The old kingdom of formality

It's been long time ago we have turned our backs to Mathematics. "The problem of these boys is that they don't know Math", a professor confessed during one of my exams years ago. J.A., who holds two PhD, one in Chemistry and another in Physics, used to say: "The key is Math; you know Math and you can learn everything". But still, Mathematics has been so coarsely mistreated and simplified. No one seems willing to pay the price of formality and full-understanding. I think it was Einstein who said that "calculations should not be be more complex than what they actually are... Nor simpler either". On page 14 of his Hydrodynamic Stability, P.G. Drazin states: "To illustrate some of the mechanisms and concepts of stability we shall now work through a classic problem that demands little mathematics". Alas! A "classic problem that demands little mathematics"... But on which I struggle like a poppy in a water spring. 

It caught my attention the formal definitions of stability provided by Darzin. Apparently, the mathematical concept was laid out by someone called Alexander Lyapounov, a Russian mathematician (1857-1918): “we say a basic flow is stable if, for any ε > 0, there exists some positive number δ (depending upon ε) such that if the norm of u(x,0) – U(x,0) or the norm of p(x,0) – P(x,0), etc. is < δ, then such norms are < ε for all times, t > 0”. This stuff involving positive, but small numbers ε and δ, sounds familiar. From an early age we learned the  formal definition of a continuous function in a point x0 belonging to an open interval (a,b), for example, as: for any positive ε, as small as desired, there exists another positive real number, δ,  such as if the norm of x-x0 < δ, then the norm f(x)-f(x0) < ε. In fact, there is an -apparently- interesting review of the evolution of the concept of continuity in the Spanish manuals of high-school level from 1940 to 2000 here.

The point I wish to make is this: it was a time of gold when, just like life, formality was a kingdom to master and aspire to. A time of gold, all spurn today.