I said:
>> An exponential curve like (n^X) is not a polynomial. If n is only slightly greater >> than 1 then at small values of X it does not even look like a curve, it looks >> almost like a straight horizontal line, at medium values of X it looks almost like >> a geometric curve, at large values of X it looks almost like a straight vertical line.
Hal said:
>think what Emlyn meant was that this is all depending on your scale factors.
>By appropriately zooming and scaling you can make each portion of
>an exponential curve look like every other portion.
If you magnify enough you can make any continuos curve look like a flat line. Perhaps the speed of intelligence could be thought of as magnification, if so then intelligence need not ever experience a singularity because as events speed up subjective time slows down and the singularity like a horizon would keep receding into the distance.
>There is no intrinsic difference between the horizontal and vertical-looking
>parts of the curve.
Well, in my example the different parts of the function would certainly have a different slope and a different derivative, that seems intrinsic to me. One particular exponential function out of an infinite number do what you want , the function y = e^x because the rate of change with respect to the x direction is the function itself.
>No, by your definitions Moore's Law would be exponential. What you
>are calling geometric is a polynomial growth curve, whose doubling time
>slows down. Exponential growth has a constant doubling time; Moore's
>law says that density doubles every 18 months (or 12 or 24 months by
>some definitions).
I have no definition of Moore's Law, only the shape of the curve describing the number of transistors that can be put on a chip over time is important. I would maintain that we don't know the shape of that curve for a long enough time and with sufficient precision to know if it's polynomial or exponential. And even it we did we wouldn't know if it would continue.
> Of course this kind of curve fitting does not have strong predictive value
I agree. I can see no physical reason that Nanotechnology won't be developed someday and I think that increasing intelligence will increase the rate of change of events exponentially, but I'm skeptical that we can figure out when this will happen by looking at existing curves and extrapolating.
John K Clark jonkc@att.net