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

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:
On Dec 3, 8:07 am, Angus Rodgers <twir...@xxxxxxxxxxx> wrote:
On Mon, 03 Dec 2007 04:24:27 -0500, quasi

I don't know how to explain my point better. Rather than try
to do so (which would only lead to further possibly unreadable
verbiage), may I simply ask how /you/ think of the meaning of
(for example) the statement "either x > 1 or x <= 1", where x
is a variable, which has been introduced in an informal proof,
and you are still in the middle of the proof? No-one is asking
for this statement to be frozen, quantified, and then assigned a
truth value!

I can't understand what is bothering you about such
a statement. I would say that of course it has
a truth value. And if it is a valid proof, then
that truth value better be "T".

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.

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.

So again I ask you, what problem do you have with
it? Why do you not accept that statement to be
true of all real numbers?

- Randy
.

Relevant Pages

  • Re: How to Become a Christian, Version 1.01
    ... Sorry this bothers you. ... > This clearly indicates that your actions are irritating to lots of people, ... Given the fact that those who are irritated by the truth tend to be rather ... Sorry being openly Christian bothers you. ...
    (sci.med)
  • Re: How to Become a Christian, Version 1.01
    ... Sorry this bothers you. ... > This clearly indicates that your actions are irritating to lots of people, ... Given the fact that those who are irritated by the truth tend to be rather ... Sorry being openly Christian bothers you. ...
    (sci.med.cardiology)
  • Re: Obamacare reduced to one line
    ... It still bothers me that he wanted to kill Granny, ... change his mind? ... Your link actually provides some truth! ...
    (alt.support.diabetes)
  • Re: good Cholesterol
    ... >> that bothers you. ... > In truth, neither your arrogance nor your insincere apology bother me. ... The query was about increasing HDL and not losing weight. ... Andrew B. Chung, MD/PhD ...
    (sci.med.cardiology)
  • Re: A quote (and question) about intuitionism
    ... |Is that a definition of truth? ... then, according to the meaning, the statement is sometimes ... So for me the evidence I have for FLT is ... If I understand correctly, constructivists, roughly speaking, ...
    (sci.math)