Well, for me both examples are totally unreadable. I think math's is the most unreadable notation ever and even a cross between PERL and Brainfuck would be better. Mathematicians are masochists, and I refuse to follow their lead. I prefer meaningful variable names, good use of horizontal and vertical whitespace, context-free grammars, and the like. So maybe this is the difference and the reason for me perceiving the notations as more or less equivalent: I'm not biased, as most people, in favor of infix, but rather biased the other way.

Anyway, let's try doing something with your postfix example (I hope it displays alright; also, I think you made a mistake in your translation to postfix, but I left it as it is):

    c_z   x *
    s_z   y *
        s_y *
    c_y   z *
    +
        c_x *

    c_z   y *
    s_z   x *
    -
        s_x *
    -

        d_z =

Now, this is much more readable than unformatted infix version you give and that's before factoring this into smaller parts. I didn't read that much of Forth, but I'm 97% sure that it would be factored into 3 or 4 words, I think. And it also has an advantage that you read the operation in order they're going to be executed, while with infix you need to read the entire expression to know which computation occurs first.

Of course, it's only more readable for me, with my particular background knowledge and habits; I'm not saying this is or should be the same for anyone else. But, if there are people who see one form as more readable and people who see the other form as more readable, then I think that's a solid argument in favor of both forms not being drastically, qualitatively different.

The problems you point to are definitely real; it's true "the operator doesn't show which operands it has" by default (for example) and you need to go out of your way to show it. But infix notation has it's problems too, and you need to work around them as well. Like operator precedence, which is frankly a terrible idea.

> Postfix also suffers from not making it clear how many operands a given operator takes. With infix, this is always clear.

No, not always. J has words which take one argument on the left and, for example, two on the right. Or three. Or a variable number of arguments on the left (granted, they are `tied` with ` word, but still) and nothing on the right... And Ken Iverson says it's very readable! (To him, at least).

To summarize: I'm still not convinced that there is a major and unfixable difference in readability between the notations, and I still think you can make all 3 notations as readable as any other.

If you take a look at my reply to TazeTSchnitzel you'll see that I "solved" (I think? at least tried to) this problem with indentation (generally whitespace. You can delimit expressions in many ways.

And in Forth case, you can very easily define your own delimiters:

    : [ ;
    : ] ;

which should make you able to write:

    [ [ 1 2 + ] [ 3 4 + ] * ] . 
    21  ok

(tested with gforth and works [EDIT: but of course breaks Forth! Both [ and ] are already defined, so in practice you'd rather choose another chars])

I mean, there is no rule saying that postfix notation cannot provide grouping constructs. I still fail to see a fundamental difference here :)

BTW, it's not going as fast as I'd like, but I managed to parse TXR man page and use it for displaying docs along auto-completion: https://github.com/piotrklibert/txr-mode/blob/master/screen....

I think I'll be able to find some time this weekend (or next weekend) to clean up the code and make it usable for others as well :)