[FOM] Shapiro on natural and formal languages

[Vladimir Sazonov]

It seems I agree almost with everything written by
Arnon Avron (hi, Arnon!) in his posting on
Stewart Shapiro's book "Foundations Without Foundationalism",
until this place:

> 4) One final note: there are plenty of geometrical ways of reasoning 
> that can easily be visualized and understood, but are extremely difficult  
> to be translated into any natural language. In my opinion
> this fact strongly supports my belief that
> reasoning comes before languages, 

Oops! Are not these geometrical ways of reasoning done in a version
of a formal language? Are not these geometrical figures and
manipulations with them in geometrical proofs something like
figures of logical inference? I cannot imagine a reasoning
outside of a language (probably in a wide sense), that is - as a
pure(??) reasoning. Of course, *rigorous* reasoning uses a *formal*
language (in that or other way). Is not "formal" [language, etc.]
derivative from "form" [possibly geometrical one]?


and that *all* languages
> (whether "natural" or "formal") just model reasoning 

model *pure* reasoning existing independently or before any language?


(only
> formal languages usually do it much better). 

Do we *model* or *formalize* (i.e. express in a formal language)?
Is there anything to model here, in this context? You wrote yourself
that only formal language can describe some concepts. Therefore,
before formalizing we have only a vague idea (like a germ) of what
should be formalized. Otherwise, what is the reason to formalize
what was clear enough beforehand, as expressed in an informal way?
I understand formalization just as a specific process of growing
up of a "germ of thought", of expressing what did not and could
not exist as a full-fledged concept before expressing it formally.
This seems something different from modeling, although formal
languages can be used for modeling too. Say, the "language" of
differential equations can model the movement of planets.

However, the main point of my notes is not about the difference
between formalization and modeling. These are sufficiently close 
concepts, although in some contexts one term is more preferable
than the other. To model something it should exist. I just do not
believe in pure, language-free existence of reasoning.


There is no reason therefore
> to attach any priority here to natural languages. On the contrary. 

Here I almost agree again, except that I want to make some
clarification; it seems there is no real point of disagreement
with Arnon here.

The initial vague idea is usually expressed in a natural language.
(For natural numbers this is "one, two, many". Anything more advanced
in arithmetics, like decimal notation or induction axiom,  appears
only together with and due to some formalization - explicit or
implicit.) Therefore, natural language is the *first* to start
with, *but not the primary* language to express some kind of things.

Here I assume for simplicity that (any) natural language does not
contain formal languages as its fragments. Although I like the idea
on a growing in time, or indefinite natural language which might
incorporate some formal languages too. Is "twenty three" (not so
primitive as "many") formal or informal expression in English?
(The difference between "formal" and "informal" is not crystal clear,
but it is clear enough to make a distinction.)


Vladimir Sazonov