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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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, 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.
.
.
