From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/179 Path: news.gmane.org!not-for-mail From: Bill Lawvere Newsgroups: gmane.science.mathematics.categories Subject: Re: Functions in programming Date: Wed, 18 Mar 2009 11:34:27 -0400 (EDT) Message-ID: Reply-To: Bill Lawvere NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1237470512 12120 80.91.229.12 (19 Mar 2009 13:48:32 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 19 Mar 2009 13:48:32 +0000 (UTC) To: Robin Cockett , categories@mta.ca Original-X-From: categories@mta.ca Thu Mar 19 14:49:47 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1LkId1-0005l3-0j for gsmc-categories@m.gmane.org; Thu, 19 Mar 2009 14:49:47 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LkHvo-0003rl-Vs for categories-list@mta.ca; Thu, 19 Mar 2009 10:05:09 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:179 Archived-At: 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 > > > >