Ihor Kendiukhov

The problem was Euclid's fifth postulate, the parallel postulate, which states, in one of its equivalent formulations, that through any point not on a given line, there is exactly one line parallel to the given one.

For over 2,000 years, mathematicians had felt that something was off about this postulate.

The other, four, were short, crisp, self-evident.

You can draw a straight line between any two points, you can extend a line indefinitely, you can draw a circle with any centre and radius, all right angles are equal.

The fifth postulate, by contrast, was long, complicated, and felt more like a theorem that ought to be provable from the others than a foundational assumption standing on its own.

Generation after generation of mathematicians attempted to derive it from the remaining, four, and failed.

Farkas Boliai begged his son to stay away.

J.A.

Noes ignored his father's advice, but not in the way Farkas feared.

Instead of trying to prove the postulate, he asked a question that turned the entire enterprise upside down.

What happens if the postulate is simply false?

What if you can draw more than one parallel line through a point?

Rather than deriving a contradiction, which would have constituted a proof of the fifth postulate by Reduccio, he found something a perfectly consistent geometry, as internally coherent as Euclid's, just describing a different kind of space.

Loboshevsky independently reached the same conclusion around the same time.

The parallel postulate was not wrong, exactly, but it was not necessary.

It was one choice among several, and the other choices led to geometries that were not merely logically valid but turned out, a century later, to describe the actual physical universe better than Euclid's flat space ever could.

Roughly two centuries later, people were discussing decision theories and axioms of expected utility.

The standard argument went roughly like this.

Rational agents must maximize expected utility.

The von Neumann-Morgenstern theorem proves it.