Towards a Better Notation for Mathematics (2010) [pdf]
As the author admits in his update, this document is not very interesting. When I saw the title I was mainly interested in perhaps a formalism of mathematical exposition and proof notation.<p>The problem with the author&#x27;s proposals is this: he proposes replacing well-known, well-distinguished symbols (alphabets and digits) with symbols that reflect a certain geometric interpretation.<p>First, the geometric symbols are difficult to read, and a small error (either in printing or in reading) can mean a huge change in meaning.<p>Second, the obvious criticism that they are too radical a departure from the current convention for them to be widely adopted.<p>Most salient, however, is that they offer no benefit to the development of mathematics. The mathematical enterprise is based on abstraction, generalization, and symbolic manipulation. The symbols proposed reflect only the most banal representation of the mathematical objects they discuss.<p>For example, he discusses representing the natural numbers with a number line. But this is an elementary schooler&#x27;s notion of natural numbers. Mathematicians would like to discuss other representations and constructions of the natural numbers, say, their Peano axiomatization, as Church numerals, as an algebraic structure, as indices, as a countable set, and many more. Hence it is honestly more convenient, and less intrusive to the discussion of novel mathematical ideas and perspectives, to simply use a flavorless symbol of the alphabet, rather than one that is associated with a particular geometric interpretation.<p>Finally, I&#x27;d like to point out that while many things have changed in mathematics since ancient times, we still follow Aristotle in using letters to denote mathematical objects, and Euclid&#x27;s rigorous exposition and proof, and actually use them today with far more frequency than they ever did.