Geometry, in its original and most basic sense, is the study of distance relationships in the physical world. The concepts of point and straight line are what geometry starts with, but geometric reasoning itself does not tell us *which* line between two given points is a straight line. Whether we define straight lines ostensively, as was necessary in Ancient Greece, or we define them by reference to light rays, as some do today, the concept of straight line is a foundation block of and, therefore, must precede all other geometric analysis.
This is why the basic properties of straight lines constitute the postulates, not the theorems, of geometry. Such postulates are not self-evident primaries. They are inductions from experience in the physical world. If, for instance, we were to construct a very large triangle and measure the sum of its interior angles to be greater than 180 degrees, we cannot just reject the result out of hand because it violates Euclid's postulates. We might have to question these postulates.
If we do in fact define straight lines using light rays, we find that certain triangles wouldn't add to 180 degrees, depending on their proximity to massive objects. This is what is meant by the idea that the universe is non-Euclidean. It is ultimately a matter of physics, because physics is the study of external objects in general (including those used to define straight lines).
Nevertheless, based on the totality of the physical evidence (including observations of quantum non-locality and also subtler indications provided by modern quantum field theories), I do not personally believe that the universe is non-Euclidean. It looks more likely that using light to define straight lines is defective. But this is, again, a question of physics -- not a question to be resolved by scrutinizing standard geometrical concepts.
Thursday, February 7, 2008
Subscribe to:
Post Comments (Atom)
2 comments:
You do not use light to define straight lines: you define a straight line as a geodesic, the minimum distance beetwen two points; and the light always follows the geodesic, so is the easier instrument you can use to find this geodesic/straight line.
There is little doubt about the non-euclidity of the Universe (at least at a local level): General Relativity is today one of the most highly confirmed theories in Science.
Your is a very interesting blog. I'll keep on reading :)
But how do you define the length of a line connecting two points? As the sum of infinitesimal line elements along the line. And what is the length of each line element? Roughly, the time it takes light to traverse the line element.
I doubt that the physical world is actually non-Euclidean in a fundamental way because EPR/Bell experiments may, in principle, make possible a determination of absolute simultaneity. And they definitely demonstrate faster-than-light causation, which indicates that Lorentz symmetry is probably emergent, not a fundamental symmetry. But this takes us into a set of different issues.
Post a Comment