Re: The Law of the Excluded Middle again (long)

On Mon, 3 Dec 2007 20:09:02 -0800 (PST), Randy Poe
<poespam-trap@xxxxxxxxx> wrote:

On Dec 3, 6:32 pm, Angus Rodgers <twir...@xxxxxxxxxxx> wrote:
On Mon, 3 Dec 2007 14:12:38 -0800 (PST), Randy Poe
<poespam-t...@xxxxxxxxx> wrote:

Why do you think we can't say "either x>1 or x<=1"
in a proof? If x is a real number, there aren't
any other possibilities.

We are apparently in heated agreement! Nothing at all bothers
me about such a statement occurring in a proof! But apparently,
according to constructivists such as Keith Ramsay and galathaea,
it /ought/ to bother me; and, as both those people are better
mathematicians than I am, it bothers me that they think that it
should bother me, when it doesn't.

Somehow I think you're seriously mangling something
perfectly reasonable other people are saying.

Oh dear! That's always possible, I suppose. And given that it's
now after 4 a.m. over here, and I'm dog-tired, I am very probably
going to mangle something reasonable you've said! Please accept
my apologies in advance.

Am I right in inferring that you think I have made constructivism
seem unreasonable? (I certainly wouldn't claim to have done so
using rational argument, and I don't think I have done so using
devious misrepresentation either.)

But apparently I should instead think of a hypothetical situation
in which someone (not necessarily a single individual, but some
kind of generalised subject) has actually constructed some numbers
x, y and z. In such a situation, it is not possible to say with
certainty either that x > 1 or that x <= 1, because either of
these statements would require a proof, which simply might not
be available (to the person or "subject" in question).

Yes, it is possible to say that. For every real number,
either x > 1, or x <= 1. There are no real numbers
which fail to meet one of these conditions. That
requires no proof. It is in fact a restatement of
the trichotomy axiom.

So you are directly contradicting what you said
above. I asked you what problem you had with the
statement "either x > 1 or x <= 1". You said you
had no problem with it. A short paragraph later,
you say you can't accept it as true without "proof".
That constitutes a problem.

Perhaps, but it is a problem very easily solved by the simple
devices of repetition and underlining (in a monospaced font):

On Dec 3, 6:32 pm, Angus Rodgers <twir...@xxxxxxxxxxx> wrote:

But apparently I should instead think [...]
^^^^^^^^^^ ^^^^^^

The whole paragraph was a paraphrase of what I have been told
by certain other people.

It was not an expression of my own views, which, so far as I
can tell, coincide with yours.

You can check this. You quoted the whole paragraph above, and
I have retained your quotation.

Re-reading it, I can see that it was ambiguous. The second
sentence might well not have been understood to be qualified
by the opening phrase (underlined above). I apologise for
the ambiguity; but I think you should also apologise for not
seeing that it could be read either way, and cutting me some
slack.

This is hard work! :-)

So again I ask you, what problem do you have with it?

And I say again, none.
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.

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