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A lot of these things were established long before mathematicians started talking about functions, let alone give a robust definition of what a function is. It

A lot of these things were established long before mathematicians started talking about functions, let alone give a robust definition of what a function is. It may have obscured the fact that these things are very similar to each other. At the point when there was an effort to formalize these things ideas like RPN starting popping up. Mathematics also deals with a much more flexible medium. It's written on a plane, in 2 dimensions, whereas programmers confine themselves to just one. Mathematicians are also free to make up their own notation and symbols, whereas programmers are confined to a single alphabet and most often can't extend the syntax. EDIT: Case in point: some mathematicians argued for a while if abs (absolute value) is a function. Some argued that it's not, because you need two formulas to define it. When somebody pointed out that (for real numbers) it's just "sqrt(x^2)", the first ones agreed. Sounds silly in retrospect now that we have a set-theoretic definition, but at the time it wasn't obvious and all these functions looked very different from each other.

nybble41 7y ago
> Case in point: some mathematicians argued for a while if abs (absolute value) is a function. Some argued that it's not, because you need two formulas to define it. When somebody pointed out that (for real numbers) it's just "sqrt(x^2)", the first ones agreed. Isn't that a circular definition, though? sqrt(x²) = ±x, which isn't a function since there are two values in the domain for each value in the range (other than zero). The version they're equating with abs(x) is the absolute value of the square root, or abs(x) = abs(sqrt(x²)), which is true but wouldn't help prove that abs(x) is a function. Of course, the underlying problem was the premise that a function must be defined by exactly one formula.
- abjKT26nO8 7y ago
  When people take the square root of a number during working with real numbers, they almost always mean the non-negative root. It's a convention. Only when they start considering complex numbers, this convention breaks, and the square root ceases to be a function - even according to the modern definition -, because "non-negative" still doesn't narrow it down to a single number in the general case.
  - nybble41 7y ago
    Isn't that basically what I said? The convention is that sqrt(x) is generally read as abs(sqrt(x)). The definition is that sqrt(x²) = x, which has positive and negative solutions in x for any x² > 0. You can choose to ignore the negative solutions (or the positive solutions) to make it a function, but I wouldn't consider that any simpler or closer to a single formula than the piecewise-defined version of abs(x). It's an arbitrary restriction—much like the mistaken idea that a function must be defined by exactly one formula. Why not just say that abs(x) = ±x, ignoring the negative solutions? If you'll accept "the non-negative square root of x²" then I see no reason to reject "the non-negative component of ±x". Both are single formulas with positive and negative solutions combined with a qualifier rejecting the negative solutions.
    - abjKT26nO8 7y ago
      > The definition is that sqrt(x²) = x, which has positive and negative solutions in x for any x² > 0. When people write "sqrt", they mean the function, or, "principal square root". "A square root" is a different thing. Saying "the definition" makes sense only within the context where the definition is established, otherwise we have to fallback to the common usage. Wittgenstein would laugh at this conversation. Also, I'm not arguing for this convention ("it needs to have a single formula to be a function"). And when you start substituting symbols, almost nothing withstands this test. Because then you can always always twist the formula into several cases. And generally thinking about functions as something that is defined by formulas is a very limiting view. Almost all functions cannot be defined by formulas.

nybble41 7y ago

> Case in point: some mathematicians argued for a while if abs (absolute value) is a function. Some argued that it's not, because you need two formulas to define it. When somebody pointed out that (for real numbers) it's just "sqrt(x^2)", the first ones agreed. Isn't that a circular definition, though? sqrt(x²) = ±x, which isn't a function since there are two values in the domain for each value in the range (other than zero). The version they're equating with abs(x) is the absolute value of the square root, or abs(x) = abs(sqrt(x²)), which is true but wouldn't help prove that abs(x) is a function. Of course, the underlying problem was the premise that a function must be defined by exactly one formula.

abjKT26nO8 7y ago
When people take the square root of a number during working with real numbers, they almost always mean the non-negative root. It's a convention. Only when they start considering complex numbers, this convention breaks, and the square root ceases to be a function - even according to the modern definition -, because "non-negative" still doesn't narrow it down to a single number in the general case.
- nybble41 7y ago
  Isn't that basically what I said? The convention is that sqrt(x) is generally read as abs(sqrt(x)). The definition is that sqrt(x²) = x, which has positive and negative solutions in x for any x² > 0. You can choose to ignore the negative solutions (or the positive solutions) to make it a function, but I wouldn't consider that any simpler or closer to a single formula than the piecewise-defined version of abs(x). It's an arbitrary restriction—much like the mistaken idea that a function must be defined by exactly one formula. Why not just say that abs(x) = ±x, ignoring the negative solutions? If you'll accept "the non-negative square root of x²" then I see no reason to reject "the non-negative component of ±x". Both are single formulas with positive and negative solutions combined with a qualifier rejecting the negative solutions.
  - abjKT26nO8 7y ago
    > The definition is that sqrt(x²) = x, which has positive and negative solutions in x for any x² > 0. When people write "sqrt", they mean the function, or, "principal square root". "A square root" is a different thing. Saying "the definition" makes sense only within the context where the definition is established, otherwise we have to fallback to the common usage. Wittgenstein would laugh at this conversation. Also, I'm not arguing for this convention ("it needs to have a single formula to be a function"). And when you start substituting symbols, almost nothing withstands this test. Because then you can always always twist the formula into several cases. And generally thinking about functions as something that is defined by formulas is a very limiting view. Almost all functions cannot be defined by formulas.