Re: Proposal: 6NF

On Oct 24, 9:04 am, "Aloha Kakuikanu" <aloha.kakuik..._at_yahoo.com> wrote:
>
> with diagonals as equivalence classes...

And, furthermore, an adaptation of Euclid's algorithm that we could call, with perfect accuracy and more than a trace of humor, Greatest Common Subtrahend, that we could use to normalize the (nat, nat) pairs such that at least one of the nats was zero, in *exactly* the same way as we use Greatest Common Divisor to normalize the (int, int) pairs that make a rational to be relatively prime. (In fact, if we're feeling particularly ambitious, we can define a *single* normalizing algorithm and parameterize it with either the Predecessor relation or the Subtraction relation as the inverses of the iterated {successor, sum} that constructed the sets in the first place.)

But even if we do this, woo hoo. The constructionist viewpoint is not the only viewpoint, nor even a definition of the number in question. It's just that: a construction. We're not bound by it in deciding identity.

Tossing out the reals for the much-easier-to-handle rationals, (which are identically useful for the purposes of arguing the construction argument,) we have to ask, if we assume the naturals (and the successor function) and construct the integers from that, such that there is an integer two constructed as (2,0), (or, more descriptively, "(2-0)"), and then construct the rationals from pairs of integers, such that we have a two in the form of (2,1) or more descriptively (2/1), vs. alternatively assume the integers and construct the rationals from the integers such that we have a rational (2/1) built out of assumed integers instead of constructed-from-naturals integers, who gives a damn anyway? (I *dare* you to diagram that sentence.) There is no meaningful difference between the two; they are alternate, equivalent constructions. Do we now have, like, nine different kinds of two? No. There's Only One Two, which is exactly the slogan of a local TV news channel from years ago.

If we're going to try to say that these various twos are all different in anything *except* their construction, then I'd say we're in a heap o' trouble, hoss. If you define various rational arithmetic functions + - *, they're all going to give the same answers for calculations done with two either way, so what was the point again? And when we come to the case of divide, they *won't* produce the same answer, which shows that they weren't the same function in the first place, so we have different functions and that way we get the same answer again anyway.

The argument that alternative constructions of two make for different numbers (or even that there are unqiue constructions) is bogus.

Marshall Received on Wed Oct 25 2006 - 07:45:28 CEST