Re: double declaration (OT)

From: Chris Uppal (chris.uppal_at_metagnostic.REMOVE-THIS.org)
Date: 03/01/05 

Date: Tue, 1 Mar 2005 08:08:32 -0000

John C. Bollinger wrote:

> In my experience (and I am not a mathematician, though I do hold a
> degree in the subject) [... ]

Snap ! In fact I can be a touch more precise than that -- specifically, I am
not a /pure/ mathematician ;-)

> [ ...] the integers are widely described, treated,
> considered, etc. by mathematicians as a subset of the reals. It has
> been some time since my formal training, but I am sure I can turn up
> examples if I look.

Oh, I'm sure I could find examples, especially in 1st-year texts (or lower) of
statements like:
    N > Q > R > C
(where the letters are their double forms, and ">" is used for
is-a-sub-set-of -- where's Unicode when you need it ?). But such texts /are/
introductory, and they are being sloppy with the terminology (for good
reasons).

> Moreover, I reject the proposition that absence of a construction for
> the reals independent of the integers would imply that the integers are
> not also reals. There is no circular definition there, because the
> integers can be constructed without reliance on the (rest of) the reals.

I'm sorry, I don't follow your argument here. If the integers /are/ reals,
then the set of integers cannot be defined without the set of reals. For the
same reason as if you don't know what a blaskoop is, then you can't make any
sense of the subset of the blaskoopii that are called the brindleshooms.

Perhaps you are taking a more realist stance than I care to ? If the reals and
integers are "just given" somehow and the job of the foundational mathematician
is only to /find/ them, then I can see that you might want to say that if you
have "found" the integers, but not (yet) the reals, that doesn't mean that they
are different kinds of thing.

> I'm not sure that the integers can be constructed independently of
> the whole numbers, though -- are the whole numbers then not integers?

Matter of definition, but not in my book. (Though I don't claim that the
distinction is important for any purpose /except/ formal rigour)

> Or consider the "extended real numbers" constructed by adding positive
> and negative infinity to the reals. Are the real numbers not also
> extended reals?

Again, it depends on how you construct it. If you construct your extended R by
adding two (otherwise unknown) elements to it, then I'd say that you are
talking about subsetting. If, on the other hand, you construct your extended
set independently of R, and then show that it has a subset that is isomorphic
to R, but with two extra elements, then I wouldn't say they were the same.

> > An
> > alternative way of putting it is to point out that integers and reals
> > occur at diferent levels of discourse.
>
> I'm afraid I'm not sure what you mean by that.

An ill-choose phrase, I'm afraid. All I was meaning was that you can build the
integers (or whole numbers) and enjoy reasoned and valuable discourse about
them without ever bringing in the reals. To bring in the reals involves
building another layer of machinery -- much akin to layering in software.

> > (As an aside -- and /way/ off-topic -- the reason I care about this is
> > that I'm not convinced that the real number line is even a coherent
> > concept (it's Just Too F***ing Big), whereas I have no problems with
> > the integers.)
>
> I'm surprised to hear that from you. Not that you should be disallowed
> conceptual uncertainty -- far from it -- but that it should be a basis
> or impetus for your argument.

Well, I'm not really as uncertain as I sounded (I was understating, as I have
been known to do on occasion ;-). But I certainly don't claim that a disquiet
with the formal properties of R either motivates or supports my position -- I
was just mentioning why I wouldn't ever just shrug and say "they're isomorphic,
so what does it matter ?". Unlike Patricia, I don't have two "modes", one
where I'm being formal and care about the distinction, and one where I'm being
less formal (about this issue) and don't.

    -- chris

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