The Law of the Excluded Middle again (long)

If this post seems odd, please think of it as being about a
"dog that didn't bark in the night", and the reason for it
will become clear.

A few weeks ago, in the thread "A quote (and question) about
intuitionism" (which I have temporarily let drop, with the
ball still in my court) I said I had intended to pick some
simple everyday example of a mathematical argument in which
the Law of the Excluded Middle is used.

Such a simple everyday argument has just presented itself.

A recent thread asked for a "slick" proof that x^y + y^x > 1
for all positive real numbers x and y; and a reference was
given to a proof of this by Wade Ramey in sci.math in 2003.
I wondered about an extension to more than two variables (I
see this was also asked in the earlier thread), and found a
simple deduction of the case of 3 variables from the known
case of 2 variables. I posted about this, and Wade gave a
neat solution, similar to my own.

Absolutely no deep ideas were involved in any of this, and it
was little more than a matter of idle curiosity (at least on
my part). At most, when I was about to drop the matter out of
a lack of real interest, I piqued my interest again by saying
to myself, silently, "Well, the infimum of the set of numbers
x^y + y^z + z^x certainly /exists/, even though I have no idea
how to /construct/ it", and this made me curious enough to see
if I could form any idea of what the number actually was. (It
soon enough turned out to be 1.)

Again, nothing remarkable about this. But that is just what is
wanted: a simple, unremarkable example of everyday mathematical
reasoning in action, using the Law of the Excluded Middle. We
should be able to use it to examine the role of the LEM, without
distraction (caused either by complications in the mathematics,
or by an absence of any real bread-and-butter mathematics to
think about).

In the present case, the LEM is used in at least two places.
(That is, explicitly; probably it is used implicitly as well.)

We first argue that if any of x, y, or z is >= 1, then the sum
x^y + y^z + z^x is > 1. So we say that it suffices to consider
only the case where all of x, y and z are in (0, 1) (although in
fact the argument more naturally considers them all to be in the
half-open interval (0, 1]).

Either way, we are appealing to the LEM, by saying "either
(x >= 1 or y >= 1 or z >= 1) or (x < 1 and y < 1 and z < 1)"
(and then deducing the same conclusion in both cases).

We then make a further division into cases, say (as in Wade's
argument) into the cases x <= y and x >= y. (As before, there is
no need to use a sharper and less symmetrical alternative x > y.)

If x <= y, then z^x >= z^y, therefore y^z + z^x >= y^z + z^y. If
x >= y, then ... actually, I think the more complicated argument
that I used actually is needed at this point! (I assumed, without
examining it closely, that I had complicated things unnecessarily.)

Unfortunately this now become a less childishly simple example
than I wanted it to be, but it is still elementary, and far from
complicated. (Also, it is still realistic. Or rather, it is real,
in the sense that it was used in practice, not just concocted for
philosophical purposes.)

One way to phrase the argument is "either 0 < x <= y <= z <= 1
or 0 < z <= y <= x <= 1 or ..." (four other cases, which can
then be reduced to these two "by symmetry", i.e. by renaming
the variables).

(Again the argument most naturally uses overlapping cases, but
again this doesn't affect the essential point at issue.)

If 0 < x <= y <= z <= 1, we can argue y^z + z^x >= x^z + z^x,
or y^z + z^x >= y^z + z^y. Either way, we can use the known
result x^y + y^x > 1 (with renaming of variables) to deduce
that, in this case, x^y + y^z + z^x > 1.

If 0 < z <= y <= x <= 1, we can argue x^y + y^z >= x^y + y^x,
or x^y + y^z >= z^y + y^z, and deduce the same conclusion in
this case as well.

(I am spelling this out laboriously in order to avoid making
any slip like the one that I, and apparently also Wade, made
earlier.)

The point I am making with this post might either turn out to
be something objective, or merely something about a lack in
myself as a student; I am not yet in a position to say which
it is; but in either case (am I appealing to the LEM here as
well?!), it is worth spelling it out (at least so that I can
learn something).

I pride myself on having a reasonably well-functioning bull***
detector, and the point is that (like "the dog that didn't bark
in the night") it didn't give the slightest peep, all the time I
was working on this problem, reading Wade's solution, and seeing
what I could learn by doing a "post mortem" on the problem. It
was only after it was all over (I thought!) that the point about
the Law of the Excluded Middle occurred to me.

Can someone explain to me why my bull*** detector /should/ have
emitted a loud warning (about the use of the LEM in this argument),
and perhaps how I could train/tune my bull*** detector to work
better in the future? Or, is it possible that there is simply
nothing wrong with the argument? That is what I still believe,
but I am open to correction.

That last question raises another (quite distinct) point (which
has also come up in recent discussions about the philosophy of
mathematics in sci.math): of course it /is/ still perfectly
possible that there is something wrong with the argument!

I think we have already had an example of this, when Wade gave a
version of the argument that was simpler than mine, and I just
assumed (and so, I guess, did Wade) that it was equally valid,
and that my argument had been overcomplicated. Nevertheless we
were both probably "right" to assume that there was nothing
essentially wrong with it, and that the problem was solved.

So what do I mean when I say, so confidently, that there is
"nothing wrong" with the argument (and even that there was nothing
"essentially" wrong with it before, even though I now think that
there /was/ an error)? Am I (as galathaea might perhaps suggest)
claiming something like Papal infallibility? Am I about to start
spluttering about the need to crucify the vandals who dare lay
siege to Rome? Am I a "fundamentalist"? Am I even claiming to
be "omniscient"?

Hardly! Please believe me when I say that yesterday (and indeed
today) I could hardly have been in a less confident or self-
aggrandising mood; all I did was cheer myself up in a depression
by working at a simple and unimportant bit of mathematics. I'm
now making a song and dance about this near-triviality because of
the existence of philosophies of mathematics that would drain all
meaning from this sort of activity (so that it wouldn't cheer me
up for long, if at all).

I mean it is my (fallible, uncertain) opinion that the argument
does not contain any logical fallacy. (In particular, I don't think
that the Law of the Excluded Middle, when applied to mathematics,
is a logical fallacy.) As a background to this, the belief that an
argument is valid causes, psychologically, a sense of certainty.

Also, again as a background, there is a sense that the question
of the validity or invalidity of arguments (expressed initially
in informal terms, as is the example I have given) can be expressed
as a question as to whether the argument does or does not follow
certain "rules", which can be described. I seriously have my doubts
about this (but I admit at once that this may be just ignorance on
my part), just as I do not believe at all that questions of morality
reduce to obeying or following rules (although some vague notion of
universalisability surely does have some importance, and I must admit
I have scarcely begun to examine this, and haven't read Kant, etc.).

I'm sure much could be written about these general background questions
of psychology and philosophy (and I would be extremely interested), but
I don't want to digress too far, for the moment, from the question of
the validity of the "Law" of the Excluded Middle. (The scare quotes are
to remind us of these background connections with other questions.)

It's not that I don't see all sorts of difficulties in the foundations
of mathematics. When I was an undergraduate student (so long ago that
I can scarcely believe it ever happened), I felt uneasy about lots of
things, but could see no way of discussing them or otherwise coming to
grips with them, so I just tried to deal with them by ignoring them.

And indeed, there is wisdom in that approach. (Cauchy: "Carry on, and
faith will come." - Can someone authenticate this quotation?) I have
been doing something like it for the last three years: studying maths
in a quiet, mainstream way, and trying (when I am subscribed here) not
to be /too/ distracted by all the foundational threads that so often
erupt in sci.math. All my difficulties will have to be dealt with
eventually, but I am in no hurry to do it, knowing the difficulty of
the task.

It's just that worrying about the validity of the Law of the Excluded
Middle has /never/ been among my difficulties! (Not in pure maths,
that is. In applied maths, it is a quite different story, but applied
maths is a minefield for me, and this difficulty is only one of many.)

What has become a difficulty now is that the Law of the Excluded Middle
is seriously worried about by people whom I respect (partly because
they know vastly more maths than I do, and are probably more talented
as well), and because I am now trying to learn how other mathematicians
think, I am faced with trying to learn to have the same doubts and see
the same problems as these people see. And I am coming up blank.

This won't stop me trying to learn mathematics; nor will it stop me
"believing" in the validity of a simple and elementary proof like the
one I have just been labouring over (perhaps very annoyingly, I can't
tell); but it is now a genuine problem in my path (along with genuine
mathematical problems themselves).

Just as I don't think "crank" posters should always be vilified for
what seems like (and may be) arrogance, I hope I won't be vilified
by disbelievers in the LEM for what seems like (and for all I know
may be) a presumption of omniscience, or an unthinking dogmatism. I
am saying what I think - believing, of course, that what I think is
true, is true! - but not in order to shove it down anybody's throat,
and not to protect it from criticism, but rather to open it up to
criticism, and thus either to make it stronger (if it deserves to
be), or else to abandon it (if it deserves /that/ fate, instead).

I think Bertrand Russell said something like, "The virtue of a logical
argument is not that it compels certainty, but that it suggests doubt."
(Again, can someone find the exact quotation for me?) I want what I
write to be taken in something like that spirit. That is, I am saying,
"This is what I believe. I think it is true and important. Moreover,
it matters a lot to me personally. Can you show me where I am wrong,
or is it possible that I am right?"
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.