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I'd like to get comprehensive cost-benefit analysis of using infix notation and postfix/Polish notation in arithmetic for functions (I consider equality differently, since it maps to truth values, instead of numbers, and qualifies as a predicate). What else would it behoove me to consider? I have this much so far:

Infix notation:

Advantages:

  1. Commonality of expression.
  2. We don't need separation symbols for numbers, since symbols for functions effectively do this.
  3. We'd don't have to have a clear idea about the arity of functional symbols before using them.

Disadvantages:

  1. Lack of ability to use the rule of replacement mechanically.
  2. Possible, and arguably actual, ambiguous expressions.
  3. We need an "order of operations", or something similar, to prevent ambiguity.
  4. It takes time and effort to learn about the infinity of arithmetical expressions, in the sense that seeing that from any arithmetical expression and closure, we can have arithmetical expressions beyond that of any given length.

Posfix notation:

Advantages:

  1. Can mechanically use the rule of replacement.
  2. So far as I can tell, it comes as consonant with how we practically use mathematics in everyday life, e. g. when we buy something with say x dollars which costs y dollars, we then figure out how much x and y differ by. Or if we start off walking at time j and finish walking at time i without a stopwatch, then we figure out how long we've walked for by figuring out how much j and i differ by.
  3. We can quickly learn about the infinity of arithmetical expressions.

Disadvantages:

  1. For clarity, we require separation symbols or spaces using Hindu-Arabic numerals.
  2. Not often used by mathematical and logical writers at present.
  3. To keep things clear, it comes as important to recognize the arity of functional symbols before we use them.

What advantages and disadvantages have I missed?

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    Some comments: postfix notation is asymmetrical so I think it takes a human more thinking and memory to parse the larger expressions (which numbers are being added/subtracted/multiplied to which other numbers in what order?), and the information that operations can be interpreted as encoding often have no chronology or asymmetry. Ambiguity is sometimes relative to context and experience (a logician regularly concerned about wildly implausible interpretations of expressions is probably not the norm and can be neglected in the discussion of mathematics notation at large), and where it exists...2012-10-20
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    ... it is easily mitigated by the occasional use of parentheses (which are *very* nice things when a balance is found in their usage: they exhibit where and how clumping and patterns occur in expressions). Most operations used are only unary or binary; otherwise we tend to think in terms of functions and can use parentheses and subscripted indices to count the number of arguments. I haven't the faintest idea what "the infinity of arithmetical expressions" means. One must also realize pros/cons can be categorized as either short-term (particular to the learning curve) or long-term (systemic).2012-10-20
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    People often tend to draw inferences conceptually rather than purely through symbols devoid of thoughts about meaning and content, so I don't see how lack of mechanical ability to use rule of replacement, as you see it, is problematic. This doesn't even touch on how differing notations in arithmetic could affect the rest of modern algebra notation. Ultimately, as usual, I find the most disagreeable aspect of your opinions is that you're more interested (in a *de facto* sense at least) in readability by, say, computers than you are by other human beings.2012-10-20
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    @anon Have you done any experiments on say 4 year olds who have have little to no experience with either postfix or infix notation? I know that I find 2+4*3 less readable than 2,4,3*+. "The infinity of arithmetical expressions" refers to the fact that there exists an infinity of formulas in arithmetic... in binary with just addition we have 00+, 01+, 10+, 11+, 000++, 001++, 010++, etc. If you have the rule of replacement in place, from one single formula you can tell that there exist an infinity of formulas, just by using the rule of replacement over and over again.2012-10-20
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    There is the lambda-calculus notation, which views "+" as a function of one number, which returns a function of one variable. So roughly, in normal functional notation, $+\,3\,4$ would be interpreted as $(+(3))(4)$. That is, "plus" is applied to $3$ and we get a function returned, and then that function is applied to $4$. Lambda calculus looks like infix notation, but it is a little deeper - the idea that all functions have one parameter, but can return another function...2013-03-20

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