[FOM] Re: Shapiro on natural and formal languages

[Timothy Y. Chow]

Arnon Avron <aa at tau.ac.il> wrote:
>  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.
[...]
> 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.

I don't know exactly what Shapiro's views are.  I am guessing, though, 
that he may be saying no more than that informal concepts of "proof," 
"integer," and so forth necessarily precede their formal counterparts. 
When we write down a formal language, it is not created in a vacuum but
is supposed to capture certain key aspects of pre-existing mathematical 
discourse.

I can see someone not bothering to draw a sharp distinction between 
natural language itself and the kinds of reasoning that we carry out using 
natural language, especially when making a somewhat casual statement that 
is not going to be explicitly defended in detail.  So if Shapiro uses
the term "natural language" to refer to what I have called "informal 
concepts," then maybe his view is not as "completely opposite" to yours
as it would be if one took his use of the term "natural language" more 
literally.

Tim