Sunday, 17 July 2011

Euclid's fifth axiom?

I'm (re)reading a book by one of my favourite authors, David Berlinski, called Infinite Ascent: a short history of mathematics, where he's talking about Euclidean geometry and he presents Euclid's fifth axiom on parallel lines as a "worm" in an otherwise unassailable system:

"There remains the fifth axiom of Eulcid's system, and with the fifth, that worm. It is a worm that may now be seen wriggling in words due to the eighteenth-century Scottish mathematician John Playfair:

5. Through a point outside a given line, one and only one line may be drawn parallel to that line."

Now, history has given us Riemann and Lobachevsky and their great works on non-Euclidean geometries that provide an answer in the negative to the fifth: there are no parallel lines in spherical and hyperbolic geometries (respectively). But these geometries describe space that is not flat.

Berlinski again: "One straight line; one exterior point; and only one line through the point and parallel to the given line. Yet the picture corresponding to the parallel postulate does not cancel a sense of mathematical unease. In some very obscure way, the axiom contains an assumption that it does not entirely succeed in conveying. Parallel lines and a point in space - clear enough. And the picture that results - clear enough as well. There is yet something odd and unresolved about the picture of those jaunty parallel lines, its visual plausibility depending entirely on the assumption that the space in which they are embedded is flat."

Here is a graphic taken from Wikipedia entry on non-Euclidean geometry to illustrate:

I have a little table saw to cut balsa wood and it has, like other regular table saws, a slide that is inserted parallel to the blade to make cuts of any angle. Now, the slide on my little table saw has an adjustable compass with the zero mark on top of the semi-circle and going 90 degrees either way to the base of the semi-circle (much like the middle figure above, Euclidean-wise). There is only one point on the compass where a parallel cut can be made and that is the 90 degree mark - every other angle than 90 either way would eventually result in an intersection to the line otherwise.

Now, following the tradition of the Greeks, the construction of plane-geometric shapes with straight lines and curves using only a compass and a straight-edge, I would say that the construction of parallel lines is more a theorem derivable from the first four postulates of Euclid because it (the parallel line) is constructed only from the combination of the more primative elements of straight lines and circles - ie, the parallel line is not a self-evident statement but a logical consequence of the first four postulates regarding points, lines and curves.

It is still beautiful nonetheless - the fifth statement - for it, like the other postulates, has spawned great insights into the nature of space and its various descriptions which would not have come about in so natural a way without it. It is not so much a "worm", I'd say; but - again, perverting Berlinski's beautiful imageries of his excellent book - it is the forked tongue of the devil with powerfully pregnant possibilities embedded in its use.


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