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

[Harvey Friedman]

Before I finished proof reading this reply to Arnon Avron
http://www.cs.nyu.edu/pipermail/fom/2006-January/009614.html

I saw replies http://www.cs.nyu.edu/pipermail/fom/2006-January/009618.html
and http://www.cs.nyu.edu/pipermail/fom/2006-January/009620.html

I generally subscribe to these two replies.

You will see that my own reply is of a somewhat different and totally
compatible character, emphasizing different points.

**********************************

On 1/26/06 12:45 PM, "aa at post.tau.ac.il" <aa at post.tau.ac.il> wrote:

> On January 21 Friedman sent a very polemical posting to FOM, in which,
> ironically, he attacked "polemicists" concerning FOM.

There is nothing polemical about it. It merely reports that

a. The usual polemical f.o.m. writings that claim some special restriction
on the validity of mathematical methods are unconvincing because they don't
indicate what is wrong with the methods that they restrict, and usually
contain claims, implicit or explicit, that nothing useful or important or
natural or "good" can come out of using the 'banned' methods.

b. Sure, they dwell on features of the mathematical methods being 'banned'
that are not shared by the 'certified' methods. But that is far from
pointing to any good reason why these 'banned' methods are inappropriate.

c. This does not imply any particular conviction on my part that high
powered methods are "correct" or "consistent" or whatever. It is merely a
rather skeptical assessment of the usual kind of polemics that 'ban' what is
considered to be inappropriate methods.

d. In particular, 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.

e. Nowhere do I claim that item d is an argument for the validity or
consistency of the productive but 'banned' methods.

f. But d refutes the claim that is often made, or implied, in the kind of
polemics I am talking about. It also reveals costs associated with taking a
negative polemical position against the 'banned' methods.

g. What is really new is the new and growing level of evidence for d.

> The polemical nature of Friedman's posting becomes apparent
> when he writes:
> 
>> 4. The polemical nature of this setup becomes apparent because of the
>> absence of any convincing explanation as to what is "wrong" with the
>> disallowed methods, and the absence of any convincing explanation of what
>> "good" mathematics is.
> 
> I guess that by "absence of any convincing explanation" Friedman means
> "absence of any explanation that might convince Friedman".

>So this
> polemic argument is practically tautological, since I cannot imagine
> any explanation whatsoever that might possibly convince Friedman
> about points of view he does not already accept...

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.

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.

>  Now the real situation is the opposite
> of what Friedman is describing. It is Friedman who fails to give
> any convincing explanation why the highly dubious methods he accepts
> should be accepted.

I haven't accepted any methods in the writings you are referring to.

>Thus the only "argument" that Friedman
> provides for using axioms of strong infinity is that there are
> certain arithmetical statements  that cannot "naturally"
> be proved without them, and some "core" mathematicians
> were kind enough to tell him that they find these statements  interesting.

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.

> (I am really sorry to say that these efforts of Friedman
> to get a "Kosher" stamp from "core" mathematicians look pathetic
> to me, especially when I recall that Friedman is the one who
> has introduced the very important criterion of g.i.i. into the discussions
> in FOM!). 

I have my own impressions of what is or is not natural mathematically. I
check with famous and not famous core mathematicians for three reasons:

a. To check my impressions with theirs. Generally, there has been a lot of
agreement, and this is valuable for me to know. In particular, and what is
particularly valuable, is that when I see that something at a certain level
of naturalness has been improved to a higher level of naturalness, then
these mathematicians usually agree.

b. Other people may not have reliable instincts about this. So if famous
mathematician X says something or other is "reasonably natural" or "natural"
or, even stronger, "beautiful", and the statement lies in their area of
expertise - or even when it does not lie in their area - then this has the
effect of educating people without reliable instincts in a way that I cannot
do on my own.

c. They may suggest something that is related, that allows me to get even
more "natural" examples. Indications are that this is very likely to happen
with these new Pi01 sentences. I already have had a lot of leads for better
results, coming out of talking to core mathematicians, and mathematicians
way outside logic.

>So assume that these statements of Friedman  are indeed "natural",
> "interesting" "good mathematics" or whatever other attributes Friedman
> would like to assign them, and assume that indeed every "natural" proof of
> them
> would require some axiom of strong infinity (or really the assumption
> that adding such an axiom to ZF does not lead to contradictions). So what???

That shows the cost of 'banning' these methods. This is an essential,
relevant, finding.

> Will all these facts make the truth of those statements any more absolutely
> certain so to make them entitled to be called  *mathematical theorems* ??

I regard that as beyond the scope of anything I have written that you are
referring to. I strictly follow the practice of labeling anything proved in
ZFC as a Theorem, and anything proved using large cardinals as a
Proposition. In fact, if a Theorem is proved using a substantial amount of
set theory, then I will note this (e.g., the power set of the reals and
higher). 

> Well - only to the degree that  we can say that the consistency
> of the relevant axioms of strong infinity are entitled to be called
> mathematical theorems (the truth of which is 100% certain).
> Friedman's arguments are just begging the question here, and
> (like "proofs" of the existence of God) are able to convince
> only those who are already believers!

"Friedman's arguments" doesn't refer to anything, because I didn't make any
such arguments at all in the writings to which you refer.
> 
> Actually, the claim about "absence of any convincing explanation" is the
> mild part of Friedman Polemics.

This claim I certainly did make, and continue to make.

>Much worse is the grotesque picture
> he made of predicativists like myself. From Friedman's message
> I have at last understood that I belong to a group of people whose main
> joy in life is to look for areas and methods of mathematics that
> we might be able to cheerfully reject (for no other reason
> except that we find perverse enjoyment in doing so).

I made it clear that I am talking about a certain kind of polemic that
states, implicitly or explicitly, that the 'banned' methods have no cost.
I.e., do not deprive us of "good" mathematical information.

If you write a "banning polemic" that acknowledges that there is a cost -
that 'good' mathematical information will be lost, of the most concrete kind
- then 

i. You become immune to the force of the f.o.m. results that I cite.

ii. You deprive yourself of a principal argument that is commonly used to
justify, or justify the interest in, or gain adherents on the fence, to
"banning".

> Unfortunately
> for us, because of  Friedman there is no safe place for us anymore
> to practice our  favorites activity of attacking mathematics
> and its methods. Even the areas we were sure to be safe
> for doing our evil (like axioms of strong infinity) are not really
> safe for us as long as Friedman is around!

That is absolutely correct. So people will have to abandon this argument
that the 'banned' stuff is not useful.

Of course, if you are not one of these people, then you don't have this
particular difficulty. However, you are deprived of a main argument that is
commonly used in favor of 'banning'.
> 
> So Harvey, it is about a time to tell you a secret. Predicativists
> like me are not looking at all for areas and methods of mathematics
> which we can safely attack. Actually, we are looking to expand as
> far as possible  those areas and methods of mathematics
> that we can safely *apply*!

Then you don't have this particular difficulty. I never mentioned you by
name in these postings of mine.

On the other hand, 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.  
> 
>...
> 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.

The fact (not established yet, but will be) that one cannot even restrict to
integers below 8!! and be disentangled from measurable cardinals - for the
purposes of 'good', 'valuable', 'natural', mathematics - is something that
is very relevant. If one person sees no relevance, surely some of their
adversaries in the discussions will make use of its relevance, in the
debate. 

>... For the time being it suffices to note that
> it should be clear to anyone who understands logic that the results
> mentioned by Friedman in his latest postings simply
> have no significance for predicativists.

You might think that they have no relevance, but your adversaries in the
debate not only might think them relevant, but might very skillfully and
effectively use that relevance in a debate with you. In fact, they will.

> The fact that it pays to believe that something is true
> (and this is what all the results described by Friedman amount to)
> does not makes that something certainly true - unless one is ready
> to cheat himself/herself  (which I am not).

I never addressed this matter.

>This does not
> mean that these results have no value. Predicativists do not
> identify "valuable" with "certain", and there are parts of mathematics
> that are definitely less than certain, but still quite valuable.
> But whatever value these results of Friedman have, they are
> totally irrelevant to the predicativist agenda.

I repeat the above starting with "You".

If you want to sharply limit the debate to "certainty" then *perhaps*, if
you are careful, you can avoid such f.o.m. results of the kind I presented.

However, I have no doubt that a whole bunch of other f.o.m. results can be
obtained that will cause great difficulties with polemics like "such and
such is certain" but "such and such is not certain". Who knows? I might be
able, in the future, to write about "certainty polemics" in a similar way.

Harvey Friedman