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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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closed as off-topic by Grigory M, Hakim, Davide Giraudo, Jean-Claude Arbaut, Sami Ben Romdhane May 27 at 21:36

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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... –  anon Oct 20 '12 at 3:51
    
... 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). –  anon Oct 20 '12 at 3:52
    
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. –  anon Oct 20 '12 at 3:55
    
@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. –  Doug Spoonwood Oct 20 '12 at 4:57
    
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... –  Thomas Andrews Mar 20 '13 at 4:59

2 Answers 2

I disagree with much of your analysis, but I think you are missing by far the most important aspect: the ease by which a human can look at a formula and understand what it says.

For example, compare

$$ 2 x^2 + 3 x + 1 $$

with

$$ 2 x 2 \hat{\ } \cdot 3 x \cdot 1 + +$$

There are many aspects to standard notation for arithmetic that help make reading easy for humans. The spatial proximity of an operator to what it operates on helps link them together. The visual grouping between terms helps pick out components. To get an idea of the importance of spacing, look at

$$ 2 x + 3 y + 4 z $$ $$ 2\,\; x +\;\! 3\,\;y\;\!+\;\!4\,z$$ $$ 2 \cdot x + 3 \cdot y + 4 \cdot z $$ $$2 \times x + 3 \times y + 4 \times z $$

In the first of these, it is easy to visually pick out the individual terms ($2x$, $3y$, and $4z$) and recognize they are being added together. This becomes progressively more difficult as you go down the line. And if we used postfix notation

$$ 2\: x \times 3\: y \times 4 \:z \times ++ $$

maybe with some training one can pick out the three terms quickly, but one cannot recognize how they are be combined without scanning all the way to the end of the expression. A task that is much more difficult if this is part of a larger expression.

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I find "2x2 ^⋅3x⋅1++" easier to read than "2x2+3x+1", and I qualify as a human. "In the first of these, it is easy to visually pick out the individual terms (2x, 3y, and 4z) and recognize they are being added together." I don't follow. How does 2x qualify as a term? 2x does not qualify as a constant, nor a variable, nor a function. "2x×3y×4z×++" does not have three terms. It has six terms: "2", "x", "3", "y", "4" and "z". Also, since you have singled out "2x", "3y", and "4z", why do you not also single out "3y×4z×+" since it gets added to "2x×"? Interesting point about spacing. –  Doug Spoonwood Oct 20 '12 at 5:13
    
My experience is that humans can't read mathematical expressions without training. They're generally trained with infix expressions. So your claim about the ease with which humans can read infix carries about as much weight as an English speaker claiming that SVO order is easier to understand. –  Dan Piponi Jun 11 at 4:37

I think the biggest issue would be association of the mathematical operation to be executed on any two numbers or variables. As the equation becomes much larger it will be more difficult to relate the operand to the variables. As you will need to scan back and forth, plus you would need to keep track of each operand as it is used. I think spatial relation becomes and important thing here. Disambiguation of the equations in either notation have their drawbacks. But if you natively read and speak a language that is read from left to right it is easier to interpret infix notation, and use an order of operations.

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