There is no truth – truth died with Riemann

(written by Lawrence Krubner, however indented passages are often quotes)

A great essay on the confidence that preceded Riemann, and the confusion that came after him. Personally, I like living in this new universe, where math is a language to be used expressively, rather than an absolute description of the truth.

For two centuries after Newton, phenomenal science aspired to the kind of rigor and purity that seemed to be embodied in mathematics. The metaphysical situation seemed simple; mathematics embodied perfect a-priori knowledge, those sciences able to most mathematicize themselves were the most successful at phenomenal prediction; perfect knowledge would therefore consist of a mathematical formalism, arrived at by science and embracing all of reality, that would ground a-posteriori empirical understanding in a-priori rational logic. It was in this spirit that Condorcet dared to imagine describing the entire universe as a mutually-solving set of partial differential equations.

The first cracks in this inspiring picture appeared in the latter half of the 19th century when Riemann and Lobachevsky independently proved that Euclid’s Axiom of Parallels could be replaced by alternatives which yielded consistent geometries. Riemann’s geometry was modeled on a sphere, Lobachevsky’s on a hyperboloid of rotation.

The impact of this discovery has been obscured by later and greater upheavals, but at the time it broke on the intellectual world like a thunderbolt. For the existence of mutually inconsistent axiom systems for geometry, any of which could be modeled in the phenomenal universe, called the whole relationship between mathematics and physical theory into question.

When there was only Euclid, there was only one possible geometry. One could believe that the Euclidean axioms constituted a kind of perfect a-priori knowledge about geometry in the phenomenal world. But suddenly we had three geometries, an embarrassment of metaphysical riches.

For how were we to choose between the axioms of plane, spherical, and hyperbolic geometry as a description of “real” geometry? Because all three are consistent, we couldn’t choose on any a-priori basis — the choice had to become empirical, based on their predictive power for a given situation.

Of course, physical theorists had long been accustomed to choosing formalisms to fit a scientific problem. But it had been widely, if unconsciously, assumed that the need to do so ad hoc was a function of human ignorance; that, given good enough mathematics and logic, we could deduce the correct choice from first principles, producing a-priori descriptions of reality to be confirmed, as an afterthought, by empirical check.

But now, the Euclidean geometry that had been considered the model for axiomatic perfection in mathematics for over two thousand years, had been dethroned. If one could not know something as fundamental as the geometry of space a-priori, what hope was there for a purely “rational” theory encompassing all of nature? Psychologically, Riemann/Lobachevsky struck at the very heart of the enterprise of mathematics as it was then conceived.

Furthermore, Riemann/Lobachevsky called the nature of mathematical intuition into question. It had been easy to believe implicitly that mathematical intuition was a form of perception — a glimpse of the Platonic noumena behind reality. But with two other geometries jostling Euclid, nobody knew for sure what the noumena looked like any more!

Mathematicians responded to this dual problem with an increase in rigor, by trying to apply the axiomatic method throughout mathematics. It was gradually realized that the belief in mathematical intuition as a kind of perception of a noumenal world had encouraged sloppiness; proofs in the pre-axiomatic period often relied on shared intuitions about mathematical “reality” that could no longer be considered automatically valid.