Part 28 (1/2)

The rate of change of world knowledge is proportional to the velocity of computation:

(2)[image]

Subst.i.tuting (1) into (2) gives:

(3)[image]

The solution to this is:

(4)[image]

and W W grows exponentially with time (e is the base of the natural logarithms ). grows exponentially with time (e is the base of the natural logarithms ).

The data that I've gathered shows that there is exponential growth in the rate of (exponent for) exponential growth (we doubled computer power every three years early in the twentieth century and every two years in the middle of the century, and are doubling it everyone year now). The exponentially growing power of technology results in exponential growth of the economy. This can be observed going back at least a century. Interestingly, recessions, including the Great Depression, can be modeled as a fairly weak cycle on top of the underlying exponential growth. In each case, the economy ”snaps back” to where it would have been had the recession/depression never existed in the first place. We can see even more rapid exponential growth in specific industries tied to the exponentially growing technologies, such as the computer industry.

If we factor in the exponentially growing resources for computation, we can see the source for the second level of exponential growth.

Once again we have:

(5)[image]

But now we include the fact that the resources deployed for computation, N N, are also growing exponentially:

(6)[image]

The rate of change of world knowledge is now proportional to the product of the velocity of computation and the deployed resources:

(7)[image]

Subst.i.tuting (5) and (6) into (7) we get:

(8)[image]

The solution to this is:

(9)[image]

and world knowledge acc.u.mulates at a double exponential rate.

Now let's consider some real-world data. In chapter 3, I estimated the computational capacity of the human brain, based on the requirements for functional simulation of all brain regions, to be approximately 1016 cps. Simulating the salient nonlinearities in every neuron and interneuronal connection would require a higher level of computing: 10 cps. Simulating the salient nonlinearities in every neuron and interneuronal connection would require a higher level of computing: 1011 neurons times an average 10 neurons times an average 103 connections per neuron (with the calculations taking place primarily in the connections) times 10 connections per neuron (with the calculations taking place primarily in the connections) times 102 transactions per second times 10 transactions per second times 103 calculations per transaction-a total of about 10 calculations per transaction-a total of about 1019 cps. The a.n.a.lysis below a.s.sumes the level for functional simulation (10 cps. The a.n.a.lysis below a.s.sumes the level for functional simulation (1016 cps). cps).