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Old 10-11-2008, 05:20 AM   #127 (permalink)
ygolo
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OK. So you arent saying "Everything is computing?"

There are two main problems with your argument you presented.

1) "Everything is made up of functions" is vague. How, for instance, are quarks "made up of functions?"

2) It is Mathematically provable that there exist mathematically well-defined functions that are not computable.

Here is the proof:

Take any real number a, define f_a(x)=a, for all real-numbers x. f_a is a well defined mathematical function, no matter what the selection of a. This means, since there are uncountably many real numbers, there are uncountably many well-defined functions.

However, by the Church-Turing Thesis each computable function is computed by some Turing machine. A Turing machine can be represented as a finite string of symbols over a finite alphabet of symbols. The set of all finite strings over a finite alphabet is countable, so the set of all Turing machines is countable, which means the set of all computable functions is countable.

This means there exists uncountably many uncomputable well-defined functions.

What if one of those well-defined functions is a function describing the human reasoning process?

A pithy way to summarize the this view-point is to say:
The world (and the mind) are analog, while computation (along with the deductive processes that spawn it) is digital.

If you are going to change the what you are calling "computing" from something equivalent to a Turing machine of some-sort, to something else, that needs to be made clear.

Because your argument is not valid based on what the standard definition of computing is.

(as an interesting side note, look up the Busy Beaver Function as a specific example of an uncomputable but well-defined function, its cool).
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