Re: Functions in programming

[Bill Lawvere <wlawvere@buffalo.edu>]

Question: Do the various programming languages explicitly
implement the indispensible ingredient of categorical
semantics, that every map has a unique codomain?

I don't know the technical meaning of "lazy"; was it an
attempt to avoid the processing speed and ram needed to
take account of the composition with inclusion maps,
etcetera?

Bill

************************************************************
F. William Lawvere, Professor emeritus
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
************************************************************



On Tue, 17 Mar 2009, Robin Cockett wrote:

> 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
>
>
>
>