[FOM] The lure of the infinite

[Harvey Friedman]

The idea that "impredicative definitions do not have any philosophical
basis" represents a misuse of the word "philosophical".

In fact, Weaver has already given a philosophical basis along these lines:

The sets of integers are already there, independently of human construction.
Therefore we have full comprehension for sets of integers. This is in
complete analogy to full comprehension for all sets of: integers of
magnitude at most a given integer n."

I know full well how to complain about this, and almost anything else. I
don't accept it or reject it. It is obviously philosophically coherent, far
more so than any complicated exposition of predicativity getting to at least
Gamma_0.  

On 2/16/06 5:34 AM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:

> Could I just say that I am not actually calling for working
> with more "reliable" methods.  Perhaps I haven't made that
> sufficiently clear yet.

You claim that the use of impredicative methods "has no philosophical
basis", whereas "predicative methods do".

Are you forcefully claiming, then, that you do NOT want to DISCOURAGE
mathematicians from doing mathematics that "has no philosophical basis?" Is
there anything wrong with doing mathematics that "has no philosophical
basis"? Is there anything BETTER about mathematics that "has a philosophical
basis" than mathematics that "has no philosophical basis? Is one better
mathematics than the other? Is one better intellectual activity than the
other?

There are now several important examples of countable theorems that can only
be proved using impredicative methods, and in one of these cases I alluded
to involving a Fiedls Medalist, strongly impredicative methods are used on
essentially a daily basis as common fare. I already studied (previously to
my contact with him/her) the obvious general formulation, which I discussed
with that Fields Medalist as fairly encapsulating the fundamental principle,
and showed it to require substantial impredicative methods (Pi11-CA0).
Furthermore, this is only the most logically low level proof! Their proof is
rather heavily into the impredicative comprehension axiom - not even that!
The impredicative form involving (for all integers n)(therexists set of
integers x)(A(n,x))!!  And all this, without being embarrassed or ashamed or
non confident or disgusted or unhappy!!!! What do you make of that?

This is on top of Kruskal's theorem and the graph minor theorem.

Now there is no doubt that when I point out the difference, logically,
between these kinds of things that THEY do and more usual things, they do
get the point, as well as the connections I show them between these issues
and concreteness. 

Invariably, they get at least somewhat interested, intellectually, in these
distinctions. 

But their reaction is NEVER: oh my God! I am doing something philosophically
incoherent, and I must take this account in my publications and research
life.

Yes, I can get them somewhat interested in whether more special cases
require impredicative arguments, and whether or not there might be a
difference in algorithmic aspects, and growth rates, etcetera.

In other words, their attitude, informed by f.o.m., is, for all practical
purposes, similar to mine. There are very interesting levels, some levels
higher than others in various senses, and it is interesting to know what
level one is at, and how it relates to concrete aspects - such as algorithms
and growth rates.

Of course, they are still going to be even more interested in their main
research, rather than logical points, regardless of how exquisite and deep
that they sometimes acknowledge that they are.

> There seems to be a naturalistic fallacy --- inference of
> "ought" from "is" --- here.  Weaver says that only predicative
> mathematics has a clear philosophical basis, therefore Weaver
> wants people to stop doing impredicative mathematics.

Then what is the significance of "not having a clear philosophical basis?"
And what about in the classroom and thesis advising?

We all seem to agree that the DE FACTO GOLD STANDRARD currently for
mathematical proof is:

ZFC.

Do you endorse that standard? Or do you want to suggest that the Annals of
Mathematics and the Transactions of the AMS, etcetera, change that standard?

Do you recommend that the Fields Medalist and his colleagues put a footnote,
when they publish, to the effect that

*The proof of this result uses methods for which there is no philosophical
basis. The proofs of the remaining results only use methods for which there
is a philosophical basis.

Specifically, when the occasion arises,

**This proof uses the least upper bound principle, which is, unfortunately,
known not to have a philosophical basis.
> 
> This would be a fairly pointless position to advocate because
> most mathematicians already do effectively restrict themselves
> to predicative mathematics, even though they are generally not
> aware of this.  (A question could be raised as to whether or
> not they would be better served by working in some explicitly
> predicative system.  I'll return to that in a later message.)

So when they deviate from that standard, should that be noted? Should
reviews of their work negatively comment on this aspect? Dr. XXX then proves
a theorem with methods that have no philosophical basis. He then returns to
adhering to methods with a philosophical basis. We encourage and admire that
return.
 
> Set theorists do not restrict themselves to predicative
> mathematics.  I personally think that set theory is very
> interesting.  

What then is your interest in set theory?

> Harvey Friedman talks about my desire to "marginalize"
> set theory.  Perhaps there is a fear that regardless of
> my personal motivation, once the word gets out that core
> mathematics can be done predicatively set theorists will be
> in danger of being marginalized by the mainstream mathematics
> community.  

I already address this issue by spending 40 years identifying and verifying
various counterexamples to various forms of this. I do the best I can,
which, form many points of view is not much, and from other points of view,
is much.

>May I be frank: set theory is already marginalized
> by the mainstream community.

Whatever this means, it is because the amount of set theory actually used
for relatively concrete mathematics is right now quite small. IT IS NOT
BECAUSE SET THEORY IS REGARDED AS HAVING NO PHILOSOPHICAL BASIS.

However, this is obviously not going to stay this way over reasonably
nontrivial periods of time. Current mathematics has only a negligible amount
of man/woman years in it, and is completely unrepresentative of the future.

I have no doubt that I have set in motion what needs to be set in motion for
a complete change in this respect. BRT certainly will have various desired
effect, as well as more recent things with other desired effects.

It has proved extremely difficult and deep to find out where substantial
amounts of set theory are needed for concrete normal and core mathematics.

However, the difficulty is intimately connected, of course, with the number
of man/woman years devoted to this enterprise. The number is 40 in my case,
and not too much more from other investigators. Want to help?

If I can do what I do with 40 man years, what can be done with, say, a
reasonable number of man/woman years like 10^6?

I would hope that 10^6 man/woman years of others would get a lot farther
than 40 man years from me!!

>Not in the sense that core
> mathematicians look down on set theorists (maybe some do, but
> not many), but in the sense that they see axiomatic set theory
> as not being of any use in helping to solve the kinds of problems
> they want to solve.

Over reasonable time frames, this is almost certainly false. BRT and
extensions will be adopted as something that people will come to want to do.
I have no doubt about this. And large cardinals are necessary to do
absolutely fundamental things in BRT. Also, it will become clear that
absolutely fundamental things in graph theory require large cardinals.

This will spread all through core mathematics, as there is a fundamental
principle that I am moving towards, that is interesting and important
everywhere.

> Davis talks about "so long as mathematicians
> can obtain worthwhile results using set-theoretic methods" but I
> think most mathematicians would be unable to give even one example
> of a worthwhile result in their field that was obtained using
> set-theoretic methods.

I already gave the story of basic stuff used every day in countable
mathematics, which uses set theoretic methods. I reduced the methods to
Pi11-CA0, but it cannot be further reduced (I don't know this yet for
special cases, but it is likely to be not reducible even for moderately
special cases). These people used far more impredicativity than they had to
- without embarrassment!

To be sure, the situation will look a lot different even by 2100. Hopefully
by 2010.

>Pinning one's hopes on the prospect of
> large cardinals being the vehicle for mainstreaming set theory
> seems to me not realistic.

It is near certain. Everything points to it. It is already here with serious
impredicativity.

>... the regions of mathematics that really have this flavor
> of solidity actually rest on a firm philosophical basis, namely
> predicativism.  The part that doesn't have this flavor (do large
> cardinals exist, is the continuum hypothesis true or false) does
> not rest on a firm philosophical basis.

Your "firm philosophical basis" is an illusion, as other subscribers on the
f.o.m. have recently pointed out. You are just picking a particular place to
start and stop. There are others that are equally philosophically coherent.
Generally, even more philosophically coherent.

> I find this explanation intellectually satisfying and I think
> the foundations community has dropped the ball in a major way
> by marginalizing predicativism.

You are not going to be able to argue effectively against the idea that
there is a remarkably robust hierarchy from EFA to j:V into V, and
furthermore, results coming out suggest that every non ad hoc level is
realized on the button by statements in normal mathematics - perhaps core
mathematics - even at the level of Pi01. This is the appropriate productive,
informative, and stunning way to look at the situation we are discussing.
Not to go back to old fashioned polemics about what "has a philosophical
basis" and what "doesn't have a philosophical basis", with its polemics and
double standards. 

(Having said this, I have no doubt that a major breakthrough that is heavily
philosophical and important for these f.o.m. issues under debate, is
possible. But I haven't seen anything on the horizon like that.)

What is most likely: only after we come to understand this robustness, and
how it interacts with normal concrete mathematics, will we then PERHAPS be
able to draw any meaningful philosophical conclusions of the kind under
discussion.

Harvey Friedman