Nothing is Clear-Cut (e.g. Math)

3/20/2017
Monday
Afternoon
Partly Cloudy, but Chilling

Of course, nothing is clear-cut. Nothing fits nicely into boxes/categories/sets we create for them, but rather things are blurry, multifaceted, and they fall into many different groups. Here is a specific example related to, you guessed it, math.

Consider the fact that the derivatives of the sum of two functions is the sum of the derivatives of the composite functions. Stated a bit more symbolically, given two differentiable functions f and g, we see

(f+g)'(x)=f'(x)+g'(x).

A high-school calculus student would see this and recognize "a-ha! this is the Sum Rule!" On, say, the AP Calculus AB exam, this would be sufficient reasoning behind this fact; it was dictated in the class, and the students have memorized it, either through brute force or through much practice of finding derivatives of functions.

Of course, there is different justification for this fact from different mathematical perspectives. For example, consider Real Analysis:  calculus' rigorous, proof-based older brother. A mathematical analyst will tell you "well, the definition of the derivative is given by a limit, and since we can add limits, we can add derivatives." Moreover, the analyst could relate these limits back to sequences, and reason that the Sum Rule follows from the arithmetic of limits of functions, which follows from the arithmetic of limits of sequences.

From another perspective, consider Linear Algebra. A linear algebraist would tell you that the set of real-valued, differentiable functions form a vector space with the operations of function addition and multiplication of real numbers. Therefore, such a mathematician would say that the Sum Rule follows from the fact that the vector space of differentiable functions is closed under addition.

This is something about math that I like; you learn one thing, and it pops up everywhere, in a bunch of different fields of math. I suppose this tells us that math is less like a library, each field belonging to a bookshelf, each bookshelf containing an exhaustive list of the ideas in the field. Rather, math is like a swamp, where concepts lurk beneath the fields we've constructed.

Think different. Think DASANI.