[FOM] Shapiro on natural and formal languages

[Arnon Avron]

Not long ago I read Stewart Shapiro's book "Foundations Without 
Foundationalism". It is a very interesting, well-written book, which
does  very nice and objective job in presenting mathematical facts concerning 
logics (especially higher-order logics) and their implications to  
different approaches to the foundations of Mathematics. My own main
interest was in the last chapter, concerning "competitors" to second-order
logic, but I enjoyed the whole book (most of this last chapter, by the way, 
is reproduced, with minor changes, as the second chapter of Vol. 1 of the
new edition of the "Handbook of Philosophical Logic").

  There is however one central thesis in this book which Shapiro repeats
several times in it, and he seems to take it for granted, as no
justification of this thesis is given. The thesis (as I understand it)
is that natural languages have priority over any formal language, and real
proofs and arguments are only those that are done in natural languages. 
Thus at the very first page of his book Shapiro writes:

"The languages of full logics are, at least in part, mathematical models
of fragments of ordinary natural languages, like English, or perhaps 
ordinary languages augmented with expressions used in mathematics. The
latter may be called `natural languages of mathematics'. For emphasis,
or to avoid confusion, the language of a full logic is sometimes
called a {\em formal language}".

Shapiro then continues (still at P. 1):

"As a mathematical model, there is always a gap between the language of a 
logic and its natural language counterpart. The fit between model and 
modeled can be good or bad, useful or misleading, for whatever purpose
at hand"

Shapiro takes this thesis (which he does not even state as as thesis) as
crucial for his book. Thus P.1 of the book ends with the following observation:

"Many of the considerations of this book concern the match, or lack of match,
between natural languages and the formal languages of various logics".

This thesis  has surprised me. I have always 
taken for granted the complete opposite: that both natural languages
and formal languages are used to model valid reasoning and arguments,
and that formal languages are designed precisely because natural languages
fail to do it adequately. In other words: formal languages provide a much
better framework for  modeling  valid reasoning than that
provided by natural languages. I was even more surprised by the fact that 
Shapiro did not even bother to defend this thesis which he himself described
as central for his considerations. How can he expect his later arguments
to be convincing for someone like me, who does not accept the very first 
thesis? Do I belong here to a very small minority that can be disregarded?
Well, this is what I would like to know from others members of FOM.

Let me explain now why I think that (what I call here) Shapiro's thesis 
is actually unacceptable:

1) Let me start with the question: what does Shapiro exactly mean
by saying that formal languages model ordinary natural languages?
What natural language? There is English (the example given by Shapiro),
Hebrew (my own natural language) and several hundreds other "natural
languages". Does Shapiro mean their union? intersection? or maybe
(as I suspect) he does not care, because he takes for granted that
all natural languages are "equivalent" in some sense, and so it does 
not matter which one is chosen? Well, even if this is true (which I doubt) -
it is an empirical matter that one should not base his/her philosophy
of mathematics on. But the more important issue is that this equivalence,
if it exists, depends on *translations* from one "natural language" to
another. But Shapiro himself writes (again at the very first page of that book)
that "there is no acclaimed criterion for what counts as a good, or even
acceptable, `translation'" (Shapiro has in mind here translation from
a "natural language" to a "formal language", but the same applies 
of course to translations from one "natural language" to another). 
Hence each "natural language" can, according to Shapiro himself, 
at most  *model* arguments when those arguments are originally formulated
in another "natural language" - exactly what formal languages do
(according to Shapiro). It follows that unless we give some "natural language"
a priority over the others (which we cant), all we have are just models
that model each other, but nothing that is really modelled...

  I suspect that what Shapiro really has in mind is an ideal, nonexistent
"natural language", of which the actual ones are just shadows. Well, if this
is the case, I would like to hear it.

  A related question is what exactly count as a "natural language"? Is 
Esperanto a "natural language"? Is any language spoken by some group
of people a "natural language"? (how big should the group be?
Is it inconceivable that some group of logicians would speak in some
"formal language" while developing some branch of Mathematics? Some
books I know come very close to this!). Does by "natural language"
one mean only "natural language" spoken now (and in the way they are spoken 
now)? I strongly doubt that counterparts of all the mathematical theorems 
we now find in books could have been formulated in ancient Hebrew or Greek...

2) After discussing the doubtful meaning of a "natural language" in
Shapiro's thesis, it is time to note that whatever counts
as a "natural language", no present text of mathematics is written
in such a "natural language". Shapiro is aware of this, of course, and so 
he corrected his thesis by referring ("perhaps") to `natural languages 
of mathematics', i.e. `ordinary languages augmented with expressions 
used in mathematics'. Now what expressions? certainly variables are
used in any present text of mathematics. So `natural languages
of mathematics' include variables. They certainly include also
symbols like `=" and `<". I guess also symbols like \sum for summation,
or the integration symbol \int,  are included in `natural languages
of mathematics'. What about \vee, \forall and the other standard
connectives and quantifiers? I would really like to hear what possible 
argument can be given for accepting \int and \sum as "natural", but
rejecting \forall! But if all the symbols used in First-order
Peano Arithmetics are part of a `natural language of mathematics'
then this formal language is actually a *fragment* of a 
`natural language of mathematics', and as such it simply model...
itself (according to Shapiro's thesis).  If Shapiro accepts this
then his thesis, whatever its meaning is, becomes trivial and unintersting
(I suspect he does not, but I don't see on what possible ground - unless
he wishes to claim that the grammar of this fragment is too simple
to count as natural, and only if this symbols are used according
to some very complicated grammar we can talk about a related fragment
of natural language..).  

3) Programming languages are another type of formal languages, used
to describe algorithms. I am curious to know if here too Shapiro
would claim that what they do is to "model" descriptions of algorithms
in natural languages. It is well known that there are many
algorithms that are practically impossible to describe in
a natural language like English, and so their descriptions 
in texts are done in  languages which are very close to actual 
programming languages (or sometimes *are* actual programming languages). 

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, and that *all* languages
(whether "natural" or "formal") just model reasoning (only
formal languages usually do it much better). There is no reason therefore
to attach any priority here to natural languages. On the contrary. 

Arnon Avron