Re: [Haskell-cafe] Category Theory woes
Nick, Actually, clopen is a set that is both closed and open. Not one that is neither. Except in the case of half-open intervals, I can't remember talking much in topology about sets with a partial boundary. Category theory-wise. No one seems to have mentioned MacLane's "Categories for the Working Mathematician." Although, I don't seem to recall instant enlightenment when I picked it up. Mike - Original Message - From: "Nick Rudnick" To: "Michael Matsko" Cc: haskell-cafe@haskell.org Sent: Thursday, February 18, 2010 4:54:03 PM GMT -05:00 US/Canada Eastern Subject: Re: [Haskell-cafe] Category Theory woes Hi Mike, of course... But in the same spirit, one could introduce a straightforward extension, «partially bordered», which would be as least as good as «clopen»... ;-) I must admit we've come a little off the topic -- how to introduce to category theory. The intent was to present some examples that mathematical terminology culture is not that exemplary as one should expect, but to motivate an open discussion about how one might «rename refactor» category theory (of 2:48 PM). I would be very interested in other people's proposals... :-) Michael Matsko wrote: Nick, That is correct. An open set contains no point on its boundary. A closed set contains its boundary, i.e. for a closed set c, Closure(c) = c. Note that for a general set, which is neither closed or open (say the half closed interval (0,1]), may contain points on its boundary. Every set contains its interior, which is the part of the set without its boundary and is contained in its closure - for a given set x, Interior(x) is a subset of x is a subset of Closure(x). Mike ----- Original Message - From: "Nick Rudnick" To: "Michael Matsko" Cc: haskell-cafe@haskell.org Sent: Thursday, February 18, 2010 3:15:49 PM GMT -05:00 US/Canada Eastern Subject: Re: Fwd: [Haskell-cafe] Category Theory woes Hi Mike, so an open set does not contain elements constituting a border/boundary of it, does it? But a closed set does, doesn't it? Cheers, Nick Michael Matsko wrote: - Forwarded Message - From: "Michael Matsko" To: "Nick Rudnick" Sent: Thursday, February 18, 2010 2:16:18 PM GMT -05:00 US/Canada Eastern Subject: Re: [Haskell-cafe] Category Theory woes Gregg, Topologically speaking, the border of an open set is called the boundary of the set. The boundary is defined as the closure of the set minus the set itself. As an example consider the open interval (0,1) on the real line. The closure of the set is [0,1], the closed interval on 0, 1. The boundary would be the points 0 and 1. Mike Matsko - Original Message - From: "Nick Rudnick" To: "Gregg Reynolds" Cc: "Haskell Café List" Sent: Thursday, February 18, 2010 1:55:31 PM GMT -05:00 US/Canada Eastern Subject: Re: [Haskell-cafe] Category Theory woes Gregg Reynolds wrote: On Thu, Feb 18, 2010 at 7:48 AM, Nick Rudnick < joerg.rudn...@t-online.de > wrote: IM(H??)O, a really introductive book on category theory still is to be written -- if category theory is really that fundamental (what I believe, due to its lifting of restrictions usually implicit at 'orthodox maths'), than it should find a reflection in our every day's common sense, shouldn't it? Goldblatt works for me. Accidentially, I have Goldblatt here, although I didn't read it before -- you agree with me it's far away from every day's common sense, even for a hobby coder?? I mean, this is not «Head first categories», is it? ;-)) With «every day's common sense» I did not mean «a mathematician's every day's common sense», but that of, e.g., a housewife or a child... But I have became curious now for Goldblatt... * the definition of open/closed sets in topology with the boundary elements of a closed set to considerable extent regardable as facing to an «outside» (so that reversing these terms could even appear more intuitive, or «bordered» instead of closed and «unbordered» instead of open), Both have a border, just in different places. Which elements form the border of an open set?? As an example, let's play a little: Arrows: Arrows are more fundamental than objects, in fact, categories may be defined with arrows only. Although I like the term arrow (more than 'morphism'), I intuitively would find the term «reference» less contradictive with the actual intention, as this term Arrows don't refer. A *referrer* (object) refers to a *referee* (object) by a *reference* (arrow). Categories: In every day's language, a category is a completely different thing, without the least Not necesssarily (for Kantians, Aristoteleans?) Are you sure...?? See
Re: [Haskell-cafe] Category Theory woes
Nick, That is correct. An open set contains no point on its boundary. A closed set contains its boundary, i.e. for a closed set c, Closure(c) = c. Note that for a general set, which is neither closed or open (say the half closed interval (0,1]), may contain points on its boundary. Every set contains its interior, which is the part of the set without its boundary and is contained in its closure - for a given set x, Interior(x) is a subset of x is a subset of Closure(x). Mike - Original Message - From: "Nick Rudnick" To: "Michael Matsko" Cc: haskell-cafe@haskell.org Sent: Thursday, February 18, 2010 3:15:49 PM GMT -05:00 US/Canada Eastern Subject: Re: Fwd: [Haskell-cafe] Category Theory woes Hi Mike, so an open set does not contain elements constituting a border/boundary of it, does it? But a closed set does, doesn't it? Cheers, Nick Michael Matsko wrote: - Forwarded Message ----- From: "Michael Matsko" To: "Nick Rudnick" Sent: Thursday, February 18, 2010 2:16:18 PM GMT -05:00 US/Canada Eastern Subject: Re: [Haskell-cafe] Category Theory woes Gregg, Topologically speaking, the border of an open set is called the boundary of the set. The boundary is defined as the closure of the set minus the set itself. As an example consider the open interval (0,1) on the real line. The closure of the set is [0,1], the closed interval on 0, 1. The boundary would be the points 0 and 1. Mike Matsko - Original Message - From: "Nick Rudnick" To: "Gregg Reynolds" Cc: "Haskell Café List" Sent: Thursday, February 18, 2010 1:55:31 PM GMT -05:00 US/Canada Eastern Subject: Re: [Haskell-cafe] Category Theory woes Gregg Reynolds wrote: On Thu, Feb 18, 2010 at 7:48 AM, Nick Rudnick < joerg.rudn...@t-online.de > wrote: IM(H??)O, a really introductive book on category theory still is to be written -- if category theory is really that fundamental (what I believe, due to its lifting of restrictions usually implicit at 'orthodox maths'), than it should find a reflection in our every day's common sense, shouldn't it? Goldblatt works for me. Accidentially, I have Goldblatt here, although I didn't read it before -- you agree with me it's far away from every day's common sense, even for a hobby coder?? I mean, this is not «Head first categories», is it? ;-)) With «every day's common sense» I did not mean «a mathematician's every day's common sense», but that of, e.g., a housewife or a child... But I have became curious now for Goldblatt... * the definition of open/closed sets in topology with the boundary elements of a closed set to considerable extent regardable as facing to an «outside» (so that reversing these terms could even appear more intuitive, or «bordered» instead of closed and «unbordered» instead of open), Both have a border, just in different places. Which elements form the border of an open set?? As an example, let's play a little: Arrows: Arrows are more fundamental than objects, in fact, categories may be defined with arrows only. Although I like the term arrow (more than 'morphism'), I intuitively would find the term «reference» less contradictive with the actual intention, as this term Arrows don't refer. A *referrer* (object) refers to a *referee* (object) by a *reference* (arrow). Categories: In every day's language, a category is a completely different thing, without the least Not necesssarily (for Kantians, Aristoteleans?) Are you sure...?? See http://en.wikipedia.org/wiki/Categories_(Aristotle) ... If memory serves, MacLane says somewhere that he and Eilenberg picked the term "category" as an explicit play on the same term in philosophy. In general I find mathematical terminology well-chosen and revealing, if one takes the trouble to do a little digging. If you want to know what terminological chaos really looks like try linguistics. ;-) For linguistics, granted... In regard of «a little digging», don't you think terminology work takes a great share, especially at interdisciplinary efforts? Wouldn't it be great to be able to drop, say 20% or even more, of such efforts and be able to progress more fluidly ? -g ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Fwd: [Haskell-cafe] Category Theory woes
- Forwarded Message - From: "Michael Matsko" To: "Nick Rudnick" Sent: Thursday, February 18, 2010 2:16:18 PM GMT -05:00 US/Canada Eastern Subject: Re: [Haskell-cafe] Category Theory woes Gregg, Topologically speaking, the border of an open set is called the boundary of the set. The boundary is defined as the closure of the set minus the set itself. As an example consider the open interval (0,1) on the real line. The closure of the set is [0,1], the closed interval on 0, 1. The boundary would be the points 0 and 1. Mike Matsko - Original Message - From: "Nick Rudnick" To: "Gregg Reynolds" Cc: "Haskell Café List" Sent: Thursday, February 18, 2010 1:55:31 PM GMT -05:00 US/Canada Eastern Subject: Re: [Haskell-cafe] Category Theory woes Gregg Reynolds wrote: On Thu, Feb 18, 2010 at 7:48 AM, Nick Rudnick < joerg.rudn...@t-online.de > wrote: IM(H??)O, a really introductive book on category theory still is to be written -- if category theory is really that fundamental (what I believe, due to its lifting of restrictions usually implicit at 'orthodox maths'), than it should find a reflection in our every day's common sense, shouldn't it? Goldblatt works for me. Accidentially, I have Goldblatt here, although I didn't read it before -- you agree with me it's far away from every day's common sense, even for a hobby coder?? I mean, this is not «Head first categories», is it? ;-)) With «every day's common sense» I did not mean «a mathematician's every day's common sense», but that of, e.g., a housewife or a child... But I have became curious now for Goldblatt... * the definition of open/closed sets in topology with the boundary elements of a closed set to considerable extent regardable as facing to an «outside» (so that reversing these terms could even appear more intuitive, or «bordered» instead of closed and «unbordered» instead of open), Both have a border, just in different places. Which elements form the border of an open set?? As an example, let's play a little: Arrows: Arrows are more fundamental than objects, in fact, categories may be defined with arrows only. Although I like the term arrow (more than 'morphism'), I intuitively would find the term «reference» less contradictive with the actual intention, as this term Arrows don't refer. A *referrer* (object) refers to a *referee* (object) by a *reference* (arrow). Categories: In every day's language, a category is a completely different thing, without the least Not necesssarily (for Kantians, Aristoteleans?) Are you sure...?? See http://en.wikipedia.org/wiki/Categories_(Aristotle) ... If memory serves, MacLane says somewhere that he and Eilenberg picked the term "category" as an explicit play on the same term in philosophy. In general I find mathematical terminology well-chosen and revealing, if one takes the trouble to do a little digging. If you want to know what terminological chaos really looks like try linguistics. ;-) For linguistics, granted... In regard of «a little digging», don't you think terminology work takes a great share, especially at interdisciplinary efforts? Wouldn't it be great to be able to drop, say 20% or even more, of such efforts and be able to progress more fluidly ? -g ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Looking for the fastest Haskell primes algorithm
You might want to look at Pari/GP ( http://pari.math.u-bordeaux.fr/ ) for ideas of what kind of functions to supply. Also, as a source of ideas for algorithms. Mike Matsko - Original Message - From: "Max Rabkin" To: "Andrew Wagner" Cc: "R.A. Niemeijer" , haskell-cafe@haskell.org Sent: Tuesday, April 14, 2009 12:35:19 PM GMT -05:00 US/Canada Eastern Subject: Re: [Haskell-cafe] Looking for the fastest Haskell primes algorithm On Tue, Apr 14, 2009 at 2:47 PM, Andrew Wagner wrote: > Some other ideas for things to put in this package possibly: > is_prime :: Int -> Bool I'd also add isProbablePrime using a Miller-Rabin test or somesuch, for use with large numbers. It'd have to be in a monad which supplies randomness, of course. But to start with, I'd just package what I had and put it on Hackage. --Max ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] mathematical notation and functional programming
Also, Walter Noll of Carnegie Mellon Univ. wrote a book, "Finite-Dimensional Spaces"in 1987 which basically presented undergraduate math in a notationally and conceptually unified manner. Some of the notation and terminology was strange, but consistent. Mike Matsko - Original Message - From: Fritz Ruehr <[EMAIL PROTECTED]> Date: Friday, January 28, 2005 3:10 pm Subject: Re: [Haskell-cafe] mathematical notation and functional programming > Well, I don't know about modern works which might appeal to > knowledge > of FP languages, but there is a well-known, 2-volume work by Cajori: > > Cajori, F., A History of Mathematical Notations, The Open > Court > Publishing Company, Chicago, 1929 (Available from Dover). > > I know it through Ken Iverson (may he rest in peace), the creator > of > APL. (Dr. Iverson's own notations were not to everyone's taste, but > I > think they were a bigger influence on Backus and the recent wave of > FP > than is generally acknowledged.) > > APL *did* have "implicit maps and zipWiths" in the sense that > scalar > functions would be automatically extended to vectors (and similarly > for > higher dimensions). I think my PhD advisor, Satish Thatte, did some > work on extending this sort of "notational abuse" to Hindley-Milner > systems, but I don't have the citations at hand. > > OK then, googling on Cajori yields this quote from a math history > site: > "He almost single-handedly created the history of mathematics as > an > academic subject in the United States >and, particularly with his book on the history of mathematical > notation, he is still one of the most quoted >historians of mathematics today." > > More googling on "mathematical notation" reveals that there *are* > people concerned about these issues, Steven Wolfram being an > easily-recognized example (he refers to Cajori's work). > > -- Fritz > > On Jan 27, 2005, at 12:14 PM, Henning Thielemann wrote: > > > I wonder if > > mathematical notation is subject of a mathematical branch and > whether> there are papers about this topic, e.g. how one can > improve common > > mathematical notation with the knowledge of functional languages. > > ___ > Haskell-Cafe mailing list > Haskell-Cafe@haskell.org > http://www.haskell.org/mailman/listinfo/haskell-cafe > ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Matroids in Haskell
Dimitri Matriods are generalization of vector spaces. Basically, they are defined by a set of linear dependence axioms and basis exchange properties. Oxley's "Matriod Theory" is the standard reference. There are a multitude of equivalent formulations. Mike Matsko - Original Message - From: Dmitri Pissarenko <[EMAIL PROTECTED]> Date: Friday, January 14, 2005 2:00 pm Subject: Re: [Haskell-cafe] Matroids in Haskell > Hello! > > Gerhard Navratil wrote: > > Recently I had a course on matroids and would like to investigate > the> topic a little further. Did anybody write (or start writing) a > > Haskell-implementation for matroids? > > What is a matroid? > > Thanks > > Dmitri Pissarenko > -- > Dmitri Pissarenko > Software Engineer > http://dapissarenko.com > ___ > Haskell-Cafe mailing list > Haskell-Cafe@haskell.org > http://www.haskell.org/mailman/listinfo/haskell-cafe > ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
