Re: Peano's space-filling curve

From: John Morgan (john.morgan_at_REMOVECAPSataraxie.fr)
Date: 06/09/04 

Date: Wed, 9 Jun 2004 10:57:10 +0200

David C. Ullrich <ullrich@math.okstate.edu> wrote in message
news:q3v8c0lnvlqvboc9htnmrb17mhopr2ddgp@4ax.com...
> On Mon, 7 Jun 2004 14:42:44 +0200, "John Morgan"
> <john.morgan@REMOVECAPSataraxie.fr> wrote:
>
> >
> >David C. Ullrich <ullrich@math.okstate.edu> wrote in
message
> >news:l354c0974tb47j6gtbn56lk09gmdopqfgk@4ax.com...
> >> On Sat, 5 Jun 2004 13:50:34 +0200, "John Morgan"
> >> <john.morgan@REMOVECAPSataraxie.fr> wrote:
> >
> Yes, I mean the point (1/3,2/3). It's this sort of thing
> that leads to the exasperation you perceive.

You'll have to forgive me for that. I get carried away by my
own incompetence sometimes, with some rather pathetic
sarcasm resulting. At least you accept that I am serious
about wanting to know. Thanks.

>
> I can _prove_ it passes through every point of the
> square. Proving this in a way that you're going to
> find accessible seems like it would take really
> really a lot of space. I'll be happy to show you
> the proof some day, when you can tell me the
> definitions of the following three words/phrases:
> "continuous", "compact", "uniformly convergent",
> and when you understand the definitions well
> enough that you can give me correct proofs of
> the following two trivial facts: (i) a uniform
> limit of continuous functions is continuous,
> (ii) the image of a compact set under a
> continuous function is compact.
>
> Let me know when you feel you're ready to
> study the proof.

When I read the above I was willing to give it a go. But a
couple of other posts have just come in that were, well not
to put too fine a point on it, illuminating. I need to read
'em and sleep. The information overload I've laboured under
for a month may be about to bear fruit and if this turns out
to be an 'aha' moment I can stop pestering you.

<snip>

> >Simply, because your generalisation "very clearly"
> >actually applies to the limited perspective that
> you and your fellow mathies have.
>
> Uh, it's not that simple. For example, many people have
> said very clearly that the curve passes through every
> point of the square. "The curve passes through every
> point of the square" is perfectly _clear_, even to
> someone who doesn't know the secret handshake. But
> no matter how many times people say the curve passes
> through every point of the square you continue to
> make statements about the points of the square that
> it doesn't pass through. That's a mystery that has
> nothing to do with limited perspectives.

When the car salesman says," It's a nice little runner, sir"
and I find I can't start the engine, and he tells me that
when I do start the engine I'll discover that it's a nice
little runner, should I part with my money at this point? I
reckon you, me and everybody else would reply instead,"But
the engine won't start"

> Please note that I'm _not_ saying that you should
> _believe_ us because we're Mathematicians! Simply
> not believing the things we say is in some sense
> a reasonable option. But if you simply didn't
> believe us it's hard to see why you keep asking
> us about these things...

Ahh. There's the rub, and time for explanations. You
of all people certainly deserve them.

Primarily, because I want to understand. Because
it (maths) fascinates me. The way its total non-worldliness,
nevertheless, invades every aspect of the real world. Also
it bugs me that there are things that I don't know about or
can't comprehend. And the only kind of knowledge I can have
any respect for is that which I understand.

I know my "knowledge" in the final analysis is superficial,
a short or long chain of syllogism that leads from premisses
to conclusion, from axioms to theorem (or worse, those
convoluted tautologies that seduce us into believing we
really know our arse from breakfast-time). I'm reassured
that such "knowledge" has pushed my species to the point
where it can take the whole universe. If we survive the war
against ourselves, that is.

With regard to the particulars, I will be satisfied to
understand the Peano curve in my own terms. That is, to make
up a (probably idiosyncratic) chain of reasoning that
squares with everything you have told me about it. I have no
delusions about becoming a mathy, but I've learned quite a
bit in the last four weeks.

Why am I telling you all this. Because you, David, and a few
others have stuck with me through thick and thin. It would
have been pretty easy to ignore me - along with my fruitless
scientific hunt for falsification and accompanying valueless
petulance - but you didn't, and I sincerely thank you for
that. If it really is that 'aha' moment coming up, you won't
hear from me again. Bet you're relieved, eh?

<snip>

> - neither statement is obvious, they
> both require non-obvious proof.

No chance for me, then!

> Like if someone tells you a tomato exists you assume
> that that implies that a red tomato does not exist.
> Makes perfect sense.

Not quite over yet, is it. Well, it's more like this. If
someone tells me a tomato exists, at this point I don't
bother to think further about the existence or otherwise of
different kinds or varieties of salad vegetables in what may
be an unfamiliar universe of discourse to me, unless I'm
specifically instructed to do so.

Cheers,

John
But I mean no harm, nor put fault
On anyone who lives in a vault
But it's all right ma, if I can please him
- Bob Dylan

P.s. I see you and Dan Grubb have begun discussing some
things of which I cannot possibly gain knowledge of, or even
perhaps comprehension of, in any reasonable time span - I'm
much older than I feel :-( Nevertheless, I hope to see my
name in the Acknowledgements, if a paper follows your
discussions ;-))))

J

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