prologue
The importance of nomenclature, notation, and language as tools of thought has long been recognized.
In chemistry and in botany, for example, the establishment of systems of nomenclature by Lavoisier and Linnaeus did much to stimulate and to channel later investigation.
Concerning language, George Boole in his Laws of Thought asserted:
that language is an instrument of human reason, and not merely a medium for the expression of thought, is a truth generally admitted.
Mathematical notation provides perhaps the best-known and best-developed example of language used consciously as a tool of thought.
Recognition of the important role of notation in mathematics is clear from the quotations from mathematicians given in Cajori’s History of Mathematical Notations. They are well worth reading in full, but the following excerpts suggest the tone:
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.
The quantity of meaning compressed into small space by algebraic signs, is another circumstance that facilitates the reasonings we are accustomed to carry on by their aid.
Nevertheless, mathematical notation has serious deficiencies. In particular, it lacks universality, and must be interpreted differently according to the topic, according to the author, and even according to the immediate context.
Programming languages, because they were designed for the purpose of directing computers, offer important advantages as tools of thought.
authorship
So began KenIverson, inventor of APL, progenitor of J, and winner of the Turing Award, in his essay Notation as a Tool of Thought .