E.W.Dijkstra Archive: The notational conventions I adopted, and why (EWD 1300)

EWD 1300

The notational conventions I adopted, and why

At a given moment, the concept of polite mathematics emerged, the underlying idea of which is that, even if you have only 60 readers, it pays to spend an hour if by doing so you can save your average reader a minute. By inventing an idealized “average reader”, we could translate most of the lofty, human goal of politeness into more or less formal criteria we could apply to our texts. This note is devoted to the resulting notational and stylistic conventions that were adopted as the years went by.

We don’t want to baffle or puzzle our readers, in particular it should be clear what has to be done to check our argument and it should be possible to do so without pencil and paper. This dictates small, explicit steps. On the other hand it is well known that brevity is the leading characteristic of mathematical elegance, and some fear that this ideal excludes the small, explicit steps, but one of the joys of my professional life has been the discovery that this fear is unfounded, for brevity can be achieved without committing the sin of omission.

I should point out that my ideal of crisp clarity is not universally shared. Some consider the puzzles that are created by their omissions as spicy challenges, without which their texts would be boring; others shun clarity lest their work is considered trivial.

As time went by, we accepted as challenges to avoid pulling rabbits out of the magician’s hat. There is a mathematical style in which proofs are presented as strings of unmotivated tricks that miraculously do the job, but we found greater intellectual satisfaction in showing how each next step in the argument, if not actually forced, is at least something sweetly reasonable to try. Another reason for avoiding rabbits as much as possible was that we did not want to teach proofs, we wanted to teach proof design. Eventually, expelling rabbits became another joy of my professional life.

I should point out that also my ideal of proofs without rabbits is not universally shared. Some authors believed that, in order to keep the reader awake, one has to tickle him with surprises. Others believe that simulated sparks of divine inspiration are essential for earning the respect of their readership.

As a final influence I must mention our desire to let the symbols do the work —more precisely: as much of the work as profitably possible—. The intuitive mathematician feels that he understands what he is talking about and uses formulae primarily to summarize situations and relations in to him familiar universes. When he seems to derive one formula from another, the transformations he allows are those that seem to be true in the universe he has in mind. The formalist, however, prefers to manipulate his formulae, temporarily ignoring all interpretations they might admit, the rules for the permissible symbol manipulations being formulated in terms of those symbols: the formalist calculates with uninterpreted formulae. While the average intuitive mathematician is perfectly happy with semantically ambiguous formulae “because he knows what is meant”, the formalist obviously insists on an unambiguous formalism.

Saying that the preferences of the formalist are not universally shared, would be the understatement of the year: intuitive mathematicians can hate more formally inclined colleagues with the same visceral hatred as Republicans can hate Democratic presidents with.

Infix notation and expressions

Without much hesitation I have decided to stick to the usual infix operators. They have the disadvantage of requiring parentheses to distinguish (p+q)/r from p+(q/r) which prefix notation would have rendered as /+pqr and +p/qr respectively.

Remark One of the names of prefix or Polish notation is “parenthesis-free” notation, but that is a little bit misleading, for though no longer needed, the parentheses are still permissible: we could have written /(+pq)r and +p(/qr) respectively. (End of Remark).

A little hesitance was induced when my late colleague Wouter Peremans remarked in passing that infix notation was of course “a notational relic”. It made me realize why I like it so much for associative operators: it allows us to write p+q+r without being forced to choose between (p+q)+r and p+(q+r); in prefix notation, the choice between ++pqr and +p+qr would have been unavoidable.

The main reason to stick to the infix notation for the usual operators was, of course, that we are so terribly used to it, but the associativity does play an honest role: I did introduce —with great satisfaction!— infix operators for what I used to denote by max(x, y) and gcd(x, y) , and find that I don’t introduce anymore infix operators that are not associative.

Priority rules, i.e., rules of relative binding power, are the standard way of reducing the...