[FOM] Platonism and Formalism

Vladimir Sazonov V.Sazonov at csc.liv.ac.uk
Mon Sep 22 14:50:34 EDT 2003

Karlis Podnieks wrote:
> ----- Original Message -----
> From: "Torkel Franzen" <torkel at sm.luth.se>
> To: <fom at cs.nyu.edu>
> Cc: <torkel at sm.luth.se>
> Sent: Saturday, September 06, 2003 8:35 AM
> Subject: Re: [FOM] Platonism and Formalism (a reply to Podnieks)
> > ...
> >We must examine how people actually work, how they argue, how
> > they apply mathematics, what kind of considerations seem to guide
> > their thinking in practice.
> > ...
> > A consistently non-Platonistic point of view with regard
> > to mathematics is much like a consistently skeptical view of human
> > knowledge: it has a certain appeal to the intellect, but it has no
> > apparent relation to how people, including professed non-Platonists or
> > skeptics, actually go about their business.

I have already replied to this.

> >
> > ---
> > Torkel Franzen
> I tried to explain this "practical" aspect of the mathematical Platonism in
> my old paper "Platonism, Intuition, and the Nature of Mathematics"
> (http://www.ltn.lv/~podnieks/gt1.html):
> FOR HUMANS, Platonist thinking is the best way of working with stable
> self-contained systems (the "true" subject of mathematics - at least, for
> me). Thus, a correct philosophical position of a mathematician should be:
> a) Platonism - on working days - when I'm doing mathematics (otherwise, my
> "doing" will be inefficient),
> b) Advanced Formalism - on weekends - when I'm thinking "about" mathematics
> (otherwise, I will end up in mysticism).
> (Of course, the initial version of this aphorism is due to Reuben Hersh).
> For me, such a position is neither schizophrenia, nor perversity - it is
> determined by the very nature of mathematics. 

I agree with Karlis Podnieks, to a certain degree, because I 
feel WHAT is behind of his views. I always sympathized with his 
writings on f.o.m. because of this feeling. However, I would 
present things somewhat differently. 

I do not think that Karlis intends to say that Advanced Formalism 
on weekends consists only of formalisms without any intuition 
on which they are based. 

What Karlis calls "Platonism" on working days is just a kind 
of intuition related with the classical logic. (The proper 
Platonism is, of course, a wrong idea.) Given a classical 
first-order theory, we just imagine a world of abstract objects 
with operations and relations over them as if it exists. We also 
imagine that each sentence of this theory as interpreted in 
this world and as having a "truth" value. I use "" because 
everything is imagined. This is not a real world where some 
sentences may be true or false. Also our intuition or imagination 
is, by the very nature of these concepts, vague and unstable. 
Thus, there is no good philosophical reason to think that any 
such imaginary world is rigidly fixed for all mathematicians 
and independent on the theory considered. Saying the converse 
means essentially a wrong understanding of what is the intuition 
in general. 

Then, I think, there is no essential difference between working 
days and weekends, except that on weekends we are more relaxed 
and reflecting on what happened on working days. Both intuition 
and formalisms are under our consideration permanently. 
On working days it is quite normal for any mathematician to ask 
himself: I have an IDEA; how to FORMALIZE it? It does not matter 
that mathematicians usually do not explicitly work with formal 
systems as logicians describe them. They say "FORMALIZE", and 
it is clear that this is really about a formalization. We strictly 
distinguish between INFORMAL (intuitive) and FORMAL (rigorous) 
reasoning. Both are used almost simultaneously or interactively 
on our working days. 

Moreover, on working days, any new, deep, revolutionary mathematical 
concept can hardly arise without some reflection on the nature 
of mathematics, how any ideas could be formalized and what 
does it mean to formalize. 

The kind of intuition described above is related with classical logic. 
But we know that there is also intuitionistic mathematics based on 
a different intuition. The fact that intuitionistic logic is 
formally reducible to classical, say, via Kripke models plays here 
not so essential role. Even when working classically with, say, 
Heyting algebras we need to use some additional non-classical intuition. 

I recall a paper by Dana Scott where he advised something like this: 
Do not listen to those who advise you that if anything is reducible 
to classical logic then the corresponding intuition is reduced too. 
I hope that I formulated the main point correctly. 

Thus, in general, there is no need to mention Platonism, truth or 
anything more or less analogous. We rather should use most general 
terms like idea, intuition or imagination because we cannot predict 
what particular kind of formalisms and intuitions can be considered in 
mathematics. In principle, some formalisms and intuitions could even 
be not reducible to classical logic. 

As an illustration, consider the following axioms on the "set" F 
(which is not a set in the traditional set theory) of Feasible Numbers: 

(1) 0,1 \in F,
(2) F + F \subseteq F (in particular, F has no last number), and 
(3) \forall n \in F (log_2 log_2 n < 10). 

The first two axiom say that 0 and 1 are feasible, and feasible 
numbers are closed under the addition operations. Some evident 
axioms like 0 < n + 1, n < n + 1, transitivity of <, etc. are 
omitted for the brevity. The axiom (3) is true in the sense that 
it can be experimentally confirmed for all n physically(!!) 
written in the unary notation |||...|| or 0+1+...+1. It says 
quite reasonably that all feasible numbers, although having no 
last number, are less than 2^{1024}. That is why it is interesting 
and intriguing to take it as an axiom. 

In the framework of the classical logic, we can derive in this theory 
a contradiction by a feasible(!) proof (not so trivially, although 
also not so difficult for those who is acquainted with intractability 
of cut elimination). Then, the problem is to find a natural 
(feasibly) consistent formalization, i.e., actually, a logical system 
for the above axioms. Some restricted version of classical logic 
does work. But during consideration of the resulting formal theory 
together with underlying intuition (here omitted) it becomes clear 
that both are irreducible to the classical ones. For example, 
adding one more operation 2*n and the axiom 2*n = n + n makes 
this theory feasibly contradictory even in this logic. 

Note, that somewhat analogous considerations on feasible numbers 
in the framework of classical logic have been done by Pohit Parikh. 
The idea of feasible consistency seems was first explicitly 
explored by him for a coherent formalization of an interesting 
intuition. I omit here any further comparisons. 

> And this is why I would ask
> FOMers having bigger mathematical intuitions than my own to try a high level
> ("Corfield style") explanation of the last remaining Platonist illusion:
> Which properties of structures and methods used in mathematics and
> metamathematics are leading to the illusion that the natural number system
> is
> a stable and unique mathematical structure that exists independently of any
> axioms and cannot be defined by using axioms?

It is really one of the most crucial questions on the foundations 
and philosophy of mathematics. 

Vladimir Sazonov

> Best wishes,
> Karlis.Podnieks at mii.lu.lv
> www.ltn.lv/~podnieks
> Institute of Mathematics and Computer Science
> University of Latvia

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