[FOM] polemics restricting methods

[Harvey Friedman]

On 2/2/06 3:55 AM, "Arnon Avron" <aa at tau.ac.il> wrote:

>> This does not imply any particular conviction on my part that high
>> powered methods are "correct" or "consistent" or whatever.
> 
> So you share my skepticism about certain high powered methods
> or not? You confuse me here.

It would be more productive for you to start a new thread, instead of
complaining about postings that do not address such issues.
> 
>> d. results over 40 years (and longer) strongly indicate that
>> the stronger the methods, the more information one gets of
>> a "good" kind  that one cannot get otherwise.
> 
> This is obvious, and one needs 40 seconds to understand this,
> not 40 years. 

Incorrect. To indicate how nonobvious this is, quite the opposite is
generally believed by the mathematics community. In particular, that from
substantial set theoretic methods, one cannot get "real concrete
mathematical conclusions of the kind we are interested in."

For example, see what happens when you suggest that their students (or them)
take the time to learn about axioms of set theory, new and old.

>If one cannot prove more using stronger methods,
> than those methods are by definition not really stronger.

Incorrect. The strength may not make any difference for "real" mathematics.
In fact, that is the generally prevailing opinion among mathematicians.

> As for the ""good" kind" (which I guess is your main point) -
> "good" here is something completely subjective, and without
> some objective criteria, claims involving it have no content.

Incorrect. *Good kind* is certainly not "completely subjective".
Mathematicians make these judgments all the time in the choice of theorems
to prove and conjectures to make.

>and without
> some objective criteria, claims involving it have no content.

Incorrect. Almost no intellectual activity is based on objective criteria in
the sense that I think you mean. Yet it is absurd to say that almost no
intellectual actively has no content. This seems to me to a common fallacy.

Taking the point much further, does music have no content?

>  A related point: probabilistic methods can provide even more
> information about concrete, practical questions (like: is this
> number prime?) that one practically (and sometimes even in principle)
> cannot get otherwise.

Actually, you have glossed over an interesting problem. I do not know of a
single case of a mathematical fact that has been established
probabilistically,   that we know - or even believe - cannot be established
by "normal" methods. This is in sharp contrast with the situations that we
are talking about. 

>So are we going to accept them as
> mathematical proofs (as M. Rabin is actually advocating)?

It would be more productive for you to start a new thread, instead of
complaining about postings that do not address such issues.

> And what about inductive methods? All of us have no real
> doubt that P is not equal to NP on the basis of very  convincing
> inductive data. 

Incorrect. In my P = NP lecture, I quoted the Gasarch survey which lists
opinions about P = NP, including several prominent people who disbelieve P =
NP. Have you seen it?

>Are we going to accept it as a mathematical
> theorem because we cannot get this very important information
> (and certainly "good" mathematics)
> otherwise (and maybe we'll never be able to)?

We do not know that we cannot obtain it otherwise. We do not really know if
P = NP, as there are several very prominent people who have real doubts
about it.
 
>> e. Nowhere do I claim that item d is an argument for the validity or
>> consistency of the productive but 'banned' methods.
> 
> If so, then *what* after all is your claim about these methods? Are they
> valid or not? Both you and Joe Shipman seem to say "we do not
> claim that these methods are mathematically certain, but we better
> accept them as mathematically certain". The logic here is beyond me.

It would be more productive for you to start a new thread, instead of
complaining about postings that do not address such issues.
> 
>> f. But d refutes the claim that is often made, or implied,
>> in the kind of  polemics I am talking about.
> 
> Can you give an example of a predicativist (or someone with
> closer tendencies) who have made the claims you are
> "refuting", and where have these claims been made?

I regard it as implicit throughout, e.g., in light of Weyl's reworking of
analysis under the restrictions he imposes. To what extent it is explicit, I
don't know, because I don't have time for historical work.

However, judging by conversations with contemporary mathematicians,
especially the late Halsey Royden, the idea that "you can't get anything
concretely interesting from heavily set theoretic methods (that you can't
get otherwise)" is very much ingrained. I would find it incredible if it
wasn't also very much ingrained in the thinking of Weyl and Poincare and
others, who went further to recommend banning in print - much further than
these contemporary mathematicians I talk to.

I also have no doubt that the realization that one loses Kruskal's theorem
and the graph minor theorem and also various finite forms would have had a
profound effect on the thinking of Weyl and Poincare and many others. How
they would resolve the conflict - between continuing to ban those methods
and not wanting to discard important, beautiful, natural, generally
accepted, mathematics - I am not sure.

But they undoubtedly would have regarded this new information (from the
1980s) as absolutely crucial for any informed discussion of predicativity.
> 
>> g. What is really new is the new and growing level of evidence for d.
> 
> Already Goedel incompleteness theorem provide enough evidence
> for d. 

Incorrect. It has become clear over the decades since the 30s that arbitrary
formal sentences, even in restricted languages, are FAR too broad to be the
basis for discussions of incompleteness issues.

When that is realized, one begins to see why mathematicians, who blissfully
ignore the commonly quoted independence results, see gigantic qualitative
differences between the usual examples and what they do for a living.

>Consistency claims of axioms of strong infinity are certainly
> interesting and important Pi-0_1 claims (in my opinion:
> more interesting and important than all the equivalents
> you have found for them) that cannot (if true) be decided
> even in ZF. 

AS STATEMENT IN: analysis, topology, complex variables, differential
geometry, number theory, ergodic theory, probability theory, finite
combinatorics, and, yes, finite graph theory, consistency statements about
formal systems have no interest whatsoever.

One judge of this matter is mathematicians themselves in these various
areas. 

So you can present, say, Con(ZFC + Mahlo cardinals) to, say, an analyst, and
I can present, say, the latest to finite graph theorists. I wish you the
best of luck...

>In predicative mathematics one cannot (most
> probably) prove even Con(ZF), which is an extremely important
> proposition. For making the philosophical (?) point you are
> trying to make these facts are more than enough. But
> the whole point is of no relevance to the question of
> mathematical validity.

It would be more productive for you to start a new thread, instead of
complaining about postings that do not address such issues.
 
>> My own point of view is that we don't know nearly enough basic
>> information about f.o.m. to make any kind of decision about
>> the ultimate status of the  mathematical methods in question.
>>  I am merely trying to make a start at filling in at least some
>> of the basic  f.o.m. information needed to make informed
>> decisions about the status of the
>> mathematical methods in question.
> 
> The only new information that might be relevant is a mathematical
> proof that a certain dubious principle is equivalent to (or follows
> from) some other, more acceptablre principle.

Can you justify this claim that "the only new..."? I certainly do not
believe this statement.

>A case in point is
> the well-ordering principle that Cantor wanted at a certain point
> to accept as an axiom, but had no ground for doing so. The status
> of this principle changes when it was inferred from a much more
> intuitive principle (that is obviously valid for anyone who is
> a platonist): AC. I dont see that any of your results are of this
> nature. 

In this totally different direction, I had recently mentioned on the FOM my
work on Reflection in the form of Ternary System Theory. Here large cardinal
principles follow from very basic looking reflection principles without any
significant set theoretic infrastructure.

>On the contrary: if I understand you correctly, your only
> ground for accepting your combinatorial propositions as true is that
> they follow from some  "natural" axioms of strong inaccessibility
> (more accurately: from the assumption that those axioms are consistent),
> not the other way around. So if new relevant data for *FOM* is what you seek,
> you are making great efforts in a wrong direction.

It would be more productive for you to start a new thread, instead of
complaining about postings that do not address such issues."

> Another important observation: as you write yourself in this
> paragraph, the road you take is a never-ending road.

Mathematics and human intellectual activity is a never-ending road.

The vision is that there will be a catch up, in the sense that the natural,
interesting, concrete mathematics that comes only from expanding the usual
axioms becomes a significant fraction of the mathematics that is done -
especially if depth is factored into the mix.

I.e., in most concrete mathematical contexts, one can get stronger, deeper,
information, of a valued kind, only by going well beyond ZFC. This won't be
the exception - it will be the general rule.

Things I am doing right now but have not completed, seem to suggest this as
a real possibility. However, we are still far from achieving this overhaul
of mathematical activity.

 >According
> to your way of thought, mathematicians will need accepting
> *ever* stronger methods.

j:V into V is enough on my plate. I am nowhere near that now. Didn't Godel
spoke of a never ending (or open ended) series of extensions?

>Whatever asoteric axioms you persuade
> mathematicians that they need for "good" mathematics, you will need to start
> all over again the efforts of convincing "core mathematicians"
> that Goedel theorems are still relevant, and even stronger axioms
> should be sought.

There will be this vast array of beautiful valuable exciting concrete
mathematics, using all kinds of levels of logical strength all the way from
very modest up through, say, j:V into V.

There will be a uniform method of stating additional information of a valued
kind in nearly every concrete mathematical context, that can only be
obtained by going far beyond the usual axioms.

This is a steady relentless development, which will not require any
backtracking.
>> 
>> I haven't accepted any methods in the writings you are referring to.
> 
> So again: are your efforts not in order to persuade the mathematical
> community that some methods (that you are not accepting yourself?)
> should be accepted? This, I believe, is what you stated very strongly
> in your postings, especially the polemical one I have responded to.

As I said many times, I am developing crucial information relevant to any
informed discussion of these f.o.m. matters. I do not think that we have
anything close to sufficient basic f.o.m. information to do things like
"decide once and for all the status of various proposed axioms, etc."

>> I'll make your point stronger: I didn't provide ANY argument for using
>> axioms of strong infinity at all. I merely claimed results
>> that indicate
>> what you can get with them that you cannot get without them.
> 
> And there is no goal behind declaring  these results on FOM, and
> no moral to infer from them? Well, I am relieved to learn this.

I have been declaring results on the FOM practically since the inception in
1997. Strangely, you didn't complain over the years. I don't see anything
different now. I always was trying to assemble new relevant information
needed for informed f.o.m. discussion.
> 
> ...if we keep waiting
> forever to be informed by new f.o.m. results we shall never
> make any progress. However, I do agree that at any time we can
> benefit from new f.o.m. results, and deepen our understanding
> on the basis of them.

You have my permission to discuss f.o.m. matters without waiting for my
results. 

Look at how informed f.o.m. discussion today would be different without
Godel's incompleteness theorems.
> 
> One last thing: at a newer message Friedman asked me to explain
> in FOM why certain mathematical methods have different status
> than others (this is my formulation of the challenge, not his). I'll do so
> sometimes in the future.

I said "ban". I freely acknowledged that it makes sense to talk about
differences between stronger methods and weaker methods. Nevertheless, I
look forward to what you write.

Harvey Friedman