Contamination. Reasoning is a sequence of deductions. Suppose
that we use the axiom of choice to prove Theorem A, and put Theorem B after
Theorem A. To say Theorem B is independent of the axiom of choice, we must show
that every statement in the proof of Theorem B does not use Theorem A. This
tracing and screening process is very annoying.
Clearly, the proof of Urysohn's
metrization theorem in [Wil, p.166, Theorem 23.1] does not use Tichonov's
theorem [Po3, p.78, Theorem 5] or even the concept of compactness. However,
according to [Po3, p.85, l.19-l.22], metrization relies on [Po3, p.81, Theorem
6]. It is troublesome to verify that every statement in [Po3, p.81, l.22-p.82,
l.25] does not use [Po3, p.78, Theorem 5]. If some statements did use [Po3,
p.78, Theorem 5], could we save the situation by proving
"A countable product of compact spaces
is compact" without using the axiom of
choice? The side questions of this kind may bother us endlessly.
Indirectness. [Wil, p.166, Theorem 23.1] shows that it is
unnecessary to prove Urysohn's
metrization theorem via paracompactness (see [Dug, p.194, Theorem 9.1]).
How to increase effectiveness.
Algebraic methods.
Direct product.
In terms of effectiveness, algebraic structure is the flagship
of all math structures. If we take care of the leading structure, the remaining structures
will follow suit [Po3, p.121, Theorem 13].
Discrete normal subgroups [Po3, p.130, Theorem 15].
Reducing a problem from global to local [Po3, p.97, E)].
The advantage of adopting the viewpoint of effectiveness.
(Regular Lie subgroups) A theorem of "if
and only if " type has two directions.
We often wonder which one is more useful. By adopting the viewpoint of
effectiveness, we may easily find the correct direction (see [Olv, p.18,
l.8-l.25]).
What does effectiveness mean in mathematics?
We would like to find the most efficient method to check a
definition under special conditions [Dug, p.268, Theorem 7.2].
Antithesis. A text usually teaches us the most efficient algorithm for a
solution. However, based on the spirit of the method, we may have many
options to obtain the solution. For example, matrix inverse [Rut, p.35, l.-3].
Identification. An effective description should be as definite
and specific as possible. [Dug, p.272, l.1-l.6] shows the difference between
metrizability and actual construction of metric d^{+} from d.
Effectiveness sets the stage for complete understanding. When we
study two topologies on a set, the first step is to show that they are
different. However, our ultimate goal is to understand the two topologies
completely. The metric topology in [Dug, p.271, l.2] is crystal clear. We may
use [Dug, p.258, Ex.1 & p.99, Ex.3] to prove that the c-topology is not
discrete. However, for the c-topology, this description is far from
satisfactory. We would like to find a more effective way to characterize the
c-topology [Dug, p.271, l.3] even though it would be a digression from the main
task of showing the difference
between the two topologies.
When we say that a generalization includes all the cases, it does not mean
that the generalization simultaneously preserves the effectiveness of each case.
The lemma in [Dug, p.280, l.15-l.16] can be proved via the binomial theorem or
Fejér's theorem. Thus we may use the
binomial series to prove Stone's
theorem [Dug, p.282, Theorem 3.3 or Ru1, p.152, Theorem 7.31]. Therefore, it is
only in a very loose sense that Fejér's theorem
can be considered a special case of Stone's
theorem. In other words, the vague language in the conclusion of Stone's
theorem is not effective enough to grasp the essence of Fejér's
theorem. Similarly, once we prove Stone's
theorem via Fejér's theorem, we can no
longer recover the feature of the binomial theorem.
(Method and product) A stronger hypothesis gives us a more effective
(straightforward) method to find a more effective (global instead of local) Lie
group whose Lie algebra is the given Lie algebra (Compare [Po3, p.453, Theorem
102] with [Po3, p.412, Theorem 88], see [Po3, p.453, l.-3-p.454,
l.4]).
Through application we may gain a vantage-point for effectiveness because
there are more resources available. Effectiveness is surely an ongoing trend for
the development of mathematics.
We use reduction to absurdity to prove "path-connectedness
Þ connectedness" [Dug, p.115, Theorem 5.3], so it is more straightforward and effective to check connectedness by finding a path between two points. Even though the formal structure for connectedness is firmly established [Che, p.36, Proposition 2], we prefer proving connectedness by operating explicit paths [Fom, p.14, l.6; p.15, l.-2], rather than by deducting from ineffective structure theorems [Che, p.37, l.2-l.4].
To determine the classes of a group from the formal definition of classes is tedious. In contrast, physical-symmetry considerations will greatly simplify the procedure
[Tin, p.12, l.-8-l.-6].
[Wat, p.362, l.-5] requires that J_{u}(Z)
be dominated by the leading term of its series representation. Does this requirement contradict
[Wat, p.16, (4)]? No, because in [Wat, p.362, l.-5]
the domain of Z
is bounded. This answer inspires me to review the gain and loss of the axiomatic approach
and wonder whether the current teaching practice in universities is founded on
emptiness. [Ru1, p.11, Theorem 1.36] fails to tell us how to find the lub.
[Ru1, p.77, Theorem 4.16] also fails to tell us how to find p and q.
Suppose we want to write a computer program for the following problem: if a
function f(z) and a bounded domain D are given, how do we find an upper bound
for f on D? The above two theorems are related, but they are not helpful in
solving this specific problem. A natural way to solve this problem is to express
f in a Taylor series. Because D is bounded, we may assume
|z|<1.
The leading term can be considered a dominant term.
Both [Kre, p.141, (43.2)] and [Wea1, p.100, (6)] give the differential equations of the geodesics on a surface.
Which one is better? If our goal is to prove the existence of a geodesic that
satisfies the initial conditions, then [Kre, p.141, (43.2)] is good enough.
However, If we want to design a computer program to plot the graph of the
geodesic, then [Wea1, p.100, (6)] will help us design a more effective program
because it has fewer variables than [Kre, p.141, (43.2)]. Modern mathematicians tend to ignore a solution's
effectiveness. It seems that we have to look for older mathematical literature
if we want to find more effective solutions.
Effective methods of construction.
By [Dug, p.127, Definition 6.1], X attached to Y by f is a
special case of quotient space. Conversely, a quotient space [Dug, p.125, Definition 4.1]
can be considered a degenerate case of X attached to Y by f. Thus, the two
concepts are special cases of each other through different perspectives. We
cannot determine which method of construction is more effective based on
specificity (or
generality) alone. We need to make a further comparison: the construction in
[Dug, p.127, Definition 6.1] is a refinement of [Dug, p.125, Definition 4.1].
This is because the former construction includes the latter
construction as part of the procedure, contains more tools and resources, and
produces more delicate topological structures. Similar discussion applies to homotopy [Dug, p.315, Definition 1.1] vs. H-homotopy [Dug, p.321, Definition
4.1].
When we talk about effectiveness, we must know what aspect we refer to.
Sometimes, we refer to an entire process of constructing a function's expansion
as an infinite product. If we want to construct the expansion of a given
function as an infinite product from scratch,
then [Guo, p.25, Theorem 1] is more effective than [Gon1, p.202, Theorem 3.16].
However, if the identity for the expansion is already given [Gon1, p.206,
(3.5-7)], we just want to
prove its validity, then the less effective and more general theorem [Gon1,
p.202, Theorem 3.16] can be the best choice.
Compare the proof of [Gon1, p.206, (3.5-7)] with [Guo, p.26, (2)]. [1]
A shorter proof is not necessarily a more effective proof [1].
The use of a complicated theorem can make a constructive existence less effective.
Sometimes, it is impossible for a generalized theorem to preserve the
effectiveness of a specific case.
(The Riemann-Lebesgue Lemmas)
Suppose we want to use a computer to verify [Wat1,
p.172, Theorem 9.41 (I)] for a given function. It is easier to effectively
convert the argument to a computer program using the proof of [Wat1, p.172, Theorem 9.41 (I)] than
it is to do so using the proof given in [Ru2, p.109, l.-16-l.-6].
This is
because a complicated theorem [Ru2, p.96, Theorem 4.25] is used in [Ru2, p.109,
l.-13]. In addition, the method given in [Ru2,
§5.14] cannot be used to prove the specific
case given in [Wat1, p.172,
Theorem 9.41 (II)].
The existence of a finite system of squares given in [Sak, p.35, Theorem
10.3] can be considered constructive. This is because in [Sak, Introduction
― theory of sets, §10] all the
arguments using the method of reduction to absurdity are either trivial or
geometrically obvious. A typical modern proof of [Sak, p.35, Theorem 10.3] is given by [Dug, p.185,
Theorem 4.3; p.234, Theorem 4.4]. If we desire to use compactness, we must prove
[Ru1, p.34, Theorem 2.40]. However, the proof of [Ru1, p.34, Theorem 2.40] requires a nontrivial argument
using reduction to absurdity. Thus, the existence of a finite subcover of I
given in the proof of [Ru1, p.34, Theorem 2.40] is merely a logical existence.
Consequently, the modern proof of [Sak, p.35, Theorem 10.3] using [Dug, p.185, Theorem 4.3; p.234, Theorem
4.4] is ineffective.
Remark. Topology has grown so popular that it has become a prerequisite for studying the books in complex analysis written in
the past forty years.
However, topology is not an effective tool to solve problems in analysis. Thus, ineffectiveness is deeply rooted in many math
books written in the past forty years [Con, p.28, Theorem 5.17; p.138,
Proposition 1.2; Ru2, p.285, Theorem 13.3; Sil, p.29, Theorem 2.4]. It would take a tremendous effort to overhaul the
problem of ineffectiveness.¬¬
Quantitative vs. qualitative formulations
Example. (The inverse function theorem)
[Har, p.11, Exercise 2.3] provides a quantitative formulation about the inverse
function theorem because it assigns the size of the ball D_{1} on which f is one-to-one. In contrast, [Ru1, p.193, Theorem 9.17] provides only a qualitative formulation
because it is stated in terms of open sets. By comparison, the former is a more effective
formulation.
Remark 1 (Proof of [Har, p.11, Exercise 2.3]). By [Har, p.10, Theorem 2.1], there exists a function g: {x: |x| £ b/M}
® D such that g_{ ° }f = Id.
Similarly, there exists a function h: {y: |y| £ b/(MM_{1})}
® D^{~}_{0} such that h_{ ° }g = Id.
h = h_{ ° }(g_{ ° }f) = (h_{ ° }g)_{ ° }f = f on {y: |y| £ b/(MM_{1})}.
It is unnecessary to use [Har, p.5, Theorem 2.5].
Remark 2. The inverse function theorem provides a vantage point for us to see
the insight of reason why uniqueness implies continuity [Cod, p.23, l.10-l.12].