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
Re: [Haskell-cafe] mathematical notation and functional programming
Also, Walter Noll of Carnegie Mellon Univ. wrote a book, Finite-Dimensional Spacesin 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] mathematical notation and functional programming
Things I'm unhappy about are for instance f(x) \in L(\R) where f \in L(\R) is meant F(x) = \int f(x) \dif x where x shouldn't be visible outside the integral O(n) which should be O(\n - n) (a remark by Simon Thompson in The Craft of Functional Programming) f(.) which means \x - f x or just f All of these are the same notation abuse, sometimes f x is meant to be interpreted as \x-f x In some cases it would be really tedious to add the extra lambdas, so the expression used in its definition is used to denote the function itself. a b c which is a short-cut of a b \land b c Both, ambiguity and complex notation, can lead to (human) parsing problems, which is what we are trying to minimise here. J.A. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] mathematical notation and functional programming
At 9:14 PM +0100 2005/1/27, Henning Thielemann wrote: Over the past years I became more and more aware that common mathematical notation is full of inaccuracies, abuses and stupidity. 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. Your distaste for common mathematical notation was shared by Edsger W. Dijkstra. For a sampling of his writings on the subject, visit http://www.cs.utexas.edu/users/EWD/welcome.html and search the transcriptions for notation. Or use the Advanced Search facility to pick up other forms such as notations and notational. Things I'm unhappy about are for instance f(x) \in L(\R) where f \in L(\R) is meant F(x) = \int f(x) \dif x where x shouldn't be visible outside the integral O(n) which should be O(\n - n) (a remark by Simon Thompson in The Craft of Functional Programming) I use lambda notation extensively in the part of my discrete math course that deals with complexity. Works fine. Moreover --anticipating a reply further down the list-- my students learn that O(f) for some function f denotes a set of functions, so that f = O(g), where f and g are functions of the same type, is simply a type error. I seem to recall that Donald Knuth made this point a few decades ago, but old habits die hard. a b c which is a short-cut of a b \land b c This problem has worse consequences when the operator is (=). By common convention, a = b = c is shorthand for a = b /\ b = c which is a great mistake because it obscures the extremely valuable (for equational reasoning) fact that when a, b, and c are Boolean, (=) is associative. In their textbook, _A Logical Approach to Discrete Math_, Gries and Schneider explain that (=) is conjunctional, and that an operator can't be both conjunctional and associative. Following Dijkstra, they introduce another operator, the equivalence (=) (that should display as a three-bar equality operator). It's defined only for Boolean operands, and it's not conjunctional but associative. f(.) which means \x - f x or just f All of these examples expose a common misunderstanding of functions, so I assume that the pioneers of functional programming also must have worried about common mathematical notation. As did Dijkstra, who devoted the last part of his career to the streamlining of mathematical argument. Regards, --Ham ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
