| The review distills the book's view of the difference between pure mathematics and applied mathematics. "applied" split from "pure" to meet the technical needs of the US military during WW2. My best example of the split is https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives
Wikpedia notes that "The list of unsuccessful proposed proofs started with Euler's, published in 1740,[3] although already in 1721 Bernoulli had implicitly assumed the result with no formal justification." The split between pure (Euler) and applied(Bernoulli) is already there. The result is hard to prove because it isn't actually true. A simple proof will apply to a counter example, so cannot be correct. A correct proof will have to use the additional hypotheses needed to block the counter examples, so cannot be simple. Since the human life span is 70 years, I face an urgent dilemma. Do I master the technique needed to understand the proof (fun) or do I crack on and build things (satisfaction)? Pure mathematicians are planning on constructing long and intricate chains of reasoning; a small error can get amplified into a error that matters. From a contradiction one can prove anything. Applied mathematics gets applied to engineering; build a prototype and discover problems with tolerances, material impurities, and annoying edge cases in the mathematical analysis. A error will likely show up in the prototype. Pure? Applied? It is really about the ticking of the clock. |
The example you gave concerns differentiation. Differentiation is messy in real analysis because it's messy in numerical computing. How real analysis fixes this mess parallels how numerical computing must fix the mess. How do we make differentiation - or just derivatives, perhaps - computable?
The rock-bottom condition for computability is continuity. All discontinuous functions are uncomputable. It turns out that it is sufficient, to make your theorem hold, to have the 2nd partial derivatives f_{xy} and f_{yx} be continuous. They wouldn't even be computable otherwise!
One of the proofs provided uses integration. In numerical contexts, it is integration which is considered "easy", and "differentiation" which is considered hard. This is totally backwards to symbolic calculus.
The article also mentions Distribution Theory. This is important in the theory of linear PDEs. I suspect it is implicit in the algorithmic theory as well, whether practitioners have spelled this out or not. This is a theory that makes the differentiation operator itself computable, but at the cost of making the derivatives weaker than ordinary functions. How so? On the one hand, it allows to obtain things like the Dirac delta as derivatives, but those aren't even functions. On the other hand, these objects behave like functions - let's say f(x,y) - but we can't evaluate them at points; instead, we can take their inner product with test functions, which we can use to approximate evaluation. This is important because PDE solvers may only be able to provide solutions in the weak, distribution-theoretic sense.