Postulating parallels

The world in our heads. Our heads in the world.

Geometry began as a division of geography but blossomed as a subset of philosophy. Practical people wanted to measure portions of the earth, and they did so employing lines and angles. But philosophers, devotees of pure reason, found in geometry a playground for testing their wits and reasoning abilities.

The most famous of the early geometers was Euclid, a resident of Alexandria who lived three centuries years before Christ. Euclid distilled geometry to a handful of undeniable postulates from which he reasoned his way to hundreds of theorems, many surprising. The first postulate was that a straight line can be drawn connecting any two points. The second was that a straight line can be extended to any desired length. The third was that a circle of any radius can be drawn with any point as its center. The fourth was that all right angles are equal.

These seemed so obvious that no one disputed them.

But Euclid had a fifth postulate that was different. This one said that if a straight line intersects two other straight lines so that the interior angles formed add up to less than two right angles, then the two lines will eventually intersect.

This fifth postulate came to be called the parallel postulate, because, in conjunction with the first four postulates, it was equivalent to saying that through a point not on a given line, exactly one line can be drawn that doesn't intersect the given line — that is, one line that is parallel to the given line.

In contrast to the first four postulates, which could be envisioned mentally, the fifth postulate required most people to make sketches to convince themselves of its truth. People did convince themselves, but the effort seemed excessive for a postulate, which was supposed to be self-evident.

Many geometers thought the fifth postulate ought to be provable from the first four. The search began for such a proof. Every so often someone would claim to have found a proof. Some of the claims held up for years or decades. But each time a logical hole would be discovered in the proof. The search would start anew.

Finally, two millennia after Euclid, it occurred to mathematicians that the postulate might be false. More precisely, they proposed that logically consistent systems existed in which the first four postulates were true and the fifth was not.

Proving this required actually creating such a system.

Before Euclid, this would have been nearly impossible. Geometry and geography remained attached in the minds of nearly everyone. Geometry was supposed to describe the real world, and the real world certainly behaved as though the parallel postulate was true. Even the followers of Euclid for many centuries thought they were describing the physical world, of which there was only one.

The breakthrough came in the 19th century, by which time mathematicians were growing comfortable creating imaginary worlds. Most notable was the world of imaginary numbers, starting with the square root of -1, denoted by i. This required a definite leap of imagination, in that all previous numbers, rooted in the real world, produced positive numbers when squared. The mathematicians made the leap and fashioned a world of numbers untethered from physical reality.

The geometers now did the same thing. Some changed the parallel postulate to state that through a given point not on a line many parallel lines could be drawn. Others said no parallels could be drawn.

Each version required reimagining what lines were. One version, modeling itself on the surface of a sphere, postulated that lines were the great circles on the sphere — the equivalents of meridians of longitude on the earth. Every meridian meets every other meridian at the north and south poles. Thus there are no parallel lines.

The geometries that resulted from these flights of imagination were called non-Euclidean. For a time they seemed wholly theoretical, mere caprices of the geometrical mind.

But when Albert Einstein in the early 20th century asserted that space is best understood as curved, one of the non-Euclidean geometries supplied a mathematical framework on which he built his theory of relativity.

In the age-old debate over whether mathematics is discovered or invented, the efforts to prove the parallel postulate provided evidence for both sides. On one hand, if the non-Euclidean geometers could replace the parallel postulate with postulates of their own making, this certainly sounded like mathematics as invention. Mathematics was just a game. The mathematicians chose whatever rules suited them. If they didn't like the old rules, they wrote new ones.

But when it turned out that non-Euclidean geometry described Einstein's universe, this sounded like mathematics as discovery. The universe existed with its own rules, and the job of mathematicians was to discover them.

The debate continues. So do the discoveries and/or inventions.