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

On Mon, 3 Dec 2007 14:12:38 -0800 (PST), Randy Poe
<poespam-trap@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.

A refined logical sensitivity (such as that of either of those
two posters, or that of Brouwer, Heyting et al.) which rejects
the Law of the Excluded Middle as invalid would be the "dog that
didn't bark in the night", which I keep going on about. I can't
understand what I am supposed to see as being objectionable in
the LEM. But the effort to understand that mystery obliges me
to explain my own beliefs - in a way which I find educational,
even if quasi (who is presumably not alone in his opinion, and
generally seems a reasonable person, except now in relation to
me) believes it to be merely perversely self-gratifying.

I'm truly sorry that I don't know how to write about things that
interest me without using /lots/ of words! My style has been
the subject of some complaints in other newsgroups as well as
sci.math (although usually in more friendly and less insulting
terms than those which quasi has just used). I am not just doing
it to annoy! And I am sorry that my meaning has not been plain
to you (and again, I doubt if you are alone in that).

As far as I understand it (which is not far), the reason why I
/should/ be bothered by such a statement (although I am not!)
is that you have to consider it, in its context, as applying to
what Keith Ramsay called "a hypothetical [...] situation in which
a real number x has been given. That is to say, a certain kind of
construction has been made".

I think I understand the "hypothetical situation" part (which
is more or less how I would have expressed the matter myself).

Obviously it is the "construction" part that is the issue.

I still just tend to visualise ordered triples of real numbers
(x, y, z) (such as occur in the sample proof under discussion)
as somehow objectively ranging over a region of mathematical
"space", which exists objectively. (Here I am setting aside some
worries about set theory and the definition of ordered triples ...)

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).

(Take all this with a pinch of salt, of course, because I am
essentially only repeating something I have been told but have
not yet understood.)

From this perspective, I think I can understand why my argument
about free variables and so on was a red herring (because in the
"hypothetical situation", x is no longer a "free variable", but
denotes an actually constructed real number). However, I still
have a long way to go in understanding the perspective.

I still just don't get why I should always think of real numbers,
in particular, as having actually been constructed by someone
(in an actual or hypothetical situation).

And why did you specify "informal"?

Only because the specific proof I was using as an example (from
the thread "x^y + y^z + z^x > 1") was informal. Secondarily also
because I know embarrassingly little about formal logic. (I did
learn some once, and made serious use of the Compactness Theorem
for some now-forgotten purpose, but I've now also forgotten most
of the little formal logic I once knew.) I'm trying to stick to
what I know, as a base from which to explore what I don't know.

(I think it may also be worth emphasising informality because it
draws attention to the thought processes of a human mathematician
doing the human activity of mathematics, and reduces a possible
temptation to think of a mathematical argument as some kind of
external object, perhaps consisting of marks on paper. But this
may be just another expression of my emotional aversion to formal
logic, or at least to an overemphasis on formal logic.)
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.

Relevant Pages

  • Re: open letter to the Google company, on the value of the scientific groups
    ... sum of the squares of the legs of a triangle has been proved to be equal ... You are confusing the true proposition that mathematicians ... other mathematical laws were discovered or constructed. ... just as we see when the construction ...
    (sci.lang)
  • Re: Math as Religion
    ... Gene, like Dik, knows mathematics and has helped you in your polysign ... you have NOT constructed the reals. ... the polysign construction. ... I do not respect mathematicians who practice their subject as one. ...
    (sci.math)
  • Re: Corrective interpretation of real numbers
    ... > seems easy to decide whether it exists in the construction we are ... latter take the effort to accept that their infinity and their continuum ... Mathematicians are humans with intentions and quarrels, ... Rigorosity can cause serious errors if one takes it a gospel that ...
    (sci.math)
  • Re: vote on cantor !!!
    ... I would like to present a construction here that I call the gorilla ... Hence the complications which mathematicians ... Hence all of physics should ...
    (sci.math)
  • Re: Numbers last night
    ... > The pi business was indeed rediculous and the consultant ... > literary/dramatic licence but that bothers me. ... and mathematicians really are amazed at how many areas of ...
    (rec.arts.tv)