[FOM] predicative foundations

[Harvey Friedman]

On 2/7/06 8:37 AM, "Nik Weaver" <nweaver at math.wustl.edu> wrote:

> 
> It seems that this debate on predicativism has largely run
> its course, but I have one or two points to add.

The debate on predicativity hasn't properly begun.

In particular, there are claims, particularly by Avron, and implicitly by
you, that predicativity has some special place in the robust hierarchy of
logical strengths ranging from EFA through j:V into V. In fact, such a
special place that it has even been suggested that it should be identified
with "certainty".

The debate on predicativism will begin as soon as some proponent of a
special place for it gives some credible reason why it has such a special
place. This hasn't yet occurred.

Proponents of predicativity generally speak of an obvious absolute
objectivity of the natural number system, and all first order sentences in
the semiring of natural languages.

I don't see anything particularly compelling about that view either  - it is
simply one particularly stopping place in the robust hierarchy of
commitments one may wish to make.

It is subject to criticisms that it is too strong, and also subject to
criticisms that it is too weak. Just like any of a very large number of
stopping places. 

>Much of
> the previous discussion seems to me not very substantial.
> I think it largely stemmed from very confused ideas about
> predicativism, such as that we are out to "ban" impredicative
> mathematics, or to classify various kinds of mathematics as
> "good" or "bad", etc.

Who is "we"? Have you any comments about Pollard's recent
http://www.cs.nyu.edu/pipermail/fom/2006-February/009698.html ?
> 
> (I would have liked to join in the discussion earlier, if only
> to express my thorough approval for Arnon Avron's position,
> but I was on a trip and didn't have the opportunity.)

Avron wrote a lot recently, much of it rebutted by Steel and me. What
approval do you have for his positions?

The most dubious claim is that of a special status of predicativity among
the fine grained hierarchy from EFA to j:V into V.
 have very little support?

>However, we now know that virtually all ordinary mathematics
> is predicatively justified --- say, 95% of all theorems that appear in
> the Annals of Mathematics, or 99% if we exclude the rare set theory/
> logic paper.  

The same can be said of systems substantially weaker than predicativity -
although the specific numbers you cite could use some investigation.

E.g., what about a system corresponding to ACA or ACA0? Also 95%?

I think the percentage is also very large for RCA and RCA0.

>Conversely, 99% of what gets lost in the world of
> predicativism is mathematics that ordinary working mathematicians
> would regard as set-theoretically pathological.

Of course, the exceptions here are what is important. E.g., Kruskal's
theorem and the graph minor theorem. Also lub stuff may be regarded as
*visibly out of the ordinary* in some sense, but not "set theoretically
pathological". 

E.g., I would doubt that the word "pathological" would really be used by
most for, say, "every uncountable closed set of reals contains a Cantor
set". 

I have no doubt that the situation will look entirely different by the end
of the century. At some point, there will be a special kind of breakthrough
that shows how to use higher principles in a huge variety of contexts to
gain sharper information of a valued kind.

>That's why after
> decades of eclipse predicativism is reemerging as a major foundational
> stance.

This I wasn't aware of.
> 
> Harvey Friedman wrote (referring to his own work)
> 
>> ... is there STILL some way of siphoning off these new results
>> as not good mathematics, that can be distinguished from real
>> mathematics? ... I can imagine experiments like this that just
>> might convince a very wide range of people that they can't tell
>> the difference in terms of "naturalness", "beauty", etc.
> 
> I am not interested in classifying anything as "not good" or
> "not real mathematics", but I do think there is a reasonably
> well-defined notion of core mathematics.

Does this include finite graph theory?

The situation is getting more blurred, as there are a growing number of
connections now between graph theory and what everybody now calls
"core mathematics". This has been said to me by "core mathematicians" here.

>And probably a better
> criterion than "beauty" or "naturalness" for membership in this
> class is "having essential connections with other core mathematics"
> --- at least, this is a criterion that core mathematicians are
> more likely to accept.

There is, separately, the criteria of natural, beautiful, etc., that I have
pointed to, and which core mathematicians have used, in connection with
mathematics that is definitely not "core mathematics". I.e., even core
mathematicians readily make these evaluations (sometimes positively) about
work in mathematics generally.

In fact, these other notions are much more independent of perhaps transient
sociological features.

Also most mathematics done in the entire world is outside "core
mathematics". E.g., there is a huge volume of finite graph theory done in
computer science and engineering departments, and industry, and government.

>On that count the results in question are
> unsuccessful at the moment, although it is obviously premature to
> draw any definite conclusions about this.

They are improving steadily as "natural, beautiful, etc." in finite graph
theory. 

As I have said before, once I get to the bottom of what is going on, in its
most elementally clear form, I expect it to be transferred to a huge range
of contexts in a completely natural and uniform way - and that includes a
huge range of contexts in "core mathematics".

> Another basic classification is "can it be concretely visualized
> or does it depend on vague metaphysical speculation involving a
> ghostly `universe of sets'?"  It is a remarkable fact that this
> distinction corresponds so exactly to the preceding core/not core
> distinction.  

It doesn't correspond at all. In particular, the recent independent
statements can obviously "be concretely visualized" and in no way shape or
form, "depends on vague metaphysical speculation involving a ghostly
universe of sets". 

>The correspondence is so good that Hilbert's point
> about predicativism not fitting ordinary mathematical practice
> can now be turned into a point in predicativism's favor: it is
> by a wide margin the foundational stance which *best* fits
> ordinary mathematical practice.  I'll post a specific example
> of this in a separate message.
> 
It doesn't fit very well. For instance, see my remark above about ACA. Or
even ACA0. In addition, the percentage handled in RCA0 like systems is also
very high. 

If you go to branches of mathematics, in particular finite mathematics
including especially number theory, then EFA = exponential function
arithmetic, corresponds incomparably better. For a discussion, see

http://www.andrew.cmu.edu/user/avigad/
Number theory and elementary arithmetic
(Philosophia Mathematica 11:257-284, 2003)
Abtract: html , Paper: pdf .

Harvey Friedman