Marty's Singularity Blog
The Singularity is happening now!
October 27, 2010 at 6:00 pm
filed under General, Singularity
Tagged Singularity
The previous post was about the law of diminishing returns from the field of economics. We saw that its exact definition contains the stipulation that all inputs except one are constrained, and we saw how it has been distorted into the idea that increased application of effort and skill leads to diminishing returns.
Actually the world does not work like this at all. Processes in the real world are not constrained so that only one input can be increased. Imagine you are the farmer from the previous post and you increase the amount of land you are farming and the number of workers and you study and experiment to find the proper amount of fertilizer. Your output would explode.
Time for a little mathematics. Two types of progressions are arithmetic and geometric. An arithmetic progression is one in which that same number is added to each succeeding term. For example, 1, 3, 5, 7, 9, …. The number 2 is added to each term and we have the progression of odd numbers. In a geometric progression each term is multiplied by the same number. For example, 1, 2, 4, 8, 16, …. Here each term is doubled or multiplied by 2. At the low end there seems to be little difference in the two progressions. But consider this: the tenth term in the arithmetic progression would be 21 while the tenth term in the geometric progression would be 512. The eleventh terms would be 23 and 1024. As you can see, the differences are large and getting larger.
Geometric progressions are a subset of a class of mathematical functions called exponential functions. These functions all involve exponents. For example, our doubling geometric progression above could be written as
y = 2(n-1) where n is an integer which is the number of the term in the progression.
Exponential functions in general are not limited to integers so the general form is
y = ex where e is the base of the natural logarithms, approximately 2.718.
If x is positive then the exponential function is an increasing one; if x is negative a decreasing one. One thing that all increasing exponential functions share is that as x increases the value of y tends to explode.
How does this relate to the Singularity? If we look at history, as Ray Kurzweil has in his book The Singularity Is Near, we can see that science and technology have followed an exponential path, especially over the last 300 years or so since Isaac Newton, and even more so over the last hundred years or so since Albert Einstein. Not only are the levels of science and technology advancing exponentially, but the rate at which they are advancing is also increasing exponentially. Like all increasing exponential functions this has led to an explosion. Indeed, explosion is a very apt term for what happened in technology in the 20th Century: radio, television, nuclear power, aviation, space travel, computers, integrated circuits, to name just a few. As we will see in the next post, the continuing explosion promises to be even more profound.
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Mathematics Revision
on December 7, 2010 at 8:25 pm
Types of Sequences…
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