Re: Functions in programming

[Robin Cockett <robin@ucalgary.ca>]

Vaughan Pratt wrote:
> The categories mailing list is a good one for this sort of discussion ...
So here is some (gentle) push-back for Vaughan ...

BTW thank you everyone for pointing out that /everything/ I said was
wrong/right!   There really are underlying serious pedagogical and
practical issue behind this ... typified by the comment.

> Thorsten Altenkirch wrote:
> > There really shouldn't be a difference between the functions in
> > Mathematics and in Computer Science, especially functional programming.
>
The fact that nothing is quite what it "should be" in what has become
the/a leading functional language is  bothersome on the one hand for
students struggling to develop a (unified) mathematical view and,
simultaneously  exciting for researchers who now have to find out which
(!?!@!) category one is actually working in ... the fact that the answer
is not an entirely simple (I do like Vaughan's "nuanced"!) is cause
simultaneously for (pedagogical) concern and (researcher) delight.

This reflects a general tension between semantics and implementation and
the tussle over which is to be the cart and which is to be the horse.
As it happens (I recall) one of the motivations behind Haskell was to
produce a /lazy/ functional language and so a significant focus was
actually on the implementation side ... perhaps at some semantical cost?

Vaughan comments:
> There should be differences within Mathematics and within Computer
> Science, and therefore between them.
I confess -- in this context -- what springs (uncalled) to mind is the
(modified) comment of Chairman Mao: Computer Science is the continuation
of Mathematics by other means!   ... and sometimes the balance between
what /should/ be done and what /can/ be done is pushed too far.  At what
stage this becomes a "bug" -- as Thorsten bluntly puts it -- definitely
should be up for debate.  And there is no doubt in my mind that in
making this judgment the clarity of the underlying (categorical)
semantics adds an important perspective .... and even should be
prescriptive.

Semantics does have some "real" effects: the semantics that a programmer
has in mind and what is actually implemented by a language/API can be
rather different ... and this can become particularly subtle as
languages become more abstract (and peculiar) and are built on top of
each other.  Through these gaps can lie some very unexpected behaviors!

Vaughan comments:
> 2.  One should not assume that mathematics has the answer to every
> practical problem.
Oft quoted John Arbuthnot commented (some time ago!):
"There are very few things which we know, which are not capable of being
reduced to a Mathematical Reasoning; and when they cannot it's a sign
our knowledge of them is very small and confused; and when a
Mathematical Reasoning can be had it's as great a folly to make use of
any other, as to grope for a thing in the dark, when you have a Candle
standing by you."

It really is hard to say more :-)  ... but maybe "candle" should be
replaced "compact florescent light bulb"?

-robin