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I hear the notion of a point being the most primitive notion in geometry. But to talk about a point, one needs to think of a space of some sort. Only then, the point can be understood as a position within it. This also mirrors set theory nicely since space can be thought of as a set and a set is the primitive notion in set theory. But are there other problems that arise by taking space as the primitive notion in geometry?

  • I would suggest that what is primitive in set-theory is not a set but its background, the blank page on which the Venn diagram in drawn. Making sets fundamental to set theory is what Russell failed to do. – user20253 Mar 20 at 11:08
  • Thanks Peter, what I mean by a primitive notion is an object that can be taken for granted without appeal to another object in the theory. How did Russell fail to do that for sets? – csp2018 Mar 20 at 11:14
  • Well, I suppose we can take anything for granted as we wish. But it is not possible to reduce every set to one set. Sets must be reduced to the blank piece of paper. If we take for granted a primitive object we will have to call it a turtle and have a pile of them. Just as a point needs a space a set needs a background. The problem with taking space as primitive is that it is extended, thus implies time, division, multiplicity (of locations) etc. A metaphysical theory must reduce extension or remain non-reductive. I like your question because it raises these fundamental issues. . – user20253 Mar 20 at 11:29
  • Hmmm, how I'm thinking of it considers those implications of space as properties of space which feels inherent to space. To think of a point as a position feels like a non inherent property outside of the notion of a point which needs the object, space. Also, isn't the blank paper also a set within which we confine our discussion without discussing things outside of it? – csp2018 Mar 20 at 11:45
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    What one needs to think and understand is quite different from what one needs to talk formally. As you can see from Hilbert's axiomatization, one does not need any spaces to talk plane geometry, the primitive notions are only "lines" and "points" (plus three relations, incidence, betweenness and congruence), which can be renamed into "desks" and "chairs" for all it matters. This does mirror set theory, but not the way you think. Sets are analogs of points, not spaces, and the only primitive relation is element-of. – Conifold Mar 20 at 23:21
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There are many geometries. Most of us are familiar with the Euclidean, having Descartes' coordinate grid superimposed on it. This grid naturally leads us to think in terms of points. Set theorists are apt to follow that convention. But the real world is closer on a large scale to the Minkowski spacetime geometry of Relativity theory, modified on a minute scale by quantum theory - we doubt whether the idea of a zero-dimensional point can have any physical meaning.

In projective geometry, both point and line are equally primitive. A point may be defined as the set of lines which meet in it, known as a "pencil" of lines. Dually a line may be defined as the set of points which lie on it, known as a "range" of points. Ideas such as plane and space arise by allowing a point which is not on a given line, or dually a line which is not in a given point, and similar axioms. These dualities arise as a theorem of projective geometry, so when you have allowed or proved one statement you have also allowed or proved the dual statement. One can therefore derive a set-theoretic model of projective geometry using lines as a primitive instead of points.

The Euclidean and non-Euclidean geometries (elliptic, hyperbolic, affine) may be derived from projective geometry in various ways, but they all break projective duality, i.e. duality is not a theorem any more but just arises in certain circumstances.

So the answer depends very much on the particular geometry or class of geometries you are thinking of.

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