Metaphysics of Potential Infinity

exanews_at_gmail.com
Date: 01/21/05 

Date: 20 Jan 2005 19:54:03 -0800

As a careless reply, r.e.s. wrote:

:>>> you can program a machine such that it can ask for
:>>> more memory from us when it needs it, with a suitable
:>>> architecture you can add as many memory devices as needed.
:
: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
:As many memory devices as needed.
:
:> in a finite space universe like ours, no machine can have
:> infinite extent, and an AGS is no exception. It can at most
:> grow until it becomes impossible to enlarge it any further
:> physically.
:
:NOT as many memory devices as needed.

In which r.e.s. covertly implies that to become "unbounded" you also
have to become "infinite", which shows his complete lack of knowledge
of the distinction between potential and actual infinity. That is why
he made a terrible interpretation of his selected quotes from Turing.

Unbounded means potentially infinite, and is quite a different beast
than actual infinity.

Knowing a little math does not automatically mean that you have any
clue about philosophy. Everybody here knows a lot of useful and useless
mathematics.

Sorry r.e.s. You've made it clear that you think unbounded means
actually infinite in an actually finite universe (ours!). Many people
including me would regard that as a conceptual error. Try cleaning your
conceptual lenses before you start arguing with somebody who cares
about concepts.

Again. Let me ask you in an easy way without getting too technical (in
the phil. sense).

Imagine a possible world A that consists of 10 cells.

0 1 2 3 ... 9

There is a process X which starts with 1 cell at 0, and grows from left
to right until it fills all the ten cells of the universe A. It could
be anything, such as painting them red. (Let's say initially they were
black)

Are you saying that the extent of this process is not unbounded? Then,
you are saying that "unbounded" refers to an immaterial property.
(Since no other process can extend beyond the extent of process X)

The problem with immaterial properties is that they do not exist. The
physicalist objection would be that such a property would be on the
same ontological level as "the infinite love of an angel", e.g. an
immaterial property, a fiction.

The obvious philosophical fact for such a world is, if "unbounded" is
to have *any* meaning at all, it means growing indefinitely. It is no
matter that eventually the process cannot grow beyond the limits of
this world, for NOTHING in this world can do such.

Again, let me repeat to you very clearly. The "bounds of the universe"
is different from the "bounds of something in the universe". Beyond the
bounds of a universe, NOTHING exists.

That is exactly why some physicists suggest that it is nonsense to talk
of "the cause of the big-bang", since NOTHING exists beyond the bounds
of our space-time. (Note that you would have to challenge this very
famous reasoning, if you had to challenge my refined exposition. That
is how Hawking could have upset the Pope if he wanted to.)

Therefore, the process X is UNBOUNDED but NOT INFINITE in this FINITE
universe. It is UNBOUNDED as far as ANYTHING can become unbounded in A.

This is in fact the definition of POTENTIALLY INFINITE.

I don't think you know what potentially infinite means. So I will have
to explain it to you. Potentially infinite is best described as a
metaphysical statement involving possible worlds.

In order to make this comprehensible to those who are not
philosophically inclined, I will give a definition for the "space"
aspect of infinity (excluding time, for instance), but keep in mind
that it would have to be more general.

def. (Potential Infinity) A process is potentially infinite, if it has
the (strong) tendency to become infinite in an infinite world. In a
finite world, it has the (strong) tendency to occupy the world.

In other words, potential infinity requires a machine that would not
stop growing if it could, provided we are taking the spatial sense of
infinity as above. That is the common element in either an infinite or
finite world. So, it is the only thing that is necessarily true about
being potentially infinite.

[Of course, the set of integers corresponds nicely to these
metaphysical explanations. By definition, every integer has finite
extent, but you can always construct a larger integer from a given
integer. This is an indefinite growing process.]

On the other hand, actual infinity can occur only in an infinite world.
That is no surprise I suppose. Of course, one can (arguably!) imagine a
possible world that is actually infinite.

It is utter nonsense to suggest that the red-painting process X has a
useful bound, it grows indefinitely, up to the physical limits.

It is that notion which I expounded upon and labeled "practically
unbounded", which means that the only upper bound you can give to such
a process is trivially huge: it does not make any sense to call it
bounded.

A memory system can clearly use the material of several solar systems
with sufficiently advanced technology to store bits. Personally, I
cannot imagine where it would have to stop. Therefore, I think it's
adequate to call such a thing practically unbounded.

The problem with your language is that you fail to distinguish between
theoretical entities and real-world entities. You commit no mistake in
mathematics, but only in an elementary conceptual task.

In a finite universe, it is nonsense to talk of the existence of
infinite objects to describe a physical concept such as unbounded.
(Mind you, every useful concept has to be physical!) There are no such
things in a finite universe. At most, there are indefinitely growing
processes as the one mentioned, X, and even those will have to stop
before they involve each and every "atom" in such a universe. Hence, my
term "practically unbounded" to underline the lack of a useful upper
bound to the extent of such things.

[As Wolf mentioned, the surface of a sphere (itself!) is perfectly
unbounded , but finite. That is a nice example to the necessary
distinction in our conception.]

If you did not understand all the text up to now, here is a slightly
alternative summary.

1. In a physical world, a property that would require an object to
extend beyond the physical limits of this world would be an absurd
property since you are asking for the undoable.
2. If unbounded meant extending beyond the physical limits of this
universe, then NOTHING in this universe could be unbounded. This does
not make sense, because EVERYTHING (e.g. every process) in the world
shares the same physical limits. Then, "unbounded" could not become a
physical concept in this world, because it can not distinguish one
physical object from another. It is rather like "soul" in the religious
sense, then. Not acceptable to a philosopher, that is bringing back
dualism, by talking about immaterial things and properties that DO NOT
EXTEND IN SPACE.

I have very little faith that you will come to appreciate a
contemporary physicalist's point of view, but you must try to
understand that this goes a long way from what you have been taught as
philosophy of mathematics.

Regards, 

--
Eray Ozkural

Relevant Pages

  • Re: Reading Dawkins ... was "Scrump"
    ... such an explanation now, so an alternative would have to be ... Some physicists, by the way, would like to find an explanation for the ... postulating a God to explain the universe gives ... you an infinite regress--unless your method for stopping the regress ...
    (alt.usage.english)
  • Re: Can any old earther refute common genetic ancestry?
    ... variability to infinite variability. ... We live in a slightly bio-friendly universe. ...
    (talk.origins)
  • Re: Reading Dawkins ... was "Scrump"
    ... Some physicists, by the way, would like to find an explanation for the ... postulating a God to explain the universe gives ... you an infinite regress--unless your method for stopping the regress ... maybe) and the tithe part is harder still. ...
    (alt.usage.english)
  • Re: Logarithm of transfinite numbers
    ... Tony Orlow wrote: ... But since you criticize other mathematics on ... widespread usage of the Leibniz notation, how in the limit via infinite ... The universe is infinite, infinite sets are equivalent. ...
    (sci.math)
  • Re: ping Frez
    ... physicists have had a particularly difficult relationship with the notion of boundlessness. ... The general theory of relativity predicts that singularities in which physical properties become infinite occur in the centre of black holes and in the Big Bang that kicked our Universe into existence. ... Unless we are missing something of fundamental importance, these observations indicate that our expanding Universe is about 14 billion years old, contains copious quantities of dark matter in some unidentified form, and is expanding at an accelerating rate. ...
    (alt.sports.football.pro.ne-patriots)
Loading