4 ms·

>Why do mathematicians need to be told what notation to use by non-mathematicians when their notations (note the plural) are in wide, and apparently successful,

by konkin 5y ago

>Why do mathematicians need to be told what notation to use by non-mathematicians when their notations (note the plural) are in wide, and apparently successful, use today? The same reason why geometers needed to be told to start using Arabic numerals by algebraists: mathematics is asking new questions and the old tools aren't good enough. That we even need to make a notational distinction between operators like addition and operators like high order partial derivatives should show that clearly enough. >How would Polish notation help, for instance, with things like describing set membership or logical relations? How is this more readable (using ASCII): Start working on longer expressions and the lack of parenthesis becomes a godsend. -> -> a b c and -> a -> b c can be parsed in only one way, by comparison with infix notation you need miscellaneous parenthesis to show the way you want a -> b -> c to be parsed: (a -> b) -> c and a -> (b -> c). Or you need to memorize an arbitrary number of rules which increase exponentially with each new operator you add. > Which provides a convention sense of the relationship that the former does not? The second one can be "read" in order: a implies b. The first cannot. In English and only for binary operators, and even then you can just as easily understand "add 5 and 6" as you can "5 plus 6". How would you use infix notation to express a ternary or higher operator? The notation for the definite integral needs a super and sub script as well as a dummy variable for an operator that is fundamentally a ternary one. By comparison in prefix notation it becomes definite_integral upper_limit lower_limit function.

Jtsummers 5y ago
> The same reason why geometers needed to be told to start using Arabic numerals by algebraists. Mathematics is changing and the old tools aren't good enough to answer current questions. At least those were other mathematicians who could understand the domain and the importance of notations for the work. Hah, no, "-> -> a b c" is not easy to parse, except for a computer. going are people not to read easily. It's a silly argument that prefix is fundamentally easier for humans when it disrupts the flow even in your example. Even using the words, "implies implies a b c", does not parse well as a reader, even if it can only be parsed one way. If you go to a higher arity operation, switch to function (which is prefix, like polish) notation if it makes sense. But even then, we have things like the integral which uses physical placement to take (typically) up to 3 arguments: b ∫ f(x) dx a With other similar operations. And if order of operations is not clear from an expression, then yes, use parentheses. That's what they have been used for for quite a while.
- konkin 5y ago
  >Hah, no, "-> -> a b c" is not easy to parse, except for a computer. going are people not to read easily. It's a silly argument that prefix is fundamentally easier for humans when it disrupts the flow even in your example. The only argument I'm seeing here is that you can't parse prefix notation and won't put the effort into trying it, which is the same argument people used against Arabic numerals in the middle ages. A prefix only language is fundamentally easier to read and write to anyone who hasn't spend two decades learning the mishmash of pre, in, post, and upside-down fix that mathematics currently uses. >But even then, we have things like the integral which uses physical placement to take (typically) up to 3 arguments: Using four variables. It's amazing to me people defend this type of mess as something meaningful instead of a terrible historical contingency which we should get rid of as quickly as we can. In prefix notation you can just use "integral upper lower function" to convey the same information in pure text without the need for inventing TeX and vector displays.
  - Jtsummers 5y ago
    Try out your notation, which I will agree can be unambiguous in a technical sense, but will quickly become unreadable for actual human beings, on a few modest examples. = 1 + square sin x square cos x -- a trig identity / + - b √ - square b * * 4 a c * 2 a ^ oops, ambiguous, you need a new symbol for negation Note as well, in that second example, the great distance between the division operation and the divisor. This does not lend itself to readability by people, they have to keep the entire stack of operations in their head. Ok, so you switch the notation to be more tree-like and use indentation (how I write complex Lisp math expressions): v still ambiguous, what alternate symbol can we use? / + - b √ - square b * * 4 a c * 2 a Ok, somewhat clearer, but hardly "easy" to read for a person. At this point you may as well just put back the parentheses (and be back at Lisp s-exprs, which as you noted earlier is not well-suited to handwriting). The mathematical expression is clearer for people. It requires them to keep less in their head at any one point in the reading. _________ -b + √b^2 - 4ac --------------- 2a I'm not saying this is perfect, there is room for improvement in this form, but it is going to be better for the average math literate person (by which I mean, someone who has at least covered high school algebra). Even if you taught prefix notation from the start, the user of that notation still has to keep more in their head as they try to parse it.

by konkin 5y ago

Jtsummers 5y ago

> The same reason why geometers needed to be told to start using Arabic numerals by algebraists. Mathematics is changing and the old tools aren't good enough to answer current questions. At least those were other mathematicians who could understand the domain and the importance of notations for the work. Hah, no, "-> -> a b c" is not easy to parse, except for a computer. going are people not to read easily. It's a silly argument that prefix is fundamentally easier for humans when it disrupts the flow even in your example. Even using the words, "implies implies a b c", does not parse well as a reader, even if it can only be parsed one way. If you go to a higher arity operation, switch to function (which is prefix, like polish) notation if it makes sense. But even then, we have things like the integral which uses physical placement to take (typically) up to 3 arguments: b ∫ f(x) dx a With other similar operations. And if order of operations is not clear from an expression, then yes, use parentheses. That's what they have been used for for quite a while.

konkin 5y ago
>Hah, no, "-> -> a b c" is not easy to parse, except for a computer. going are people not to read easily. It's a silly argument that prefix is fundamentally easier for humans when it disrupts the flow even in your example. The only argument I'm seeing here is that you can't parse prefix notation and won't put the effort into trying it, which is the same argument people used against Arabic numerals in the middle ages. A prefix only language is fundamentally easier to read and write to anyone who hasn't spend two decades learning the mishmash of pre, in, post, and upside-down fix that mathematics currently uses. >But even then, we have things like the integral which uses physical placement to take (typically) up to 3 arguments: Using four variables. It's amazing to me people defend this type of mess as something meaningful instead of a terrible historical contingency which we should get rid of as quickly as we can. In prefix notation you can just use "integral upper lower function" to convey the same information in pure text without the need for inventing TeX and vector displays.
- Jtsummers 5y ago
  Try out your notation, which I will agree can be unambiguous in a technical sense, but will quickly become unreadable for actual human beings, on a few modest examples. = 1 + square sin x square cos x -- a trig identity / + - b √ - square b * * 4 a c * 2 a ^ oops, ambiguous, you need a new symbol for negation Note as well, in that second example, the great distance between the division operation and the divisor. This does not lend itself to readability by people, they have to keep the entire stack of operations in their head. Ok, so you switch the notation to be more tree-like and use indentation (how I write complex Lisp math expressions): v still ambiguous, what alternate symbol can we use? / + - b √ - square b * * 4 a c * 2 a Ok, somewhat clearer, but hardly "easy" to read for a person. At this point you may as well just put back the parentheses (and be back at Lisp s-exprs, which as you noted earlier is not well-suited to handwriting). The mathematical expression is clearer for people. It requires them to keep less in their head at any one point in the reading. _________ -b + √b^2 - 4ac --------------- 2a I'm not saying this is perfect, there is room for improvement in this form, but it is going to be better for the average math literate person (by which I mean, someone who has at least covered high school algebra). Even if you taught prefix notation from the start, the user of that notation still has to keep more in their head as they try to parse it.