In the following post (q.v.)
we examined some related ideas -- analogy, generalization, abstraction -- that characterize the practice of doing math. Herewith some further terminology along similar lines:
Geometry (especially differential geometry) clarifies, codifies, and then generalizes ideas arising from our intuitions about certain aspects of the world.
The theory of differentiable manifolds is a natural result of extending and clarifying notions already familiar from multivariable calculus.
-- Jeffrey Lee, Manifolds and Differential Geometry (2009), p. xi - 1
I find it unsatisfactory to “classify” partial differential equations: this is possible in two variables, but creates the false impression that there is some kind of general and useful classification scheme available in general.
-- Lawrence Evans, Partial Differential Equations (1998, 2nd. edn. 2010), p. xix
In contrast to ordinary differential equtions, there is no unified theory of partial differential equations. Some equations have their own theories, while others have no theory at all. The reason for this complexity is a more complicated geometry. In the case of an ordinary differential equation, a locally integrable vector field (that is, one having integral curves) is defined on a manifold. For a partial differential equation, a subspace of the tangent space of dimension greater than 1 is defined at each point of the manifold. As is known, even a field of two-dimensional planes in three-dimensional space is in general not integrable.
-- Vladimir I. Arnold, Lectures on Partial Differential Equations (Russian edition 1997; English translation 2004), p.1
His Berkeley colleague concurs:
There is no general theory known concerning the solvability of all partial differential equations. Such a theory is extremely unlikely to exist, given the rich variety of physical, geometric, and probabilistic phenomena which can be modeled by PDE.
-- Lawrence Evans, Partial Differential Equations (1998, 2nd. edn. 2010), p. 3
This shows a becoming modesty, as against the media-physicists “quest” for a “Theory of Everything” -- there may be no ToE even for PDE’s. Yet the reason he cites for their presumable non-existence seems weak, compared with that given by Arnold: the unexpectedly rich variety of applications of this or that area of mathematics is precisely what gave rise to the marveling at “the unreasonable effectiveness of mathematics”.
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In this post
we described the undergraduate ‘crush’ upon abstraction, almost for its own sake. Herewith a caution, from a very old hand in the game.
Instead of the principle of maximal generality that is usual in mathematical books, the author has attempted to adhere to the principle of minimal generality, according to which every idea should first be clearly understood in the simplest situation; only then can the method developed be extended to more complicated cases.
-- Vladimir I. Arnold, Lectures on Partial Differential Equations (Russian edition 1997; English translation 2004), Preface to the second Russian edition
Indeed, the point is not one merely of “simple” versus “difficult”: rather, the “simplest situation” he is referring to is typically the motivating example of the theory. Thus, the notion of a Boolean semi-ring was inspired by the facts about the Natural Numbers.
He goes on:
Although it is usually simpler to prove a general fact than to prove numerous special cases of it, for a student the content of a mathematical theory is never larger than the set of examples that are thoroughly understood. That is why it is examples and ideas, rather than general theorems and axioms, that form the basis of this book.
Bringing it all back home:
We could perhaps refer to the fact that both these statements have already been proved in Chaper III … but we prefer to prove them here without getting involved ... with other more general problems.
-- A. D. Aleksandrov, “Non-Euclidean Geometry”, in: Aleksandrov et al, eds, Mathematics: Its Content, Methods, and Meaning (publication in the original Russian: 1956; Eng. tr. publ. 1963), III.125.
There is wisdom in this -- not as regards the essence of mathematics in its Platonic sphere, but as regards mathematical truth proportionate to our understanding.
As: You, the father, could say to your two-year-old, who has just snatched a cookie from the trembling fingers of his little sister: “No! Bad! No steal cookie!” -- Or you could say: “Ahh, my young fellow! A perfect illustration of the application of the Categorical Imperative, presented to the world by Emmanuel Kant. Thus, let us take as given, that ….”
Again, the dialectic, or at least the give-and-take:
Too large a generalisation leads to mere barrenness. It is the large generalisation, limited by a happy particularity, which is the fruitful conception.
-- Alfred North Whitehead, quoted in James R. Newman, ed. World of Mathematics (1956), p. 411
A sharper form of generality is duality. In its full precision, this is a concept by itself, and deserving a separate essay. But in the following informal treatment, the term is introduced as a sort of way-station:
Before leaving 1-forms, we digress to point out that there exists a form of duality between the analysis and the geometrical notions …
Curves: γ is closed iff ∂ γ = 0
1-forms: ω is closed iff dω = 0
-- Creighton Buck, Advanced Calculus (1956, 3rd edn. 1978), p. 506
And likewise for ‘bounding’ vs. ‘exact’.
Now -- this might strike you as striking for the wrong reason: ‘closed’ means something different in either case, as do curly-d and d; these terms and symbols were chosen with insight aforethought, and in themselves indicate nothing. The real meat comes in the theorems, e.g. every closed 1-form is exact iff every closed curve is bounding.