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In retrospective of history of mathematics, I am trying to reconstruct the answers to following fundamental questions:

  1. Who proved first in a certain geometry that "The shortest distance between two points is a straight line."? Please provide proof or give precise reference thereto.
  2. Is the statement "The shortest distance between two points is a straight line." within an Euclidean geometry an axiom or a theorem?
  3. Does Euclidean geometry (and/or any other geometry) define (per axiom), what a "point", what a "line", and what a "straight line" to be?

Remarks:

  • The statement above could have been of course formulated over time in different ways, for instance instead of the notion "distance" I also recognized "path" and "curve".
  • Precise references, explanation and proofs needed, non-trivial answers appreciated.

Thank you in advance for your support.

Mauro ALLEGRANZA
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al-Hwarizmi
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  • I wonder if this question is more suited to https://hsm.stackexchange.com/ –  Nov 14 '21 at 09:19
  • @StinkingBishop Indeed, it would be interesting to set the first question also to there. However, I really aim at focusing on math and non-trivial answers. I know setting such questions in other forums rather would end up in foggy narratives and philosophical debates. The questions here are pure of mathematics and mathematical formalism. Please kindly be focused. – al-Hwarizmi Nov 14 '21 at 09:45
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    Triangular inequality (proved by Euclid in the Elements) is the first theorem that comes to mind. – Intelligenti pauca Nov 14 '21 at 15:14
  • Statement (1) isn't true in the taxi cab metric. – Michael Hoppe Nov 14 '21 at 20:07
  • I'm not well-versed in the history side of things, but here is a sort of intuitive proof: https://ibb.co/K9Jjy2S – TheBestMagician Nov 15 '21 at 04:04
  • In the original Euclid's Greek text there is no "distance". – Mauro ALLEGRANZA Nov 15 '21 at 07:47
  • "point" and "line" are undefined in the modern sense, because the definitions provided are quite useless and puzzling. According to scholars, they are probably later additions to the original text. – Mauro ALLEGRANZA Nov 15 '21 at 07:48

1 Answers1

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In his On the Sphere and Cylinder I, Archimedes gives some definitions, and then some assumptions. His first assumption reads (translated into English):

"Of all lines which have the same extremities the straight line is the least." [translated by Thomas Heath, emphasis as given by Heath]

For "extremities," read "endpoints"; for "least," read "shortest." The notion that a straight line is the shortest distance between two points is often ascribed to Archimedes on the basis of this assumption. Given that, I suppose we might call this an axiom, rather than a theorem, and therefore it would not admit of a proof.

In Euclidean geometry, terms such as "point" and "line" are often called "undefined," meaning that they are defined by their behavior in the axioms, rather than explicitly in some kind of glossary.

The idea of a "straight line" (as opposed to a line that is not straight) on the other hand, appears to me to have been assumed as obvious by many of the ancient Greeks. For instance, Archimedes (in the same work) uses "straight line" in a few of the definitions, but nowhere defines it or states what conditions it must satisfy. Euclid says in Definition 4 of his Elements that

"A straight line is a line which lies evenly with the points on itself." [again, translated by Thomas Heath]

I think you'll agree that this is not exactly satisfying as a definition.

Brian Tung
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    I think Archimedes assumption CAN be proved in the framework of Euclidean geometry, if the length of a curve is defined (as it was customary) as the limit of the length of a polygonal. By triangular inequality one can easily show that segment $AB$ is shorter than any other polygonal connecting $A$ to $B$. – Intelligenti pauca Nov 15 '21 at 14:56
  • @Intelligentipauca: That's a fair point—do you have any background on when and where such a definition might have been made? – Brian Tung Nov 15 '21 at 21:28
  • I don't know of any explicit case. But just think of the way Archimedes approximated the length of a circle by inscribed and circumscribed polygons to see that this was quite common. – Intelligenti pauca Nov 15 '21 at 23:16
  • @Intelligentipauca: Oh, I agree. I don't think they were in any confusion about the practical considerations behind measuring arc lengths (at least for "well-behaved" curves), but they probably did not have the same rigorous perspective that modern mathematics has. – Brian Tung Nov 15 '21 at 23:32
  • That's true, of course. But I think Archimedes (or some other ancient geometer) could have proved that assumption, starting with some definition for the length of a curve, if he thought that was worthwhile. The tools for doing that were all there. – Intelligenti pauca Nov 15 '21 at 23:45