[FOM] Formalism/Platonism

[Vladimir Sazonov]

A belated reply to Friedman

Harvey Friedman wrote:
> 
> I would like to start a thread here on the FOM in which
> 
> 1. Descriptions of various kinds of formalism and Platonism are spelled out
> and analyzed.
> 
> 2. Arguments for and against these positions are spelled out and analyzed.
> 
> 3. New research projects suggested by 1,2 are spelled out, and perhaps to
> some extent carried out.
> 
> Formalism/Platonism is an old classical topic, but I believe that when
> considered in the light of modern f.o.m., this can be very fruitful.
> 
> I think we should first try to deal with extremes of the two. Let us call
> them ultraformalism and ultraplatonism.

I do not understand what is "ultra" here. I believe in the possibility 
of consistent formalist position unlike the platonist one. The latter 
I am considering as just a wrong idea. I hope you do not consider any 
consistent position as "ultra". 


> 
> In the interests of productivity, I don't think we should - at least at
> first - say too much about formalist positions that are so extreme that they
> deny any sense to the meaningfulness of, say, bit strings of length 2, or
> for that matter, bits. Or deny any sense to standard mathematical discourse
> about bit strings of length 2, say with the use of the connective "and".
> That may already be interesting, but is so different than what happens when
> you accept a very minimal amount of apparatus that it would make the
> discussion too unfocused, too unruly, and probably totally unproductive.
> Such truly extreme positions are far beyond those taken by anyone publicly
> on the list, including Sazonov.

Aha, there are therefore some suspicions concerning me. Thanks for not 
including me in the category of "ultra" (if I understand you correctly). 
Anyway, for a more "healthy" discussion, I agree that it is better to
not 
touch upon such questions like whether "3" exists or whether "2+3=5" is 
absolutely true. Although a consistent formalist position should assume 
something about this, and there is nothing "ultra" here as well. 

> 
> So the framework that I am thinking about for spelling out kinds of
> formalist positions is reasonably standard. There is an acceptance of the
> objectivity of lots of mathematical statements about lots of mathematical
> entities. Perhaps not too many statements about too many entities. But
> nonetheless a good healthy number.

Frankly, I would not start this way because this is actually about 
mathematical truth (of an elementary character) which itself is a 
controversial topic, but let it be. 

> 
> The general formalist idea is that, beyond a certain amount - perhaps
> minimal amount - of very concrete contexts, mathematics becomes a kind of
> game with prescribed rules in the form of formal systems. Under this view,
> mathematical activity  - in the sense of finished mathematical productivity
> - consists only in episodic announcements that one has "won the game" in the
> form of having a derivation of some formalized statement or other, and
> particularly when pressed, actually producing the claimed derivation in
> print or nowadays, by electronic communication. Under the formalist view,
> the mathematical record consists of such produced derivations.

I cannot say for all formalists, but I would consider *such* 
formalist view as something "ultra". Even that Fields medalist 
about whom you wrote, I believe, had some intuitions and 
imaginations on what he proved. I think he exaggerated a little 
describing his position on mathematics as a pure game. The only 
(or the main) difference with platonists consists in considering 
this intuition and imaginary worlds as inevitably vague, even 
if the axioms give us a full illusion that everything essential 
is fixed by them. That the intuition on these imaginary objects 
exists does not mean that these "objects" also exist. Only some 
kind of belief may lead us to think that these object exist. 
But my main question is:

Why do we need any beliefs at all in science. Science is based 
on something different. The only argument for existing 
mathematical objects is a belief. That is not a rational answer 
to the question "what does it mean that such objects exist 
objectively (somebody believes, somebody not; voting 
is not a scientific method as well) and that each (formal) 
question on them has a definite truth values. All of this 
is outside of science. 

Science is based on facts and theories about these facts. 
Mathematical facts are formal (rigorous) proofs of theorems. 
This is the only solid ground I see to stay in any discussion 
on the nature of mathematics. This is the reason why I am a 
formalist. Another, less solid, but quite real ground, is our 
intuitions and imaginations related with formalisms we are 
considering. That is essentially all we need to develop a 
consistent formalist view on mathematics without invoking 
any beliefs or mysticism. (I am grateful to Allen Hazen 
for his comments on platonism and mysticism and for his note 
that at least mysticism should be outside of science; I will 
think about his notes on mysticism more.)

> 
> Under the formalist view, a conjecture is nothing more than a prediction
> that someone will be able to find a derivation of the associated
> formalization within the prescribed rules - within some prescribed amount of
> time.

It seems I agree, provided some intuition and imagination 
are supporting this prediction. 

> 
> Note that if we all agree on what the prescribed rules are, 

How it is possible not agree? If I assert that I derived a 
theorem in such and such formal system then anybody *who is 
interested* MUST follow these (or equivalent) rules. 
***There is no choice at all***. 


then it is not
> clear whether there is any practical difference between the formalist and
> the non formalist (a non formalist may or may not even have a position, may
> or may not be a Platonist, etcetera). It would seem that both would operate
> the same way.

Yes, in the working days all mathematicians behave in the same way. 

> 
> However, this is not quite true. The formalist and the non formalist are
> likely to differ about what they think is important to get derivations of.
> This point has previously been made on the FOM by John Steel.

I think, to anyone what he/she thinks in weekends on mathematics 
has influence in choosing the direction of research and therefore 
on the formalisms and derivations obtained in working days. 


> 
> So I would like to pin this down. Steel, a card carrying ultraplatonist, is
> deeply interested in certain mathematical questions that do not seem to be
> so easily characterized in terms of playing a game. E.g., Steel is
> interested in the continuum hypothesis. He is not interested in working on
> the continuum hypothesis in any particular formal system, as the formal
> systems that Steel is interested in are well known to leave the continuum
> hypothesis untouched.


I am not acquainted with the research of Professor Steel just because 
I cannot be interested in everything, but I believe that he as a 
mathematician eventually *proves* his theorems in a suitable formal 
systems. Am in not right? 


> 
> Instead, Steel is interested in settling the continuum hypothesis - although
> this is not his main research interest at the moment. This seems to defy the
> model of mathematical activity that the ultraformalist has. So perhaps the
> ultraformalist would say that this research interest of Steel's is
> meaningless? incomprehensible? wrong-headed? what?

If he is trying to find a natural, coherent (formal) version 
of set theory where CH will be resolved, why it is against 
formalism? I do not understand. This could be done, in principle, 
by formalist, too. 

> 
> Steel is also interested in settling the relationship between various large
> set theoretic statements; e.g., strongly compact cardinals and supercompact
> cardinals. E.g.,
> 
> is ZFC + "there exists a supercompact cardinal" interpretable in ZFC +
> "there exists a strongly compact cardinal"?
> 
> This is an arithmetical statement that Steel suspects is decided within very
> weak fragments of PA, so weak that even Sazonov would almost certainly
> accept a likely proof of this statement or its negation.
> 
> QUESTION: Does Sazonov accept the proof that, e.g., ZFC is interpretable in
> ZF?

It is strange question for me. I wrote so many times that I 
(as a formalist and because I am formalist) accept ANY formal 
system which is based on an interesting intuition and any 
derivation in this system. My personal interest to weaker 
systems is like interest of anybody else to strongest versions 
of set theory. If I would have more time I would be interested 
in many various formalisms. I NEVER said that some formal 
systems are unacceptable for me in principle. The other question 
is whether it is interesting to me or whether I consider such 
and such direction of research as promising. And the weakness 
of fragments of PA does not matter for me in this context. 

I am interested in weak fragments of PA by a reason not 
related with the above considerations. 

I WOULD NEVER SAY THAT PROOFS IN SUCH AND SUCH SYSTEM 
ARE UNACCEPTABLE BY ME BY SOME HIGHEST REASONS. 

Should I write this even in bigger letters to fix clearly 
this my position? 


> 
> However, as Steel suggested in his recent posting, maybe the crucial
> difference between he and Sazonov is that Sazonov would be very
> disinterested - on principle - in a question like
> 
> is ZFC + "there exists a supercompact cardinal" interpretable in ZFC +
> "there exists a strongly compact cardinal"?

Why "very disinterested"? If I would have more time and more 
abilities, I would be interesting in everything what is 
happening in mathematics. However, I have not so much time, 
and I should choose what seems to me more interesting. 

Of course, I should add that I doubt that CH can be resolved 
in any natural way. I think that CH is a typical example of 
strong undecidability which can be found in any sufficiently 
non-trivial and strong theory, even in PA. This corresponds 
well with my formalist views. [Please do not consider me as 
asserting that I am expert on CH in any sense! These are only 
my feelings.]


> 
> regardless of how concrete the (proof of the) answer is.
> 
> One challenge to the ultraformalist is: how are we to choose the 'rules of
> the game'? On what basis? Have you, the ultraformalist, accepted the 'rules
> of the game', or at least some of the rules of the game, and in what sense
> accepted?

It is unclear to me who is ultraformalist according your 
classification. I consider myself just a consistent (what 
does not mean "ultra") formalist. To be inconsistent would 
be just unreasonable and indefensible. At least, I am trying 
to be consistent by presenting a sufficiently full picture 
of views related with formalism. It seems to me that I am 
also presenting something new. Time is going ahead. We should 
add at least something new to the old views. 


> 
> I am not a Platonist. I am not a formalist. I am an investigator.
> 
> For me, the crucial issue is: what kind of work in f.o.m. will shed light -
> not necessarily settle - this dispute? Or at least, what kind of work in
> f.o.m. will be deeply relevant to this dispute?

Formalist views seems to me open a wider horizon. 
Platonist views do the opposite. 

However, let me read other postings. I had and have almost 
no time even for reading. I am glad to see so many postings 
around Platonism/Formalism. 

> 
> Fortunately I think I have some promising ideas about this. But I will stop
> here to see how things progress.

I am interested to read about this if you already did not present 
these ideas. 


Vladimir Sazonov

> 
> Harvey Friedman