Formalism in Differential Equations
- The Advantages and disadvantages of formalism.
- 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.
- Formalism lacks expediency: The formalization of contraction fails to
fit the global purpose [Pon, p.168, l.8].
- 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].
- To emphasize ideas rather than logic, we should eliminate formalism the best
we can by the following methods:
- Understanding a concept beyond its formal definition.
- Choosing a heuristic rather than a dogmatic approach (The integral of
differential forms). Not finished.
- Introducing the concept at the right time and in its original
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,
- 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,
- We would like to understand the roles that a theorem plays throughout the
Example. Implicit function theorem ÷
p.58, Theorem 15] ÷ [Pon, pp.182-183, (B)].
- 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.
- 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