Lagrange's equations written the way he derides are a perfect example. His need to think about what spaces the symbols live in based on their context in the expression is typical when translating math to code. He should be talking about about what kind of type inference engine one needs to allow the original notation and end up with his positional notation internally.

In fact, differential geometry has been described as the study of objects "invariant under a change of notation" playing on the lack of clarity in this field, along with the standard description of a tensor as an object "invariant under a change of coordinates".

See, for example, the introduction to this piece: http://www.math.jhu.edu/~yrubinst/RiemGeom/Thesis-pp-23-30-a...

That said, notations here really correspond to different ways of thinking and are important and shouldn't be ridden.

Edit: Notation

The story is either in "Surely You're Joking, Mr. Feynman" or "What Do You Care What Other People Think?". I forget which stories are in which book. Both are great though.

     I thought my symbols were just as good, if not better, than the regular

-- Surely You're Joking, Mr Feynman (http://www.gorgorat.com/)

That said, I'm inclined to say that at some point it makes sense to break backwards compatibility in favour of a better system.

Here's a fun one: pi is wrong! http://www.math.utah.edu/~palais/pi.pdf

I hope not.

http://www.xibalba.demon.co.uk/jbr/ranto/

A Nobel laureate in physics who grew up in a non-English-speaking country recommends learning English as a key to physics research.

http://www.phys.uu.nl/~thooft/theorist.html

A true expert in Esperanto, the linguist and psychologist Claude Piron, debunked that article years ago:

http://claudepiron.free.fr/articlesenanglais/why.htm

I don't do enough math to have a useful opinion, but it strikes me that something as simple as 1 + 1 = 2 is ambiguous (does f = x+x or x+1?) and using unambiguous notation would be tedious and useless.

Unless you're trying to turn it into a program. I sometimes think of programming as the opposite of math.

Guy Steele's ‘Fortress’ language is headed in that direction.