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:
- Commonality of expression.
- We don't need separation symbols for numbers, since symbols for functions effectively do this.
- We'd don't have to have a clear idea about the arity of functional symbols before using them.
- Lack of ability to use the rule of replacement mechanically.
- Possible, and arguably actual, ambiguous expressions.
- We need an "order of operations", or something similar, to prevent ambiguity.
- 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.
- Can mechanically use the rule of replacement.
- 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.
- We can quickly learn about the infinity of arithmetical expressions.
- For clarity, we require separation symbols or spaces using Hindu-Arabic numerals.
- Not often used by mathematical and logical writers at present.
- 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?