What are some examples of mathematics proven to be important and functional to other science but seems completely useless when first came out?

According to Einstein’s biographer, the day before Einstein finally honored that agreed upon last dance with the Grim Reaper, he lifted his trembling hand. You see, to the bitter end, Einstein was determined to read “the mind of God,” i.e., in a mathematical kind of way. Of course, as his hero Galileo noted, “The book is written in mathematical language.” Or as Paul Erdős put it, “The Book” contains all the best and most elegant proofs for mathematical theorems.

Metaphors notwithstanding, Einstein, just hours before his death, pointed to his equations, while lamenting to his son, “If only I had more mathematics.”

One can’t blame Einstein for having exhibited dogged persistence to his dying day. I suppose, it’s safe to say that Einstein’s wistful utterance stemmed from his suspicion that his mathematical shortcomings had finally caught up with him, as the remaining grains of sand in his hourglass dwindled away. After all, just fifty years before, Einstein called on his old pal Marcel Grossmann—a mathematician—who in effect helped ol’ Albert revolutionize the world of physics.

“Grossmann,” pleaded Einstein, “you’ve got to help me or I will go crazy!”

Just a decade before, when Einstein routinely skipped math class at the Polytechnic, it was Grossmann’s notes that came to the rescue. And given that history seems to have a penchant for rendering encore performances, it was Grossmann who recused Einstein in his finest hour.

Einstein, nearly on the verge of a nervous breakdown, explained to Grossmann that he was in dire need of a mathematical system. By no other means could he generalize his then special theory of relativity. You see, Einstein’s genius lay in his uncanny “intuitive feel for the order lying behind the appearance.” His skills as a mathematician, however, left much to be desired.

Of course, just as Ramanujan towered above all others in the domain of intuition yet struggled mightily to render rigorous proofs, the same held true for Einstein.

Grossmann mulled over the problem for a bit. After pouring over the mathematical literature of the day, Grossmann came back with a mathematical giant whose name and hypothesis still reign supreme. Bernhard Riemann. Non-Euclidean geometry proved to be the answer to the Einsteinian Sphinx!

“I am now working exclusively on the gravitation problem and I believe that, with the help of a mathematician friend here, I will overcome all difficulties,” Einstein wrote to a fellow physicist pal. “I have gained enormous respect for mathematics, whose more subtle parts I considered until now, in my ignorance, as pure luxury!”

Ah, the folly of youth! And as they say: the rest is history!

Riemann invented the system of geometry just for the heck of it. He was, after all, a pure mathematician. Sure, Gauss, among others, had developed some semblance of alternative geometries to that of what had long served as the gold standard, Euclidian geometry, but Riemann took it well beyond.

Riemann’s grand insight came by way of dropping the infamous parallel postulate. The result? he crafted an ingenious way to account for a surface irrespective of how its geometry altered, even if its variation spanned the gamut of spherical to hyperbolic to flat and so forth.

In short, Riemann’s insight into accounting for distances that spanned points in space, irrespective of how whimsically curved and distorted, proved key to Einstein’s general relativity. Once Einstein had in hand Riemann’s metric tensor, i.e., “vectors on steroids,” he in effect had the tool of the gods. Thereafter Einstein merely had to calculate the distance whereby points were separated in space-time.

It was the general covariance of Riemannian tensors that proved key. “The central idea of general relativity is that gravity arises from the curvature of spacetime,” noted physicist James Hartle. “Gravity is geometry.”

In short, what started out as a pure mathematician dropping the parallel postulate and extending the groundwork laid by the likes of the “Prince of Mathematics,” and seemingly had no correspondence whatsoever to the real world, in the end proved to be the missing ingredient to history’s most famous theory.

G.H. Hardy put it best:

…there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, […] the mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as ‘real’, but […] [a physicist] is trying to correlate the incoherent body of crude fact confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics.

Credit: Genius Turner