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One realises how confusing and complicated are the standard notations when one implements a programming language. You would prefer Lisp/Scheme or RPN notations,

One realises how confusing and complicated are the standard notations when one implements a programming language. You would prefer Lisp/Scheme or RPN notations, but they both scare away a lot of people, who prefer "natural" notations. People don't realise that they probably hurt them as much as they help them.

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 ago
    You'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 ago
    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 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 ago
      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.
      - j2kun 12y ago
        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. [edit sp]

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 ago
  You'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 ago
  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 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 ago
    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.
    - j2kun 12y ago
      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. [edit sp]