Re: Classical and Modern mathematics and (over?)specialization.
- From: hrubin@xxxxxxxxxxxxxxxxxxxx (Herman Rubin)
- Date: 22 Jun 2009 15:41:34 -0400
In article <1041459.13470.1245691369070.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Bacle <bacle@xxxxxxxxx> wrote:
Hi:
As a first-year student, I was surprised to see the
degree of (what I thought was ) over-specialization
in many students.
I implicitly assumed that the goal of learning math
was to have a broad base of knowledge, and this is
what I will try to do, unless something convinces
me otherwise. Still, I see people who get their
degrees in , e.g., algebra in two years, but
do not know what an open set is , nor the definition
of a Cauchy sequence, let alone basic results/defs.
related to them, like the intersection of open sets
being open, or that a space is complete if all
Cauchy seqs. converge ; I am not talking the fancier,
more advanced results, but what I think are
issues. Is this a good thing (to be fair, these
people do know their areas of specialty extremely
well)?
I have been told that this trend is more of a
modern trend, where much of the subject is
"black-boxed" , and the details don't matter,
only the overall larger result, and what can
be proven with it : we can use some theorems
on e.g.,differential forms, without understanding at
a basic level what a differential form is.
Is this trend the result of the explosion
of knowledge that makes it difficult to know
well even a small area, or is there something
else going on?.
Just wanted to know other's thoughts.
You are absolutely right. The same holds for analysts
who do not know topology, or even just knowing metric
topology; that a space is metrizable means something,
but it is rare that the metric is meaningful. But how
many mathematicians know about the basics? How many
even know the Peano Postulates for the non-negative
integers, and that they are categorical?
Even worse are those who go in for "interdisciplinary"
stuff, where they do no understand enough of the field to
which mathematics is applied, nor of the basic mathematical
concepts, to be able to properly formulate their problems.
There is no such thing as "biomathematics" or "biostatistics".
One can apply mathematics or statistics to biological
models, which require an understanding of the basic concepts.
The concepts are not taught until enough cookbook has been
taught to make the concepts much harder to understand, and
the cookbook is largely unnecessary. The biologist needs
to know what a derivative is, and what an integral is, and
then he can formulate the differential equations, or the
integral equations, needed. He does not need to know how
to compute derivatives and integrals; this can be left to
the mathematician or the computer program. The same holds
for biostatistics; I have seen biologists using statistical
procedures which are totally inappropriate to their problems,
because they know the procedures as coming down from God.
The user of a branch of knowledge needs to know the basic
concepts of "tool" fields, NOT how to compute answers, FIRST.
In this way, the problems will be stated correctly. If the
methods of getting answers are taught, problems are likely to
be formulated so that easy computations can get wrong answers.
I am often requested to repost my five commandments. These are
posted here without exegesis.
For the client:
1. Thou shalt know that thou must make assumptions.
2. Thou shalt not believe thy assumptions.
For the consultant:
3. Thou shalt not make thy client's assumptions for him.
4. Thou shalt inform thy client of the consequences
of his assumptions.
For the person who is both (e. g., a biostatistician or psychometrician):
5. Thou shalt keep thy roles distinct, lest thou violate
some of the other commandments.
The consultant is obligated to point out how their assumptions affect
their views of their domain; this is in the 4-th commandment. But the
consultant should be very careful in the assumption-making process not
to intrude beyond possibly pointing out that certain assumptions make
large differences, while others do not. A good example here is regression
analysis, where often normality has little effect, but the linearity of
the model is of great importance. Thus, it is very important for the
client to have to justify transformations.
There are, unfortunately, many fields in which much of the activity
consists of using statistical procedures without regard for any assumptions.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.
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