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Re: Landau letter, Re: Mathematica as a New Approach...
*To*: mathgroup at smc.vnet.net
*Subject*: [mg127999] Re: Landau letter, Re: Mathematica as a New Approach...
*From*: Andrzej Kozlowski <akozlowski at gmail.com>
*Date*: Sat, 8 Sep 2012 03:07:28 -0400 (EDT)
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On 1 Sep 2012, at 08:29, John Doty <noqsiaerospace at gmail.com> wrote:
> I also note that experimental psychologists, notably Macfarlane and
Tolman back in the 1930s (!), have established that even laboratory
animals are capable of constructing abstract models, specifically "maps"
of mazes. Their experimental results are inconsistent with the idea that
in learning a maze, the animals merely learn the sequence of steps
needed to solve it. Abstraction is a useful mental ability that is
hardly confined to mathematicians.
I am quite prepared to believe that "abstraction" can be very useful to
"laboratory animals" - for example, it may be useful in getting the food
that placed at the exit of a maze by an "experimental psychologist" or
even better (though no doubt less frequently) getting out altogether out
of the laboratory (alive). Moreover, it strikes me that the kind of
"abstraction" these clever animals are capable of is closer to what the
"average man in the street" is capable in mathematics than the latter is
to the true mathematical geniuses. If we look at any history of
mathematics we will find that it is rather like the history of art - the
contribution made by "ordinary men" or even "ordinary mathematicians"
hardly features in it. It is only a very tiny fraction of mankind that
is is actually responsible for virtually all the advanced mathematics
that exists today. So it seems to me that the really interesting
question is not how people learned that 1 + 1 is 2 but how people like
Grisha Perelman come to exist, do we need more of them and we get more
of them, for example, with the help of Mathematica or Conrad Wolfram's
"Computer Based Math Education"?
We are told that evolution has something to do with the development of
mathematics. No doubt at some level it is true, but hardly at a level we
would find interesting today. It is claimed that "mathematical ability
is useful". Sure, but for how long in human history has it been more
useful than, say, powerful muscles or running or throwing abilities?
Historical examples, e.g. Archimedes and the Roman soldier or Abel's
death in abject poverty are not terribly encouraging. Perhaps people
with high mathematical abilities are then better able to attract the
opposite sex (which would give them an "evolutionary advantage")? I can
almost hear the bitter laughter of my departmental colleagues. So what
exactly is the evolutionary path from a near "laboratory animal" to
Riemann or Perelman? It does not seem, I think, to lead via the
ordinary man, at whom Conrad's educational ideas are addressed.
Personally, I do have an answer to this that satisfies myself, but it is
not as entertaining as the ones that have already appeared in this
thread so I will keep it to myself. But I have no doubt that better
teaching methods (if they are indeed better) will not make the slightest
impact on the number not only go mathematical geniuses but also on the
number of good, professional mathematicians who sometimes manage to make
some impact on the footnotes of the history of mathematics.
So, since this thread has nothing at all to do with the education of
"real mathematicians" (and since it is obviously they are unlikely to be
converted into something else by arguments such as that what they do is
"meaningless" and irritating for certain non-mathematicians) the real
question seems to be: do we really need to make many people better at
the "other kind" of mathematics? Or, is an increased competence in
using Mathematica people with little mathematical interest or aptitude
going to be of a serious benefit to them or to the rest of society?
Well, I think the jury is still out on this one. I can see one obvious
benefit: to Wolfram Research. Also, perhaps to people like myself, who
sometimes get asked to teach this sort of thing. But if this is going to
happen at the expense of turning people away from non-mathematical
subjects where their real interests and talents like to make them
second-rate computer mathematicians, I think they and society will be
the poorer for this. In fact, I have yet to see any convincing argument
that more mathematicians (even of the very best kind by any definition)
is what society needs (people who think so should reflect on the fact
that the Soviet Union, particularly the university of Moscow, had a
fantastic school of mathematics, both pure and applied, and how much
good it did to it).
Of course I am not addressing the one serious issue that was raised by
the original poster - the proper mathematical education for many or most
experimental scientists, including physicists. In this respect, I
actually agree with Alexei. But as for the attempts to extend the
discussion beyond its original scope.
Andrzej Kozlowski
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