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Comment by haberman | Hacker News Reader

Why then has math (which is read and written only by humans) used infix notation for hundreds of years, whereas prefix/postfix notation were only developed in the 20th century and today are used only by Computer Scientists?

You don't have to know an entire operator precedence table to read and write idiomatic infix-notation code. Precedence is defined such that common expressions evaluate as people intuitively expect (a notable counterexample is "x & y == z" in C). Parentheses are always available to clarify more complicated expressions.

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Humans who do a lot of math switch notations when convenient. For example, for addition we'll sometimes put a summation sign in prefix notation. For division we like to put the numerator above the denominator, a notation that's inconvenient in a programming language.

Come to think of it, humans usually add and subtract by stacking numbers vertically. I don't think you can point at infix notation as "the" human-friendly notation.

Have you seriously seen (non-computer) people write simple arithmetic as + 1 3 ?

This feels like a discussion based in fiction...

I've seen lots of people doing 1 enter 2 + on their HP calculators. Using them with RPL, reverse polish lisp.

Postfix notation is popular in the financial industries.

One could have made the same argument in favour of Roman numerals, in the face of the less familiar Arabic numerals

Math was written on paper long before there were computers. As a result, the use of infix notation was an act of necessity not a calculated decision. Now that we have computers and keyboards we should use prefix notation.

if you look at math on paper, that's definitely not 'infix'.

So you think we should teach in schools (from an example above):

(/ (+ (- b) (sqrt (- (* b b) (* 4 a c)))) (* 2 a)) ?

Seems like a great idea to me.

I take it you think there is something intrinsically wrong with that idea?

Certainly, I would hope it would have been obvious, but we might each have different opinions of obvious!

I find it very hard, without bracket counting, to see exactly what the '+' and '/' bind to. With the more traditional:

(-b + sqrt(bb-4ac)) / (2a)

I find in only a glance I can tell what everything is binding to.

> I find in only a glance I can tell what everything is binding to.

It must be nice to live in a world with only 4 infix operators and expressions that have only 3 infix operators.

For example, lots of folks think that sqrt should be a prefix operator, not yet another function. I suppose you're going to assume that the top bar will serve as parentheses.

BTW "-b + sqrt(bb-4ac) / 2a" is the interesting expression. Is it "(-b + sqrt(bb-4ac)) / (2a)" or "-b + (sqrt(bb-4ac)) / 2a)" And, are you certain what "bb-4ac" means? (There's at least one major language where it doesn't mean "(bb)-(4ac)".)

We aren't talking about programming languages. We are talking about teaching math in school. In most of school mathematics, there are only 4 infix operators (well, and also the comparison operators).

> We are talking about teaching math in school. In most of school mathematics, there are only 4 infix operators (well, and also the comparison operators).

And that's how the exceptions swallow the rule. And, it's also how we get infix programming languages where that's definitely not true, and so on. Where should we make the switch?

Also, only four? What about set operations?

I would teach it like this:

  (/ (- (sqrt (discriminant a b c)) b)
     (* 2 a))

Why exactly is it a necessity to use infix on paper? And what exactly is the argument for using prefix just because we have computers? I'd argue quite the opposite, computers give use even more convenience to use whatever we like. I think it is a rather arbitrary choice, but it may relate to the prevalence of subject-verb-object in (spoken) languages; i.e., operators act like a verbs.

It's even stronger, in that mathematics generalized arithmetic algebra into groups, fields, rings and other things I don't understand. Examples of specific algebras include: boolean, relational and Kleene (aka regular expressions).

Other notations are used, but with a frequency similar to pre-fix (lisp) and post-fix (forth). "Associativity" (not affected by order of evaluation) only makes sense for in-fix.

But it really could just be familiarity, I guess. I can't see how to determine it either way. But regardless of the cause, there's overwhelming evidence that people, in fact, prefer in-fix.

> But regardless of the cause, there's overwhelming evidence that people, in fact, prefer in-fix.

How many people have seen anything other than in-fix? Of those, how many got a fair shot at an alternative?

> You don't have to know an entire operator precedence table to read and write idiomatic infix-notation code.

If it's "idiomatic", why is there such disagreement?

> Why then has math (which is read and written only by humans)

Convention has a lot of value. That said, mathematicians don't have to worry about getting things wrong. It's just paper, and they're happy to let humans fix up the errors.

> Parentheses are always available to clarify more complicated expressions.

Unnecessary parentheses are how humans deal with the fact that they can't handle infix.

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