Before digital computing took over completely, analog computing was dominant for a short while. An analog computer is based on the creation of a model which represents the problem to be solved. But mathematical models of problems can be created of ( almost ) any problem and these models can be implemented on a digital computer. A digital computer is nothing more than a convenient, fast, Turing Machine or equivalent thereof, i.e. the URM or Abacus. And from Mathematical Logic ( Goedel ) we know that these systems have its limitations. It is theoretically impossible to create a program that solves all mathematical problems. - But physicists and biologists say ( and why should we disagree? ) that we -are- computer ( brain ) controlled machines.
Is that a paradox? Humans can do more than computers, we can solve mathematical problems, in fact we -created- the concept of a 'Turing Machine'. This leads us to Roger Penrose. In The Emperor's New Mind, 1999 he claims that artificial intelligence in computers is impossible. He argued that the human brain must exploit a type of physics that he described as 'non-computable'. By this he means beyond algorithmic computing, and thus digital computing.
A picture that keeps fascinating me is that of a predator bird flying high over its prey before, at a carefully -chosen- moment, it makes the dive and following kill. And this is all done with a tiny bird brain. The best comparable thing made by humans thus far is the drone. A huge flying case loaded with bombs operated by a battery of digital computers assisted by human -computers-. Although humans have created a model of a flying bird, it is operated by a human computer on the ground.
Analog computers were special purpose computers, designed to solve one specific problem. A predator bird will never be able to learn new behavior, it cannot be trained to live with chickens. Not immediately anyaway, if ´evolution´ made the bird.
Let me summarize before this turns into a rant.
- There are other models of computing than the Turing machine, i.e. analog computing, brain computing.
- Digital computing is superior over analog computing, brain computing is superior over digital computing.
- Analog and digital computing are human creations we fully understand.
- We don't understand brain computing (yet?).
- Mathematical logic and computability theory study algorithmic ( digital ) computing.
Goedels theorems are somewhat like Russell's paradox in set theory. Goedel's incompleteness theorems are statements about logic and number theory deduced in and with the rules of logic.
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