[FOM] The irrelevance of Friedman's polemics and results

[Arnon Avron]

This is a partial reply to Friedman's message from Thu, 26 Jan 2006:

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

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

 
> 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.  

> 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? The only
related claim that I know of is Feferman's thesis that all scientifically
applicable mathematics can be developed within predicative mathematics.
However, I dont recall him identifying scientifically applicable
mathematics with "good" mathematics. Your "d." does not refute 
Feferman's thesis (how can it, when the content of "d." is almost trivially
true?). You are "refuting" a claim that to the best of my knowledge
was never made by predicativists, nor implied by them (unless one
identifies "good" mathematics with absolutely certain mathematics).

  Let me add here that in my opinion even Sol's concern about
scientifically applicable mathematics has gone too far. In principle,
a scientific theory should be taken as a whole, and there is no 
reason to demand one component of it (the mathematical one) to be
more reliable and certain than other parts. So it is quite reasonable
to employ in science mathematical methods that are not certain in
the pure, ideal mathematical sense (including even differentials,
diverging seies, and other means that are actually used). I dont 
really doubt the consistency of some axioms of strong infinity anymore 
than I doubt the proposition that we all will die one day (I believe in 
both propositions for very similar reasons), so if some physical theory 
uses this consistency as an accepted assumption, and this would be 
proved productive in expermints and applications - this is 
very OK with me (I am sure that someone will indeed attemp to do so one day,
but I am not sure about how successful such an attempt will be).
On the other hand, if certain mathematical methods are included 
as a component in a big diversity of scientific theories, than it
is very useful to know that these methods are absolutely reliable:
it means that problems with the theories should be attributed to
other components. Hence it is quite important and rational to
show that a great part of scientifically applicable mathematics is 
predicatively justified,  but that does not mean that we 
should show that *every*
piece of mathematics that some scientist has tried to apply
or will try to do so in the future should be predicatively justified.

> It also reveals costs associated with taking a
> negative polemical position against the 'banned' methods.

Seeking validity and certainty is of course costly. Are you
suggesting that we should pretend that something is certainly 
valid only because otherwise we'll pay a price? 

  In any case, the "cost" (for the notion of a mathematical
proof) of accepting something as valid 
and certain only because it is convenient to do so, not because
we really recognize it as such, is MUCH MUCH higher than the price
of not being able to decide with certainty some mathematical 
theorems, interesting or beautiful or "good" as they may be.
 
> g. What is really new is the new and growing level of evidence for d.

Already Goedel incompleteness theorem provide enough evidence
for d. 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. 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.

> 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. 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. 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.

 
> Of course, the results that I seek and have obtained tend 
> to tilt in the  direction of being in favor of accepting 
> ever stronger methods - but only in
> the sense that they tend to demolish a key argument made for the
> restricters: that nothing "good" can come of using the 
> 'banned' methods.

Again I have no idea who has made this kind of claims. The 
"key argument" is not that "nothing good" can come of using the 
"banned' methods, but that the most important parts of "good" mathematics 
(which includes all of the important mathematics done before
the 20th century and much more) do not need them. In other words: 
nobody claims that there is no cost at all. The claim is that (surprisingly)
the cost is much lower than one might have expected - and *this*
claim has by now been demonstrated!
 
  Another important observation: as you write yourself in this
paragraph, the road you take is a never-ending road. According
to your way of thought, mathematicians will need accepting
*ever* stronger methods. 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. Since I dont think that you think that 
looking for new axioms should be the only mathematical activity,
you would allow "core mathematicians" to stop from time to time
and actually *do* mathematics, even though they will necessary
do so  in a fragment which is insufficient for all "good", "natural", 
mathematics. Now it seems to me that according to your views, 
any future stopping point of this type
will be arbitrary. But if so - why not choosing a good point 
which is *not* arbitrary (like predicative mathematics) and devote 
most of our efforts  to it? or at least wait until "core mathematicians"
come themselves across problems in which they need stronger methods,
instead of ever producing for them  artificial "natural propositions"?

> 
> 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.
 
> 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. 
 
> you are deprived of a very commonly used argument in
> favor of your 'banning'. You will have to rest your case 
> for 'banning' on  other arguments.  

First of all, claims about the strength of the safe methods
are not used as arguments for banning other methods. Thus
nobody would use the fact that every construction of points
with a ruler and a compass can be done by a compass alone
as a reason to ban the use of a ruler. "Banning" (which
actually is only "classifying as less safe") can and should be 
done on a completely different ground, and the "cost" is
irrelevant. One cannot turn an unsafe method to a safe one
only on the ground that one wishes that method to be safe...
However, it is certainly good to know that so much can be done
using safe methods! 


> > So for me the most crucial problem of FOM is: is there absolute
> > truth in mathematics, and if there is - what theorems of mathematics
> > can truthfully and safely be taken as meaningful and *certainly true*.
> > Predicativism (at least for me) is all about this question.
> 
> Any new discussion of this issue needs to be informed 
> by new f.o.m. results.

I am not sure about the  "needs" part, because 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. 
 
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.
 

Arnon Avron