> Have you ever attempted to write Lisp on a blackboard
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.
There are two situations which black/white boards are awesome for.
-Giving lectures in subjects which desperately need drawing and writing. Look at what happens at 1:16:27 here
https://www.youtube.com/watch?v=BPSEpDq6QYc
The speaker can just go draw a picture, in response to a question. I know of no alternative which can do something like that nearly as well.
-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 :)
Look, I have nothing against blackboards, just as I have nothing against horses. Horses are really handy in some situations. If you're in the wilderness and you need to cross a stream, a horse can be just the thing. There's no technology that can compete with a horse in that case.
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.
The great thing is we're not constraining our notation. As reikonomusha said, the standard notation is easier to read. Other notation is better for programming or certain things, and that's what we use there.
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:
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.
Ok, I can agree that we shouldn't design notation around their limitations. And I do like what they do in SICM. But, I'm just trying to root for the point up-thread:
>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
To constrain your infrastructure according to the needs of a computer is also silly. Imagine if, in order to hum a tune, one needed to write sheet music using a programming language. Or if every spoken conversation were halted the instant a word is used incorrectly.
Mathematics (and a lecture on mathematics) is closer in nature to a conversation than a road.
Those are stupidly irrelevant examples. Nobody's suggesting that you would need to have a Coq parser between your keyboard and your display. You can type an incomplete or invalid expression just as easily as you can write one, but only with a computer can you have your statements automatically and reliably checked and errors flagged in realtime.
Essentially, that is the argument. If the point is to allow students to manipulate the lecture as data, then it must be error-free. That is literally putting a parser between the lecturer and the students.
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.
Enabling multiple users to interact with the equations in realtime as they are being written is hardly the only possible benefit of using computers to communicate math, and even so it only requires that the equations be tokenized to be manipulable, not that the whole expression be completely error-free and the parser be running in an enforcing mode.
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.
Computers can have pen input too for situations where that's more efficient, and for less expense than that lecture hall's complicated apparatus of multiple sliding blackboards to get around the finite drawing area limitation that computers don't have.
Blackboards don't generally crash, or have parse errors, or have encoding errors, or have usability problems. If you have chalk and a blackboard and have some semblance of an ability to write, you can use it to its fullest extent.
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?
> 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
This is more of a problem with people or time constraints, not the medium.
Computers nowadays don't generally crash either. And when you say "Blackboards don't have parse errors" what you mean is that blackboard never tell you when you've made a mistake (because they can't). That's not a feature. It's easy to program a computer not to tell you when it detects an error. But there's a reason this is not often done: detecting errors automatically is tremendously useful, especially when you're doing math.
Blackboards might be obsolute (especially given whiteboards), but free form written math is not. You run into many of the same constraints regardless of if you are writing on a blackboard/whiteboard/paper/tablet.
Different communication mediums are better / worse for different tasks. Lisp might be great for formally writing something down, where you want to have zero ambiguity. That's not the way human conversation works though, we are always eliding details based on the context in order to communicate quickly.
The blackboard is no different - it's not the perfect way to communicate ideas but it let's us communicate ideas quickly.
The horse thing seems like a bad analogy. Whether you're using a blackboard or some fruity iPad or whatever it is, it still must go through your eyes before you read it. So you don't "optimize for blackboards", optimize for people is what you do.
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.
Going forward, the ideal notation would be suitable for both digital and analog formats while being concise but comprehensible.
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).
Mathematicians still fight college building committees and IT people to get blackboards in newly-built classrooms. (Why IT people? Dust + classroom computer system/projector/etc.)
I have no hard data on this, but dry-erase markers seem to be used up more quickly than chalk.
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.
Crayola chalk is horrid. I gladly pay premium for well-made chalk (and I suspect most mathematicians prefer high quality chalk), which makes white board markers significantly cheaper.
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.
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.