Paper of the Week: Notation as a Tool of Thought

David albert circle David Albert

This is part of our “Paper of the Week” series. For more info, check out our introductory blog post.

This week’s paper is Notation as a Tool of Thought by Kenneth Iverson. Iverson originally presented this paper as his Turing Award Lecture in 1979. It was published in the August 1980 issue of Communications of the ACM. The paper serves both as a primer on Iverson’s ideas about notation and an introduction to the APL programming language.

If you’re living in the United States, this is a perfect paper to read on your trip home for Thanksgiving. If you’re not living in the US, it’s a perfect paper to read any time this week.

Notation as a Tool of Thought was submitted by Hacker School resident R0ml, who shared the following:

The paper argues for the importance of constructing a more sophisticated form of programming language than any of the ones which have achieved popularity. The paper is easily a hundred years ahead of its time – in that after reading this paper, you look at existing programming languages and realize how crude they are compared to the ones which will have to be developed in order to synthesize the mathematical and linguistic aspects of programming. Mathematica (or the Wolfram Language) is the only programming language in use today which implements these ideas.

Iverson’s paper doesn’t have an abstract, so here’s an excerpt from the introduction:

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 A 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.—A.N. Whitehead

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.—Charles Babbage

Read Along

Read Along is a way for you to participate in Paper of the Week. If you want to take part, all you have to do is read the paper, make something small in response (code or prose), and email us a link to what you made by noon Eastern Time next Monday.

Last week’s paper, The Power of Two Random Choices: A Survey of Techniques and Results, was submitted by Hacker School alum Dan Luu. This week, Dan submitted a Read Along to his own submission about cache eviction and random choice.

You might notice that Dan wrote his submission before last week’s PotW was published. We think that’s great! If you’ve made something in the past that’s based on one of our papers, we’d love to see your submission too.

Happy Thanksgiving (and happy reading)!