Notation And Thinking at Dick Lipton’s Blog
My colleague Dick Lipton writes wonderful blog posts, and he just wrote one this last week on the role of notation in our thinking, a mathematical/computational take on the Whorfian Hypothesis. The below quote (attributed to Alfred North Whitehead) is actually from the commentary on the post, and captures the issues succinctly.
An important issue that Dick left unconsidered in his piece (but is raised in the commentary) is the up-take cost of a new notation. There is a learning cost in developing the abstraction and learning the mapping that this symbol stands for that. Much of Dick’s post is on APL which is a wonderful example of exactly this point. APL is very compact and powerful — but it uses an unusual symbol set, and requires the newbie to learn vector operations as a way of describing computations. That’s a valuable perspective to learn, but it comes with a cost. In the end, all educational decisions are economic: Is the power and brevity in notation worth the cost to learn?
It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.