Mathematics and Reality

Much of this essay covers advanced topics, but you don't have to understand all the math to appreciate the essay. This is the way math should be taught.

From the introduction: This essay is about mathematics and reality, how they relate and differ. I always feel the strength and imagination inherent within mathematics is abandoned by its terse presentation. Few students are ever aware of just how capable math is at handling what we see before our eyes and within our minds. Sure, we can calculate the trajectory of a comet with stunning accuracy, but to consider ourselves, our universe, and all we see before us—reality, and not simply manifestations of it—in a purely abstract mathematical framework is considered beyond the scope of mathematics and an arrogant breach into the realm of philosophy. But reality in disregard of our experiences—a reality stripped to the mathematics we use to understand it—suddenly transforms into a pristine and integrated place. Forces such as electromagnetism become “connections” on “principle bundles.” These are rigorously defined mathematical concepts independent of physical underpinnings. Matter itself, as corporeal as we imagine it, becomes as abstract as the forces governing it. As ethereal as space and time, matter corresponds to “sections” of an “associated vector bundle,” and mass to a particles’ “drag” through a “Higgs field.” These claims are not infallible (likely, they don’t even make sense, though that is not the intent), but they are what the Standard Model of particle physics asserts.