[FOM] The Natural Language Thesis

[Arnon Avron]

The discussions concerning the "Formalization Thesis" reveal that
many of us still hold a sort of "natural language thesis" (NLT). 
According to this  thesis, natural languages have priority 
over any formal language, and real theorems,
proofs, and arguments are only those that are formulated and
done in natural languages (or what was called by some,
without any further explanation, "ordinary mathematics").  
Accordingly, they see "formalization" as a "translation" of
"ordinary mathematics" into a formal language, rather than
what it really is:  Expressing mathematical theorems
and proofs in a precise, objective way, using precisely
defined languages that have been especially *designed* for
that purpose.

This NLT is  explicitly formulated (but not as a 
thesis, but as something self-evident) in the first page of 
Shapiro's book "Foundations Without Foundationalism" as follows:

"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 in 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"

As I have already said above, I have always 
taken for granted the complete opposite: that both natural languages
and formal languages are used to express mathematical facts
and 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  expressing  mathematical facts in a precise way
and for modeling  valid reasoning than that provided by natural languages. 

Let me explain now why I think that NLT is totally unacceptable 
(explaining why formal languages are actually *superior* will be 
delayed to another posting):

1) Let me start with the question: what exactly is meant
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 the thesis refer to their union? intersection? or maybe
(as I suspect) the supporters of NLT do not care, because they take
it 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 at best an empirical fact. 
But the more important issue is that this equivalence,
if it exists, depends on *translations* from one "natural language" to
another. But it is a major argument of the supporters of NLT
that "there is no acclaimed criterion for what counts as a good, or even
acceptable, `translation'" (they have 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 this very argument,
at most  *model* arguments when those arguments are originally formulated
in another "natural language" - exactly what formal languages do
according to NLT. It follows that unless we give some 
"natural language" a priority over the others (which we can't), all 
we have are just models that model each other, but nothing that is 
really modelled... (needless to say, this was the point I was trying
to make in my posting on the "Hebrew-English" thesis).

  I suspect that what the supporters of NLT really have 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? And
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...
And I know from my own experience that it would be 
practically impossible for us today to understand many mathematical 
theorems in their original formulation,  without translating them
first at least into the formal algebraic language used now in 
high schools for expressing basic identities.
 
2) After discussing the doubtful meaning of a "natural language" in
the above 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". Accordingly , in the above mentioned book 
Shapiro reformulated this thesis by referring  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 current
`natural languages of mathematics' include variables. They certainly include 
also symbols like `=" and `<". I guess that symbols like \sum for summation,
or the integration symbol \int,  are also 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
denying this status from  \forall (but still allowing of course
the use of the words "for all" as "natural"...)!
But if all the symbols used in First-order
Peano Arithmetics or ZFC are part of a `natural language of mathematics'
then these formal languages are actually  *fragments* of a 
`natural language of mathematics', and as such 
they simply model ... themselves (according to the corrected NLT).  If so,
then the thesis, whatever its meaning is, becomes trivial and uninteresting
(I suspect that supporters of NLT would not
accept this, but I don't see on what possible ground - unless
they wish to claim that the grammar of these fragments are too simple
to count as natural, and only if this symbols are used according
to some very complicated (not really known) 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 people
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 discrete language, either "natural" 
or formal. In my opinion
this fact strongly supports my belief that
reasoning comes before languages, and that *all* discrete languages
(whether "natural" or "formal") just model reasoning (only
in my opinion formal languages usually do it much better).
There is no reason to attach any priority here to natural 
languages. On the contrary. 

Two last comments. 
  First, what I wrote is a reaction to some of the 
reactions to Tim's FT - those that concentrate on 
the general possibility of formulating mathematics in formal languages and 
theories. Tim's original thesis was more specific, referring explicitly
to ZFC, and its validity is a different matter (about which
I'll write in another posting). 
  Second: the above is an edited new version
of a small article I have already posted to FOM about 3 years ago, 
with very few reactions then.

Arnon Avron