Re: Functions in programming

[Bill Lawvere <wlawvere@buffalo.edu>]

Thanks for the several interesting replies.
I think I am learning what "lazy" means, but
there seems to be embedded in it a presupposition
that the order in which new expressions are
introduced in a computation is something that
one does not carefully plan. It still seems to
me to be possible that the developing languages
are partly the legacy of an epoch when computational
power was qualitatively less than now.

I was especially heartened to see the lack of
support for St. John's position "in the beginning
was the word", especially in view of the recent
attempts by physicists to revive that idealist
position. Many of us have long instinctively
believed that language should fit concepts
and concepts should fit reality.

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 Fri, 20 Mar 2009, Robin Cockett wrote:

> Peter Selinger wrote:
>> I hope that these examples help to dispell the view that untyped
>> languages, or lazy languages, are some kind of "practical hack"
>> invented by programmers and implementors that don't really fit any
>> mathematical model. Or, as Robin put it, "whose relation to standard
>> mathematical settings is surprisingly remote". Actually, the contrary
>> is very much true, and category theory is a large part of the reason.
>> [The same cannot, of course, be said for such languages as C, Fortran,
>> Perl, or Visual Basic, which actually *are* practical hacks with no
>> mathematical model. That is precisely why theoreticians study Haskell
>> or ML or Scheme or the lambda calculus or Agda or Charity and so
>> forth.]
>>
> Well I certainly did not intend to imply (or I think say) that lazy
> functional languages (Haskell) or other strict languages (the ML family)
> are hacks!  Absolutely not -- gosh I would loose my job!!!
>
> Clearly I should be clearer on what I did not say!
>
> For example I should be clear that things "whose relation to standard
> mathematical settings is surprisingly remote" can be (in fact usually
> are) extremely mathematical and important to the development of
> mathematical thought!   When I said that producing something from
> nothing  "may be a bit of a shock to a mathematician" this was not
> saying that, therefore it is not mathematical: the fact that Fermat's
> last theorem was proven may also have shocked a good few mathematicians
> who had not expected it.  In this case it is a shock in the sense that
> naively a mathematician may not have considered carefully what setting
> he was actually in ...
>
> I certainly did want to emphasize the sense in which a program language
> can be an exploration of a particular perspective (Mathematics by other
> means).   Furthermore, that there is a creative tension between
> implementation (which, in case anyone thinks otherwise, is -- if done
> well -- /very/ mathematical: for example in the case of the lambda
> calculus rests on the now extensive theory of rewriting) and semantics,
> which I view as prescriptive mathematics (typified by set theory and
> logic of the 1900s) in which one asserts far reaching axioms hoping
> madly that they are "right" -- or at least useful.
>
> Computer Scientists probably understand this best as simply the
> difference between a bottom-up and top-down approach -- played out
> perhaps on a slightly broader canvas -- and would instantly recognize
> the importance of both (as I am sure would mathematicians).
>
> Of course, it is nice if the two approaches meet.  (And if I did not
> know Peter better I might think that he is  saying that the whole
> business is nicely sewn up ...  nothing could be further from the
> truth!  Of course he didn't say this. It only may have sounded like he
> did! ;-) )
>
> If you will now forgive me for being a little colorful to reemphasize a
> point  ... (then that is definitely it from me!)
>
> The prescriptive approach, of course, is extremely arrogant: it says
> "this is what I want" and "this is how the world should be".  It was
> typified by the development of set theory whose the aim was to a pin
> down exactly what God had intended once and for all.   Of course, the
> real danger to this, much more serious than getting it wrong -- you can
> always fix that -- is that God is actually playing with a number of
> settings and hasn't really settled on which one he likes best!   The
> Greeks understood this perfectly 8-)       .
>
> This is where Category Theory slips into the picture: it is clearly
> stupid to study just one setting (unless God makes up his mind suddenly)
> instead lets just study them all!  Now in the process of dredging
> through God's basement (where he keeps his discarded ideas) it would
> definitely be rather disheartening if we found the lambda calculus
> (after all that is where HE keeps his discarded ideas)!  So let us
> suppose that this does not happen ... but  that we do find ("hiding
> under the wreckage of set theory" Paul Taylor quickly adds) a really
> nice implementable category which has a simple axiomatization and
> resembles the sort of naive mathematics that everyone uses.  Wouldn't it
> just be tempting to pull it out and implement it anyway?  ... and,
> strictly as a market ploy of course, put about that God had had an off
> day and had made a mistake?
>
> The point is that even if there is a base computational reality, we are
> in the altogether strange situation that we don't actually have to
> accept it!  Computation, just like mathematics, is open to
> prescription.  We can still choose to work in more restrictive settings
> where getting around (in some ways) may be easier (or safer).  And this
> is a important aspect of programming language and semantics research
> ...  in particular, being Turing complete is /not /an absolute
> requirement of a programming language (it is certainly something which
> one can ask about a language!).
>
> Far from the semantical/prescriptive approach to computation being
> dead,  it is more alive than ever ...  this is not an argument against
> computability, however, it IS an argument for the important role of
> Category Theory in Computer Science for helping us to understand the
> relationship between settings we might want to implement (... on which I
> think Peter agrees!)
>
> -robin