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When in history did the notion of space, geometric space appear? I. e. when in history geometric space was treated or thought of as a whole, as the site in which all geometric objects exist? When I think of Euclid's Elements, I have the impression that it just treats relations between segments, areas; straight lines, planes (?), but doesn't treat space as a whole, as the site in which all geometric objects exist. (I am assuming the whole content in Euclid's Elements is what I said it is, but I have never read it thoroughly.)

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    $\begingroup$ IMO you are right: there is no definition of "space" (maybe neither use of the word) in the text of the Elements: we may say that for Euclid geometry is the study of geometrical figures. Having said that, what do you mean with "treat space as a whole" ? $\endgroup$ Jun 21 at 9:05
  • $\begingroup$ If by a "space" you mean a "set" then the answer is "early 20th century," either Tarski or Birkhoff. $\endgroup$ Jun 21 at 12:35
  • $\begingroup$ When I say "space as a whole", I mean an object by itsel, an entity by itself, a totality, and when considered like that, we can think of it as an object having properties, or a structure corresponding to itself not to its components. For example, natural numbers were not considered as a whole, as a totality before set theory, and when considered like that, we can think of it having a structure: it is a monoid, for exaemple. Now as to a geometric space, I have the impression that a geometric space, or the space, was considered a totality before set theory... $\endgroup$ Jun 21 at 19:38
  • $\begingroup$ ... because Postulate 5 was intensely studied, and this might lead us to think of it not as a propierty of lines but of space. $\endgroup$ Jun 21 at 19:40
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It is the opposite: for Euclid there is nothing but the (2- or 3-)space. All the other things (points, lines, planes, etc.) live there.

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  • $\begingroup$ Any (translated) quote in Euclid's Elements to see that? $\endgroup$ Jun 20 at 15:18
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    $\begingroup$ @QuiqueRuiz: If you read the book, you can see that clearly. It would be instructive if you can provide a quote showing the opposite. $\endgroup$
    – Mark Sapir
    Jun 20 at 18:35
  • $\begingroup$ Your answer surprises me, because Greeks tried to avoid infinity, and if one considers a space, a geometric space as a whole, one must think of if as being infinite. I know it is considered that the infinitude of prime numbers is proved in Euclid's Elements, but one can say that what is proved is that if one has a finite list of prime numbres, one can find a new prime number which is not in the list. That's why I asked when in history a geometric space was treated as a whole. Maybe I should say "studied". $\endgroup$ Jun 20 at 21:27
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    $\begingroup$ @QuiqueRuiz: Euclid does not claim infinitude of the 2- or 3-space anywhere except in Postulate 5: that a line can be continued indefinitely. That is why Postulate 5 was considered suspicious. In the definitions of the circle, he is talking about "plane figures". But there is no doubt, that points and lines live in a plane too. $\endgroup$
    – Mark Sapir
    Jun 20 at 21:45
  • $\begingroup$ @QuiqueRuiz: One can claim that Euclid studies the plane and the 3-space in Elements. But he never claims it himself. $\endgroup$
    – Mark Sapir
    Jun 20 at 21:48
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In general, Euclid's definitions talk about points and lines having a relationship to a plane but he does not say they need a plane in order to live. The only line he says needs a plane is the circle.

I think he had a geometric conception of space as a whole, but it differs from the modern conception of space where a plane or surface are needed for points or lines to live. The latter are called manifolds, but Euclidean geometry, when understood properly, is not a flat manifold because it is not a manifold.

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