Syntax Vs. Notation

Jon Sterling made the following point on Mastodon:

https://types.pl/@jonmsterling@mathstodon.xyz/113963468931231759

Well, syntax is a sound and complete formal representation of an object that has a universal property (e.g. initial algebra, etc.). Notation is the stuff you write down… Notation is usually very distant from the structure it refers to, because it leaves out details or makes various affordances. When reading informal mathematics, the notation is not part of the thing being described but should instead be seen as implicit instructions for your brain to recontruct the "actual" thing being described.

For this reason, notation and syntax are orthogonal—sometimes notation is "instructions to reconstruct a piece of syntax", but more often than not, notation is "instructions to reconstruct a piece of semantics". When you use nice notations, for example, to write about fields in algebra, you are not saying "Interpret this stuff in the formal language of fields, and then use the universal homomorphism from syntax to semantics to interpret it in my chosen field" — and this is very lucky, indeed, because there is no initial object in the category of fields. Instead, nice notation about fields is interpreted directly in the field you are talking about, and this is not a formal process but a presupposition of the human reading/writing process.

I don't mean to fixate on kinds of theory where you don't have initial models etc., but rather to highlight the different roles of notation and syntax. The syntax is part of the mathematics, the notation is not: it is eliminated as soon as you have read the sentence.