[FOM] FOM currents [A.Robinson, Platonism and Formalism]

[Vladimir Sazonov]

Harvey Friedman wrote:
> 

"Dean Buckner" <Dean.Buckner at btopenworld.com> wrote

> >> [A] completely standard piece of mathematics that you forced
> >> ordinary language considerations on, is that "in any sequence of sets of
> >> integers, some set of integers is missing."
> >
> > This is a standard and elementary theorem of ordinary language, too!  

My Reply to Dean Buckner:

There could be NO theorems "of ordinary language". Theorems may by 
only in mathematics and in a suitable mathematical language 
(formalism). If somebody (does not matter who) proves a theorem 
(if it is really a theorem), he/she is at this moment a mathematician. 


Why
> > should I make complaints about it?  My complaint is about the idea that any
> > set of integers constitutes ALL integers.


As I do not know the context, I cannot comment on this. 


Friedman: 
> 
> The set of all integers has every integer in it (as an element). Yes, there
> are some people who are uncomfortable with this idea. 


that the set of natural numbers is existing (actually or potentially) 
in an absolute way? 


Friedman: 
> Not Cantor, and not
> the overwhelming number of mathematicians. But certainly there are some.



E.g. Abraham Robinson. Recently his formalist position was mentioned 
by Haim Gaifman with reference to his  1964 paper``Formalism 1964'' 
in {\it Logic Methodology and Philosophy of Science}, Y. Bar-Hillel 
ed., North-Holland 1965, pp.228-246.

>From that context it was unclear to me what exactly was the position 
of Abraham Robinson. Now I have got this paper and can cite it 
with some my own comments in-between (and with no pretension 
on a most adequate presentation of the ideas of Robinson; 
at least not everything in this paper is sufficiently clear 
for me): 


Abraham Robinson:
"My position concerning the foundations of Mathematics is based on 
the following two main points or principles. 

(i) Infinite totalities do not exist in any sense of the world (i.e., 
either really or ideally). More precisely, any mention, or purported 
mention, of infinite totalities is, literally, {\em meaningless}. 

(ii) Nevertheless, we should act {\em as if} infinite totalities 
really existed. [VS - we can imagine them without any beliefs 
in their absolute existence.]

..........

K.G\"odel, who may be regarded as the outstanding platonist of our 
time, has emphasized the similarity between the investigation of 
physical objects on one hand and of mathematical objects on the 
other. ... I am in sympathy with this point of view to a very 
limited extent. 

[VS - I recently argued in FOM what is the difference between physical 
objects, even sush abstract as electrons, and mathematical ones.]

......

By contrast, I feel quite unable to grasp the idea of an actual 
infinite totality. 

.....

It follows that I must regard a theory which refers to an 
infinite totality as {\em meaningless} ...

.....

An opponent to my position might put....because my brain suffers 
from a particular limitation.

.....


As far as I know, ***only a small minority of mathematicians, 
even those with platonist views*** [VS - my emphasis], accept 
the idea that there may be mathematical facts which are {em true} 
but unknowable. 

[VS - does not it mean that the overwhelming majority of 
mathematicians are just formalists in their "working days"?] 

.....


At any rate, it seems to me that the present situation in Set Theory
favors the Formalist. 

.......


...the formalist holds that direct interpretability is not a 
necessary condition for the acceptability of a mathematical theory. 

.....

... it is a fact that the organized world of Pure mathematics 
is regulated to a very large extent by our ***vague intuitive ideas*** 
[VS - my emphasis] on mathematical beauty and pure mathematical 
importance. 

.....

To {\em understand} a theory means to be able to follow its logical 
development and not, necessarily, to interpret, or give a denotation 
for its individual terms. [VS - I understand - any precise or even 
only "unique" platonistic denotation is unnecessary; only a vague 
idea suffices.]


In Hilbert's view the formal or uninterpreted part of a theory 
belonged entirely to Mathematics. 

........

Since the formalist cannot rely on semantic interpretation of 
his theories [VS - I think, because it is based only on our vague 
intuition], their complete formalization is essential to him. 
This is part of the original formalist approach due to Hilbert."

etc. 



Vladimir Sazonov:

Abraham Robinson is very polite. Nevertheless, he used the quite 
strong word "meaningless". Also, I do not think that his "brain 
suffers from a particular limitation". 


By the way, was not this doubt of Robinson in meaningfulness 
of the unique N his starting point to invention of Nonstandard 
Analysis? Does anybody know how have he came to this idea? 
It does not matter that he used quite standard mathematical 
tools (ultraproducts or whatever else). 


Friedman [VS - again]: 
> Not Cantor, and not
> the overwhelming number of mathematicians. But certainly there are some.
> 
> E.g., Sazonov. 

I ("together" with A.Robinson) already commented above on
"overwhelming".

I feel myself very comfortable in this company of David Hilbert 
and Abraham Robinson. What is very strange to me that the greatest 
Kurt Godel was a Platonist. When first I studied his mathematical 
results I had no idea about his general views and was highly 
surprised later. It seems to me that his (working days) results 
contradict to his (weekend) philosophy. 


> But note that Sazonov's postings have some connection with
> the foundations of mathematics, 

Thanks for not considering my efforts as irrelevant. 


> even though - in my opinion - he has not
> convinced anyone to be any more skeptical about the set of all integers than
> they were before they read his postings.


Who knows? There are some people who explicitly in FOM or 
privately wrote or told me on analogous views. Not so many 
people are writing to FOM. Those who have some superbeliefs 
cannot be convinced. What I consider important, that MEANINGLESS 
or MYSTICAL views in science or in foundation of science should 
meet some counteraction. They are not so harmless as it could seem. 


Even if to put aside the question of the vagueness of N 
(whether it is understood potentially or actually), WHAT DOES 
IT MEAN that it exists UNIQUELY. 

"What does it mean" is the most important scientific question 
as we know this from Albert Einstein and other greatest scientists. 
We are, indeed, using this question everyday, although in some 
specific cases, related with our beliefs, forget. 


What is still unclear to me, is your own, Professor Friedman, 
opinion. You seems to deviate from some of the above questions. 
The reason looks good - let people discuss, the concrete 
mathematical activity is more important. But what is YOUR 
personal opinion on mysticism in the form of Platonism? 

By the way, I recall some your quite concrete arithmetical 
sentences in terms of sin and exponential which you predicted 
will be never proved or disproved in any natural extension 
of PA (that is, even by using some strong extensions of ZFC). 
[I do not remember precisely, but this seems was your point.]
This looks rather plausible. Large cardinals can hardly have 
any relation to resolving such kind of statements. 

Also, some people believe that P=?NP will be eventually resolved 
in PA. But let us assume temporary that it is not the case. 
Will then large cardinals settle this problem? The character of this 
problem is such that this problem is hardly related in any way 
to stronger versions of ZFC. On the other hand, this (or a very close)
problem can be formulated in a rather weak fragments of PA, or 
just as a finite (non-asymptotic) form, like the problem "who wins 
in 8x8 Chess game?". 

We could take, for example, a finite, may be second-order, 
(recursive) arithmetic of a segment {0,1,...,1000} 
or something like this and ask whether second order quantifiers 
can be eliminated. Who knows, may be this is a direction where 
this problem will be properly understood and settled down? 

(Here we may need a formal theory feasible numbers because 
we should consider only feasible proofs to demonstrate the 
possibility or inpossibility of feasible second order 
quantifier elimination or some independence or non-constructibility 
result. Say, to demonstrate that whites in Chess have no 
constructive winning strategy we cannot avoid some formalized 
feasibility concept.) 

P=?NP might be quite independent of PA (like CH on ZFC), but 
resolvable (in some way) in quite different approaches assuming 
some new views on natural numbers. This might be considered as 
a fantasy. But even your own, somewhat artificial but convincing, 
candidates (with sin, etc.) on undecidable arithmetical statements 
show that it is quite imaginable that such effects like independence 
of CH are possible in PA without any hope to be resolved forever 
(except as trivially taking them as axioms). 

Should we still believe in the uniqueness of N? 
Or, is it worth to continue to believe? Why to believe at all 
in anything in mathematics or in science? This seems to me 
a very inappropriate term and way of behavior in science. 

Likewise the term "realism" (equal? to Platonism) in the philosophy 
of mathematics is highly misleading. I consider myself a realist 
in the normal meaning of this word, and my realism has nothing 
general with "realism" = "Platonism" how philosophers are using it. 
The very occupation of this quite respectable term in quite 
opposite way demonstrates that this philosophy is wrong. 
Terminology should help to understanding, not to disappointment. 
Does not it mean that philosophers just have no clear idea 
on the proper meaning of this word, or rather try to use it 
to make it easier to convince us in their wrong ideas. 

I refer to your examples (arithmetical candidates on a strongest 
independence). What is YOUR opinion? Why have you presented them 
at all? 
 

> Harvey Friedman

Best regards, 

Vladimir Sazonov