On 05 May 2014, at 14:27, Craig Weinberg wrote:
On Saturday, May 3, 2014 3:53:48 PM UTC-4, Bruno Marchal wrote:
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.
Only because you have an a priori expectation of sequence which can
be inferred. Otherwise nothing is defined and you have only
unrelated statements. You need sense to draw them together and match
your intuition.
No. Logic is the art of making derivation without sense. That is even
why so many people think that a machine which can reason is just doing
syntactical manipulation without understanding, and at the low level,
that's correct.
A derivation, in a formal theory, is valid or non valid, independently
of any of its possible interpretation (all those terms are well
defined).
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.
But "eventually" means that you must follow a sequence of steps to
do your defining. You smuggle the expectation for sequence in from
the start.
Hmm, ... I will not insist here, as this will be the object to the
next post in the math thread.
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.
"Derive" requires sequence and sense.
Not at all.
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.
The theory is that logic and arithmetic are particular continuations
of sense, not the other way around.
Sense is a vague term. Not two human being understand it in the same
way. It is a bit like God. Important notion, but hardly usable in
theories.
Before arithmetic can exist, there must exist a sense of expectation
for counting. Counting includes a sense of recursive steps as well
as sequence, comparison, memory, change, digits, etc. It cannot be
primitive as it is a manipulation of attention.
Not at all. More in the math thread, but you might need to reread all
posts.
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.
There are other lamps...other keys.
Yes, that's the point.
Bruno
Craig
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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