Re: Problem with `big oh' estimates in number theory

From: Angus Rodgers (angus_prune_at_bigfoot.com)
Date: 02/09/05 

Date: Wed, 09 Feb 2005 17:29:34 +0000

On Wed, 09 Feb 2005 16:29:51 +0000, Robin Chapman
<rjc@ivorynospamtower.freeserve.co.uk> wrote:

>Angus Rodgers wrote:
>
>> This isn't just some `obvious' point, which he expects
>> the mathematically mature reader to work out for him/
>> herself.
>
>I think it is ....

Oh, well, I've always known I didn't have that `maturity'
thing! (Only age, and lots of it!)

But - taking my life in my hands, I know! - I will continue
to question whether it can possibly be a good idea to use
a notation O() when you have to check it every single time
it's used, in order to determine what the lower bound x_0
happens to be in each case, when the whole point of the
O() notation - I thought! - was to enable you to *forget*
about the values of M and x_0 when you *don't* need to
know them.

I think it's a pain, and a quite unnecessary one, to boot.

Remember, you have to question each use of the O() notation,
*and* trace it back through all the intermediate results
being appealed to, checking (perhaps on paper, perhaps in
your head) what value of x_0 *could have* been recorded at
each stage, if only the author had thought/bothered to do
so. And I can see no compensating advantage whatsoever,
except: (1) adherence to tradition; and (2) the fact that
the O() notation certainly *looks* neat, and sticking an
extra subscript on it (for the value of x_0 when needed)
messes it up no end. But that's a price I'd be willing
to pay, for the simple (and surely obvious) gain in sheer
reliability of reasoning. Stick in the value of x_0, and
you can much more easily tell whether something is a valid
proof or not. Leave it out, and your program code (so to
speak) suddenly acquires all sorts of confusing global
references, violating the `black box' integrity of the
theorems to which you are *apparently* (but no longer
actually) appealing.

If it's OK to drop the value of x_0 even when it *is*
needed, why don't we randomly do this with other bits
of information?

The value of x_0 is needed - you're not disputing that -
so why isn't it in the book? Is the reader required to
pencil in all the required values, for future reference?
If so, why not just print them there in the first place?

I don't mean to sound so indignant, by the way. I really
like Apostol as an author - honestly! But it's irksome to
have identified a problem in somebody's writing (and spent
hours and hours thinking it through, and double-checking),
only to be told that it's not - it's a problem in my reading!

Why should the reader be expected to jump through so many
hoops, when: (1) it is not normal mathematical practice to
require this (normally you state precisely the results you
plan later to use, and throw away only your temporary,
intermediate results and notations); and (2) it looks for
all the world like a minor *slip* on the part of the author,
which was never intended to be turned into a holy *rule* for
the poor old reader? (i.e. me - condemned as a poor reader,
and now feeling old!)

I think I'll go and lie down.

>> If that were the case, then filling in the
>> missing steps of the argument would not have required
>> a strengthening of the conclusion of a preceding theorem!

What he said. :) 

-- 
Angus Rodgers
(angus_prune@ eats spam; reply to angusrod@)
Contains mild peril

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