Kaba
Posts:
289
Registered:
5/23/11
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Re: Maths pedagaogy
Posted:
Mar 18, 2013 3:43 PM
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17.3.2013 19:09, David C. Ullrich wrote:
> There are better reasons. Not that the writer doesn't have the
> energy to write out the trivial proof, but that it's "obvious"
> that the book will be a better book, easier to read(!), if
> the triviality is omitted.
>
> To give a silly example, suppose we're proving something,
> and we need the fact that the sum of two even integers
> is even. Which of the following seems best?
--8x-- (an example of a proof on too low an abstraction level)
> Ok, that's a little silly. But I hope the point is clear.
> If one includes all the details then the book will
> be much too long, and more important, the book
> will be much harder to read! Leaving the
> honestly trivial parts to the reader makes it
> easier for the reader to see what the main
> _points_ to the argument are.
To say it otherwise, you need to choose the abstraction level correctly.
The amount of information that needs to be assumed necessarily increases
as the abstraction level increases. This is something that you, quasi,
Schmuel and I fully agree on (this post is also a reply to them on this
subject).
The detail you illustrate in your (admittedly amusing:) proof is not the
detail I mean.
Providing a detailed proof is an orthogonal task to choosing the
abstraction level. Whatever your abstraction level, by being detailed I
mean that you need to be precise _at that level_, not levels below it.
For example, if I were writing a proof on the non-degenerateness of
bilinear forms (see the correspondence with quasi), then providing a
detailed proof means that I will do so on the level bilinear spaces, not
on the level of vector spaces or set-equality.
>The difference between
> the two books is just that in one case the
> author's decision on this issue matches your
> preference much better than in the other book
I disagree. These are matters of style (in the sense of Strunk & White),
and style has a lot of commonly applicable rules; it's not just my
preference. They all aim at aiding the reader for easier understanding.
The difference between the same text with good and bad style is
dramatic. Lang has good content in his book. He could make the book much
better by improving on his style.
Wouldn't you say that Lee has a good style?
> There _are_ _many_ details left to the reader in the book
> you say you like!
Lee chooses his abstraction level very well, and is detailed in his
proofs in the sense described above.
--
http://kaba.hilvi.org
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