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

In article <d87t9j$1d8$1$8300dec7@xxxxxxxxxxxxxxxx>,
Colin Fine <news@xxxxxxxxxx> wrote:
>Herman Rubin wrote:
>> In article <d82psf$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.

As of this time, it is not a full language, in that one has
to use an external language before one can get to its powers
of self-reference. One can ask this about "natural" languages;
how do we get started? The amount of natural language needed
to understand the syntax of mathematics is quite small.

>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).

This is unclear. There are types of "objects" described;
is it not semantics to say that one object is of a given
type, and therefore satisfies all the results, and obeys
the restrictions, of that type?

If I am told something is a group, I do not rush to look
up one of the characterizations, but I can use anything I
know about groups for that thing.

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

No, the semantic assignment of mathematical objects to
"real world" objects is linguistic, not mathematical.
This, and translating the mathematical results back, is
the very important linguistic use of mathematical notation.

....


>>>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.

Sometimes they are, but there are alternate choices. The
point of parsimony is that less has to be proved. As for
elegance, that is purely a matter of taste.

>> 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.

Before one can get the answers, the questions have to be asked.
Asking the questions is communication.

One of the letters to the editor in the _New York Times_ stated
that he did not "use" algebra that much, but wished he had
"learned" probability and statistics. Now there is no way that
I can teach probability reasonably to someone who cannot use
algebra as a language; whether or not a problem can be solved
is quite irrelevant. Similarly, to understand statistics, one
must use probability as a language. In many, if not most,
practical problems of statistics, one must ask a computer
appropriately to get the answer, so again it is language which
is the important part.

>>>[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).

Do you think this is just Europeanized? It was true in
ancient Chinese, and also in India, Sanskrit is still
considered to be THE language. The Prakrit languages were
considered to be inferior.

It seems that in Rome, after the conquest of Greece, a
Roman was not considered educated if he did not speak
"correct" Greek.

>> 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


--
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
.

Relevant Pages

  • Re: History of French
    ... Mathematics and statistics are what are called "context free". ... The language has a rigid grammar and a SMALL ... Herman Rubin, Department of Statistics, Purdue University ...
    (sci.lang)
  • Re: Universal grammar
    ... One times infinite equals two times ... is a "wormhole" between mathematics ruled by the basic formula ... of accompanying neuronal network computing, ... of today, however, you can never really tame language. ...
    (sci.lang)
  • Berlinski paper presented 1985 at Applied systems analysis
    ... Complexity, Language and Life: Mathematical Approaches, ... This paper explores the idea that life comprises a language-like ...
    (talk.origins)
  • Re: Warning to newbies
    ... believed that the language had to allow changes to evaluation order to ... optimized in the limited context of the optimizer you are concerning ... NAN is a floating point number ... to follow the laws of finite mathematics, ...
    (comp.lang.c)
  • Re: Cantors Theory: Mathematical creationism
    ... Science is just another religion in that sense in that it's just another ... It came from mathematics. ... a big part of the foundation of math. ... what is said with language? ...
    (sci.math)