Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.
|
|
|
|
Matheology § 099
Posted:
Jul 25, 2012 8:41 AM
|
|
Matheology § 099
Alexander Zenkin: OPEN LETTER
TO: The Bulletin of Symbolic Logic
CC: The International Mathematical Union
Dear Professor Blass,
As you certainly guessed, the question is not only about a publication
of my comments "Whether the Lord Exists in G. Cantor's Transfinite
'Paradise'" to the scandally-known, quasi-"pedagogical" W. Hodges'
paper "An Editor Recalls Some Hopeless Papers" [...] in your BSL-
journal. The question is about much more important problems. I believe
that BSL-papers like the W. Hodges' one are a dangerous phenomenon
from scientific, educational, and social points of view. There are the
following reasons to state that.
1. The main conclusion of the W. Hodges' paper that "there is nothing
wrong with Cantor's argument" is wrong fundamentally and therefore,
having been proclaimed in the BSL, such a conclusion disorients a wide
scientific community (pedagogical, mathematical, logical,
philosophical, cognitive psychological, etc. ones) as to one of the
most important problems of the humankind culture as a whole - the
problem on the veritable nature of Infinity.
2. The high symbolic logic level of the BSL-publications is a
recognized standard of a meta-mathematical thinking and an attractive
pattern for young generations of meta-mathematicians and symbolic
logicians. However a one-sided publishing BSL-policy (not to publish
points of view differing from the traditional set theoretical opinion)
deprives the young generation of the democratic right to make
independently its own scientific choice between two historical,
contradictory points of view as to the true nature of Infinity: i.e.,
between the today traditional Cantor's and all modern axiomatic set
theory's opinion, on the one hand, and the opinion of Aristotle,
Leibniz, Kant, Gauss, Cauchy, Kronecker, Hermite, Poincare, Bair,
Borel, Brouwer, Wittgenstein, Weil, Luzin, Quine, and today - Sol.
Feferman, Ja. Peregrin, V. Turchin, P. Vopenka, etc. [...]
3. One of the most important reasons for my flat objection against the
W. Hodges' and similar meta-mathematical papers is their deforming
influence on mathematical education and their dangerous social
consequences as a whole. As far back as the middle of 50s of the XX
century, the outstanding American mathematician, John von Neumann
[...] warned: "Too much formalization and symbolization in the theory
of mathematics is dangerous for the healthy development of the science
of mathematics".
In the beginning of the 60s, a large group (about 75) of outstanding
mathematicians of America and Canada (including Richard Bellman of
Rand Corporation, Richard Courant of New York University, ?.?. Pollak
of Bell Telephone Laboratories, George Polya of Stanford University,
Andre Weil of Institute for Advanced Study, and others) tried to
attract attention of mathematical community to the same problem - to
the danger to provoke a stable disgust of pupils, students and their
parents (who, by the way, are today's Presidents, Government-men,
Congressmen, Government ministers, etc.) to mathematics by means of a
premature, excessive, deterrent, and simply thoughtless formalization
of mathematical education.
In their known Memorandum "ON ??? MATHEMATICS CURRICULUM OF THE HIGH
SCHOOL" (American Mathematical Monthly, 1962, March, 189-193) they, in
particular, wrote.
"It would [...] b? ? tragedy if the curriculum reform [...] should be
misdirected and the golden opportunity wasted. There are,
unfortunately, factors and forces in the current scene which may lead
us astray. [...] premature formalization may lead to sterility;
premature introduction of abstractions meets resistance especially
from critical minds who, before accepting an abstraction, wish to know
why it is relevant and how it could be used.
In its cultural significance as well as in its practical use,
mathematics is linked to the other sciences and the other sciences are
linked to mathematics, which is their language and their essential
instrument. Mathematics separated from the other sciences loses one of
its most important sources of interest and motivation. [...]
We wish especially that the new curricula should reflect more the
connection between mathematics and science and carefully heed the
distinction between matters logically prior and matters which should
have priority in teaching. Only in this way can we hope that the basic
values of mathematics, its real meaning, purpose, and usefulness will
be made accessible to all students [...]"
In conclusion, they again accentuate and expressed their "concern
about ? trend to excessive emphasis on abstraction in the teaching of
mathematics"
As the posterior history showed, this very serious, very anxious, and
high professional warning of outstanding mathematicians of the middle
of the XX c. as to the danger of the "excessive formalization and
symbolization of mathematical education" was not heard.
Today the situation is further aggravated. The Vice-President of the
International Mathematical Union, Academician of the Russian Academy
of Sciences, outstanding mathematician and mathematical educator,
professor Vladimir I. Arnold of Steklov Mathematical Institute
(Moscow) in his numerous papers of the last decade again tries hard to
attract attention of mathematicians and educational community to the
catastrophic situation in modern mathematics and mathematical
education. The main reason is the same one ? a (today already) global
super-formalization or, using his term, ?bourbakization? [...] of the
modern mathematics as a whole (see, e.g., V.I. Arnold, "International
Mathematical Congress in Berlin." [...])
?Our brain, - writes Arnold, - has two halves: one <the left-
hemisphere> is responsible for the multiplication of polynomials and
languages <i.e., for the abstract, formal, rational thinking>, and the
other half <the right-hemisphere> is responsible for orientation of
figures in space and all the things important in real life <i.e., for
the intuitive, visual, creative thinking>. Mathematics is geometry
when you have to use both halves.
In the middle of the XX Century, a [...] mafia of ?left-hemispheric?
mathematicians could exclude geometry from mathematical education
(firstly in France, and then in other countries), replaced all
informal aspects of this discipline by a training in a formal
manipulation with abstract ?notions? <i.e., with empty names, terms,
symbols, etc.>. All geometry, and consequently all connection of
mathematics with the real world and with other sciences was excluded
from mathematical education.
Such the ?abstract? description of mathematics is unfit neither for
education, nor for any practical applications.
?Compelling miserable schoolboys/girls to learn such <formalized
mathematics>, ?left-hemispherical criminals? created a modern
distinctly negative attitude of society and governments to
mathematics.
?The aversion to mathematics which government ministers, exposed to
such the experience of such the education in school, have is a healthy
and valid reaction. Unfortunately, this their disgust spreads on all
mathematics without exclusions, and that can kill it as a whole?.
... these ?left-hemispherical invalids? were able to cultivate whole
generations of mathematicians that don?t understand any other approach
to mathematics and are able only to teach next generations by the same
way.
? It is awful to think what kind of pressure the Bourbakists put on
(evidently nonsilly) students to reduce them to formal machines! This
kind of formalized education is completely useless for any practical
problem and even dangerous, leading to Chernobyl-type accidents.
[...]
? Modern formalized (bourbakized) education in mathematics is an exact
antithesis for teaching the critical thinking and the true scientific
foundations of mathematics. Such the mathematical education is
dangerous for a humankind as a whole?.
The ?clinical? picture of the ?bourbakism? drawn by Prof. Arnold
verily can be called a mental Acquired Immunodeficiency Syndrome of
brain, i.e., shortly a MENTAL-AIDS. As an experience testifies, the
MENTAL-AIDS is a very infectious illness which affects especially easy
an unprotected kid?s brain, unfortunately, without any perspective to
get well: as far back as XVIII c. the great English philosopher G.
Berkeley said that "a human-being mind, immersed in high level
abstractions from a young age, loses an adequate perception of the
real world to its adult age".
I shouldn't be surprised if many parents of modern schoolboys/girls
and students would like to bring an action against modern Cantorians
and their official "scientific" communities and journals because of
their deliberate cultivation and propagation of such the dreadful
infectious social disease as that MENTAL-AIDS.
However that may be, today, in the very beginning of the XXI Century,
we have a much more painful diagnosis concerning prospects of modern
mathematics and mathematical education.
I state and can prove that the main historical source of this
dangerous social illness is just the modern cantorianism with its pure
abstract, ambitious transfinite constructions with an empty ontology,
based upon the only Cantor's theorem on the uncountability of real
numbers. [...]
I am sure that there is a lot of judicious mathematicians and simply
provident parents of future mathematicians of genius who would not
like that their children became "formal machines" used to execute
criminal, terroristic, anti-human "deductive" plans. I hope to have
their active support.
Nobody, including the BSL-team, will save the Cantor's transfinite
"paradise".
Sincerely yours,
Alexander Zenkin [...]
P.S.1. I have attentively read the enclosed BSL-reviewer's report
[...] and regret too that the report reviewer distorted the sense of
the Comments-1 deliberately, high professionally and fundamentally,
and misled you and the mathematical and symbolic logic community as
to
the important problems touched upon in the Comments-1. [...]
P.S.2. [...] my system VISAD (for VISual Anaysis of Data), based on
the Cognitive Computer Graphics (CCG) conception, has fulfilled a
comparative analysis of CCG-images of the W. Hodges' paper text and
the anonymous BSL-reviewer's report text and has established that the
both authors are the same face <are as like as two peas>. - It is
quite interesting result from the professional scientific ethics point
of view, is not so?
http://www.ccas.ru/alexzen/papers/OPEN_LETTER-2_to_the_BSL.doc
Regards, WM
|
|
|
|
