On Thu, Jan 12, 2012 Bruno Marchal <marc...@ulb.ac.be> wrote:
> I am not entirely sure what you mean by computable numbers (I guess you
> mean function).
A computable number is a number that can be approximated by a computable
function, and a computable function is a function that can be evaluated
with a mechanical device given unlimited time and storage space. Turing's
famous 1936 paper where among other things he introduced the idea of what
we now call a "Turing Machine" was called:
"On Computable Numbers, with an Application to the Entscheidungsproblem".
Turing showed that a very few real numbers, like the integers and the
rational numbers, have formulas to calculate their value as closely as
you'd like, but for the vast majority of numbers there is no way to do
this. There are a few more numbers like PI that are computable with
algorithms like PI= (4/1)-(4/3)+(4/5)-(4/7)+(4/9).... , but for most
numbers there is nothing like that and no way to approximate their value.
In fact he showed that almost all the numbers on the real number line are
non-computable. There are LITERALLY infinitely more non-computable numbers
than there are computable numbers; Turing proved that these numbers exist
but ironically, despite their ubiquitous nature, neither Turing nor anybody
else can unambiguously point to a single one of these numbers because there
is no way to derive such a number from the numbers that we can point to,
the computable numbers.
So numbers, at least the numbers we or computers can use, cannot be the
only fundamental thing, non-computable numbers must be too. My point was
that if there are 2 general classes of fundamental things that can not be
simplified then there might be more. I think the intelligence-consciousness
link is a third fundamental thing, but unlike Turing I can not prove it.
And there may be fundamental things that we can never prove are
fundamental, truth and proof are not the same thing.
> > We can even ascribe it [consciousness] a role (explaining its Darwinian
There is no way consciousness can have a direct Darwinian advantage so it
must be a byproduct of something that does have that virtue, and the
obvious candidate is intelligence.
> like relative universal self-speeding.
I don't know what that means.
> I suggest that the quantum nature of the observable reality might reflect
> the discovery that we are in that 'digital matrix'.
I don't know if that's true or not, but I do know that if I get too close
to even the most beautiful and detailed picture on my computer screen I
start to see individual pixels; and sometimes late at night I speculate
that somebody made a programing mistake and tried to divide by zero at the
singularity in the center of a Black Hole.
> I think that here you miss the UDA point.
That is entirely possible because I am unable to follow what you call your
dovetailing argument; I really don't think you have stated it as clearly as
John K Clark
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