On 04 Oct 2011, at 21:59, benjayk wrote:
Bruno Marchal wrote:
Well, we don't need concrete *physical* objects, necessarily, but
"mental" objects, for example measurement. What do numbers mean
concrete object, or measurement? What does 1+1=2 mean if there
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.
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
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).
we can say 1+2=3 is 3 just because we defined numbers in the way
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
My point is that we can't derive something about the fundamental
things just by adhering to our own definitions of what numbers are,
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
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
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:
Lol, the funny thing is that in your explantion you used concrete
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
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:
On the contrary: to understand arithmetic, is quasi-equivalent with
the understanding that a statement like 17 is prime, is independent
all concrete situation, in which 17 might be represented.
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
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
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
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
Show that they exist
beyond that, and then the further reasoning can be taken more
numbers, and abstract machines exist just as ideas, everything
them will be further ideas. You can't unambigously conclude from
something about reality.
Reality is an idea itself.
Bruno Marchal wrote:
No. Otherwise we would understand 0, 1, 2, before we understood "one
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
all computations are equivalent.
I think that 0, 1, 2, and many others are far more simple
than any something you can multiply them by.
something", which clearly is not true.
This does not follows.
We understand 1 through "one apple",
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
more complex to understand, because we mentally have to add the
order for the numbers to have meaning beyond intellectual
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.
Bruno Marchal wrote:
Of course an asteroid won't influence that the number 17 has no non-
divisor, because we defined the numbers in a way so that the number
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
the number 17 would get a non trivial divisor?
That does not make sense, I think.
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
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
So, the number 17 is always prime because we defined numbers in the
I define some other number system of natural numbers where I just
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.
You are just
asserting the truth of you own axioms when you say that number 17 is
which is as good as saying my axiom is "everything goes" and I
that that you are in reality living inside the belly of an invisible
Yes, my proposal of declaring 17 to not be prime is ridiculous,
doesn't fit with our conceptions of what properties numbers ought to
or ought to be able to have. But these conceptions come from our sense
perceptions, and imagination, were we can count and measure things.
you want to apply numbers to the fundamental realtiy, which as such
obviously is not countable, nor measurable, your natural numbers are
weird as mine, because they both miss the point that reality is not
Of course we can do a lot of interpretation to rescue our theory, for
example by interpreting something beyond numbers into numbers via
then we could as well just use our capability of interpretation and
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.
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