Feedback on Doron Zeilberger's Opinion 65

Feedback by Tim Gowers

From W.T.Gowers at dpmms dot cam dot ac dot uk  Tue Jun 21 13:41:17 2005
On Tue, 21 Jun 2005, Tim Gowers wrote:

 
Dear Doron,
 
I've just looked at your opinions page for the first time for a
while, and read your article on two pedagogical principles. I 
was particularly interested in the first, because as a result
of editing the Princeton Companion I have become incredibly 
conscious of it myself -- I'm tempted to say that I discovered
it independently. Of course, it doesn't bother me that Gelfand
got there first -- it is SO clearly correct that it would be a
miracle if I had not been anticipated. Instead, we have the
depressing miracle that something so obvious should be practised
by such a small percentage of mathematicians. I feel quite evangelistic
about this, and have already started a one-man (except that now
I see that you are an ally) campaign to publicize the principle.
For example, a few weeks ago I was asked to give a talk about
the Princeton Companion, and EXAMPLES FIRST was one of the main
themes (which I illustrated by an example first: I gave a ridiculous
and unmemorable definition of a "C-space" which was in fact a
mathematical model of a car, and as soon as the word "car" was
uttered, the definition was magically easier to remember).
 
I had always been aware, of course, of the value of giving the
simplest non-trivial example. The thing that has really struck
me is the value of giving it FIRST. I think it is very important
to stress that this is an independently important part of the
Gelfand principle (or else, if you were not including it, a 
separate and equally important principle).

Here is my "proof" that it is better to start with concrete 
examples and proceed to abstract definitions than it is to begin
with the abstract definitions. If you give the example first, then it
is easy for the reader to understand, so not much effort is needed
to remember anything. Then, when you are presented with the abstract
definition, you have a mental picture of an example, so the various
components of the abstract definition become labels that you attach
to this picture. If, on the other hand, you give the abstract definition
first, then the components are meaningless, so you have no choice but
to memorize them as if you were learning Chinese vocabulary or something.
Then when you see the example, you have to go back and see how this
meaningless stuff does in fact mean something. But that effort of
memorization should have been unnecessary!

I have a theory about why it is customary to do things the wrong
way round. Suppose, for example (again -- it's very important that 
I should stick to my principles) that you want to explain what a
Lie group is. What could be more natural than to start with the
words, "A Lie group is"? But if you do that, then you are more or
less forced to give the abstract definition. The naturalness of the
wrong approach means that the examples-first principle is a habit 
you have to get into rather than something that should happen with 
no effort at all. That is why I think it should be publicized as 
much as possible.

I have just written a long article that is largely on how we memorize
mathematics. When I feel happy with it I'll put it up on my website.
It deals with some of these questions.
 
The main thing I wanted to ask was whether you know the history of
the Gelfand principle. Has he ever gone into print with it? If there
is something I could refer to then I'd like to do so.
 
Best wishes,
 
Tim
 
PS I think I agree with the second principle too, but I feel less
passionately about it.

Feedback by Volodia Retakh

Tue Jun 21 13:01:53 2005
From: Vladimir Retakh vretakh at math dot rutgers dot edu
To: Doron Zeilberger 
cc: W.T.Gowers at dpmms dot cam dot ac dot uk


Dear Doron,

I cannot say whether the Gelfand priciple was ever stated in a written
form. However, I remember that Gelfand asked Joseph (Iosif)
Bernstein what is the main thing Joseph learned from Gelfand.
Bernstein replied; "That Mathematics must be done on simplest
examples". Then Gelfand asked: "So, have you learned it?"
and Bernstein replied: "Not entirely; whenever I could find
a simple example, you would find an even simpler one."

All the best,

Volodia

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