[Edu-sig] nouns and verbs

[Edward Cherlin]

On Tue, Aug 5, 2008 at 10:32 PM, Yoshiki Ohshima <yoshiki at vpri.org> wrote:
> At Sun, 3 Aug 2008 20:03:03 -0700,
> michel paul wrote:
>>
>> In secondary math classes we often say "Math is a language", but we really don't teach it that way.
>>
>> The closest we get to that is calling the comparison operators 'verbs' and the various kinds of values that can be
>> combined into expressions 'nouns'.
>
>  I enjoyed reading your lines of thought, and Edward has a good
> observation.  But I also have to point out that when people say "math
> is a language", it means that Math is a language to describe what it
> can describe well.  But trying to make an analogy to English doesn't
> get you go too far.  After all, why does it have to have anything to
> do with the English syntax?  It is not a great language to express
> what you would like to do over weekend either.
>
>  The "language-ness" is not in whether it has verbs and nouns, but
> the relationship between the target concept (Idea) and the description
> to mean it, and also something to "think in".

Most of math uses quite other forms of language, including equations
and relations. It turns out that math for imperative programming is
the kind that makes the best use of the noun-verb-adverb-pronoun
family of distinctions. On the other hand, there are math languages
important for computing that are analogous to quite different human
languages. Among them is relational algebra, which can express all
standard relational database operations, and can be expressed quite
directly in Lojban, which has relation words but no separate nouns,
verbs, or adjectives. There are also declarative languages such as
Prolog that do not specify how to carry out a computation, but do give
sets of constraints on the solution that a Prolog engine can process.

The question is not, Which language is best? but, Which is best for
this purpose? Which is not only a question of inherent mathematics,
but of external network effects and ecologies and of available
hardware.

>  And, the language-ness is not in these mathematical symbols and
> syntax, either.  It would be possible to write equations in
> English-like syntax (like your "sum of 2 and 3" example).  But the
> aspiration of preciseness compactness tends to favor a simpler and
> less ambiguious notation.

The solution of the cubic equation was discovered and presented in
this sort of language. Florian Cajori wrote an excellent History of
Mathematical Notation that talks about the relationship between
notations and discoveries in considerable detail.

>  So, it would be appropriate to say "math is a language for of
> physics" but saying "math is a language" doesn't sound like a complete
> sentence to me.  "Is math a language of math?" would be an interesting
> question^^;

There are many languages in math.

>  Now, computer languages are like mathematics, but much more complex
> in many ways.  It is built on top of some axioms, but the set of
> axioms tends to be very big.  The notation is less ambiguous than
> typical mathematics one because one of the intended readers of the
> notation is the computer.

Actually, to the mathematician, programming is a fairly simple concept
that can be expressed in several different ways as the working out of
only two basic concepts, such as the S and K combinators (Unlambda or
J), or Lambda expressions and application (LISP and many related
languages). Most programming languages have a good deal of unneeded
and counterproductive complexity added on, like C++.

To the non-mathematician, these simpler solutions seem harder than
memorizing the complex syntax of conventional languages, as was often
borne in upon Computer Scientist Edsger Dijkstra. He spent much of his
career trying to make programming easier to do well, and was regularly
told by practitioners that he had made it harder instead.

The same principle applies with even greater force in education.
"Don't do us no favors," teachers seem to say. "if you make it so that
we can really teach this stuff, then we will all have to go learn it
ourselves, and we can't." This is a delusion in a way, but not the
delusion of the teachersthemselves. It is a delusion enforced by the
social system they work in. Like Ethiopian teachers treating questions
from students as personal insults, until they get XOs. There
experience suggests that there is hope for the profession as a whole.

> -- Yoshiki
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