Parallel lines

We continue to wonder what good can come of trying to do what can’t be done.

Last week we suggested that a useful book might be written on what positive results have come out of attempts to do the mathematically impossible.  We haven’t the background in math, or the history of math, to attempt it ourselves, and freely pass on the idea to anyone so equipped.  However, we do know of several long-standing problems that provoked much activity, and in the end turned out to be impossible.  One of these is the Parallel Postulate.

Ancient Greek mathematics originated the idea of the rigorous proof, that is, the deduction of things that must be true from a limited number of assumptions.  In this it differed from those of other cultures, which were geared toward producing results that were close enough for practical purposes.  It was epitomized by the work of Euclid, who set out in his Elements thirteen volumes of results, many of them far from obvious, based on five postulates.  Four are relatively simple and intuitive.  The last, the Parallel Postulate, has been troubling since the beginning.  It says, in a modern formulation: “Given a line and a point not on that line.  There is one and only one line through the point parallel to the given line.”  (Parallel means the two lines never meet.)

Modern geometry students have to learn quite a few more postulates than five.  Mostly this is to save time.  Almost all of them can actually be proved from previous postulates, but the proofs are often long and tedious.  For over two thousand years a large fraction of working geometers thought that the Fifth Postulate was of this sort, and tried to prove it from the other four.  There are many famous names in this group, and much ingenuity, insight and sheer hard work was brought to bear on the problem.  Alas, there was always an unjustified step, or an assumption that turned out to be equivalent to the Parallel Postulate, rendering the argument circular.  Now, if you set out to draw a parallel line according to the given prescription with pencil and paper, you would only ever find one.  But we are not doing experimental or Babylonian mathematics; we are following in the absolute Greek tradition.

Then, in the nineteenth century, a few mathematicians wondered (independently), what if it’s not true?  What if, for instance, you can draw more than one parallel line through our given point?  This Alternate Postulate did not result in a contradiction.  In fact it led to a rich and varied sort of alternate universe, where the angles in a triangle add up to less than 180 degrees and where there is no such thing as a similar triangle.  Eventually it was worked out that hyperbolic geometry (as it is called) is exactly as consistent as Euclidean geometry, as is elliptic geometry (where there are no parallel lines through an external point).  It was a revolution in mathematics.  “Geometry” itself had to be redefined, as was (indeed) “mathematics.”  It’s hard for those not familiar with the field to realize what a profound change that meant.

All because something was impossible. . .