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

In article <d82psf$qgi$1$8302bc10@xxxxxxxxxxxxxxxx>,
Colin Fine <news@xxxxxxxxxx> wrote:
>Herman Rubin wrote:
>> In article <3gjd96Fck07bU1@xxxxxxxxxxxxxx>,
>> Harlan Messinger <hmessinger.removethis@xxxxxxxxxxx> wrote:

>>>Greg wrote:

>>>>Des Small wrote:


>>>>>It seems that langwidge, for Herman, properly aspires to the condition
>>>>>of mathematics.


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

>> 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.

>>>>As I read postings in this thread, I keep trying to deduce what the
>>>>posters think the difference is between natural language and
>>>>mathematics.


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


>> Most of mathematics, in use, is language. I have
>> frequently stated that I would like students to be able to
>> formulate their problems, NOT to compute the answers. In
>> the practical problems of today, such as medical treatment,
>> the best-formulated problems are only going to be solved by
>> computers.

>You seem to be contradicting your earlier claims. Look through any
>mathematical paper and you will find large sections which are not
>language but mathematical notation: I thought this was your argument.

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.

>>>And then there's Herman's lament that natural language isn't as precise
>>>as mathematics (by which, yes, I understand him to mean "mathematical
>>>representation"). Well, that *is* a *difference* between them, isn't it?
>>>Natural language relies on human psychology for interpretation.
>>>Mathematical representation doesn't.

>> It is unclear how much "human psychology" is needed for
>> interpretation of mathematics. Mathematical concepts are
>> not understood as collections of words, but as more than
>> that. One problem is that some concepts seem to have more
>> than one intensional attribute; for simple things like the
>> non-negative integers, the two basic ones are quite distinct,
>> and there are related versions of them. Students are taught
>> HOW to add, but not what addition means, and what it means
>> differs in the different intensional meanings.

>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.

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.

>[In another mail:]

>> One of the major problems is that most people seem to think
>> that mathematics consists of means of calculating answers,
>> and that language consists of some mysterious means of
>> putting sounds together in a "natural" manner. That most
>> "advanced" languages are also written, and that this means
>> of communication is widely used does not seem to some to be
>> real language, only spoken.

>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.

>> 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.

>[from yet another posting:}
>> Possibly in high school, the various students came from
>> regions speaking essentially the same dialect, in which
>> little, if anything, precise was discussed, other than
>> possibly spelling of words, or the application of
>> computational rules, without any understanding.

>Here you go again with the patronising assumption that if they are not
>speaking a standard dialect then they are incapable of precision.

>> In college, this is not the case. Communication is
>> needed, and precise communication. Also, even in high
>> school or elementary school, few examinations are oral.

I see these students, and assure you that the communication
gap is that great.

>> A possible reason Dean Fish had the success he had is
>> that children pick up a good deal of grammar before
>> they learn too much vocabulary, so they still retain
>> some of the structure of their native language even
>> after the schools had ignored it for all those years.
>This is nonsense. Children learn vocabulary at a phenomenal rate, during
>the time they are learning grammar and for some time after.
>Most people throughout the world retain more or less all of the
>structure of their native language throughout their lives, unless they
>move away from other speakers of it.

Is the rate "phenomenal"? Comparing the vocabulary expected
after my one year of college French with the vocabulary of an
8 year old, I would have to say that mine was greater. And
yet this was obtained in a course meeting three times a week
for one hour, with a requirement of reading and reporting on
600 pages outside of text material.

I can still understand slow speech in that language and others
which I know less well. An amusing incident occurred here; at
a meeting in Germany, one of the speakers, an American, gave
his talk in German. I had no problem understanding him, but
the native Germans would not have if he had not given the
technical terms in English.

>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?

>Colin


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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