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Re: Landau letter, Re: Mathematica as a New Approach...

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  • Subject: [mg127999] Re: Landau letter, Re: Mathematica as a New Approach...
  • From: Andrzej Kozlowski <akozlowski at>
  • 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> 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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