History and Philosophy of Mathematic Reform
– Dr. William Lawvere, InterDisciplinary Conference, Windsor, February 10, 1996
–
For a long time calculus teachers have been demanding
better textbooks. In recent months, after much fanfare, a few
books claiming to respond to this demand have been launched. But
many teachers are saying already that this "new" is very similar
to the old that worked so poorly, except now with much more and
elaborate and subtle attempts to persuade the student (and the
teacher) that some kind of "understanding" is going on, even
while all the more resolutely concealing the essential
principles.
Also in recent months, a journal for the guidance of college
math teachers carried a lead editorial aiming to refute the "myth
of scientific literacy". Under the banner of this "scientific
literacy" a major curriculumrestructuring over the past decades
had been justified, which many scientists judged to be a forced
wateringdown of content, aiming at producing students who had no
understanding of science but who would be able to speak and write
about it all the same. However now we are told that most students
should not achieve even this "literacy", but only a muchreduced
goal of "awareness". A current example (from the journal of a
teachers union) of what such awareness means is the ability to
just recognize terms such as "chaos theory" and "paradigm shift"
and on that basis submit to whatever postmodernist
reorganization is being proposed.
The claim that we are now in a "postmodern" and "playful" age
is being put forward, by university centers for pedagogy, as a
basis for replacing the learning of mathematics with "an inquiry
into our habits". In an attempt to enlist some professors for
this transformation, the old ecoterrorist claims, that
mathematical thinking supports military technology and
destruction of the environment, and that we "are" our
mathematics, have been resurrected.
Philosophical basis for these counterreform "reforms" is also
being offered in recent months. In U.S. and Canadian journals for
college math teachers, it is revealed that mathematical ideas are
social in character and that such social rituals constitute a
third category of being, and we are assured that curriculum comes
between the individual and the collective. At least two recent
mathematics publications have praised the philosophers Kuhn,
Lakatos, Feyerabend, and Popper, one saying that the philosophy
of mathematics needs some great innovators in their mould, the
other calling them exemplary "philosophers of Irrationalism
".
The relation in 1996 between teaching of mathematics and
philosophy of mathematics is only the latest stage in a long
process. Three hundred years ago the Leibniz rule for the rate of
change of a product, and Newton's theorem that the rate of change
of the area under a curve is the height of the curve, were made
explicit and publicized. This calculus was developed by Bernoulli
and Euler, Cauchy and Maxwell into the universal instrument for
designing engines, ships, electric power and communication
systems, etc. Yet after three hundred years, most people,
including most people who actually build those wonders, are still
in no position to challenge these designs on their own grounds
because knowledge of the instrument has been denied. Even more
bizarre: the lack of comprehension is shared even by most
people who have been through courses which the State has gone to
considerable effort to provide.
There do exist professors who believe that most students are
incapable and unwilling to learn any serious subject. But already
around 1750 in Milan, Italy, Maria Agnesi wrote and printed a
textbook, based on the premise that all Italian youth could and
should learn calculus. Her enlightened vision is yet to be
realized (in Italy or elsewhere). What happened?
After the French revolution had introduced the decimalized
system, it was necessary to recalculate the trigonometric and
logarithmic tables used in construction, navigation, etc. A few
years ago the following true story about this was widely
publicized by a giant multinational computer corporation: An
accomplished engineer called Prony was appointed to organize the
large task of calculation. Borrowing from Adam Smith, he divided
the personnel into three levels: Level A consisted of a few
mathematicians who were able to invent appropriate formulas,
Level B of a somewhat larger group of people who were able to
convert the formulas into algorithms, but a much larger group C
of men actually carried out the algorithms by adding and
multiplying. The point which the computer corporation found
worthy of resurrecting two hundred years later was this: "It was
found that the work went more smoothly if those in group C knew
no mathematics."
In Britain in the 1830's millions of people were in motion,
demanding democracy. Among the measures for quelling the demands
of this Chartist movement, the Privy Council created for the
first time a system of statesupported schools and
teachertraining and inspection, to "introduce order and
discipline into the workingclass population when older methods
of wielding authority had broken down." (Encyclopedia Brittanica)
The education provided by these new schools is consciously two
tiered: for example one of the defining documents requires that
"Arithmetic is the Logic of the poor". In the school systems
modeled on that, it is often apparent that while teachers are
struggling to teach enough, administrative expenditures and
regulations have the larger aim of insuring that we don't teach
too much.
In the period before 1848 there was optimism for the
possibility of general enlightenment. For example, the Danish
physicist Oersted, who discovered an important principle relating
electricity and magnetism, set up an institution to make it known
to all. The German mathematician Grassmann, who in 1844 published
a new theory and method in geometry which now is becoming widely
used by physicists, was actually a highschool teacher who
insisted that his new dialectical philosophy was at least as
important, since it was directed explicitly at assisting students
to learn and understand. However, in the 1870's, when one of his
followers published a book showing in detail how Grassmann's
methods could be used to teach not only geometry, but also to
introduce calculus in highschool, he received a very scathing
review and condemnation for suggesting such an upset in the
Prussian order of things; the author of the review, Felix Klein,
was later the official representative of that Empire at the
World's Fair held in conjunction with the opening of
Rockefeller's University of Chicago.
Following the Privy Council's lead there has been developed a
body of technique, a sort of prizefighter's technique for
occasionally appearing to give in to the demands for reform while
actually thereby directing our energies to serve an opposite aim.
For example, forty years after Grassmann's death the Prussian
establishment decided to make him "a great German soul" but he
was portrayed in pragmatist journals as a philosophical idealist.
In 1908 Lenin defended Grassmannn's materialist philosophy from
this unwarranted distortion, and also remarked, in connection
with some proposals to introduce higher mathematics into the
schools, that it was surely not being done in order to deepen and
broaden the knowledge of science, but rather to provide a basis
for the promotion of idealist philosophy.
Indeed, the popularizers of pragmatism and the organizers of
collegiate math teaching were closely associated for many years,
and the leading circles of philosophy, such as the Gifford
Lectures in Scotland and the Silliman Lectures in Yale, began to
systematically misuse mathematics and especially their audience's
ignorance of mathematics. The Prime Minister of the British
empire (who was nicknamed "bloody Balfour" for his suppression
of the Irish and later would be famous for his declaration in
support of Zionism) who wrote several books on philosophy, was
also known for his Education Act which reorganized the high
schools. Balfour stated in one of his Gifford lectures "I wish I
were a mathematician".
Indeed being known as a mathematician became a road to
historical recognition as a philosopher. For example Bertrand
Russell's opinions on everything became soughtfor and he even
eventually received the Nobel Prize, partly because of his
notoriety as a mathematician; through clever wordplay he devised
a new branch of philosophy known as "foundations of mathematics"
whose only role is to give mathematics permission to exist, and
which must be written in symbols different from the usual
mathematical ones. The latter ruse he had learned from Peano,
whose followers had proudly produced a highschool text written
entirely in symbols, in order to dispel any false idea that with
"numbers", "lines", or "space" we are really referring to
anything; amazingly, this work was advertised as a clarification
of Grassmann.
Perhaps the bestknown 20th century figure who consciously
guided his actions by the pragmatic philosophy was Mussolini; but
probably as important was John Dewey, whose teachings and
organizations had a tremendous influence on education throughout
the world. He was occasionally quite clear about the direction of
his reform; for example, in China in 1919 he gave a course at a
college for teachers, in which he enunciated his infamous
principle:
"Teach the child, not the science"!
Of course conscientious teachers through the centuries have
done both: the acquisition of some portion of the accumulated
knowledge of humanity (science) is the purpose of the child's
presence in school, but the teacher endeavors to guide this
acquisition with due regard for each child's particular
situation. Why then Dewey's prohibition of the teaching of the
knowledge? In China he compared the alleged "authoritarianism" of
science with the recentlyoverthrown imperial regime, and since
then the broadbrush charge of "authoritarianism" has been used
thousands of times as a pretext to eliminate from school systems
the teaching of the deductive aspect in geometry, of the
grammatical parts of speech, of the diagramming of sentences,
etc. Indeed, now many college teachers of mathematics analyze
that a large part of the difficulties which are had by students
fresh from highschool are not due to mathematics itself, but due
to their first real encounter with the requirement that ordinary
language be used in a precise way. Dewey's powerful principle of
"teaching the child, not the science" has many corollaries, such
as the antichild theory that "learning is fun", and ultimately
the logic whereby jokes take the place of reasoning; certainly
the principle includes the injunction often addressed to pupils:
"Say it in your own words". This injunction is very attractive to
teachers, who know that understanding requires a conscious act by
the individual. However, the whole atmosphere of the school often
mandates that "in your own words" should mean "as imprecisely as
possible" thus destroying the acquisition of concepts in any
usable form.
Already in the early 1900's the tradition, that school should
be devoted to learning the accumulated human knowledge, was being
eroded in another way: The steel city of Gary, Indiana was built
whole at Rockefeller's demand, the factories, workers' homes,
sidewalks, and school system. To minimize the free time of the
sons and daughters of the workers, extracurricular activities at
the schools were declared essential to the development of "the
child". This Deweyendorsed school system was studied by
administrators from all over the world who journeyed to observe
it. The whole Dewey program was styled "progressive" education,
illustrating through the use of this term the fundamental tenet
of pragmatist epistemology: truth is what you can get away
with.
In 1915 the U.S. mathematical organization split into two, one
devoted primarily to the promotion of research, and the other
supposedly devoted to the promotion of college teaching. The
latter maintained and deepened its ties with pragmatic philosophy
in 1921 when, at a meeting at Wellesly College, the widow of the
leading publicizer of pragmatism, Paul Carus (whose stated aim
was to promote religion on the basis of recent science), gave
several thousand dollars to finance a series of monographs. At
the same meeting the president of the organization gave his
address entitled: "The religion of a mathematician", consisting
of principles such as "since we know infinitesimals, we must also
know our own insignificance; since we believe in infinity, we
must also believe in an almighty; since we can imagine the fourth
dimension, we can also imagine heaven, etc." This policy of the
organization has never been repudiated, and its publications,
which aim to give guidance to college teachers, have over the
years refined to a precise art a writing style similar to that of
the Scientific American, i.e. it is presumed that the readers
will not advance from a lower to a higher level, and hence under
the guise of "popularization", all concepts are made sufficiently
imprecise so as to be unusable by anybody. The deliverer of that
presidential address was the author of one of the few texts on
the history of mathematics then available in English.
Mathematics itself is often said to have made more advances in
the 20th century than in all previous centuries. The advances
include not only the solution and formulation of difficult
problems with geometrical and other content, but also (
indispensably to that) the development of unifying concepts which
are of great simplifying and clarifying value. An opportunity
seemed to present itself around 1960 to disseminate that
simplifying and clarifying value to a vastly larger number of
students. The occasion, as I understand it, was the following:
The U.S. ruling circles, fresh from rejoicing that their friend
Khrushchev had succeeded in overthrowing the socialist system and
was transforming it into a pseudosocialist system, suddenly
realized that they were thereby also faced with a rival
superpower. This implied a certain shift of the boundary between
the B and C Levels on the Prony scale, a readjustment of the line
between "arithmetic" and "logic" on the Privy Council's
antiChartist plan: more students would have to learn more math
and science in order to counter the Sputnik threat. Whatever the
precise details of the background, the opportunity was presented
around 1960 to have university researchers directing summer
schools for eager highschool teachers, to have writing teams
producing new text books for pupils and teachers, etc. The
challenge was taken up enthusiastically by many professionals in
the spirit of letting those concepts, which had proved to be so
enlightening for them, also serve to enlighten everyone. Of
course such an undertaking requires several years of pupil
feedback and text revision (and new mathematical research!) in
order to become successful. But that stage was never reached
because the movement was discredited; the enthusiastic
professionals had underestimated the preparation of the
opposition. By an artful confusion of the meanings of words like
"foundation", the foundationalist trend (which had become entrenched
since the days when Bertrand Russell started at the London School of
Economics  LSE) insisted that the texts must be written in their
idiosyncratic
notation. And the professional schools of pedagogy (housed since
Dewey in ivory towers remote from the actual scientific
departments) took leadership of the movement from the bewildered
scientists, insuring its destruction.
It would seem that an obvious way to improve math teaching
would be to give more examples and more applications. This is
correct, of course, but to give that as a demand and stop there
was again to underestimate what we are up against. The 1970's and
80's saw the publication of many "applied" calculus texts in
which explicit principles were subordinated to problems
watereddown and distorted beyond usability from various fields.
But as many professors in those various fields understand, math
is theory. What a student needs of math in a field of application
such as chemistry, business management, etc. is to know math as
well as possible in order that the applied concepts be
approachable with little mathrelated mystery and in order that
mastery of appropriate new methods can be partly selfguided.
Isolated, particular methods learned mechanically and then
forgotten, and especially, halfbaked attempts to teach an
alleged application instead of the explicit principles of
calculus, can only negatively affect the students' ability to
apply math.
The demand for better calculus text books in the
Englishspeaking world thus began, for the above and many other
reasons, to become more threatening. The initial response of the
publishers (that they would never change their policy of
offering the next year an exact copy of the competing text which
had made most profit the previous year, only with more colors)
was met by deserved contempt. Some of the more recent offerings
are the result of multimillion dollar government
intervention.
Contrary to the portrait of the professor who views teaching
and research as inimical to each other, many see them as mutually
supportive.
Many of the ideas, which have led to long and fruitful
development by researchers, actually arose from attempts to
explain matters more clearly to students. For example, attempts
in the 1960's to provide a clearer, simpler, yet rigorous base
for understanding calculus led to a new trend of research in the
foundations of topology, logic, and analysis within which many
innovative papers and over a dozen books have now been produced.
On the other hand, research leads from time to time to new
synthesizing concepts, which clarify matters enormously for the
researchers, who then struggle to find ways to spread this
clarification to students. For example research into the
mathematical foundations of continuum thermomechanics and the
constitutive relations of materials led to new, more direct, ways
of dealing with infinitesimals, function spaces, and extensive
quantities, which are now being taught to undergraduates in some
places.
Research and teaching are of course different aspects of
endeavor, but as long as they are still alive, they have an
orientation in common, a commitment to tirelessly combating the
absence of knowledge.
History shows that the teachers, yearning for a greater
opportunity to participate in the creation and dissemination of
enlightenment, will not be satisfied by waiting for this or that
establishment entity to provide it. Not only would the
fulfillment of these needs remain forever a mere policy
objective; our enthusiasm would continue to be used as the engine
for the spread of still more pseudoknowledge and pessimism. The
problem can be taken up for solution, without milliondollar
grants, both by devising teaching materials which reflect the
actual historical development of a given field (neither repeating
some entrenched hundredyear old false summation, nor succumbing
to the ultrarevolutionary postmodernist degeneration), as well
as by making explicit the philosophy which emerges from actual
research developments of the recent decades. Collective effort is
necessary, however, to concentrate such materials and to
disseminate them so as to serve the needs of society as a
whole.
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