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