OSHER DOCTOROW wrote:
Curiously enough, I went into the super-abstract mathematics direction outside computers and only fairly recently have been coming back to the more concrete reality and some of it involves computers.
I started out in mathematics.
One of the difficulties in reducing computer programs to their real and imaginary components also occurs in non-computer mathematics and quantitative sciences, where one comes up against the fact that MULTIPLICATION of complex numbers doesn't decompose (doesn't correspond to any operation in the reals for Re(z) and Im(z)) according to any known theory accepted by the mainstream.
I don't see why this is a difficulty; is (a,b)(c,d)=(ac-bd,ad+bc) any more difficult than a mortgage repayment plan? I'd say it's quite a bit simpler - and a lot easier to implement if that's a concern. And doesn't that count as a "decomposition"? Nor do I see why it's an issue; why _should_ complex numbers or the entities of any other algebraic structure be expected to "decompose" into any operation in reals in whatever sense? What you describe sounds as though you expect "addition" in one space to act like "addition" in another, or that multiplication in one should the same thing as multiplication in the other. I'm afraid that idea of "permanence of form" went out in the middle of the nineteenth century, due to work by Boole, Hamilton, and others, along with the idea that mathematical entities had to necessarily correspond to anything in the real world. It's been a while since Euclidean geometry, for example, was something considered _a priori_ True. Where's the difficulty, for example, in a system where x+x = x*x = x for all x? Maybe we are approaching from opposite sides; at least, I don't get where you're coming from. Morgan L. Owens "(int)abs(sin(2*n*pi))"