[Edu-sig] nouns and verbs

[Edward Cherlin]

2008/8/3 michel paul <mpaul213 at gmail.com>:
> In secondary math classes we often say "Math is a language", but we really
> don't teach it that way.

Turing Award winner Ken Iverson did. He called data objects nouns,
functions verbs, identifiers pronouns, and operators (to create new
verbs from old) adverbs.

So 2 is a noun, as is '2'. 'Hello' is a noun with five member nouns.

1 2 3
4 5 6

is a noun with 2 rows and 3 columns, and thus 6 members. Its shape 2 3
can thus be regarded a a noun with 2 members.

Verbs include +-<> and many more that we usually write in non-ASCII symbols.

Iverson began using / as an adverb in the 1950s, so that +/1 2 3 puts
+ between the members of 1 2 3, giving 1+2+3, and similarly for other
verbs. Their scope has been very greatly expanded since.

I have his textbooks on Arithmetic, Algebra, and Calculus written this
way, with every math expression executable. (At One Laptop Per Child
we are thinking about going beyond text you can type in and run. We
want students to be able to click on expressions in electronic 'books'
to execute them, and to be able to tweak the parameters on graphs and
have them update. And so on.)

> 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'.
>
> So, as far as contemporary secondary math thinking is concerned, the only
> 'verbs' in 'math' are '=', '<', and '>', forms of 'to be'.
>
> However, I'd bet that a lot of students, and even teachers, would categorize
> '+', '-', '*', and '/' as 'verbs'.

or even ×÷

> After all, they seem to suggest 'action'.
>
> But is '2 + 3' a sentence or a complete thought?

In some programming languages, including all those designed by
Iverson, 2+3 is a complete imperative sentence, to which the
interpreter responds with the answer.

   2+3
5

> No, it is an expression which is equivalent in value to another expression,
> namely, '5'.

Sometimes.

> I keep thinking that, mathematically, it makes more sense to say that '2 +
> 3' is one of the already-existing partitions of '5'.
>
> Interestingly, if '2 + 3' were to be taken as a sentence, it would be
> imperative:
> "Hey you!  Yeah, you!  Find the value of 2 + 3, now!"

Indeed.

> Mathematically speaking, it would be more accurate to read '2 + 3' as 'the
> sum of 2 and 3'.
>
> Mathematically speaking, '2 + 3' is a value that does not need to emerge in
> time.  It already exists.
>
> But, computationally speaking, it actually does make sense to think of '2 +
> 3' as a process.
>
> We start with a '2', then we do something with some registers or whatever,
> and we increment '2' by '3', and we end up with '5'.

In Smalltalk we send a message to 2 telling it '+3', and the
interpreter looks up the method for 2 to carry out this request.

> And then this leads into the question of functions - are they 'nouns' or
> 'verbs'?
>
> In OO terms, functions are how objects relate to other objects.

In OO, objects have methods.

> Mathematically speaking, at least according to contemporary secondary
> textbooks, a function is a set of ordered pairs.

That's a relation. A function is a relation with no duplicate first values.

> The expression 'f(x)' is a VALUE.

The expression 'f(x)' may have a value, but is not required to do so.

> So, 'f(x)' is a 'noun'.

You have to be careful of that word 'is'. It may be that 'function' is
defined in this way, but functions are many other things besides.

> But again, computationally speaking, a function is a process.  It takes in
> some information, does something to it, and then yields some new
> information.
>
> In computational terms, a function is something we 'do' to a value to
> produce a new value.

In functional programming, a function must return a value, and not
change any values. Code that either does not return a value, or
changes values (side effects) may be called a 'procedure'. In
procedural programming, the mapping of names (pronouns) and values can
be changed by assignment within a procedure.

> I do apologize if this is not the correct venue for raising such questions,
> as it is not language specific, but I find this really interesting.  I think
> it might have a lot to do with contemporary students, and teachers!, not
> 'getting' what math really is,

My mentor in Buddhism, Rev. Jiyu Kennett, told me early on that she
had been unable to do algebra between studying it in school in the
1930s, and the time we met in 1969, because the idea of a variable
made no sense. Her teacher had tried to explain 'variable' as a number
that changes its value, which she and I both understood cannot happen.
When I explained that a variable is a name that can be given to
different numbers at different times (a pronoun) without the numbers
changing, she went away for about half an hour to try it out, and came
back saying that she could do it all now, and we didn't need to talk
about it any further.

Yale University had several math professors teaching in middle school
in New Haven in the 1960s, and reporting similar misunderstandings.
The children also frequently asked deep questions in the foundations
of mathematics that the teachers had no idea how to deal with. One of
the simpler ones is, "Is there a biggest number?" This question was a
part of the puzzle of foundations, roughly from Peano, Frege, and
Russell to Zermelo and Fraenkel, a period of more than thirty years.
If by 'number' you mean the integers, then there is no biggest number.
If you mean the Cantor transfinite cardinals, it turns out in
Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC, "The
Standard Model") that the transfinites have an upper limit. At this
point one can argue (pointlessly) whether that limit is or is not a
number, since all other numbers can be defined as sets, and the limit
cannot. ZFC is apparently consistent, and satisfies all of the
requirements of working mathematicians up to its time, but it is
deeply dissatisfying to the philosophically-minded. One can also look
for other foundations for mathematics, as is currently being done in
Topose Theory.

If you didn't understand all of that, you are in good company. Nobody
knows how this phase of it will come out, and we have even less idea
what will come after. Furthermore, it is just the same for set theory
as Bohr said of his field, "Anyone who is not shocked by quantum
theory has not understood it."

> and it might shed a lot of useful light on
> the disconnect between the current secondary math curriculum and the current
> state of computational understanding.

In the time since Turing and Goedel proved the essential
undecidability of mathematics in general and computation in
particular, we do our students a great disservice by pretending that
mathematics always gives the right answer.

> This semester, at a high school level, I do intend to teach math as a
> 'language', and I'd like to get really clear about these kinds of things.
>
> Thanks very much for any feedback,
>
> Michel Paul

I can point you to software and textbooks that will help, and put you
into contact with others who are doing this.
> _______________________________________________
> Edu-sig mailing list
> Edu-sig at python.org
> http://mail.python.org/mailman/listinfo/edu-sig
>
>



-- 
Silent Thunder [默雷/शब्दगर्ज] is my name,
And Children are my nation.
The whole world is my dwelling place,
And Truth my destination.