[FOM] afterthoughts re my reply to Tim Chow's "Re: Harvey's> effective number theorists."

[Timothy Y. Chow]

It seems to me that the discussion has degenerated into quibbling, and I 
am inclined to agree with Harvey Friedman that it does not seem to be 
moving in a productive direction.  I will try to close off the discussion 
here.

Gabriel Stolzenberg <gstolzen at math.bu.edu> wrote:
>But "pushing" suggests that use was made of "not low enough" bounds (or 
>at least techniques invented to get them) to get a low enough one.  Is 
>this what you mean?  It seems to me that you have been silent about this.

If mathematician X proves a bound and mathematician Y proves a better 
bound, it is not uncommon to say that Y pushed X's bound lower, regardless 
of whether Y explicitly cites a theorem of X or uses techniques used by X.  
That, at least, is my experience as a native English speaker.  Perhaps 
your linguistic experience differs.  But this seems to be an argument for 
the compilers of the Oxford English Dictionary, not for FOM.  I can't see 
why it matters for any of the issues we have been discussing what the 
meaning of the word "push" is.  This is why I say that the discussion is 
degenerating into quibbling.

>But my disagreement with Harvey is about the value of pursuing any old 
>bound, not taking into account the time and effort that may be required.
[...]
>Thus, according to Harvey, as I understand him, it is a good in
>itself to get ever lower upper bounds for no matter what, whether
>or not we ever get a realistic one and whether or not, in the course
>of trying, we invent techniques that help others to get one.

Consider the following two scenarios.

Scenario A.  Someone proves an extremely interesting nonconstructive 
existence theorem.  Someone else, with a reasonable amount of effort, gets 
a constructive bound, but it's very large and unhelpful.  A third person, 
with perhaps a lot of effort, explicitly uses the techniques of the second 
person but extends those techniques and arrives at a very low and useful 
bound.

Scenario B.  Someone proves an extremely interesting nonconstructive 
existence theorem.  Someone else, with an extraordinary amount of effort, 
manages to get a constructive bound, but it's astronomical and "totally 
useless."  A third person also invests an extraordinary amount of effort, 
uses totally new techniques unrelated to those of the previous workers, 
and gets a slightly better but still astronomically useless bound.  At 
this point, despite concerted attempts by many researchers, some kind of 
very difficult barrier appears to be reached, and further progress is 
stalled indefinitely.

If I understand you correctly, the work done in Scenario A is interesting, 
perhaps even *mathematically* interesting, and perhaps even 
*intrinsically* interesting, and perhaps even *fundamental*.  The second 
person in Scenario A, however, was lucky that the third person built on 
his or her techniques; otherwise, the intrinsic interest of the second 
person's work might have been cast into doubt.

The bounds obtained in Scenario B, however, are of questionable value.  
They're "any old bounds," obtained with too much time and effort, and as 
far as we can tell aren't going to lead to a realistic bound.

If this is indeed what the debate boils down to, then again it seems to me 
to have degenerated past the point of usefulness.  There's no way we can 
settle such a debate.  For what would it take to prove that the bounds in 
Scenario B are interesting?  One would have to exhibit an example of 
Scenario B, prove that it is really a "Scenario B" and not a "Scenario A" 
by making sure that the bounds are not realistic, and that they hold no 
promise of leading to realistic bounds.  Then one would have to hunt 
around for number theorists to agree that the work is *intrinsically* 
interesting.  It seems hopeless to try to find an example that is so 
paradigmatic and unequivocal that it would settle the debate, since so 
many of the desired features of the scenario are fuzzy and debatable.

For example, say we find an example of a problem where the current bounds 
are terrible and nobody knows how to improve them.  Say we poll number 
theorists as to whether they think the proof of the current bound is of 
"intrinsic" interesting and of "fundamental" importance, and miraculously 
we are able to get them all to understand the critical importance of these 
particular adjectives.  If the majority say that the result is indeed of 
intrinsic interest, that still won't settle anything, because we can just 
dismiss them as "just talking that way" since they're obviously not 
backing up their claims by publishing any further results (never mind that 
the reason is that the problem seems too difficult).  If the majority say 
that the best known results are *not* interesting, that also won't settle 
anything because they might change their minds when someone uses the 
current proof to push the bound lower, thereby putting us into Scenario A.

More to the point, I don't even see any more the purpose of trying to 
settle this debate.  Suppose we concede to you that a dead-end proof of a 
hopelessly large bound is merely "interesting," and doesn't merit the 
glamorous adjectives "intrinsic" and "fundamental."  It seems to me that 
the larger points Harvey Friedman was trying to make still hold.  It's 
still the case that Scenario A occurs frequently enough that 
mathematicians typically strive to move down that path, and are typically 
very interested in any progress down that path (provided the problem 
itself is interesting enough).  What does it matter if counterexamples 
like Scenario B show up occasionally?  We're not doing math here 
("Definition: A bound is *intrinsically* interesting if...  Theorem: Every 
bound is intrinsically interesting.  Proof:...").

As I said, I hope to close the discussion at this point.  If you still 
want to continue the discussion, then I would recommend that you take me 
as conceding, for the sake of argument, your point that some bounds might 
not be interesting, and focus instead on the larger issues, e.g., along 
the lines that Harvey Friedman recently reiterated:

>"The effective bound situation is definitely the UNIQUE place we can 
>point to now, at which proof theorists, f.o.m. people, constructivists, 
>and core mathematicians can have serious common grounds and interests. 
>This is why I continue the exchange concerning attitudes of core 
>mathematicians. The interest among core mathematicians is clearly 
>sufficient to maintain dialogs and serious interactions. This is a very 
>good thing, and minimizing it would serve no useful purpose."
>
>So I hope that subscribers will reread my original posting
>Classical/Constructive Arithmetic
>
>http://www.cs.nyu.edu/pipermail/fom/2006-March/010208.html
>
>and start a productive thread.

Tim