[FOM] On Existence of Mathematical Objects

[Vladimir Sazonov]

Dear Professor Friedman, 

I am sorry for the delay with replying to your posting. 
Just a pressure of time (teaching, etc.) and a struggle with 
English... Thank you very much for encouraging my attempts to 
present in FOM some ideas on Formalism vs. Platonism. 
Nothing very deep on "the imaginary natural number system". 
Just trying to make things clearer. Unfortunately rather long 
and repetitive.  I do not know why, but there is some problem 
of mutual understanding on this subject which seems to me so 
simple and clear. But people feel some extremism in my position... 


Harvey Friedman wrote:
> 
> Reply to Sazonov.
> 
> On 10/9/03 3:42 PM, "Vladimir Sazonov" <V.Sazonov at csc.liv.ac.uk> wrote:
> 
> > More precisely, if you will take my posting as the whole, you will
> > see that I even agree that all these fictions, as well as electrons
> > and mathematical objects has some kind of existence, but quite
> > different. When we say "exists" we should be sufficiently precise,
> > in which sense. Many of them exists only in your, my and any other
> > person's imagination and have no immediate relation to the real
> > world.
> >
> >...
> 
> Perhaps Sazonov thinks that almost everybody else is talking
> 
> *NOT about the natural number system*
> 
> but rather talking about the
> 
> **imaginary natural number system**.
> 

I do not assert about everybody's way of expressing their thought, 
but if the context permits and to simplify the language expressions 
of what we are talking about we may talk about the natural number 
system quite freely. However, I think, we should sometimes (e.g. 
when the context forces to do that) give account on what we are 
really talking about. Then I see no way as talking about the 
**imaginary natural number system**. 

Here I should cite myself (from a recent posting to FOM): 

"There is 
something general between various formalisms, say, PRA, PA, ZFC, 
etc. which allows us to use the same colloquial term "natural numbers" 
in each of them. But no more than that. How it is possible that 
colloquial becomes scientific without sufficient grounds for that?" 


By asking the above question, do you want to say that N exists in 
any absolute sense? I remember your recent posting where you wrote 
that you are not a Platonist. Have I understood you correctly? 


> Then Sazonov may agree that the axiom scheme of induction for
> 
> **imaginary natural numbers**
> 
> is highly compelling, if not evident?

It is very natural, but by no means self-evident. 
(Is it self-evident the explanation of Induction Axiom 
as an iteration of the modus ponens rule where a strong 
mixing of syntax with (informal) semantics is happening?) 
The unrestricted Comprehension Axiom seems also 
very natural. Postulating Induction Axiom or even only the 
axiom that there is no biggest number is very revolutionary 
step in the mind of everyone who consider this for the first 
time (usually with the help of school teacher as an "instructor" 
- the authority of a teacher is very high for children as 
well as for University students). It strongly changes our 
initial vague idea on natural numbers. The idea is changing 
but it still is remaining a vague one as ANY IDEA. 


> 
> >... I do not say here about imagination.
> > This is quite different thing. We can imagine anything
> > without any belief in what we imagined. All mathematics
> > can be understood and practically developed in this way.
> >
> >... The great Cantor also had some very nice idea. It proved to be
> > contradictory.
> 
> The current view of the history of set theory is that Cantor's ideas were
> NOT (as far as we know) contradictory. The current view is that the
> large/small distinction is implicit in Cantor's writings (i.e., class/set
> distinction).

My stress here was not so on the Cantor's name, but on the idea behind 
the unrestricted Comprehension Axiom. Anyway, is not it his original 
idea? He himself discovered paradoxes in his set theory and probably 
had feeling of their existence before that. 

> 
> > I do not see
> > any essential difference between this idea and the idea of natural
> > numbers.
> 
> This does not seem to be defensible. The imaginary natural number system
> enjoys some very fundamental properties that the imaginary set system does
> not.

Of course, I exaggerated a little, what is possible and even normal 
in a polemic. Saying "there is no essential difference" usually 
assumes some understanding WHAT is "essential". It should nod be 
understood literally. 

I do not see
any essential difference between this idea and the idea of natural
numbers *in the sense* that both are vague ideas, the Cantors' idea 
still exists despite some contradiction were discovered, and, in 
principle, PA also could be contradictory. There is no absolute proof 
of consistency of PA and I see no way even to imagine such a proof. 
The initial idea of Hilbert to prove consistency of PA or stronger 
theories was essentially based on doing that in something like PRA 
(a version of Hilbert's "finitism"). Even if he would succeed, 
this would be only a *relative* proof. 

> 
> >May be PA will be once shown to be contradictory, too.
> 
> Are you suggesting a fundamental defect in the imagination of working
> mathematicians?

NO!!! Working mathematicians do everything well (- a slight
exaggeration). 
At "weekends" they are sometime strange. This has no influence on
correctness and strengths of their mathematical results, however 
the direction of their research (especially in f.o.m.) may depend 
on their "weekend" philosophizing. (I guess, this was the case with 
Nonstandard Analysis of Robinson and his formalist views with "not 
understanding" what is the standard N. I repeat my queston: does 
anybody know anything on how Robinson came to this theory? May be 
Professor Martin Davis can say anything as the author of the book 
on Nonstandard Analysis which, by the way, was translated to Russian. 
I guess, Professor Vladimir Kanovei can know something.)

I do not assert that I believe that PA is inconsistent. I just do not 
know. May be we are working not there where the contradiction exists. 
Especially, the non-deterministic character of formal derivations does 
not allow us to refer to the fact that so long there was not found 
any contradiction. The contradiction (if any) may be even not so 
overcomplicated. I also do not think that our intuition supporting 
PA is the full guarantee. (Of course, it IS a kind of guarantee.) 

> 
> > It would be disaster [inconsistency in Peano Arithmetic] only for those who
> have superbeliefs.
> > Just as it was in the case with the old paradoxes in set theory.
> 
> Wouldn't various theorems in real analysis and other areas of mathematics
> have to be rewritten and retaught with negations in front of them?

Of course, in comparison with paradoxes of Cantorian Set Theory any
contradiction in such theories as ZFC and especially PA would 
probably have a very strong effect on mathematics. But I would not 
consider this as a disaster. This even might be something like a 
fresh air, a new vision. Many results will be reconsidered and 
transformed in some way. Let us assume that a contradiction will 
be found only in PA, but not in PRA. It is well known how much it 
can be done in PRA - a lot of contemporary mathematics. (However, 
I cannot consider myself a specialist here.) Even if only Bounded 
Arithmetic will be preserved, I think it will be not a disaster. 
Just a lot of work and reconsideration of our views on Mathematics. 
This could be in principle even very fruitful for Mathematics. 

Evidently, all these considerations have only *conditional* character. 

I would even say that our intuition(s) on natural numbers is in a 
rather strong coherence with the axioms of PA so that a contradiction 
in PA is hardly possible. But this is not a sufficient reason 
to *absolutely* exclude a possibility of a contradiction. 

What is so extremist in this view? It is only some reflection 
concerning the fact that we have NO (absolute) proof of consistency 
of PA and even cannot expect it. What does it mean at all the 
"absolute proof"? 

> 
> Taranovsky:
> 
> > >most fields of
> >> knowledge and practical constructions depend on arithmetic (although
> >> most practical applications can actually be carried out in weaker
> >> systems).
> 
> Sazonov:
> >
> > Yes, of course. This or other way the most important mathematical
> > considerations will survive. We will probably get a very good
> > lesson on the general philosophy of mathematics.
> 
> Why do you believe this? From your point of view, why shouldn't the
> contradiction appear in our imagining, say, all permutations of the first
> 100 natural numbers? What imaginations do you accept and what imaginations
> are you skeptical of?

As I already said, I cannot exclude this possibility, in principle. 
I touched on this subject in a context of some discussion. This was 
not my claim to say that there is some defect in Mathematics. 
But I see defects in Platonistic Philosophies of Mathematics. 

I accept ANY imaginations in mathematical thinking except those 
which have no hope to be formalized or even those which cannot 
be considered in any rational way (like platonistic imaginations). 
The key question "what does it mean" should have a reasonable answer. 
Reference to some beliefs is not such an answer. Somebody belives, 
somebody not. It is out of Science. 


> 
> Taranovsky:
> 
> >> Fortunately, we know a priori that every axiom of PA is true,
> >> and hence PA is consistent.
> 
> Savonov:
> >
> > ??????????????????????
> > ----------------------
> >
> Is every axiom of PA true in the imaginary natural number system?

Yes, (essentially) in the same way as unrestricted Comprehension 
Axiom is "true" in some imaginary universe of sets. It does not 
matter that it proved to be contradictory. It is "true" in some 
imaginary universe because we *want* it to be true. We "like" 
this axiom. Imagination is quite useful thing, but, in principle, 
not so reliable. Which ever way can we know *a priori* that every 
axiom of PA is true (and in which *precise* sense?), and that PA 
is consistent? 

What about a priori truth of *only one* Euclidean Geometry? 
(A fact on a belief from the history of mathematics.)

What about a priori truth of *absoluteness* of the physical time? 

Etc. 

(Also, what about my critical notes and other question marks to 
other claims of Taranovsky? Do you consider his claims defensible?) 

As soon as we decided to work in PA we, quite naturally, 
assume some imaginary universe of natural numbers where 
these axioms hold. Exactly in the same way as with unrestricted 
Comprehension Axiom. It is not the case that we FIRST imagine 
some universe and ONLY AFTERWARDS discover in some way that such 
and such axioms hold there. (I mean a typical case. Of course, it 
is possible that some axioms "suggest themselves" afterwards, 
like the Axiom of Choice.) Of course the real process is more 
complicated. Out intuition and axioms are developed with some 
interplaying and simultaneously. Our imagination without any 
support of some formal axioms and rules is something like amoeba. 
(I am sorry for repetitions from my previous postings.)
Only after deciding to postulate some formal rules regulating our 
amoeba-like intuition (or accepting them because of an authority 
of a teacher, as at school) our "amoeba" becomes an advanced 
and powerful "animal" with a supporting "skeleton" (formalism). 

Thus, the process of arising and developing a (new) mathematical 
intuition is going along with the process of postulating some 
appropriate formalism. 

I will finish by saying that my (formalist) views on Mathematics seem 
to me quite realistic (I mean respecting the reality) and coherent. 
The main point - excluding any mysticism. (Our intuitions are not 
mysticism, of course.) Another main point - a unique and general 
formalist view on mathematics independent on whether we are considering 
set theory or natural numbers or anything else. (Compare this with 
the view of Arnon Avron - and not only his view - who is, as I 
understand, formalist for set theory and platonist for arithmetic. 
Once I allowed some doubtful joke concerning this fact in FOM for 
which I sincerely apologize. Actually, I feel a great sympathy to 
him as a person.) 


> Harvey Friedman

Vladimir Sazonov