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It seems tempting to have a single unambiguous notation for mathematics. But In constructing such a language, one will quickly realize that doing mathematics be
by reikonomusha 12y ago
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.
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.
- lisper 12y ago> 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.
- j2kun 12y agoYou're forgetting the many populations in the world that have no access or limited access to modern technology, yet need or want to learn mathematics.
- carstimon 12y agoThere 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 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 :)
- lisper 12y agoLook, 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.
- j2kun 12y agoTo 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. [edit sp]
- 12y ago
- DanBC 12y ago> Seriously, blackboards are pretty obsolete technology. What? Blackboards are excellent. You're just working in the wrong places.
- lisper 12y agoI honestly can't tell if you're being ironic or not. But if you're not: what makes blackboards so excellent?
- DanBC 12y agoNo weird security bugs that you need to be careful of.
- duaneb 12y agoBlackboards 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.
- lisper 12y agoComputers 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.
- Chris_Newton 12y agoBlackboards don't generally crash 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?
- scott_s 12y agoLots of people, particularly people who do lots of math, still communicate with each other over a whiteboard.
- superuser2 12y agoBecause their notation is optimal on whiteboards and onerous on computer screens.
- scott_s 12y agoComputer screens are also not optimal for collaborative discussion.
- quacker 12y agoGoing 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).
- kaitai 12y agoMathematicians still fight college building committees and IT people to get blackboards in newly-built classrooms. (Why IT people? Dust + classroom computer system/projector/etc.)
- LaikaF 12y agoWhat about white boards?
- nmrm2 12y agoThere are about a million exucses (er, reasons). 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.
- quacker 12y agoI 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. [1]: http://www.amazon.com/Crayola-Non-toxic-Anti-Dust-Chalkboards-50-1402/dp/B00DZS573Y http://www.amazon.com/Crayola-Non-toxic-Anti-Dust-Chalkboard... [2]: http://www.amazon.com/Expo-Chisel-Erase-Markers-80001/dp/B00006JNK2 http://www.amazon.com/Expo-Chisel-Erase-Markers-80001/dp/B00...
- j2kun 12y agoCrayola 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.
- GFK_of_xmaspast 12y ago
- pdovy 12y agoDifferent 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.
- gizmo686 12y agoBlackboards 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.
- smorrow 12y agoThe 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. What's the equivalent in the horse/car thing?
- lisper 12y ago> 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.