Veronese’s geometry

There is a geometry little known today, developed by G. Veronese in the 19th century, that avoids the aforementioned inconsistency by design. The idea is that the continuum, e.g. a straight line, or the 3-dimensional space, is not composed of points.(1) The points do belong to the continuum though, but are not part of it. Just like the graph of a computer network: it belongs to the network, but is physically not part of it.

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Fig. 1: Point of separation.

To illustrate the idea, imagine that Fig. 1 depicts a geometrical straight line. It consists of two disjoint, touching half-lines: one is of green color, the other blue. The point is the “thing” in the middle that marks where the two are separated. Clearly, the point belongs to both half-lines, since it marks where they end. But is it also part of them? If it was part of one, then, due to the symmetry of the situation, it would be part of the other too. With that, the green and the blue half-lines could not be disjoint. Thus, the point is not part of either half-line, and as such it’s not part of the line either.

   Book 1      Book 2   
   Book 1      Book 2   
   Book 1      Book 2   
   Book 1      Book 2   
   Book 1      Book 2   
   Book 1      Book 2   
   Book 1      Book 2   
   Book 1      Book 2   
   Book 1      Book 2   
   Book 1      Book 2   
   Book 1      Book 2   

Fig. 2: Two books touching each other.

In general, the contour of a geometrical object belongs to the object, but is not part of it. Fig. 2 shows another illustrative example, where two planar books touch each other. Again, the straight line segment in the middle (would be a rectangular surface in 3 dimensions) that marks the separation between the books is not part of either book, nor it is part of the plane in which the books “live”. But it does belong to them!

In Veronese’s geometry, the continuum in question is investigated by means of a superimposed system of all points. The underlying assumption is that differences among geometrical objects manifest themselves through differences in the respective collections of points that belong to each object. That is, a geometrical object is unambiguously determined, and thus can be defined, by its points.(2)

The superimposed system of all points must be dense enough, so that applying geometrical operations cannot lead out of it. For example, ℤ3 would not be appropriate in a 3-dimensional continuum because, among other shortcomings, the straight line that passes through (0, 0, 0) and (100, 1, 0) would not intersect in ℤ3 any of the straight lines that pass through (n, 0, 0) and (n, 1, 0), where n is an integer between 1 and 99.

The ordinary analytic form of Euclidean geometry would correspond to the case where the system of all points is ℝ3, which is already dense enough. However, Veronese goes beyond that and postulates a 3-dimensional (number) system that includes infinitesimally small as well as infinitely large quantities. He also provides a method by which similar systems can be constructed for continua of up to infinitely many dimensions.

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(1) Composed of points or not? Who said what?
Hilbert
The elements of Hilbert’s geometry are points, straight lines, and planes. Any other geometrical object is defined in terms of such elements. For example, the segment (of a straight line) is defined as a “system” of two (end-)points, and the circle as a “totality” of points. Obviously, Hilbert’s goal was logical soundness rather than coherent storytelling.
Poincaré
“There is supposed to be, between the elements of the continuum, a sort of intimate bond which makes a whole of them, in which the point is not prior to the line, but the line to the point.”
Euclid
In its original form, Euclid’s geometry presents a more appealing story than that of Hilbert, albeit logically less sound. The segment and the circle are both defined as “lines”, which is in accordance with intuition. The question whether lines, surfaces, and solids are composed of points is left open.

(2) What is a geometrical object and when does a point belong to it?
A geometrical object is part of either the continuum in question or a lower-dimensional sub-continuum thereof. A point that belongs to it marks where the object or a part of it may touch another geometrical object. To be able to define geometrical objects by their points, it is important to postulate criteria as to when a collection of points belongs exactly to a part of a (sub-)continuum and when it does not.

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