This is not unlike recent discussions about the conlang Ithkuil on HN.
For better or for worse, math notation has for the most part been optimized for writing on paper or a blackboard, using context to eliminate "inessential details". Local edits are generally easy, and pieces of notation are not always so tightly coupled. (For example, we can put a "for all x" at a distance from some equation.)
Have you ever attempted to write Lisp on a blackboard, then needing to insert an expression in the middle? It's extremely difficult because you have to perform a quadratic process (erasing everything after and reindenting) to make the Lisp readable. At least this is what I've noticed in interviews. You can sometimes patch your Lisp forms with lines and arrows, but it's the quickest way to get spaghetti on your blackboard.
RPN is similarly useless for the task. RPN is exceedingly easy to write, but not so easy to read. Also, higher level mathematics becomes unwieldy in RPN.
Sussman and Wisdom in their book "Structure and Interpretation of Classical Mechanics" did take a different approach to math notation in order to make it unambiguous for that relatively small sub field. It looks like regular math notation but written in such a way to make things more easily computable. I'd say they were successful, though their work would be difficult to generalize.
What's a blackboard?
Seriously, blackboards are pretty obsolete technology. Designing a notation to optimize for blackboards in this day and age is kind of like designing roads to accommodate horses.
-Collaborating in subjects which need drawing and writing. I can stand next to someone and talk over a problem. They write some formula on the board. I insert some additional bits and pieces that they missed. We draw pictures.
I'd like to hear what you think replaces black/white boards.
The black vs. whiteboard thing is a whole different issue :)
But to constrain your infrastructure (notation in the case of mathematics, roads in the case of horses) according to the needs of a blackboard or a horse is, IMHO, a serious mistake in this day and age. If you design your roads for cars instead of horses you get tremendous productivity boosts, even as you lose the ability to deal with some edge cases.
Notice that to find an example of the real utility of a blackboard you had to bypass >95% of the lecture and go to the very end. Imagine how much better things would be if the rest of the lecture had been presented as source code that a student could analyze and manipulate and error-check using some automated tool.
Re: your last paragraph. I only went to the end because I knew that there must have been a good example in the questions. If you want I can give you examples from the middle of a talk.
Only because you're used to it. In fact, standard notation is much harder to read because it's ambiguous, often to the point of actively introducing errors. See:
http://mitpress.mit.edu/sites/default/files/titles/content/s...
> If you want I can give you examples
No, I don't dispute that blackboards are useful. What I dispute is that their utility is so high that we ought to design mathematical notation around their limitations.
>It seems tempting to have a single unambiguous notation for mathematics. But In constructing such a language, one will quickly realize that doing mathematics becomes an intensely arduous task.
When talking about math, our notation doesn't have to be precise, and that's ok.
> No, I don't dispute that blackboards are useful. Just obsolete :P
Mathematics (and a lecture on mathematics) is closer in nature to a conversation than a road.
[edit sp]
And your condescending comment does not address my point, which is that a lecture would not benefit from (and is actively harmed by) the "features" being suggested. Real time error flagging would be extremely distracting, and writing mathematics as source code would be tediously slow and again distract from the point of understanding the mathematics.
And your claim that writing mathematics as source code is too slow is very much lacking in proof. All we can say with confidence is that syntax like LaTeX markup on a standard keyboard layout is too inefficient for realtime use. This does not mean that realtime use is impossible if you allow for a more complicated IME and for a different final notation on-screen than the current standard math notation.
What? Blackboards are excellent. You're just working in the wrong places.
But they suffer from “memory exhaustion” rather easily, to the point where even several blackboards on a funky roller system might be insufficient for a one hour lecture developing a complicated proof.
And the garbage collection causes a serious interruption and sometimes misses things.
have parse errors
Your lecturers obviously had much more legible handwriting than some of mine!
have encoding errors
Well, an encoding where P, p, and ρ all occupy the same code point might be considered ill-advised, and one where m, n, r, u, v and w may variously appear distinct or not depending on the display device in use is downright mischievous.
have usability problems
Does blocking half the lecture theatre’s view every time you write up a new formula count as a usability problem?
This is more of a problem with people or time constraints, not the medium.
The blackboard is no different - it's not the perfect way to communicate ideas but it let's us communicate ideas quickly.
What's the equivalent in the horse/car thing?
I guess that depends on what you think math is for. If you think the purpose of math is merely to provide humans with intellectual stimulation then yes, it makes sense to optimize the nation for human consumption. But if you think that math is actually good for something besides being a distraction from existential despair then rendering math for human consumption might not be the thing you want to optimize for. Instead you might want to use a notation that, while it can be rendered for human consumption, isn't optimized for that, but is instead optimized for, say, automated error detection, or automated compilation into some other form, like an executable program or a design for an FPGA.
The closest I can come to an analogy in the horse/car world is that in the horse world it makes sense to dispense water in troughs to make it easy for horses to drink. But despite the fact that water and gasoline are both liquids, it might make not make sense to dispense gas in the same way you dispense water.
It's hard to hit all of those points. Lisp falls short (neither concise nor suitable for analog) as does the existing notation (not as comprehensible and not suitable for digital formats).
Chalk is also less expensive. As one data point: 48 sticks of chalk for $4.30[1] versus 12 markers for $7.74[2]. I'm not sure about the bulk-price comparison, or if colleges can get dry-erase markers for cheap. Although, many of my professors carried their own supplies as markers left in rooms were usually stolen.
There could be a health comparison to be made between dry-erase marker fumes and chalk dust, but I know nothing about it off-hand.
[1]: http://www.amazon.com/Crayola-Non-toxic-Anti-Dust-Chalkboard... [2]: http://www.amazon.com/Expo-Chisel-Erase-Markers-80001/dp/B00...
Some of them are legit: older professors often have better handwriting at blackboard; chalk is sometimes better for certain drawings; left-hand smear; etc.
I suspect the biggest reason is that, in math circles, blackboards have a much larger cool/nostagia factor.