On 10 Oct 2011, at 08:21, Stephen P. King wrote:
On 10/6/2011 12:04 PM, Bruno Marchal wrote:
On 04 Oct 2011, at 21:59, benjayk wrote:
Bruno Marchal wrote:
On 03 Oct 2011, at 21:00, benjayk wrote:
I don't see why.
Concrete objects can be helpful to grasp elementary ideas about
numbers for *some* people, but they might be embarrassing for
others.
Well, we don't need concrete *physical* objects, necessarily, but
concrete
"mental" objects, for example measurement. What do numbers mean
without any
concrete object, or measurement? What does 1+1=2 mean if there
nothing to
measure or count about the object in question?
It means that when you add the successor of zero with itself you
get the successor of one, or the successor of the successor of zero.
Bruno Marchal wrote:
The diophantine equation x^2 = 2y^2 has no solution. That fact does
not seem to me to depend on any concreteness, and I would say that
concreteness is something relative. You seem to admit that naive
materialism might be false, so why would little "concrete" pieces
on
stuff, or time, helps in understanding that no matter what: there
are
no natural numbers, different from 0, capable to satisfy the simple
equation x^2 = 2y^2.
This is just a consequence of using our definitions consistently.
Not really. In this case, we can indeed derived this from our
definitions and axioms, but this is contingent to us. The very idea
of being realist about the additive and multiplicative structure of
numbers, is that such a fact might be true independently of our
cognitive abilities.
We don't know if there is an infinity of twin primes, but we can
still believe that "God" has a definite idea on that question.
That the diophantine equation x^2 = 2y^2 has no solution, is
considered to be a discovery about natural numbers. It is not a
convention, or the result of a vote, nor of a decision. For the
early Pythagoricians that was a secret, and it seems they killed
the one who dare to make that discovery public (at least in some
legend).
Of course
we can say 1+2=3 is 3 just because we defined numbers in the way
that this
is true, without resorting to any concreteness.
Yes. Mathematical realism stems from the intuition that abstract
entities can have theor own life (relations with other abstract or
concrete entities).
My point is that we can't derive something about the fundamental
nature of
things just by adhering to our own definitions of what numbers
are, since
these ultimately are just a bunch of definitions,
You are right. We need some philosophical principles (like comp) to
understand that eventually we don't need those philosophical
principle. In the case of comp, we can understand why some
(relative) numbers will bet on it, and why some other numbers will
not. In fine, it is like with the south american, we can feel them
enough close to us to listen to them.
whereas the "actual" thing
they rely on (what numbers, or 0 and succesor actually are),
remains totally
undefined.
Not with comp. An apple becomes something very complex when defined
in pure number theory. It will involve infinite sets of long
computations, complex group of symmetries, etc. But it is definable
(in principle) from numbers (some including LUM observers).
So whatever we derive from it is just as mysterious as
consciousness, or matter, or whatever else, since the basis is
totally
undefined.
The problem does not consist in finding the ultimate definitions,
but to agree on elementary propositions, and to explain the rest,
of as much as possible from them.
Bruno Marchal wrote:
If it isn't, the whole idea of an abstract machine as an
independent existing entity goes down the drain, and with it the
consequences of COMP.
Yes. But this too me seems senseless. It like saying that we cannot
prove that 17 is really prime, we have just prove that the
fiollowing
line
.................
cannot be broken in equal non trivial parts (the trivial parts
being
the tiny . and the big ................. itself).
But we have no yet verify this for each of the following:
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
.................
etc.
On the contrary: to understand arithmetic, is quasi-equivalent with
the understanding that a statement like 17 is prime, is
independent of
all concrete situation, in which 17 might be represented.
Lol, the funny thing is that in your explantion you used concrete
things,
namely ".".
Is that a problem?
Of course concrete is relative.
I think so.
It's concreteness is not really relevant,
the point is that numbers just apply to countable or measurable
things.
Yes. The natural numbers are somehow the type of the finite
discrete or discernible entities.
Without being countable natural numbers don't even make sense.
In order for COMP to be applicable to reality, reality had to be
countable,
Raaah.... Not really. The big 3-thing *can* be countable, because
from inside it will be non countable. The important reality is not
the big 3-thing seen from outside, because no one can go there. The
"real" reality with comp is epistemological. It is the living ideas
from *inside*.
but it doesn't seem to me to be countable.
Because you are inside. (assuming comp, ...).
Abstract machines might exist, but just as ideas.
The point of platonism is that ideas, despite being epistemological
does exist, and are somehow more real than the big intellectual
construction, which in fine is shown to not really matter, and can
be very simple.
Show that they exist
beyond that, and then the further reasoning can be taken more
seriously. If
numbers, and abstract machines exist just as ideas, everything
derived from
them will be further ideas. You can't unambigously conclude from
some idea
something about reality.
Reality is an idea itself.
[SPK]
Whose "idea" exactly?
LUN's idea. (an idea occurring in the mind of the Löbian universal
numbers in virtue of their complex relations with infinities of other
numbers).
And that idea might be correct. "Idea" is not pejorative for an
objective rational immaterialist.
If there is no one to whom Reality has a meaning does it have a
meaning? No.
There are many one, and even more.
You seem to assume that meaningfulness exist in the absence of a
subject to whom that meaning obtains. That is a contradiction.
Arithmetical reality is full life and full of subjects. (even without
comp, i.e. without us being there).
Bruno Marchal wrote:
1, 2, 3,... make only sense in terms of one of something, two of
something,... OK, we could say it makes sense to have one of
nothing, two of
nothing, etc, but in this case numbers are superfluous, and all
numbers, and
all computations are equivalent.
I think that 0, 1, 2, and many others are far more simple
conceptually
than any something you can multiply them by.
No. Otherwise we would understand 0, 1, 2, before we understood
"one of
something", which clearly is not true.
This does not follows.
We understand 1 through "one apple",
etc...
It is only simpler in terms of being simpler to write down,
because we
simply eliminate the mention of the "something" that is counted.
But it is
more complex to understand, because we mentally have to add the
something in
order for the numbers to have meaning beyond intellectual
mastrubation.
What do you propose as an alternative theory?
My point is just that if we say "yes" to the doctor, then we have
literally no choice on this matter.
[SPK]
To assume Yes Doctor is to assume that the physical reality of
substitution exists.
Yes.
This existence cannot be then eliminated by some trick.
The existence is not eliminated, it is reduce to a modal existence,
whose definition is extracted from the UD argument. The math shows it
to be of the type persistent stable partially sharable deep "dream".
Bruno Marchal wrote:
But comp needs only that you belief that the elementary
arithmetical
truth does not depend on you or us (little ego).
Are you thinking that if an asteroid rips of humanity from the
cosmos,
the number 17 would get a non trivial divisor?
That does not make sense, I think.
Of course an asteroid won't influence that the number 17 has no
non-trivial
divisor, because we defined the numbers in a way so that the
number 17 is
prime, which is true regardless what happens.
All right then. That was my point.
The point is that a definition doesn't say anything beyond it's
definition.
This is deeply false. Look at the Mandelbrot set, you can intuit
that is much more than its definition. That is the base of Gödel's
discovery: the arithmetical reality is FAR beyond ANY attempt to
define it.
[SPK]
If it where not possible to define the rules of a Mandelbrot set
then there would be no Mandelbrot set. It is not more complicated
than that.
I agree. But many definable sets are not recursively enumerable. for
example the set of codes computing the factorial functions, or the set
of codes of total computable functions. Only sigma_1 are computable,
above you have many set having relevant information (about numbers)
but not at all computable, and, for machine, the arithmetical reality
is *very* rich.
So, the number 17 is always prime because we defined numbers in
the way. If
I define some other number system of natural numbers where I just
declare
that number 17 shall not be prime, then it is not prime.
No. You are just deciding to talk about something else.
Who says that your
conception of natural numbers is right, and mine is wrong?
Then you have to tell me what axioms you want me to make a change.
But you will only propose something else universal, and I have
already said that I am not sanguine about numbers in particular. I
would prefer to use the combinators, or the lambda expression, but
natural numbers are well known, and that is why I use them in this
list. The laws of mind and matter are independent of the initial
theory, once that theory verify the condition of being sigma_1
complete = sufficiently strong to represent the partial computable
functions, and to emulate the UD.
[SPK]
The fact that we can have this discussion tells us something,
but it is not that we can believe that a theory alone can justify
itself.
Comp does not justifies itself (unlike the Löbian machine, actually).
On the contrary comp justifies that you need an act of faith, you have
to bet on something impossible to justify (like a substitution level,
notably).
You are just
asserting the truth of you own axioms when you say that number 17
is prime,
which is as good as saying my axiom is "everything goes" and I
derive from
that that you are in reality living inside the belly of an
invisible pink
unicorn.
Yes, my proposal of declaring 17 to not be prime is ridiculous,
because it
doesn't fit with our conceptions of what properties numbers ought
to have,
or ought to be able to have. But these conceptions come from our
sense
perceptions, and imagination, were we can count and measure
things. So when
you want to apply numbers to the fundamental realtiy, which as such
obviously is not countable, nor measurable, your natural numbers
are as
weird as mine, because they both miss the point that reality is not
countable.
Of course we can do a lot of interpretation to rescue our theory,
for
example by interpreting something beyond numbers into numbers via
Gödel, but
then we could as well just use our capability of interpretation
and skip the
number magic.
The numbers are just more pedagogical. When you say "yes" to the
doctor he can put a java program on a disk, or a combinators, but
usually people will see only 0 and 1, and still call that a
numbers. We assume DIGITAL mechanism, and my goal is just to show
that this leads to a reversal physics/machine psychology making the
hypothesis testable. The question of using numbers or java programs
is a question of implementation and engineering, like using a mac
or a PC.
Bruno
http://iridia.ulb.ac.be/~marchal/
[SPK]
Try to state your result with no action or verb terms.
Implementation is not something that has any meaning absent some
physical process.
Implementation is definable in Peano Arithmetic. The universal number
u implements the function phi_i when for all number x we have
phi_u(<i, x>) = phi_i(x). And this definition can be written in
arithmetic (that is using only s, 0, + and * (and logical symbols).
OTOH, we do not need to assume that the physical is primitive nor
the abstract. To claim such is a straw man.
Sure. But once we say yes to the doctor, we have to take the
consequences into consideration.
Bruno
http://iridia.ulb.ac.be/~marchal/
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