From: hal@finney.org
Date: Wed Oct 13 1999 - 13:35:06 MDT
John Clark, <jonkc@worldnet.att.net>, writes:
> O'Regan, Emlyn <Emlyn.ORegan@actew.com.au> Wrote:
>
> >Reads a bit like nonsense I'm afraid. There is no flatter bit of an
> >exponential curve; it looks the same forward and back (I'm taking some
> >liberties), no matter where you stand on the curve. Multiplying by a
> >constant amount per constant time period (like multiplying transistors per
> >square inch on an IC by 2 every 18 months), says that the rate of change is
> >constant.
>
> That's a geometric curve not exponential, it can be described by a polynomial
> like (X^n) where n is a constant and X a variable.
> 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.
I 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. There is no intrinsic
difference between the horizontal and vertical-looking parts of the curve.
(Also, in my experience the term "geometric growth" is usually used
synonymously with "exponential growth".)
> Moore's law looks geometric (but we can't be certain) not exponential so it's
> spectacular but we can deal with it.
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). (BTW there was an article yesterday about Intel
predicting the upcoming failure of Moore's law.)
Exponential growth is not inherently difficult to deal with, as it always
looks the same in certain respects. There will be qualitative changes
over time but the doubling time is constant.
For exponential growth, the rate of growth must be proportional to the
growing entity itself. For Moore's law, the rate of improvement to
circuits must be proportional to the miniaturization of the available
circuits. Exponential growth is a state of constant positive feedback.
Each improvement must lead to proportionately increased speed. You are
in a sense running (growing) as fast as you can to stay in the same
place (constant doubling time). This is one reason I am skeptical of
claims that uploading or AI will lead to a singularity simply by virtue
of positive feedback. We already have positive feedback; positive
feedback leads to exponential growth, which we are already experienced
in dealing with. That's not enough for a singularity.
The real challenge is not exponential growth, but hyperbolic growth.
Hyperbolic growth has a true singularity, a point at which the curve
goes to infinity. The doubling time becomes smaller and smaller and
everything explodes.
To get hyperbolic growth, you need the rate of change to be growing
faster than the underlying technology, for example it could be growing
as the square of the technology level. As one way this could happen,
I once speculated that rate of technology/population/economic growth
should be equal to productivity times population. And if you then say
that productivity is proportional to population (because the more people
you have working on problems, the faster you will solve them) you get
that the rate of change is proportional to the square of the population.
This is the recipe for hyperbolic growth and leads to a singularity.
In fact, curve fitting for population over very long periods of time does
reveal hyperbolic growth leading to a population singularity around 2040.
(However we may have departed somewhat from this in recent years.)
Robin Hanson did an analysis of long term economic growth at
http://hanson.gmu.edu/longgrow.html, and he found that economic growth
has not quite been hyperbolic, but rather was a series of exponential
growth periods with successively faster doubling times. For several
decades we've had a doubling time for world output of about 15 years.
Extrapolating forward, it appears likely that we will move to a new and
faster doubling time of perhaps as little as 1-2 years, some time in the next
few decades. If we try to look beyond that, just a few years later the
doubling time becomes on the order of weeks, and pretty soon we are at
a singularity again.
Of course this kind of curve fitting does not have strong predictive
value, but IMO it adds credibility to projections based on expected
technology improvements like nanotech and AI. The rate of growth has
been greater than exponential over the long sweep of human history,
and there is no particular reason to expect this to change given the
prospects before us. In that case the Singularity may be more than
just a technological horizon where cumulative changes make it hard to
see the future from our perspective; it may represent a specific point
in time where things change in a fundamental and irreversible way.
Hal
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