Language as an intellectual tool: From hieroglyphics to APL

We learn elementary mathematics before understanding the source of its symbols and procedures, which therefore appear, incorrectly, to have been decreed ready-made. Language and reason are intimately related, and the embodiment of an idea in a symbol may be essential to its comprehension. APL unifies algebra into a single consistent notation; it allows us to exploit the powerful concepts of functions and operators; and it helps us to escape from the tyranny of scalars by giving us the tools to think in terms of arrays, or multiple quantity, as J. J. Sylvester so eloquently urged us to do a century ago. APL has an intellectual consistency that is a source of satisfaction and pleasure. This paper traces the history of symbols from hieroglyphics to APL.