On 05 Oct 2011, at 17:33, benjayk wrote:
meekerdb wrote:
On 10/4/2011 1:44 PM, benjayk wrote:
Bruno Marchal wrote:
Bruno Marchal wrote:
But then one 3-thing remains uncomputable, and undefined,
namely the very foundation of computations. We can define
computations in
terms of numbers relations, and we can define number relations
in
terms of
+,*,N. But what is N? It is 0 and all it's successors. But
what is
0? What
are successors? They have to remain undefined. If we define 0
as a
natural
number, natural number remains undefined. If we define 0 as
having
no
successor, successor remains undefined.
All theories are build on unprovable axioms. Just all theories.
Most scientific theories assumes the numbers, also.
But this makes not them undefinable. 0 can be defined as the
least
natural numbers, and in all models this defines it precisely.
But natural *numbers* just make sense relative to 0 and it's
successors,
because just these are the *numbers*. If you define 0 in terms of
natural
numbers, and "least" (which just makes sense relative to
numbers), you
defined them from something undefined.
So I ask you: What are natural numbers without presupposing 0
and its
successors?
This is a bit a technical question, which involves logic. With
enough
logic, 0 and s can be defined from the laws of addition and
multiplication. It is not really easy.
It is not technical at all. If you can't even explain to me what the
fundamental object of your theory is, your whole theory is
meaningless to
me.
I'd be very interested in you attempt to explain addition and
multplication
without using numbers, though.
It's easy. It's the way you explain it to children: Take those red
blocks over there and
ad them to the green blocks in this box. That's addition. Now
make all
possible
different pairs of one green block and one red block. That's
multiplication.
OK. We don't have to use numbers per se, but notions of more and
less of
something.
Anyway, we get the same problem in explaining what addition and
multiplication are in the absence of any concrete thing of which
there can
be more or less, or measurements that can be compared in terms of
more and
less.
meekerdb wrote:
Bruno Marchal wrote:
But to get the comp point, you don't need to decide what numbers
are,
you need only to agree with or just assume some principle, like 0
is
not a successor of any natural numbers, if x ≠ y then s(x) ≠
s(y),
things like that.
I agree that it is sometimes useful to assume this principle, just
as it
sometimes useful to assume that Harry Potter uses a wand. Just
because we
can usefully assume some things in some contexts, do not make them
universal
truth.
So if you want it this way, 1+1=2 is not always true, because
there might
be
other definition of natural numbers, were 1+1=&.
It's always "true" in Platonia, where "true" just means satisfying
the
axioms. In real
life it's not always true because of things like: This business is so
small we just have
one owner and one employee and 1+1=1.
Yeah, but it remains to be shown that platonia is more than just an
idea.
Physical reality is an an idea too. But as a primitive ontological
reality, it cannot even explain the belief in the physical fact by
machine. It needs a notion of body-observer which incarnate actual
infinities.
I
haven't yet seen any evidence of that.
Bruno seems to justify that by reductio ad absurdum of 1+1=2 being
dependent
on ourselves, so 1+1=2 has to be true objectively in Platonia. I
don't buy
that argument. If our mind (or an equivalent mind, say of another
species
with the same intellectual capbilites) isn't there isn't even any
meaning to
1+1=2, because there is no way to interpret the meaning in it.
This contradicts your agreement that "1+1=2" is a feature of God in a
preceding post.
Bruno
It only seems
to us to be true independently because we defined it without explicit
reference to anything outside of it. But this doesn't prove that it
is true
independently anymore than the fact that Harry Potter doesn't
mention he is
just a creation of the mind makes him exist independently of us
eternally in
Harry-Potter-land.
benjayk
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