The properties of executability and universality associated with programming languages can be combined, in a single language, with the well-known properties of mathematical notation which make it such an effective tool of thought.
The initial motive for developing APL was to provide a tool for writing and teaching. Although APL has been exploited mostly in commercial programming, I continue to believe that its most important use remains to be exploited: as a simple, precise, executable notation for the teaching of a wide range of subjects.kenneth e. iverson
I was appalled to find that the mathematical notation on which I had been raised failed to fill the needs of the courses I was assigned, and I began work on extensions to notation that might serve. In particular, I adopted the matrix algebra used in my thesis work, the systematic use of matrices and higher-dimensional arrays (almost) learned in a course in Tensor Analysis rashly taken in my third year at Queen’s, and (eventually) the notion of Operators in the sense introduced by Heaviside in his treatment of Maxwell’s equations.kenneth e. iverson
With the computer and programming languages, mathematics has newly-acquired tools, and its notation should be reviewed in the light of them. The computer may, in effect, be used as a patient, precise, and knowledgeable "native speaker" of mathematical notation.kenneth e. iverson
The difficulty, as in all this work, is to find a notation which is both concise and intelligible to at least two people of whom one may be the author.
In all engineering disciplines nowadays, software engineering excluded, there exists an established engineering process to develop a system, which is accompanied by a number of suited modeling description techniques. Software engineering, being a rather new field, has not as yet established any clear methodical guidance or a fully standardized modeling notation.
The development of Indian trigonometry , based on sine as against chord of the Greeks , a necessity for astronomical calculations with his own concise notation which expresses the full sine table in just one couplet for easy remembrance. One of the two methods suggested by him for the sine table is based on the property that the second order sine differences were proportional to sines themselves.
A class in arithmetic will be pleased to hear about the Hindoos and their invention of the "Arabic notation;" they will marvel at the thousands of years which elapsed before people had even thought of introducing into the numeral notation that Columbus-egg -- the zero; they will find it astounding that it should have taken so long to invent a notation which they themselves can now learn in a month.
Not to be overlooked is the fact that in the [Babylonian] sexagesimal notation of integers the "principle of position" was employed. Thus in 1.4 (=64)... The introduction of this principle at so early a date is the more remarkable, because in the decimal notation it was not introduced till about the fifth or sixth century after Christ.
The sexagesimal system was used also in fractions. Thus, in the Babylonian inscriptions, 1/2 and 1/3 are designated by 30 and 20, the reader being expected, in his mind, to supply the word "sixtieths." The Greek geometer Hypsicles and the Alexandrian astronomer Ptolemæus borrowed the sexagesimal notation of fractions from the Babylonians and introduced it into Greece. From that time sexagesimal fractions held almost full sway in astronomical and mathematical calculations until the sixteenth century, when they finally yielded their place to the decimal fractions.
The initial motive for developing APL was to provide a tool for writing and teaching. Although APL has been exploited mostly in commercial programming, I continue to believe that its most important use remains to be exploited: as a simple, precise, executable notation for the teaching of a wide range of subjects.
By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race.Alfred North Whitehead
Although mathematical notation undoubtedly possesses parsing rules, they are rather loose, sometimes contradictory, and seldom clearly stated. [...] The proliferation of programming languages shows no more uniformity than mathematics. Nevertheless, programming languages do bring a different perspective. [...] Because of their application to a broad range of topics, their strict grammar, and their strict interpretation, programming languages can provide new insights into mathematical notation.kenneth e. iverson