what are those?

Similarly, is any source known on why line is thought to be of no breadth? If there are any reasons here, what are those?

Do these definitions allow attaining any intended utility? Were these definitions made to attain utility as quantifying space or measuring space (thus also having origin from culture, from the need for measurement of length?), and were these definitions the only choice? If yes, how?

It seems that, these definitions as to be not those which could be accepted directly (Would one define line as to be of no breadth directly?), and on whether they had their origins in culture or from pre large reasoning, from certain needs which might have made one (Euclid or Heron of Alexandria or the one who defined it) have line as to be breadth less, or point as to be indivisible (this seems a bit acceptable?) or nothing (this might need more reasoning?).


$^1$ Euclid’s Elements Of Geometry, The Greek text of J.L. Heiberg (1883–1885) from Euclidis Elementa, edidit et Latine interpretatus est I.L. Heiberg, in aedibus B.G. Teubneri, 1883–1885 edited, and provided with a modern English translation, by Richard Fitzpatrick

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  • $\begingroup$ I will add sources which I find here. $\endgroup$ – Immortal Player Mar 21 at 12:23
  • $\begingroup$ He posed that formulation because he was smart. Or, if you prefer, it's analogous to the "spherical cow with a uniform distribution of milk" model so well-known to physicists. Math is a language, not a manifestation of reality. $\endgroup$ – Carl Witthoft Mar 21 at 12:30
  • $\begingroup$ What you are saying may be a possibility, but it seems that we can't assert it as to be case here :) $\endgroup$ – Immortal Player Mar 21 at 12:33
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    $\begingroup$ I believe Heath's edition of the Elements has commentary on these (and other) items. But Immortal Player can look them up as well as I. $\endgroup$ – Gerald Edgar Mar 21 at 12:51
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    $\begingroup$ According to some scholars the definitions are later interpolation; see L.Russo, The Definitions of Fundamental Geometric Entities Contained in Book I of Euclid's Elements (AHES, 1998) : their source is presumabbly the Definitiones by Heron of Alexandria (c.10 AD – c.70 AD). $\endgroup$ – Mauro ALLEGRANZA Mar 21 at 14:35
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As mentioned in the comments, the definitions were probably inserted by later authors, like Heron, for didactic purposes and/or for compliance with Aristotelian ideas about proper scientific exposition (which starts with "intuitive definitions" that fix the subject matter). There is plenty of textual evidence to support this, e.g. Russo discusses it at length in The Definitions of Fundamental Geometric Entities Contained in Book I of Euclid’s Elements

"Many of the ideas here reported by Heron can be found in Aristotle’s writings. Among Aristotle’s passages on this subject we may quote, e.g., Phys., IV, 11, 220a 15 ff., where, among other things, points are said extremities of lines and the analogy between point and instant of time is introduced; De Cael., III, 1, 300a 14, where the above analogy is also considered; Met., V, 6, 1016b 24–30, where that which is indivisible and has position is called point. This definition of a point as a monad having position is in fact an ancient Pythagorean definition, as we know from Proclus (95, 22).

As another example, we may recall that the definition of lines as extremities of surfaces, which is contained in a passage of Heron’s definition 2 (corresponding to definition I, 6 of the Elements), is the Platonic definition which Aristotle had criticized in Topica, VI, 6, 143 b 11. In other instances Heron may have used later sources than Euclid: for example the definition of a straight line, as we shall see in detail in Sect. 10, seems to be drawn from Archimedes."

Another, not necessarily incompatible, speculation about the functionality of the Book I definitions is given by Azzouni in Proof and Ontology in Euclidean Mathematics. It has to do with Euclid's essential use of diagrams, and the dissimilarity of his approach to modern axiomatic method, where such definitions are of no use because nothing is (supposed to be) inferred from the diagrams. This would explain why modern mathematicians are usually perplexed by these definitions, given their, along with the diagrams', "lack" of mathematical utility.

On the other hand, if one is concerned with which diagrams are admissible for demonstrations then having "fat" points, for example, has to be explicitly proscribed. Indeed, Epicureans objected to Euclid's demonstration of proposition I.1 on the basis that dimensionless points do not exist, the circles intersect over a segment, and the proof fails.

"If these uses for diagrams are backtracked to ancient Greek mathematics, an analogous role can perhaps be found for their definitions of primitive terms as well: They are heuristic guides that give the student an intuitive grasp of otherwise difficult mathematical notions. Support for this view can be found in the fact that such definitions play no role in subsequent proofs. Unfortunately, Greek mathematicians took these definitions more seriously than the above interpretation can allow: Their constant tinkering with the definitions for straight lines, points, and so on, suggests they did not see them as heuristic.

Indeed, the same point applies to their diagrams: The Greek focus on whether mechanical methods for constructing curves were admissible suggests to some that (some) Greek mathematicians had constructivist scruples about mathematical objects... It's possible, however, that constructivist scruples (at least as they appear in the tradition leading to book I) were concerned with what sorts of diagrams are admissible in proofs, and not with mathematical objects at all... My conjecture (which only historians can decisively confirm or not) is that we see in the first book of Euclid's Elements a blend of a “pictorial” proof-system (in which diagrams are an essential part of the proofs), and a language-based proof-system (in which diagrams are merely heuristic).

Alas, it is doubtful that historians can decisively confirm or infirm such conjectures.

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  • $\begingroup$ On the other hand, Euclid's purpose could have been to initiate his project with common agreed-upon logical starting points. If diagrams helped with that, then include them. It's like the explanations of words in a dictionary; they're not definitions (because circular) but help with communicating a concept. I don't think we should force modern Hilbert-like axiomatic systems into the Greek way of reasoning. $\endgroup$ – Chrystomath Mar 25 at 7:06
  • $\begingroup$ @Chrystomath I agree. Axiomatic method, as we think of it, did not come about until late 19th century, due especially to Pasch and Hilbert in geometry. I think your view is close to Azzouni's, except that the starting points (and the reasoning itself) were not necessarily logical. Greeks distinguished between logical and geometric consequences in demonstrations, with diagrammatic inferences acknowledged as irreducible to logic. Hence Euclid did not need many of Hilbert's axioms. Definitions played a role in setting up a diagrammatic system, just as informal explanations of syntax do today. $\endgroup$ – Conifold Mar 25 at 7:24
  • $\begingroup$ Agreed. By logical I was referring to the postulates, and the need to define the terms. Euclid's work is clearly inferential in nature and thus had to have a logical starting point. One can argue that it is not really logical (that the very first theorem is flawed, etc.) but the intention seems to be there. $\endgroup$ – Chrystomath Mar 26 at 6:40
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About utility, there is a record of Euclid answering this question himself. One student asked him about utility of the whole of this business (axioms, definitions, theorems, etc.) Euclid called a slave and told him "Give this student a small coin and let him go. He is looking for a "utility" everywhere, this is a wrong place for him". (This is not an exact citation of what Euclid said, but reflects the essence).

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  • $\begingroup$ If Darwin were to have known this, he might have disliked Euclid (if Euclid had really said that)? :) It seems that only those with any need would stand out for later generations? $\endgroup$ – Immortal Player Mar 22 at 3:09
  • $\begingroup$ @Immortal Player: I am not sure what Darwin has to do with the question (could you explain?). Whether Euclid really said this, of course nobody knows. We only know about Euclid what the later authors wrote about him. $\endgroup$ – Alexandre Eremenko Mar 22 at 11:48
  • $\begingroup$ "Pay him three obols, for he must profit from what he learns" is not only not a direct quote, it is a late fable that reflects neo-Platonic attitudes more than those of Euclid. It is reported by Stobaeus, just as "no royal road to geometry" is reported by Proclus, both from 5th century AD, i.e. seven centuries after. $\endgroup$ – Conifold Mar 24 at 23:47
  • $\begingroup$ @Conifold: fable or not but it answers the question. $\endgroup$ – Alexandre Eremenko Mar 25 at 1:37

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