Formalism in Differential Equations

  1. The Advantages and disadvantages of formalism.
    1. Advantage. Formalism may save us from repeating the same argument. For example, the proof patterns of [Arn1, p.276, Corollary] and [Arn1, p.277, Problem 2] are pretty much the same.
    2. Disadvantages.
      1. Formalism lacks expediency: The formalization of contraction fails to fit the global purpose [Pon, p.168, l.8].

      2. A formal theory can not be accommodated to all the variations. We prove [Mas, p.166, l.8] by using the method in the proof of [Mas, p.251, Proposition 4.2]. However, the subspace [0,1] d R does not satisfy the hypothesis of [Mas, p.251, Proposition 4.2].

  2. To emphasize ideas rather than logic, we should eliminate formalism the best we can by the following methods:

    1. Understanding a concept beyond its formal definition.
      1. Choosing a heuristic rather than a dogmatic approach (The integral of differential forms). Not finished.
      2. Introducing the concept at the right time and in its original background.
        Example. The concept of positive definiteness should not be separated from the concept of stability. Compare [Pon, pp.160-161, Theorem 7] with [Jaco, vol.2, p.186, '8].
      3. Emphasizing ideas rather than symbolism.
        Example 1. For the solutions of a linear DE of order n, it is enough to discuss the case n=2.
        Example 2. The extension to the case n>3 is purely formal [Sne, p.35, l.11-l.12].
      4. We would like to understand the roles that a theorem plays throughout the entire process.
        Example. Implicit function theorem [Wid, p.58, Theorem 15] [Pon, pp.182-183, (B)].

    2. Emphasizing methods rather than structures.
      Example 1. Arnold pinpoints the domain M [Arn1, p.275, l.9] of the contraction mapping A (Compare [Arn1, p.275, '31.7] with [Bed, p.301, '90]).
      Example 2. For solving DE's in practice, the calculation in [Arn, p.180, Lemma 1] is not so clear as that in [Bir, p.68, Lemma 2]. [Arn, p.179, '26.5] generalizes every statement in [Bir, p.68, chap.3, '6] by means of operators, domains and ranges. Thus the content is unnecessarily bloated and sophisticated [Wan3, p.106, l.-6-l.-4]. In contrast, Arnold's version blurs the procedure of finding a solution, while Birkhoff's version is customized for application.

    3. Emphasizing options rather than a rigid choice.
          A method is elastic, while a theory is not. [Sne, p.126, chap.3, sec.10] provides many choices of integral transforms to solve a PDE. In contrast, from [Joh, p.148, (2.24a,b,c)], we can not see any alternative other than using the Fourier transform.