[FOM] Platonism and Formalism

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

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:
>
> PROBLEM
> 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.