I'm sorry to nitpick, but, for some reason, Tao has one of those names that just begs for inadvertent misspellings, and it seems like a shame. His first name is "Terence".

Probably because his short name is Terry, hence the double “r”

Very interesting read. Nowhere on his desiderata does "uniqueness" come in. The rest of his examples show that it is indeed not a desirable quality, as the same abstract object benefits from different representations to make apparent its relation to other abstract concepts in each specific context. Or simply to reduce cognitive overhead.

Yeah I didn't get the point of that either. If you rearrange and solve for a different variable, for practical purposes outside of pure math(If you are thinking like a coder), you have a completely new equation that happens to be derivable. It's almost like compiling.

But since mathematicians absolutely love finding common patterns in things, maybe there's some new innovation that a representation with uniqueness would enable?

"Normal forms" are definitely a thing, most clearly in linear algebra but certainly elsewhere as well. Being able to normalize any description of an object to a single unique canonical description is incredibly useful!

But we don't always want to work with normal forms, for one reason or another, and there can be multiple kinds of normal form to choose from depending on your needs. For instance, if you do anything with something in a normal form, chances are it's no longer in a normal form! The lack of closure properties like this means you may only normalize at the very end of a series of manipulations, during which you're using a more suitable notation.

> The desiderata listed above are primarily concerned with lowering the "recurring costs", but the "one-time costs" are also a significant consideration if one is only using the mathematics from the given field X on a casual basis rather than a full-time one.

I think this is important and applies well to programming languages. Many programming techniques and styles can be extremely powerful when continuously working in the area whereas just jumping in they can be totally maddening. Sometimes this sort of thing builds up a cognitive wall to newcomers. It has its uses, but it's frustrating when you're one of those casual users having to deal with the clever ideas in the codebase.

> (Unambiguity) Every well-formed expression in the notation should have a unique mathematical interpretation in 𝑋

This is never the case in physics because the equality sign "=" is loaded and can mean equality, definition, proportionality, identity and equivalence. Physicists read the equality sign case by case, as the case may be, with one of these meanings. In scholasticism this used to be called casuistry.