4 ms·

The usage of "1 to 1" in "`map` is a 1 to 1 operation … meaning one input produces one output" is at odds with the usual mathematical usage of the term; I'm not

by JadeNB 7y ago

The usage of "1 to 1" in "`map` is a 1 to 1 operation … meaning one input produces one output" is at odds with the usual mathematical usage of the term; I'm not sure how it accords with CS usage. In fact, it seems to be referring to the `f` in `map f`, not to `map` itself. In mathematics, we say that an `f` that produces one output for one input is a function. (More generally one might consider non-function objects `f` called relations, often written infix as `x f y`; for example, we could have the relation `f = lt`, for which, say, both `1 lt 2` and `1 lt 3` hold—one input with two (or more) outputs.) If we say further of a function (or more generally of a relation) that it is one to one, we mean that different outputs produce different outputs. For example, properly speaking, one should not speak of the square-root function, since there are two real square roots of every positive real number. (What we usually call the square-root function is really the non-negative square-root function on non-negative reals.) There is a perfectly good squaring function, since every real number has a unique square, but it's not 1 to 1 on the domain of all real numbers, because different real numbers can have the same square.

deleted 7y ago
[deleted]
mrkeen 7y ago
No, it refers to map. Take a stream and count its elements: System.out.println( Stream.of(1,2,3).count()); Now map a function onto that stream, and count its elements. System.out.println( Stream.of(1,2,3).map(f).count()); For every 1 element counted in the first example, you get 1 element counted in the second example - regardless of f.
- JadeNB 7y ago
  Certainly the author is correct in the sense of "1 to 1" that just means "function" (`map` is a function, in the sense that `map f = map g` whenever `f = g`); and even in the sense of "1 to 1" that means "injective" (if `map f = map g`, then `f = g`, at least extensionally), so there's no formal sense in which the author is wrong; but I still think that the sense in which the author is right is not the intended sense. Namely, consider, to take a silly example, a function like `f x = [x, x^2 + 3]`. (I'll use Haskell notation throughout, since I'm comfortable with it.) In the formal sense this is still a function, since it produces one output, which happens to be a length-2 list, for each input; but one may also informally regard it as a one-to-two relation, in the sense that `x` is left-related to `x` but also to `x^2 + 3`. OK, so still it is true that `length $ map f [1, 2, 3] == 3`; but there's also a sense in which `map f [1, 2, 3]`, which equals `[[1, 4], [2, 7], [3, 12]]`, wants to have length 6. This is so in the sense that `concatMap f [1, 2, 3]`, which equals `[1, 4, 2, 7, 3, 12]`, does have length 6. Thus, although, again, there is no technical sense in which it is wrong to say what the author does, nonetheless, in the informal and intuitive sense, I think it is better to say that the "1-to-1-ness" (which I would prefer to call "function-ness") of `map f` is a property of `f`, or at least of how we choose to use `f`, not of `map` itself. But, again, I emphasise that I think this; though I have made a case for it, a big chunk of that case is personal preference, and so cannot, I think, be regarded as conclusively right or conclusively wrong—although I think it makes it easier to communicate with mathematicians, for whatever that's worth.

by JadeNB 7y ago

deleted 7y ago

[deleted]

mrkeen 7y ago

No, it refers to map. Take a stream and count its elements: System.out.println( Stream.of(1,2,3).count()); Now map a function onto that stream, and count its elements. System.out.println( Stream.of(1,2,3).map(f).count()); For every 1 element counted in the first example, you get 1 element counted in the second example - regardless of f.

JadeNB 7y ago
Certainly the author is correct in the sense of "1 to 1" that just means "function" (`map` is a function, in the sense that `map f = map g` whenever `f = g`); and even in the sense of "1 to 1" that means "injective" (if `map f = map g`, then `f = g`, at least extensionally), so there's no formal sense in which the author is wrong; but I still think that the sense in which the author is right is not the intended sense. Namely, consider, to take a silly example, a function like `f x = [x, x^2 + 3]`. (I'll use Haskell notation throughout, since I'm comfortable with it.) In the formal sense this is still a function, since it produces one output, which happens to be a length-2 list, for each input; but one may also informally regard it as a one-to-two relation, in the sense that `x` is left-related to `x` but also to `x^2 + 3`. OK, so still it is true that `length $ map f [1, 2, 3] == 3`; but there's also a sense in which `map f [1, 2, 3]`, which equals `[[1, 4], [2, 7], [3, 12]]`, wants to have length 6. This is so in the sense that `concatMap f [1, 2, 3]`, which equals `[1, 4, 2, 7, 3, 12]`, does have length 6. Thus, although, again, there is no technical sense in which it is wrong to say what the author does, nonetheless, in the informal and intuitive sense, I think it is better to say that the "1-to-1-ness" (which I would prefer to call "function-ness") of `map f` is a property of `f`, or at least of how we choose to use `f`, not of `map` itself. But, again, I emphasise that I think this; though I have made a case for it, a big chunk of that case is personal preference, and so cannot, I think, be regarded as conclusively right or conclusively wrong—although I think it makes it easier to communicate with mathematicians, for whatever that's worth.