Re: _Verum Et Factum Convertuntur_ (or: Surprised By Syntax)

Herman Rubin wrote: 
In article <d82psf$qgi$1$8302bc10@xxxxxxxxxxxxxxxx>,
Colin Fine  <news@xxxxxxxxxx> wrote:


			....................


Despite the apparent variety of grammatical structures, the
structure of language does not vary THAT much, which is why
we can try to understand it.


I think I would agree with that statement, though I wonder if you do appreciate how much it does vary. 


Having a familiarity with the grammars of several
Indo-European languages, and also with Hebrew, and with
"mathematical language", and some idea of what happens in
other languages, I do appreciate the details.  But I also
agree to the statement that there is an essentially common
grammar.

On a global level, I-E languages are really close relatives, even though there are a few members that have some wayward features (such as the ergative past in modern Indian languages or initial flexions in insular Celtic). And though Hebrew has some significant differences, in a number of ways it is quite similar to IE.

I still don't know what you mean by 'mathematical language'. Mathematical notation isn't language, and the language you use to talk about mathematics is English, or French, or Romanian or Chinese, even if it has some special vocabulary.

To quote Harlan:



Well, mathematics isn't language, so there's one substantial difference. Comparing natural language to mathematics is like comparing French to Poland.



.... 


Mathematical notation is PART of the language which
mathematicians use.  It is linguistic, and while they are
largely using it for exposition of mathematics, it can be
used for anything else.  A paper in physics or engineering,
or in chemistry or biology, may well use that much of what
you seem to think is restricted to mathematics.


Mathematical notation is not language in any sense known to linguistics. It has syntax, but not any of the other properties of language: in particular, it does not have semantics (we've been round this one before).

A paper in physics or other sciences may well use mathematical notation; but if it does, that part of the paper is mathematical.

....


But axiomatic formulations, proofs etc, do their best to escape from the realm of human psychology. You can argue about how far they succeed (and how far it is possible to succeed) but that is an important and fundamental difference between mathematics and language.


Not as much as you think. While the criteria for axiomatic formulations and proofs are as you say, this does not mean
that they are, or should be, that arbitrary. Any collection
of the theorems of a branch of mathematics can be taken as
"the axioms"; the ones chosen to be the axioms, and the additional results classed as basic in the field, are those
chosen to increase understanding.

I did not specify any criteria, nor did I suggest that anything about them should be arbitrary.
I'm not sure that I agree with your last sentence: sometimes they are chosen for reasons of parsimony or some notion of elegance.

In the mathematics and statistics newsgroups, I have suggested
that these not be called "definitions" but "characterizations".
At least as stated by Whitehead and Russell, there can only be
one definition, but there can be lots of characterizations.
So in topology, there are more than a dozen characterizations of
what it means for something to be a topological space. Some of
you may be familiar with the concept of an abelian group; nobody
uses Tarski's characterization as a set with a binary operation
satisfying 

	c = a - (b - (c - (a - b))).

In addition, when one proves a theorem, it is usually considered
good to have a proof which provides insight. I have criticized
the tendency of textbooks to use "cute" proofs which disguise the essence of the theorem.

The intensional approach is needed as well as the extensional.
The formal one you have stated is extensional, but the informal
intensional version is one which is needed to understand mathematics. This is why I say that one needs to understand
what addition means, not how to add.

I more or less agree with all this. I believe it has about as much relation to language as playing the piano.


[In another mail:]



I agree as to the common misapprehension about mathematics.
As to language, I believe that people are saying to you "Spoken language is also language" and you are interpreting this as "Only spoken language is language". 


Many in this group seem unable to recognize the importance of
a written language, and have so posted.

I don't agree. Many people in this group regard spoken language as primary and written as secondary, and react unfavourably to posts which seem to assume the reverse. Many get particularly contrary when a poster seems to suggest that written language is a norm or standard against which spoken language should be judged - as you appeared to do. 

The major part of mathematics is communication, and also
getting results of some kind from the communication.  Would
natural language have developed if the collections of sounds
did not get other results?  Even the minimal animal languages
have those properties, and my cat communicates by sounds.


I would not agree that the major part of mathematics is communication, though it dos depend on whether you regard 'mathematics' as primarily an abstract world to be discovered or as a social activity.
Certainly language arose because it gave practical results. While that is certainly true of some of the areas which gave rise to mathematics, it is not obviously true of mathematics as it is pursued today.


For the non-mathematician, the importance of mathematics is
communication. Real-world problems need to be precisely communicated before they can be properly approached, and
carefully treated.

I would have said that for the non-mathematician in general the importance of mathematics is either None, or That somebody can do it and get the answers. Communication is important only for the minority who are actually interested.


[from yet another posting:}


Many people throughout the world also learn a competent command of one or more standard languages, which may be very different in structure

from their native speech (even when they are dialects of the same

language). Unfortunately in Europe and Europeanised parts of the world the idea grew up in the last couple of centuries that non-standard dialects were inferior, so schools often did not so much ignore the structure of pupils' native language, but actively seek to eradicate it. (I'm not just talking about minority languages like Welsh, Breton and Catalan - I'm also talking about non-standard dialects of major languages). 


What is the difference between a dialect and a language?

There is no point in trying to answer that question in general. My point (which I believed I had made clear) was that I was talking about RP English as against Yorkshire or Cockney as well as RP English as against Welsh or Panjabi.


Colin
.

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