Guest Editorial: Should Vector Teach Mathematical Grammar?
There comes a time in the life of an old married couple when the friends of their youth reach retiring age and throw parties. The most recent we attended followed a predictable course. We fell into conversation with a retired engineer enquiring about our working lives and this elicited the fatal question, “What is so special about APL?”. Anthony, as always, fell into the trap and failed, as always, to convey the elegance of the notation for, and the implementation of, Inner Product. Our engineer thought of computers as fast adding machines with logic gates and had always been content to leave the intricacies of programming to his tradesmen in the field. He understood well enough that a specification might be wrong or incomplete but not that some of the fault might lie in the tool itself. We had no common language in which to discuss it further.
One of our daughters has an ambition to speak Spanish. Well, with a name like ours you would, wouldn’t you? Added to which she is married to a ‘Somerset Spaniard’ – born in the UK of Spanish parents and speaking Spanish at home. After several years of evening classes she decided to enroll with The Open University and do the job properly. She has a problem in common with all her fellow students of age less than forty five. “What”, she said, “is this thing called ‘a participle’?” Her difficulty is that schools stopped teaching Grammar when they abandoned Latin. Grammar is the metalanguage of natural languages. It is unnecessary for the ones you learn at your mother’s knee but to learn another in an academic environment grammar is an essential tool.
I believe our difficulty with APL is a consequence of similar ignorance of the metalanguage of mathematics. Our engineer friend is, no doubt, fluent in Calculus and several other of the mathematical dialects that are the tools of his trade but he had never been taught the grammar, the metalanguage of mathematics. Iverson is a mathematical philosopher and has spent a lifetime developing such a grammar – indeed eventually coming to conflate some of it with the grammar of a natural language, calling the result ‘J’.
Very few schools teach English grammar, none that I know of teaches mathematical grammar. Until they do our students will have as much difficulty comparing Visual Basic with APL as they have comparing English with Spanish. After all why should they try? Everyone speaks English now, don’t they?