On 02 May 2014, at 23:58, Craig Weinberg wrote:
On Friday, May 2, 2014 11:15:40 AM UTC-4, Bruno Marchal wrote:
On 01 May 2014, at 20:42, Craig Weinberg wrote:
On Friday, April 18, 2014 3:23:13 AM UTC-4, Bruno Marchal wrote:
On 16 Apr 2014, at 20:10, Craig Weinberg wrote:
What generates Platonia?
Nothing generates Platonia, although addition and multiplication
can generate the comp-relevant part of platonia, that is the UD or
equivalent.
Elementary arithmetic cannot be justified by anything less complex
(in Turing or logical sense). It is the minimum that we have to
assume to start.
Saying that elementary arithmetic is the minimum that we have to
start doesn't make sense to me. Elementary arithmetic depends on
many less complex expectations of sequence, identity, position,
motivation, etc. I keep repeating this but I don't think that you
are willing to consider it scientifically.
To define, is a reasonable precise sense, "expectations",
"sequence", "identity", "position", or "motivation" (which I doubt
is a simple notion) you need arithmetic.
How can arithmetic exist without sequence and then define sequence?
If you agree on logic and
0 ≠ s(x)
s(x) = s(y) -> x = y
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x
Then you can study how to define sequence in that theory. Gödel is the
fist who did that. He invented the "Gödel beta function", based on a
generalization of a famous chinese "lemma", about set of modular
equations in arithmetic.
Eventually (not easy exercice) you can define from the axiom and the
chine lemma a representation of the exponential function, and with its
you can define a sequence in arithmetic by using the unique
factorization of the natural numbers.
It is not the existence of arithmetic, it is the existence of 0, s(0),
etc. + the basic relation that you can derive from the axioms.
It is the same capacity to reason which tells me that 5-3=2 which
tells me that sequence can exist without arithmetic but arithmetic
cannot exist without sequence.
It is a bit imprecise. I can define sequence in *any* turing complete
language, and they are all equivalent for computationalism.
You can define a notion of sequence as primitive, instead of numbers,
yes. That is the case for LISP, somehow, which is close to combinators
and lambda calculus.
Yo have never provide any theory, so I can't figure what you talk about.
It is, I think, your unwillingness to study a bit of math and logic
which prevents you from seeing this.
Just the opposite. It is your unwillingness to question the
supremacy of math and logic which prevents you from even seeing that
there is something to question.
On the contrary I did ask people to question anything I say, which is
of the type verifiable. That's how science work.
Then it is not a question of supremacy. Only a good lamp to search the
key.
I stop when you attribute to me the contrary on point On which I
insist a lot.
Bruno
You get a lot about the numbers with few axioms written in first
order language.
I don't see why any axioms would be possible. Where do they come
from? Who is writing them?
I doubt you can define "expectation of sequence" in such a simple way.
How can you doubt it?
How will you define "sequence" without mentioning some function from
N (the set of natural numbers) to some set?
With rhythmic patterns and pointing - the way that everyone learns
to count. A horse can understand sequence without a formal
definition derived from set theory. What you are saying sounds to me
like 'you cannot make an apple unless you ask an apple pie how to do
it'.
Again, I remind you that "simple" means "simple in the 3p sharable
sense", not "simple" in the 1p personal experiential sense.
Why is that not an arbitrary bias? If I don't allow the possibility
of 3p without 1p, then simplicity can only be 1p.
All scientists agree on the arithmetic axioms,
If that's true, its an argument from authority, and it could be the
reason why all scientists fail to solve the hard problem. (which is
exactly my argument).
and I have to almost lie to myself to fake me into doubting them.
I can't remember what it was like before I learned arithmetic, but I
can still understand that we all live for years without those
notions. There is at least one culture today that has no arithmetic.
Something like "expectation" might already have a different meaning
for spiders, for different humans, etc.
Either way, it is undeniably more primitive than arithmetic in my
view.
Craig
Bruno
http://iridia.ulb.ac.be/~marchal/
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