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> The way I see it is that there's only one fundamental function, which is the exponential function, and log is its inverse. Everything else, including a^b, is

> The way I see it is that there's only one fundamental function, which is the exponential function, and log is its inverse. Everything else, including a^b, is syntactic sugar. (If you define exp on C, even sin and cos...) Why do you think this? It doesn't seem to fit historically or formally. 3^2 is a more elementary object than anything built with exp, and there's often no natural notion of exp(A) of a function A even when finite powers like A^3 are defined. exp(A) is defined with a power series that may not converge.

qsort 6y ago
Premise: Forgive the sloppiness, I have some math background but I'm not a mathematician, any correction is more than welcome. I don't really conceptualize them as the same kind of object, to be honest. I'm aware exponentiation is more fundamental, but the kind of exponentiation you are referring to is related to the intuitive concept of "do this N times", which only makes sense for positive integers. When you are talking about real numbers, the notion of "repeated multiplication" and "exponentiation" diverge, for example, (-2)^2 is well defined and equal to (-2)(-2), but (-2)^(1/2) isn't, unless you relate it to the exponential function. Since the OP was about a notation proposal for working with real numbers, in that specific context I believe the more natural interpretation is to relate everything to the complex exponential and work your way up from there.
- jessriedel 6y ago
  > the kind of exponentiation you are referring to is related to the intuitive concept of "do this N times", which only makes sense for positive integers. But it works when "this" is an action that does not have a sensible notion of being applied a fractional number of times, or an infinite number of times, which is a very large and important set of actions indeed. > When you are talking about real numbers, the notion of "repeated multiplication" and "exponentiation" diverge Yes of course, but mere continuity of the inputs doesn't pick out exp from any other base you might choose besides e. You do not have to relate it to the exp function. > Since the OP was about a notation propos... My specific objection was to your suggestion that a^b is syntactic sugar, which suggests notational convenience that does not reflect what's going on "under the hood".
  - qsort 6y ago
    > My specific objection was to your suggestion that a^b is syntactic sugar Yes, I agree that 'syntactic sugar' is not the word we are looking for here, I have no objections to your comment.

qsort 6y ago

Premise: Forgive the sloppiness, I have some math background but I'm not a mathematician, any correction is more than welcome. I don't really conceptualize them as the same kind of object, to be honest. I'm aware exponentiation is more fundamental, but the kind of exponentiation you are referring to is related to the intuitive concept of "do this N times", which only makes sense for positive integers. When you are talking about real numbers, the notion of "repeated multiplication" and "exponentiation" diverge, for example, (-2)^2 is well defined and equal to (-2)(-2), but (-2)^(1/2) isn't, unless you relate it to the exponential function. Since the OP was about a notation proposal for working with real numbers, in that specific context I believe the more natural interpretation is to relate everything to the complex exponential and work your way up from there.

jessriedel 6y ago
> the kind of exponentiation you are referring to is related to the intuitive concept of "do this N times", which only makes sense for positive integers. But it works when "this" is an action that does not have a sensible notion of being applied a fractional number of times, or an infinite number of times, which is a very large and important set of actions indeed. > When you are talking about real numbers, the notion of "repeated multiplication" and "exponentiation" diverge Yes of course, but mere continuity of the inputs doesn't pick out exp from any other base you might choose besides e. You do not have to relate it to the exp function. > Since the OP was about a notation propos... My specific objection was to your suggestion that a^b is syntactic sugar, which suggests notational convenience that does not reflect what's going on "under the hood".
- qsort 6y ago
  > My specific objection was to your suggestion that a^b is syntactic sugar Yes, I agree that 'syntactic sugar' is not the word we are looking for here, I have no objections to your comment.