On 27 Apr 2013, at 11:40, Stathis Papaioannou wrote:

On Tue, Apr 23, 2013 at 3:14 AM, Craig Weinberg <whatsons...@gmail.com> wrote:
A quote from someone on Facebook. Any comments?

"Computers can only do computations for rational numbers, not for real numbers. Every number in a computer is represented as rational. No computer can represent pi or any other real number... So even when consciousness can
be explained by computations, no computer can actually simulate it."

If it is true that you need real numbers to simulate a brain then
since real numbers are not computable the brain is not computable, and
hence consciousness is not necessarily computable (although it may
still be contingently computable). But what evidence is there that
real numbers are needed to simulate the brain?


Actually there exist notions of computable real numbers, and computable function from R to R.

For example the function y = sin(2*PI* x) is "intuitively" computable, as you can approximate as precisely as you want the input (2 * PI * i) and the corresponding output sin (2 * PI * x).

But there is no Church thesis for such notion, and there are many non equivalent definition of computability on the reals.
(I could add some nuance, here, but that's for later perhaps).

Yet, all analog machines known today, are emulable by digital machines. There would be a problem only if some real number is non computable and used in extenso by some machine. That exists ... mathematically. Some computable function of the reals can have their derivative being non computable. But in those case, the recursion theory is the same as for Turing machine with oracle, and this does not change the logic and the conceptual consequences. Nor is there any evidence that a brain uses such oracle, although it can be said that evolution uses the halting oracle, by selecting out the stopping machines (death). But that is just long term behavior of machines. It does not make us locally non emulable by computer. We already do ourself that selection for computers by buying new one, and throwing out old one ...

Bruno





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Stathis Papaioannou

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http://iridia.ulb.ac.be/~marchal/



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