Re: sin x / x tends to 1...

On Mon, 5 Sep 2005 13:50:14 +0000 (UTC), Darren J Wilkinson
<d.j.wilkinson@xxxxxxxxx> wrote:

>David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:
>> On Sun, 4 Sep 2005 20:09:00 +0000 (UTC), Darren J Wilkinson
>> >I know we've been going around the houses a bit, but I think that we may
>> >be getting to the root of my problem. Ignoring the regular/irregular
>> >polygon stuff, which we can regard as mere detail, you (and one or two
>> >other people...) are choosing to _define_ arc length as the sup over
>> >inscribed polygon lengths, knowing that the sin x / x limit immediately
>> >follows.
>>
>> Yes, the fact that sin(x)/x -> 1 follows immediately. But that's
>> not the reason "I" "chose" to define arc length that way. That
>> _is_ the _standard_ definition of arc length!
>>
>> How _should_ arc length be defined, in your opinion?
>
>You mis-understand me. I'm perfectly happy with the definition of arc
>length you adopt. But if you think about what it is saying, it is saying
>that as you make your "ruler" smaller, the discrepancy between "true"
>arc length and the length of the "ruler" will become insignificant.

It's really not quite that simple. See [1] below - I don't want to
explain here lest we lose the thread of things.

>It is
>not something you are proving - it is something that you are (quite
>reasonably) asserting. And if you think about what the sin(x)/x limit is
>saying, it is saying exactly the same thing in the special case of a
>circle.

Not really. It's saying that the length of a certain curve is
very close to the length of a certain line. But it's not saying
that that's so just because we're looking at things very closely -
it's simply not true that two curves that are very close together
have lengths that are close to equal.

>So all I'm saying is that it is a bit ingenuous to talk of
>"proving" the sin(x)/x limit when your definition of arc length is
>assuming something much stronger.

Maybe you never thought about the following: What you say here
could be said about _any_ proof in mathematics. There are always
axioms and definitions, and the fact that the theorems follow
from the axioms and definitions show that when we prove the
theorems we begin by assuming something much stronger.

>> You're
>> saying I chose that definition because it would give the
>> desired limit - how you can _think_ you know this about
>> why I did something I can't imagine, but you're wrong
>> about that. I didn't "choose" that definition at all,
>> it's the definition I _learned_.)
>
>As I've said, I think you mis-interpreted my (admittedly slightly
>provocative) choice of words.

I don't think so - you may not have said what you meant.
You said quite explicitly that the reason "I chose" that
definition was to make sin(x)/x come out right, which is
not so. You stated quite explicitly that if that hadn't
worked I would have chosen another definition. That's
also not so - you're stating things you couldn't possibly
know about why I did this and what I would have done if
that. Things which are simply not true:

>> Two questions for you: How _should_ I define arc length,
>> and how, in your opinion, would I have defined it if
>> I hadn't been determined to justify that limit?
>
>So, as I said above, I'm happy with your definition of arc length. But
>hypothetically speaking, if the sin(x)/x limit "turned out to be" (say)
>2/3 (I AM NOT SUGGESTING FOR A SECOND THAT I THINK THIS!), then you
>might (say) define arc length to be 3/2 times your definition, for
>example...

Two comments, one sort of general/philosophical and one
specific about the math. The general comment first:

Why in the world would you think that "we" would do that, instead
of just concluding that the limit of sin(x)/x was in fact 2/3?

Really. When all this started you had reasonable concerns
about circularity. But it's been shown that there's nothing
really circular about the actual math. You insist that
some circularity remains, but the circularity that remains
depends on your assumptions about people's motivations
and your assumptions about what people would have done
under other circumstances - we haven't really proved
anything, because we chose the definitions in order to
make things work out.

This is just silly. We chose the definitions because they
seemed reasonable, and they allow us to prove things that
seem like they "should" be true. If that means we haven't
really proved anything then we never prove anything in
math, ever. The fundamental theorem of calculus depends
on the fact that the real numbers are a complete ordered
field. So we haven't really proved anything when we prove
ftc, because we chose the axioms in order to allow us to
prove ftc?

If you _are_ saying that fine, it's a valid point of
view, not that there's much content to it. But I suspect
that you're saying to yourself that the bit about this
definition and sin(x)/x is different, because the proof
of one from the other is so direct. I don't see the
difference - it seems to me that you're simply not
considering the possibility that what we've learned
from all this is that the fact that sin(x)/x -> 1 is
simply not as deep as you thought - the fact that a
proof is almost trivial doesn't say to _me_ that
it doesn't really prove anything, what it says to
me is that the result that was proved is easy.

The second more mathematical comment, which come
to think of it might convince you that the proof actually
does prove something in spite of it being so clear from
the definition:

No, if it turned out that sin(x)/x -> 2/3 we would not
adjust the definition of arc length to be 2/3 of the
sup of the lengths of inscribed polygons. Why not?
Because if we did that our definition of the length
of a curve would be inconsistent with our previous
notion of the simplest possible curve, a straight
line segment!

****************

Here's the comment I didn't want to insert above.
Come to think of it it also may have some effect
in convincing you that the proof that sin(x)/x
still counts as really proving something, even
though it's so simple:

Say L is the diagonal of the unit square. Let
P_n be a staircase polygonal curve going from
the lower-left corner to the upper-right corner,
with steps of size 1/n: P_n starts in the corner,
goes up 1/n unit, right 1/n unit, up 1/n unit,
right 1/n unit, etc, until it reaches the other
corner.

So P_n tends to the diagonal as n tends to infinity;
if n is large enough you can't see the difference
without your glasses on. The diagonal has length
sqrt(2), while P_n has length 2.

So. Even with the standard definition of arc length,
it's simply not true that "if you measure something
with a small enough ruler you get a length close
to the right length". At least a sufficiently
broad interpretation of that last statement is
shown to be false by that staircase curve. So
even with the standard definition of arc length
it's _not_ obvious that sin(pi/n) is close to pi/n when
x is small, just because that line segment of length
x is very close to that segment of length sin(x).
There's still a _proof_ required. The _substance_
of the proof is that if we take n of those segments
they fit together to form a certain polygon, while
if we take n of those arcs they fit together to
form a whole circle...

It's simply not true that the definition of arc
length was motivated by a desire to make sin(x)/x
tend to 1, as you've stated. Probably what you
really meant was that the definition of arc length
was motivated by the idea that if this curve is
close to that curve then the lengths should be
close, as you more or less suggest above. It's
not implausible at least as a parable to imagine
that that _was_ the motivation, but then people
noticed that that statement is simply not true,
because of examples like that staircase. So
then people are very careful for a while, but
they finally find that with the now-standard
definition things seem to work out the way
they want.

>I don't think it is going to be productive to persue this
>further, however! ;-)
>
>Regards,


************************

David C. Ullrich
.

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