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, 2^{nd}. 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, 2^{nd}. 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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~ Commercial break ~

Relief for
beleaguered Nook lovers!

We now return you to
your regularly scheduled essay.

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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, 3^{rd}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.

.

.