Epsilontics

Now epsilontics is losing ground, and after one or two decennia we can expect the "great discovery" that, properly understood, the infinitesimal methods of a century ago are completely rigorous. Algorithms with differentials are already modern.

In principle, we should not then have made any approximation at all; it is a rule of pure mathematics, hallowed by centuries of usage, and justified by the mysteries known to undergraduates as 'epsilontics', that a convergent series is as good as a simple finite algebraic expression for a solution — which is just as well, because almost every expression that appears in any formula in physics, such as a cosine, or a logarithm, is itself only definable algebraically as the sum of an infinite series.

Both Cauchy and Abel must have been aware that epsilontics is a general tool for making rigorous proofs in analysis, but neither of them, to say nothing of their contemporaries, used it consistently.

Key abstract theorems are explained largely by physical reasoning, and are presented in the most concrete, intelligible fashion possible. Epsilontics are minimized.

One might be more precise by giving an error for equation (16.2) or by considering the much smaller errors on the exclusion of other classical properties, but these epsilontics will be omitted.

Thus, Hadamard avoids Weierstrassian epsilontics in his text for engineering students.

The sequence of Riemann sums is a constant sequence and converges to its constant value s(b) − s(a). No “epsilontics” are needed.