Re: The naive test for equality

David Cressey wrote:
...
> In order to manipulate this "single element" as data, we need a symbol for
> it, to represent it.

No, we don't need symbols to manipulate data, we need symbols, as humans, to *name* entities. So "one_half" or "0.5" are words from your chosen math dialect used to name the actual number out there. As I said before, the string of characters "123" is meaningless by itself, its menaing, as an element of integers, is realized by mapping the string to an element of the set of integers. It's very much the same as people names. You do not need the name 'John' to manipulate John out there, you can just as well use 'Hey, you' or something else depending on your need.

>
> So we choose one of the elements of the original set to stand as a
> representative of the entire set that is going to be seen as an element. In
> this case we might choose the rational with the lowest denominater, namely
> 1/2.

Right, but the reason you choose "1/2" or '0.5' is just a matter of convenience, not substance.

>
> Now, whenever we are given an unnormalized rational, such as 5/10, we ask
> the rationals engine to normalize it for us.
> The rationals engine knows the rule for normalizing, namely remove common
> factors in the numerator and denominator. So it returns 1/2, the
> normalized equivalent of 5/10.
>
> If we ask the rationals engine to normalize 1/2, it will give us back 1/2.
>
> So the process of normalizing is choosing one, out of an equivalence class,
> according to some criterion, and using the symbol that represents the
> chosen element to act as the normalized form for the entire class.

Now, that's the problem with using the word "representation". It's overloaded with multiple meaning so that often it's impossible to say what is actually meant.

You use representation, I believe, at least in two different senses:

1. A string of characters used to name an actual number. In math, this string is normally called a constant.
2. Internal number implementation by the computer, or internal representation (I assume by 'act' you meant something that the computer did with its internal number implementation).

At a logical level, it does not matter how the number is implemented and naming is a matter of convention and convenience. So instead of trying to import new words into our math vocabulary, why not use the established and simple old ones ? Why not call the string "1/2" a constant instead of something else whose meaning is obscure ambiguous ? Received on Mon Aug 08 2005 - 18:31:47 CEST