A beautiful example of the interplay between science (physics) and mathematics is the discovery of the general theory of relativity. Einstein discovered the special theory of relativity (STR) in 1905, in which space and time are intermingled. Three years later, Hermann Minkowski, a mathematician, took Einstein’s relativistic unification of space and time and abstracted it into a 4-dimensional space, now called Minkowski spacetime.
In the meantime, some mathematicians were working on the non-Euclidean geometries proposed in early 19th century. They had combined calculus and these geometries to come up with one of the most abstract branches of mathematics called differential geometry, which studied infinitesimal regions (thus calculus) of multi-dimensional spaces.
In the years after his discovery of STR, Einstein tried to invent a theory of gravity that was consistent with STR, unlike Newtonian theory of gravity which contradicted it. After many unsatisfactory attempts at directly incorporating STR within the framework of Newtonian gravity, Einstein hit upon an ingenious idea that had nothing to do with his earlier attempts! He combined Minkowski’s idea of a four-dimensional spacetime with differential geometry and invented the general theory of relativity (GTR). Thus, GTR is the result of the application of pure abstract mathematics developed by the human brain — differential geometry and Minkowsli spacetime — to Nature. And part of this pure abstract mathematics developed by the human brain — Minkowski spacetime — grew out of the study of Nature .
What is the “practical application” of the statement “mathematics, when undue emphasis is put upon the technique of calculation, suffer[s] from the same evil”? Dewey’s pragmatist in arm, William Heard Kilpatrick, indeed found a radically practical application. He was the lead author of the 1920 report, “The Problem of Mathematics in Secondary Education,” in which was stated that “algebra, geometry, and any higher math was a waste of time for most students. Advanced math, it posited, wasn’t critical for understanding greater life lessons.”
We are still suffering from the ramifications of this report, as Common Core and other attempts at improving math face resistance by parents and (some) teachers alike.
Dewey Was Wrong About Science
Dewey is even wrong about experimental aspect of science, which, unlike a theory, is closer to “applications” and the spirit of pragmatism. Laboratory activities “for their own sake” are the norm. And not only do they not “suffer from the same evil,” but are necessary for the discoveries made in science. How can the unknown sought after in a laboratory experiment be discovered if the experiment is not done for its own sake? Dewey’s emphasis on practical application renders the discovery of the unknown impossible: if it is unknown it has, by definition, no application! Ben Franklin flew his kite without any practical application in mind! Hans Christian Øersted discovered the connection between electric current and magnetic phenomena without thinking about its future application! All the experiments in electricity and magnetism of the early 19th century were done — and surprising results discovered — without knowing their potential applications!