Functions.
The term "function" got into mathematics, I was told by Prof. K.O.
May due to a misinterpretation of a proper usage by Leibnitz. Nevertheless,
it has become a fundamental concept of mathematics and whatever it is called,
it deserves better treatment. There is perhaps no better example in mathematical
education of missed opportunities than in the treatment of functions.
The attitude of mathematicians is prissy, they have been unable to adjust
to the richness of interpretations possible. It has been, for example,
quite thoroughly demonstrated in functional analysis that use of algebras in
which functions are elements constitutes a major advance in conceptualization
and efficiency. Yet, there is great reluctance to use functional
notation for functions whose values are sets or which map sets into sets
despite the fact that these are more prevalent than the element-to-element
variety of functions even in mathematics.
Geometry, analysis, algebra, topology: All of mathematics has many
examples of functions mapping sets into sets. The natural inverse of a
function not one-to-one is set-valued. In applications functions arise as
many things. For example, labelling, categorizing, implication, transformers,
maps, computing machines, are all functions in some manner of
discourse. In topology there are many functions which map sets into sets:
interior, closure, boundary, limit points of, and of course, all the functions
of set algebra such as union, intersection, complement.
To insist on complex kinds of functions which are numerical valued
when so many are present which seem simpler and better exemplify the
concept is an educational blunder of the highest magnitude. Simplifications
possible using functions have not been achieved. There is a highly
cultivated taste for the baroque, for gingerbread, in mathematics.
Simplicity, which should be mandatory when possible is not prized.
To say that the situation for nomenclature of functions is bad is
a gross understatement. The same linguistic dullness which permitted the
term "function" to be adopted maintains itself in the morass of terminology
one important instance of which I have already noted in the case of binary
operators. It is perhaps too much to ask that men who have shown themselves
unable to recognize functions right before their eyes to develop
a scheme to deal with functions.
Topology
The field of topology has become quite fashionable and it has
developed a rather large body of literature. However, the results of merit
are actually basically confined to features of geometrical spaces. The
generalizations involved in certain concepts have clarified many issues of
analysis and geometry.
The axiom system defining topological spaces is singularly
inappropriate to the intentions and activities of topologists, namely, it
is too general and it was chosen without an adequate basis of experience.
A definition of a cohesive system should not admit "sports", i.e. aspects
which require needless qualifications of theorems.
While the actual work of topologists does not support the generality
of topological spaces, there is need of much more general spaces. This
need arises because topology applies only to a small subset of phenomena of
mathematics and because it is evidently impossible to do optimal work in
topology without considering more general matters. Generalizations of
topological spaces have been proposed by Frechet, Hausdorff and others. However,
these were not taken seriously, primarily I think because applications were
sought in analysis and geometry on the same grounds for which topology best
does its job.
However, that may be, I have made a comparatively thorough analysis
of basic topological concepts and I have yet to find one which is best
explained in the framework of topological spaces. Which concepts do I mean?
I mean closure, interior, neighborhood, compactness, continuity, filter,
convergence, limit point, connectedness of sets and so on. Since each of
these concepts is basic in topological space theory, I think it is important
that they be understood. By understanding, here, I mean they should be
presentable to people who are quite uninformed about mathematics in general,
to young children and to the lay public.
Now why should I say that these concepts are not best presented
in the area which has promoted them? This is because each of these
concepts and others deserve to be related to the rest of mathematics and to
applications. I have asked several topologists what the filtering action
of a topological filter is. None seems to know and what is worse they
don't care! They accept the senselessness of notations presented to them
because so much of it is senseless. Yet. H. Cartan presumably had a
reason for his choice -- the reasonableness of which has disappeared in
formalizations. I think mathematics should make sense.
I will not go into details but I will make a few statements. The
basic idea of continuity does not find a good example in the homomorphisms
of topology. Continuity and invariance are dual aspects of the same thing -
a function is continuous with respect to whatever properties or relations
it preserves, continuity is not an intrinsic property, it is relative to
context. Every function is continuous (it preserves something); every
function is discontinuous.
A filter in a set, I define to be any device which produces an
ordered dichotomy of the set. This readily embraces topological filters
but it makes basic sense and it can be used in any mathematical enterprise
providing a richness not possible in the confines of the topological filters.
I have defined approximation spaces in a manner generalizing
topological spaces. Here it is natural to have neighborhoods of points
in one space which are in a different space. Moreover, the general idea
of approximation, like that of filter and continuity, can be explained
to anyone who knows some language. This is not to say that general
comprehension is equivalent to detailed comprehension. However, any
intelligent individual can use the general ideas to enable much more
effective specialization.
I end on a necessarily acid note. Obscurantism is practiced far
too thoroughly by topologists. If topology is important enough only to
be presented at the junior or senior level of college it is not of broad
importance. If it has deeper significance then it must be brought in
earlier. The nomenclature, the notations and the language used by
topologists are poor indices of their achievements.
Analysis and Logic
Now analysis and logic have been around in some form for some time.
Unfortunately the terminology has also been inept. I noted the faux pas
which brought "function" into mathematics before. As to words I note
"integral", "series" versus "sequence", "derivative", "functional", as
indications of a few misfits. While mathematical analysts have recognized
the value of function algebras they also have not recognized set-valued
functions which abound. The terminologies "increases indefinitely" and
"indefinite integral" are absurd. A function which assumes values on a
sequence may increase definitely but not indefinitely. "Indefinite
integral" merely means a set of functions. There is nothing indefinite
about it.
Mathematical logicians seems devoted to the assumption that any
aesthetic symbol is undesirable. What a mouthful "existential quantifiers"
is! The idea that integers are simple is an absurdity. The assumption
of rights to examine the axiom structure of geometries, for example, is
not matched by recognition that logic has a foundation in what may be
called inductive, geometric, linguistic, assumptions. Thus, try to
"prove" that two instances of a letter are the "same". Without such an
assumption which cannot be proved no substitutions and no logic or
mathematics is possible. Now this is not to say that logic is futile. On the
contrary, attempts to improve precision of description starting from
crude assumptions is valuable, just as making precision machines using
cruder machines is valuable.
Here I may mention a fantasy which has penetrated the mathematical
literature this is that empty sets are (is) unique. In the course
of my work I have seen no nationale worthy of the name for uniqueness.
I have many opportunities to use as many empty sets as there are spaces.
It is impractical, unreasonable, and unjustifiable to insist on uniqueness.
If futility and confusion are the objectives then uniqueness may be granted.
I put logic and analysis together since the initial triumph of
Newton and Leibnitz was that for the first time continuous implication
was managed in finite terms. They introduced a new generalized logic.
To have failed to recognize that analysis deals with implication systems
which are logics is an outstanding example of failure to detect patterns.
A differential or integral equation is a "logic". Note that the superior
terminology adapted by Leibnitz resulted in more rapid advance of mathematics
in continental Europe than in England.
Once I remarked to a logician friend "a function is a logic". When
asked what I meant I said "Assume x, f(x) is implied". "That's a good
idea" he said. Yet function and implication had been stored in his brain
many years without the two meeting! How, with such interpretations missed
can we claim to do justice to education?
I close with another area, so-called measure theory. How such a
fundamental concept as measure could get ensnarled in such a trivial subset
of its exemplifications is difficult to comprehend. Diameter of a
set in a metric space is a measure, geometry means measures of the earth,
a metric measures separation of two points, yet none of these satisfy
the axioms presumed to define measure. Even measure theory violates its
own requirement of countably additive set-functions with non-negative
values. Thus an exterior measure is not a measure, a vector-measure
arising in projection is not additive, the cardinal number of a set is a
measure which is not real-valued in general. Do we have to perpetuate
such ridiculous choices of terminology? I say we should not.
Relations
The status of terminology in binary relations is on a par with that
for binary operations. Most (but not all) order relations can be called
transitive but this is the least general type of relation which includes
both reflexive and irreflexive order relations. It is silly to require
reflexivity as a property of order relations when strict order relations
do not possess the property. Relations in general and binary relations
in particular deserve better treatment.
Conclusion
Let me wind up by pointing out that I do not write these jibes as
an opponent of mathematics. I feel that mathematics is important enough
not to bury it in symbolic garbage and that those who, with whatever
intentions, increase the difficulty of learning mathematics are not
taking a serious attitude toward their responsibilities. With the best
of efforts to simplify it, mathematics will remain difficult enough and
will continue to fail to achieve goals which it would be beneficial to
reach. I recommend, in short, bringing some discipline to bear in order
to better achieve social objectives, i.e., to increase the utility, the
beauty, and the enjoyability of mathematics.