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I have a question on the implications resulting from the definitions of a point and line.

In Euclid Elements a point is “that which has no part” and a line is defined to be “breadthless length" with a straight line being a line "which lies evenly with the points on itself". It is also my understanding points are also breadthless.

Question If two lines cross how can there be an intersecting point? Both lines are breadthless (no width) so how can there be a point at the intersection of the lines in the center of this $X$?

If the lines have breadth I can understand that there is an area of intersection.

Nick
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    While I don't see an issue with two breadthless objects intersecting in an object which has no part (whatever any of those phrases mean), I would also suggest that you not take 2000 year old mathematics too literally. For one thing, there are likely issues of translation. For another, Euclid was not doing mathematics in anything like the modern formalism. Euclid's approach is much more... poetic. – Xander Henderson Sep 10 '21 at 20:52
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    @Math Lover I updated the OP. The combination of intersection and no area (breadthless) was confusing me. I was thinking of a line in the physical world space where I could see it (with breadth). As the point is 0-D, if the coordinates are the same then there is an intersection. – Nick Sep 10 '21 at 21:13
  • @Xander agreed on the translation. The real only way to be exact is reading original Greek, but I think even the Heidelberg Greek version was translated from Arabic. – Nick Sep 10 '21 at 21:25
  • You are confusing real world points and lines with abstract axiomatic versions of them. The Euclid quotes are not definitions of points and lines despite appearances to the contrary and I suggest you skip over them. They were at best a clumsy attempt to avoid exactly the trap you have fallen into. Non parallel lines do intersect in a unique point in any valid axiomatisation of Euclidean geometry. – Somos Sep 10 '21 at 22:40
  • Euclid's works first became known in Europe after the dark ages in translations from Arabic. However, transcripts of the original Greek texts did survive (see https://en.wikipedia.org/wiki/Euclid%27s). I think those Greek transcripts have been the predominant source since the 16th century. So go read the Greek: it's what Euclid wrote! – Rob Arthan Sep 10 '21 at 22:59
  • @XanderHenderson: The problem does not lie in the translation. I have looked at the original Greek before just to check, and true enough Euclid was making rather meaningless definitions and axioms, even though he did an incredible job for his time. – user21820 Sep 11 '21 at 18:37
  • Nick, do not rely on Euclid's elements if you want to learn proper Euclidean geometry. Euclid was the fore-runner, but his mathematics is objectively inferior to what we have today. Don't use square-wheels if you can get round ones. – user21820 Sep 11 '21 at 18:38
  • @RobArthan your link is dead. I searched for the original Greek edition, however could only find that Heidelberg version which it’s my understanding was translated from the Arabic version, perhaps even through Latin. Appreciate any link to the original Greek – Nick Sep 12 '21 at 01:36
  • @user21820 where is the original Greek version located? I searched but can’t find it – Nick Sep 12 '21 at 01:38
  • @user21820 Is the difference between Euclid’s system and modern system that the modern system relies on the Cartesian coordinate system with a set of elements (Real, Natural, etc)? It seems the first construct in Euclid is three points without a coordinate system, whereas modern expects three points over the square grid Cart system. So the modern system has an additional dependency to function. – Nick Sep 12 '21 at 01:44
  • No. Modern Euclidean geometry does not rely on a coordinate system. If you don't know what it is, ask instead of assuming. Even wikipedia (which is often a bad place to learn things) says "Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more extensive and complete sets of axioms." and "Placing Euclidean geometry on a solid axiomatic basis was a preoccupation of mathematicians for centuries" and even has a section titled "Axiomatic formulations" mentioning Hilbert's and Tarski's. – user21820 Sep 12 '21 at 03:03
  • And I don't see any point in reading the original Greek until you have a proper foundation in basic logic (i.e. first-order logic with a practical deductive system), because it's simply impossible to have a clear grasp of what is missing until you actually know the proper basics. – user21820 Sep 12 '21 at 03:05
  • @user21820 I have read Frege and Bourbaki, so understand at a logical level everything is either constructed from logical implication or not or, which are equivalent. – Nick Sep 12 '21 at 15:12
  • Ok, but do you know a "practical deductive system" like I said above? If you don't know what that means, see this one. The reason I emphasize this is because it is hard to understand true logical rigour until you actually have seen a fully rigorous deductive system. Anyway, you can read the (likely) original Greek and a translation here. – user21820 Sep 12 '21 at 15:22
  • @user21820 most of my logic is based on Frege, in which the deductive system includes a deductive function, taking as an argument what to deduct, which leads to more true forms, which can also be deducted from. The Greek version you pointed to is included in Heath’s side by side translation. I read the Latin version was translated from Arabic. Hopefully the Greek included in Heath’s version is original. – Nick Sep 12 '21 at 16:22
  • @Nick: Well it's likely to be close to the original. Since you know Frege's system, at least you can see for yourself that my first comment here is correct (see also this). The key point is to realize that Euclid did not have much beyond propositional logic, and even the idea of using symbols in the manner of FOL is a very modern advancement. So Euclid did a good job with what little he had, but you just cannot do any rigorous Euclidean geometry unless you use a modern axiom system as I mentioned earlier. – user21820 Sep 12 '21 at 17:37
  • Apologies for the botched link. Try this: https://en.wikipedia.org/wiki/Euclid%27s_Elements. The Heiberg manuscript was made in Byzantium, but was not a translation from Arabic. PS: if you want to learn modern mathematics rather than mathematical history, then I agree with what others have said about not reading Euclid. The history is, however, fascinating. – Rob Arthan Sep 13 '21 at 22:21

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Points are also breadthless. So there is no problem. Two breadthless objects (lines) intersect into an even "smaller" breadthless object (point).

By "smaller" I mean that lines are 1-dimensional and points are of course 0-dimensional.

This is an informal answer because your question is also kind of informal.

peter.petrov
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