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.

Stephen Hawking RIP

1 day ago

Regarding our active internal analog math...

ReplyDeleteI'm a civil/environmental engineer by education but I've been working off and on on a theory which takes the tact that all abstract math symbols and expressions are secondary and arise from a handful of internal analog "math" artifacts and processes. This may not be a very polite thing to say to a mathematician, but I am wondering if you have impressions along the same line?

It turns out that we all get energy to think and do math and other things from the respiration reaction (organics + oxygen -> water + carbon dioxide +energy). And basically, what that means, if you remember your biology or organic chemistry, is, body-wide, within our cells is a ~steady creative flow of about 10^20 water molecules per second -- coming from the 160 kg of O2 we each respire each year. Generally, each water molecule is sort of tetrahedral in shape with two positive and two negative vertices and so, it turns out that there are at least six ways each water molecule can orient within an enfolding field when it first comes into being at a respiration site. That also means that a chain of n-molecules can form in 6^n different ways. Thus a sequence of 12 molecules could form in 6^12, or about 2 billion different ways. A chain of eighteen molecules could associate with 6^18 or 10^14 different impressions. Now, in this analog math theory, I am assuming that repeating vibrations in the environment ought to result in formation of similar stacks and chains of structurally coded water molecules being formed. This gets us a rather crude image of the vibrations of our internal and external environment forming an internal echo or representation within this active internal analog "math", or "language".

I say it's active because the 6^n stacks of water molecules are really also structurally coded hydrogen-bonding packets and such things, when they unfurl, are connected with and influential in protein-formation and protein-folding, which is to say, memory formation and muscle movement, which is to say, in our case, ALL human expression, perhaps beginning with our nearly universal actions and impressions of counting each of our ten fingers and ten toes, and the like.

Bizarre stuff, huh? Lots of little internal Turin devices writing out structural coded signals.

I'm wondering if mathematicians are taught this type of internal analog math as the basis of the abstract math symbols and expressions, or if they are given different associations or impressions, perhaps leaving it that there is just an uncanny (and unknown) relationship between much or all of nature and math?

Also, I vaguely see the similarity between 2^n binary or boolean math and the 6^n "multiple-state structural coding" that I've made up or stumbled onto. I expect the trend continues with starting with other polyhedra which have limited orientations "within enfolding fields" -- when a containing structure is added. My general hunch is the initial condition IS actually significant for us and we can immediately get to multiple states (relevant to ~quantum mechanics/quantum gravity) by starting with tetrahedron and adding the enfolding cube container, rather than the way it's done presently of beginning with the xyz-cubic framework and adding variants.

Initial conditions do matter in mathematics, don't they?

Best regards,

Ralph Frost

@frostscientific

http://magtet.com/images/phpshow.php

I wonder if you have seen, are influenced by "Water, the mystery", the documentary movie. -

ReplyDeletehttp://topdocumentaryfilms.com/water-great-mystery/

( More, later. )