Re: Thinking about metric spaces

On Wed, 27 Jun 2007 19:29:16 -0500, quasi <quasi@xxxxxxxx> wrote:

On Wed, 27 Jun 2007 11:25:02 -0700, bit188 <mariana@xxxxxxxxxxxxx>
wrote:

Thanks everyone.

Tommy1729: as regarding my "intentions" with this space, I was going
to... eh, *try* and define angles and trigonometry, but apparantly
that's not a good idea.

Well, it's an interesting idea.

For example, the law of cosines is one possible way.

I think you'd do better _not_ buying the taxicab book.

Play with the ideas yourself at an elementary level. So what if you
end up reinventing the wheel? At the early levels, almost anything you
try has been tried before and analyzed more completely than you
probably would do. Knowing that takes a lot of the joy out of
exploration. Of course, you should read the real theories, but on the
more experimental stuff, I say just try it. Don't even ask if it was
done before (since the answer will almost always be yes).

It's just not as much fun to read what someone else did.

Trying it yourself gives you the sense (even if illusory) of math
research -- it's exciting and builds self-confidence. Even the failed
attempts help develop critical intuition.

As an example of what I'm talking about, here's a simple question.

In a metric space X, to say 3 points a,b,c are collinear means what?

There are various possible answers, some more natural than others, but
if you haven't yet thought about or seen a discussion of this
question, it has value as a thought experiment. Will any answer you
propose be sound? Maybe, maybe not, but either way, that's the
experience you want. If the idea is sound, will it be new? Probably
not, but who cares?

quasi
.

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