I find one of the biggest mistakes programmers have about mathematical notation is that it's somehow just a terse, badly implemented programming language. But this is a very poor understanding of what mathematical notation is doing.
I think this error in thinking comes from the fact that Sigma notation can often be trivially implemented as a for loop.
Programming languages are designed to describe a specific computation, whereas mathematical notation is typically trying to describe an idea (one that might not even have a implementation!) Notation only sometimes and coincidentally describes computation as well.
The ambiguity, implied variables etc are an essential part of mathematical notation in the same way it is in common spoken language. Mathematical notation exists to help abstract and work out very hairy ideas, and often that ambiguity is necessary to show connections.
> code should be optimized for readability, not writtability
Mathematical notation is readable if you're literate in it. It takes lots of practice to become fluent in it, but once you become more familiar it's much easier to read than text (which is why it's used in the first place). Mathematical notation is an extension of mathematical writing, not computational implementation.
Reading mathematical notation is much closer to reading poetry than reading code.
I think this error in thinking comes from the fact that Sigma notation can often be trivially implemented as a for loop.
Programming languages are designed to describe a specific computation, whereas mathematical notation is typically trying to describe an idea (one that might not even have a implementation!) Notation only sometimes and coincidentally describes computation as well.
The ambiguity, implied variables etc are an essential part of mathematical notation in the same way it is in common spoken language. Mathematical notation exists to help abstract and work out very hairy ideas, and often that ambiguity is necessary to show connections.
> code should be optimized for readability, not writtability
Mathematical notation is readable if you're literate in it. It takes lots of practice to become fluent in it, but once you become more familiar it's much easier to read than text (which is why it's used in the first place). Mathematical notation is an extension of mathematical writing, not computational implementation.
Reading mathematical notation is much closer to reading poetry than reading code.