Re: Mandelbrot and Julia Sets Explored by RareEvent Theory
From Osher Doctorow
Morgan and Russell, Thanks for some interesting ideas. We may be coming from at least partly opposite directions, but that's fine. I'm 64 years old and missed the computer boat so to speak, although I can read many programs (somewhat like saying somebody can read some French but it isn't their native language or even a specialty or hobby). 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. 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. So there are no difficulties as long as one just adds and subtracts complex numbers - then there's a pretty clear correspondence to the real numbers and one could use either a complex or two real programs or a combination of two real programs. When multiplying two complex numbers, if you did manage to somehow incorporate complex multiplication into real programs you'd really be extending the real programs to complex programs because it's a known fact that complex multiplication is NOT APPLICABLE TO THE REALS EXCEPT WHEN THE IMAGINARY PARTS OF BOTH NUMBERS MULTIPLIED ARE 0. So what is surprising about Rare Event Theory (RET) insofar as it relates to complex numbers is that it seems to be able to arrive at similar results to complex numbers INCLUDING complex multiplication - but not identical results in general. In other words, RET has potential for exploring things and arriving at results that may be too complicated at present with complex multiplication - but the results may for example have an inequality in the wrong direction or something like that. RET also is useful to establish CAUSES and EXPLANATIONS which may not be so clear using complex multiplication or even complex numbers, because RET is a literally causal language (one of its forms or representations is in fuzzy multivalued logics in fact, where "a implies b" literally corresponds to 1 + b - a, which you will recognize as 1 + y - x or p1(x, y) with a, b being fuzzy multivalued real number values which are attached to the propositions. The probability- statistics meaning or translation or representation of 1 + y - x is literally: probability that x causes y. Here y < = x (is less than or equal to x). The number 1 + y - x is always between 0 and 1 when x and y are betwen 0 and 1. Thanks also to Guy for starting the comments, and let me know on fractint if you see any further ways to get some of this into computer program form or to even generalize or modify or specialize anything. Osher Doctorow
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))"
participants (2)
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MARLENE DOCTOROW -
Morgan L. Owens