So is mathematical notation not readable? On the one hand, it takes quite a while to learn how to read it. On the other hand, it can condense literally pages of text into a few equations which somebody experienced can understand almost at a glance.
The same is true for programming languages. Sure, anybody can read Python. But it's fairly verbose and attempts to mirror English. On the other hand, Haskell is somewhat terse and attempts to mirror mathematical notation. I've found that, after about the same amount of time with either language (about a year of off-and-on use for both), Haskell becomes far easier to read than Python.
So just because something is easier immediately does not mean it is any more readable. It could, in fact, be less readable and merely more familiar!
Mathematical notation is not readable, no. It requires extensive training to be able to understand it, and even experts will take quite a while to comprehend a document they are not already mostly familiar with. It is not like you cross some threshold and immediately become able to understand all mathematical notation with equal ease. There's plenty of evidence for this; recently we had this quote about the ABC theorem: "It can require a huge investment of time to understand a long and sophisticated proof, so the willingness by others to do this rests not only on the importance of the announcement but also on the track record of the authors" (http://www.nature.com/news/proof-claimed-for-deep-connection...)
> even experts will take quite a while to comprehend a document they are not already mostly familiar with
Experts don't take time to read notations. They take time to understand the full scope of consequences embedded in a line composed of such a condensed and readable notation, within the frame of an otherwise complex proof.
> Mathematical notation is not readable, no.
I beg to differ. You mention the recent possible proof of the ABC conjecture, and I happened to browse through various materials on the subject. While I am far from being able to understand the crux of the proof, I can perfectly read the various components of it. In fact the notation system is so readable it actually helps tremendously in grasping concepts previously unknown to me.
Mathematical notation is much harder to read than prose. It does a very good job of serving as a mnemonic for something you already understand, and a very poor job of explaining novel (to the reader) concepts.
It's also as bad as natural language for ambiguity and informality.
I don't think either English text or mathematical notation is a magic wand for readable programming language; It's more about reducing accidental ambiguity and the amount of mental context needed for parsing 'natural' constructs in the language.
Personally, most of the benefit I see from mathematical notation is down to pattern matching -- "Oh, this looks like a Fourier transform". The idea seems to be supported by the fact that there is a unwritten convention for which symbols to use for which concept (alpha over beta for example) and the way in which the equations are laid out.
The same is true for programming languages. Sure, anybody can read Python. But it's fairly verbose and attempts to mirror English. On the other hand, Haskell is somewhat terse and attempts to mirror mathematical notation. I've found that, after about the same amount of time with either language (about a year of off-and-on use for both), Haskell becomes far easier to read than Python.
So just because something is easier immediately does not mean it is any more readable. It could, in fact, be less readable and merely more familiar!