[FOM] Some Clarifications on Platonism and Formalism

[Vladimir Sazonov]

Dmytro Taranovsky wrote:
> 
> Podnieks:  "How is it possible that the most fundamental mathematical
> structure allows only an implicit description?"

> Formal definitions only allow descriptions of an object in terms of
> other objects.  The most fundamental concepts are simply understood
> without being formally defined.  


It is illusion that they are "simply understood without being 
formally defined."

They (and their "behavior") are formally defined (better to say - 
restricted, organized, regulated) by axioms and proof rules. 

What would be our understanding of the very elementary concepts 
of set theory if we would have NO idea on reasoning in classical 
logic (let only on examples of school geometry) and on, say, 
Extensionality Axiom or (a version of) Comprehension Axiom? 
Could you imagine this at all? 


Other concepts are defined in terms of
> the basic ones. In fact, almost all (but not all!) mathematical
> semantics can be expressed in the first order language of sets and
> membership.
> 



> 
> Harvey Friedman:  "The imaginary natural number system enjoys some very
> fundamental properties that the imaginary set system does not."
> More precisely, natural numbers are concrete.  

Is 2^1000 sufficiently concrete? In which sense it is more concrete 
than 2^{\Aleph_0}? Both are big numbers, but how big? Both do not 
exist in the sense as 3 apples exist. 

2^1000 may be considered as a representative of the set of binary 
strings of the length 1000. Each concrete such string can be 
physically presented. But what does it mean the set of ALL such 
strings? We have a lot of concrete examples, but what about ALL? 
It is definitely a vague "set", even if we can prove some theorems 
on it. I feel here a very strong analogy with 2^{\Aleph_0}. 


If n is a natural number,
> then (at least, theoretically) n can be stored and manipulated by a
> computer.

And so what? 

Yes, theoretically, especially for the case of n=2^1000 (imaginary 
"written" in unary notation system). But theoretically does not mean 
practically. 

> 
> Sazonov:  "The difference [of the study of works of fiction]
> with mathematics is only that in these stories there are no formal
> rules of behavior of these personages."
> 
> Simple axioms which are obviously true 

Nothing is obvious for me, even the modus ponens rule 
or the transitivity of implication or equality. 
Imagine a situation when a small "mistake" is accumulated 
in each step of a derivation. The longer is the derivation, 
the more "doubtful" may be the result. Consider this as 
an exercise for thought. (Something such have been written by 
Poincare.) 

As to "true", I do not understand this word in this context. 
I would prefer "natural", instead. 

> provide an almost complete theory
> of integers.  

How much "almost"? Who can measure?

Moreover, there is a unique natural way to resolve all
> natural incompletenesses in the theory of integers.  Most of the
> important incompletenesses in PA arithmetic are solved simply by
> adding:  If for all n, PA proves phi(n), then for all n phi(n).  

I do not understand these "all n". Are these n (syntactical and 
physically written) numerals 0+1+1...+1? Are you able to prove 
phi(n) for each such numeral? May be we should also prove this 
for n = 2^1000 imaginary written as a numeral? Do you realize 
that no numeral exists which would be (provably) equal to 2^1000. 
May be you mean imaginary (not physically written) numerals? 
But then, strictly speaking, they are not syntactical expressions. 
This mixes the theory with a metatheory. Does not all of this 
lead us to a vicious circle? 

But I also do not understand your argument as a whole. 


All
> known 

but what about yet unknown? 

natural arithmetical incompletenesses are solved by invoking the
> notion of a set of integers, and adding basic axioms about such sets,
> along with projective determinacy.
> 
> By contrast any attempt to formalize a work of fiction would lead to a
> long list of arbitrary axioms and plenty of natural incompletenesses,
> almost none of which are resolved by natural axioms.

Nobody wants to formalize a work of fiction, although there is 
a close analogy with formal theories. In both cases we have 
incomplete characterization of the "world" described. 

What is important about formal theories it is their potential 
applicability in (other) sciences, and the *real* mechanism 
supporting these applications is based just on these formalisms 
( + our intuition relating imaginary mathematical worlds with 
the real world). Without these formalisms (various calculi, 
algorithms) no application would be possible. 


> 
> Robinson, as quoted by Sazonov:  "As far as I know, only a small
> minority of mathematicians, even those with Platonist views, accept the
> idea that there may be mathematical facts which are true but
> unknowable."
> Only a small minority of mathematicians work in the areas where
> independence from ZFC is dominant.  Apparently, the rest do not bother
> to determine whether every mathematical incompleteness can be solved by
> human civilization.

Although the issue about majority-minority is arising time to 
time in FOM, I would prefer to discuss in different, more 
scientific terms. (Will we vote or argue?)

The point is that if you are working as mathematicians, you are 
proving theorems. As the matter of fact, according to their 
behavior, and according to the essence of mathematics, they 
are looking for proofs, not for truth. Mathematicians can 
speculate in "weekends" about truth, but only proofs (definitions, 
axioms, logical rules + some related intuition) are what they 
are really DOING. 


Also, in the areas where independence from ZFC is dominant 
the formalist view comes to the first place, just because of 
these independencies. As to arithmetic where independence 
from PA is not dominant, I already recalled in a reply to 
Friedman that some his examples are very plausible to be 
independent of PA in a very strong sense that no strong 
version of set theory would resolve them. We just should 
wait for some more natural and stronger independence results. 
However, for me the independence of Con_PA or its well-known 
(natural) combinatorial versions are quite sufficient. 

> 
> In his reply to my posting "On Platonism and Formalism"
> Sazonov wrote,  "Why do we need these subtle considerations whether the
> physical reality exists or not?"
> Such considerations are needed to dispel the belief that existence means
> physical observable existence and that references to mathematical
> objects are merely references to appropriate physical systems. 

That is, to find a way to identify physical reality with 
mathematical worlds. I already argued that the existence of 
even electrons has a sufficiently straightforward relation 
to the existence of "cobble-stones", unlike mathematical 
objects (even such as the number 2^1000). 

Of course, if it is your predominant intention to convince 
yourself in identity of physical world with mathematical you 
will always find a way. We will see below how far can you go 
in this direction. 

> Sazonov:  "There is NOTHING in the real world what would
> correspond to the real (and even natural) numbers PRECISELY."
> If the universe is infinite, then every integer is physically realized
> as the number of "cobble-stones" in a particular region of space.  

I would advise you to read my paper "On feasible numbers" 
available via my homepage. This is an attempt to formalize 
what are (feasible) numbers which can be "physically realized 
as the number of "cobble-stones" in a particular region of 
space" taking into account some (astro)physical 
***real fact*** that our universe is bounded. 

You intention to identify "imaginary" with "real" is amusing 
me more and more. 

Every
> real number is probably realized as the distance between two particular
> points.

Take two particular points on your desk and make a measurement 
of the distance between them. What "exactly" real number will 
you get? Or is it all in your imagination? Including the real 
world? 

> 
> Higher levels of the cumulative hierarchy are probably not physically
> realized, but they still exist.  They exist because they appear to exist
> and there is no evidence to the contrary.  

For me they do not appear to exist because I am a realist 
(in the normal sense of this word) and there is no evidence 
of their existence (except in my or your imagination). 

But I see strong evidence that you are not a realist (in the 
normal sense of this word). For you ***even the real world is 
imaginary and idealized*** (if you are able to "find" there 
real numbers (even \pi?). Do you recall, at least, that our 
world consists of atoms and other elementary particles? Real 
numbers serve only as a formal tool to approach some way to 
the reality (not to be included in it). May be they (in the 
current form) are even not the best tool as our world is 
discrete (and simultaneously continuous; electrons are both 
particles and waves - thus our real world even does not follow 
the classical logic). 


They are not vague because
> they can be formally defined in the first order language of set theory.

Formal (first order or even does not matter in which way 
formalized) axioms only RESTRICT or govern the behavior 
of these imaginary object. Our imagination is much more 
flexible to be FIXED in any absolute way. 

> 
> Best Wishes
> Dmytro Taranovsky
> http://web.mit.edu/dmytro/www/main.htm


Best wishes to you too, 

Vladimir Sazonov
-- 
Vladimir Sazonov                        V.Sazonov at csc.liv.ac.uk 
Department of Computer Science          tel: (+44) 0151 794-6792
University of Liverpool                 fax: (+44) 0151 794 3715
Liverpool L69 7ZF, U.K.       http://www.csc.liv.ac.uk/~sazonov