epsilontics

epsilontics (plural epsilontics)

1. (mathematics) An approach to mathematical analysis using the epsilon-delta definition of a limit, i.e. with explicit estimation of error bounds, as opposed to using infinitesimals.
   • 1973, Hans Freudenthal, Mathematics as an Educational Task, Springer Science & Business Media →ISBN, page 148

     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.

   • 1981, John M. Ziman, Puzzles, Problems and Enigmas: Occasional Pieces on the Human Aspects of Science, Cambridge University Press →ISBN, page 51

     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.

   • 2009, Detlef Laugwitz, Bernhard Riemann 1826–1866: Turning Points in the Conception of Mathematics, Springer Science & Business Media →ISBN, page 59

     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.

2. (mathematics) An instance of reasoning performed using this approach.
   • 1973, American Mathematical Society, Notices of the American Mathematical Society

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

   • 1999, Roland Omnès, Understanding Quantum Mechanics, Princeton University Press →ISBN, page 185

     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.

   • 2001, Peter Schuster, Ulrich Berger, Horst Osswald, Reuniting the Antipodes - Constructive and Nonstandard Views of the Continuum, Springer Science & Business Media →ISBN, page 137

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

   • 2007, C Haines, P Galbraith, W Blum, S Khan, Mathematical Modelling: Education, Engineering and Economics - ICTMA 12, Elsevier →ISBN, page 330

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

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