[FOM] The Gold Standard

[Harvey Friedman]

Several subscribers are continuing to express dogmatic f.o.m. views, and I
continue to point out that they have given no more or less reasons for their
dogmatism than other thinkers who push dogmatic views either lower down or
higher up, that are incompatible with theirs.

As readers have seen, there is a plethora of dogmatic f.o.m. views around,
many of which have strong and energetic adherents here on the FOM.

As I have repeatedly said, and will continue to say: we don't know nearly
enough about f.o.m. to sensibly adhere to any such dogmatic views. The
dogmatism is counterproductive, as we are in an extended EXPLORATORY phase
in f.o.m.

Adherence to dogmatic views serve mainly to slow progress in f.o.m.

As subscribers know, I will look for explanatory and illuminating theorems
and theories surrounding practically ANY f.o.m. view. That can range from
the barest kind of ultrafinistism to the highest form of Platonism.

As subscribers also know, I will argue strongly against practically any
DOGMATIC f.o.m. view.

It's all the same to me, intellectually. Why? Because we are in an
EXPLORATORY phase in f.o.m., and we know little.

Dogmatism is the enemy of progress.

On 2/22/06 1:38 AM, "Arnon Avron" <aa at tau.ac.il> wrote:

 > You are the first mathematician I ever met who claims that he feels
 > comfortable with ZFC because of the analogies with finite sets.
 > Try to ask your medalist about it.

When I mention the analogues with finite sets, every mathematician says
"aha!". I did not say that there were conscious of the reason behind their
acceptance.

In my posting http://www.cs.nyu.edu/pipermail/fom/2006-February/009969.html
I pointed out that all of the examples Avron gave of how finite set theory
differs from set theory involve transferring of statements involving
cardinality.

Avron said many times, "so what. why should infinite sets be different in
this respect from finite sets".

This amounts to a rejection of the very idea that set theory is based on a
transfer from finite set theory, right from the start.

This is entirely inappropriate in light of the fact that apparently the
analogy with finite sets is exactly what is behind the acceptance of ZFC as
the Gold Standard.

Thus we have a phenomenon. The acceptance of this Gold Standard. We seek,
among other things, to explain this phenomenon.

So Avron's suggestion to simply cut off a crucially important line of f.o.m.
research right at the start just because some of his very favorite
statements in finite set theory don't transfer to set theory, must be
rejected as unproductive and short sighted.

It is certainly true that Replacement and Foundation is used so seldom that
most mathematicians have forgotten about them. So pragmatically, they really
work in ZFC\Replacement\Foundation, and if you briefly mention what that is,
then they generally have instant recognition.

Subscribers can try out power set. Remind the working mathematician that the
power set of a finite set is a rich idea that is very concrete, with lots of
structure. E.g., the permutation group on a finite set. Then remind them
that set theory says that this is fine for any set. They will have no better
reason than that for their acceptance of the Gold Standard.

One does come across, from time to time, some finitists and even
ultrafinitists among the core mathematicians. They think of normal
mathematics with infinite objects as a kind of game. But even THEY recognize
that ZFC or ZFC\Replacement\Foundation is the BEST game in town,
foundationally.

Friedman wrote:

 >> Somehow, people grasp that an infinite set
 >> is really like an extremely large finite set.

 > Even Galileo has not graspied it, and was surprised by the paradoxes of
 > infinity.

I am talking about now, in the midst of the computer revolution, and with
the benefit of time. Besides, I would conjecture that a good reading of
history reveals at least some serious understanding of this point all the
way back to Archimedes.

 >(Also no mathematician has doubted AC for finite sets,
 > and still there was a huge debate about it. Now everybody agrees
 > that for infinite sets one needs to add it as a NEW principle).

Set theorist: axiom of choice is at least as evident as the other axioms.
Sure, you need to add, say, pairing, as a "new" principle over the other
ones, but that doesn't mean that it is problematic.

 >> And I take great pains to tell students how strong the analogies are.

Friedman wrote:

 >> Let's roll up our sleeves and get to work!

 > Fine. But I understand from this call that you admit that
 > the work has not been done yet.

I have been saying, for perhaps 40 years, that the relevant work on transfer
has not been done yet.

 >So again
 > I really don't know on what ground you base your
 > declaration about the reason that mathematicians accept ZFC
 > as the golden standard (assuming that this declaration is correct,
 > which I doubt.

You doubt that the mathematical community has come to accept ZFC as the Gold
Standard?

Or just the reason that they do?

EXPERIMENT. Present any series of reasons to mathematicians why ZFC came to
be the Gold Standard, including the transfer reason I am talking about. And
also ask them for their own reason why. Then you will see the unique power
of the transfer idea.

 > I believe most "core mathematicians" will not even
 > be able to tell what are the axioms of ZFC that
 > they are supposed to accept as the golden standard).

They will instantly recognize all of the axioms of ZFC except replacement
and foundation, if you gently remind them. (Replacement and foundation are
rarely used, and ZFC\Rep\Found is of course equiconsistent and mutually
interpretable with ZFC). Almost anybody in field X will need to be gently
reminded of basic notions in field Y, if they are working professionals.
E.g., a PDE expert may have to be gently reminded what a finitely presented
group is.

Friedman wrote:

 >> So weak that almost the entire mathematical community has accepted it
>>as the
 >> GOLD STANDARD for over 80 years.
 >
 > Maybe they do, but not because of the justification you suggested,
 > which is very weak. Worse: at least until you do the work you
 > are calling for, your justification necessarily lead to horrible
 > mistakes of students (this is not a speculation. It happens all the
time).

As I said, all f.o.m. type justifications are "extremely weak" if the
standard is: how easy is it to complain? Of course, it is just as easy to
countercomplain.

Because all justifications are so subject to complaints, practically all of
them, at least the even remotely coherent ones, demand sophisticated and
imaginative investigation. That's how progress in f.o.m. is really made.

Amazing and satisfyingly beautiful things emerge unexpectedly - often even
entirely new subjects valuable at many levels. The results can prove
relevant to a wide range of views beyond the ones that they were intended to
affect. We should be optimistic.

 > I have given already very good reasons for it. You are not convinced.
 > I am not surprised. So do not be surprised that I am not convinced
 > by your arguments (and neither are, I believe, most mathematicians.
 > Try to ask them).

As I said at the outset. I am not trying to convince you of any view. I am
only trying to convince you that your dogmatism is unwarranted and
counterproductive.

Friedman wrote:

 >> So is the notion of the powerset of a set of cardinality n based on
having
 >> an n mind ("like God or something like this"), where n = 2^2^2^2^1000?
 >
 > I totally disagree, but I am not ready to make debates about it.
 > If you really think so (I doubt)  I can do nothing about it.

I don't agree or disagree. I'm saying that you are in no position to be
dogmatic about this and related matters.

 > I can only wonder why you are trying to give justification
 > to ZFC of any kind, if you dismiss any criticism by
 > pointing out that anything can be criticized.

The acceptance of ZFC is a phenomenon, and f.o.m. studies phenomena. Also, I
am happy to prove theorems that cast doubt on dogmatic ZFC boosters.

 > However, I'll return to this type of  post-modernist arguments
 > of yours in another posting.
 >
Old fashioned dogmatic f.o.m. has been largely rejected by almost all
interested parties. It has served many useful purposes already, but its time
has passed.

Harvey Friedman