Re: Metaphysics of Potential Infinity
examachine_at_gmail.com
Date: 01/26/05
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Date: 26 Jan 2005 12:36:03 -0800
Mitch Harris wrote:
> examachine@gmail.com wrote:
>
> > Here are my answers.
> >
> >
> > For such substitution to be possible, we would require their
meanings
> > to be identical which is not the case.
>
> OK. I'm not up on Aristotle or commentaries. So I still have trouble
> in seeing the distinction.
Aristotle's views on mathematics are still relevant in my opinion. Here
is the bit on aperion.
http://plato.stanford.edu/entries/aristotle-mathematics/#12
My statements are simply a refinement of Aristotle's views, trying to
make more sense of them by observing simple things like "Everything in
a finite universe has finite extent", hopefully making it more
rigorous. (Some of the arguments in "Theories of general machines"
might look like an unnecessary work of trying to formalize simple
sentences, which is what foundations is all about in a sense)
Note that Aristotle believed in an actually infinite past, but claimed
that potential infinity is all that mathematicians will ever need. I
reuse some parts of Aristotle without change, for instance the emphasis
on material joining and division. There we see the birth of physicalism
in mathematics. (Relevant also to Aaron Sloman's philosophy of
computation)
Apparently, a mathematician who rejects the use of actual infinity is
on the same boat as Aristotle. I find it sufficient at the present to
clarify the distinction, without rejecting or accepting actual
infinity, asking what it would mean to accept or deny it. Even better,
what does potential infinity mean at all in a finite mechanical world,
e.g. digital physics?
If you asked me in a nutshell, I would say that the distinction can be
likened to the distinction between the "kinds of infinities" of Z and
R, but it's safer not to assume that it is the same thing (IMHO).
If you follow the links:
------------------------------------------------------------------------
Aristotle distinguishes two further ways of looking at an infinite
series.
1. The infinite series in potentiality: The series is not actually
ever completed. What make the series infinite is simply the fact that a
next step in the series is always possible.
2. The infinite series in actuality: We conceive of the series as
completed.
------------------------------------------------------------------------
This concept of "series" comes before modern number theory or set
theory, so it's better to think of these in their own right. But of
course we will notice, hmm, 1. looks like the infinity of Z, and 2.
looks like the infinity of R. Right of course, but let's just be
careful. (Because there is always the possibility that our set theory
might not have sufficient explanatory power - which I think is kind of
obvious _and_ necessary!)
> >>OK. Do finite things like 2 exist? 2 seems to me to be immaterial.
> >
> > Physicalist philosophy maintains that there are no things that do
not
> > extend in space
> ...
> > For a physicalist, the number 2 is merely an abstraction in your
brain,
> > as part of a larger system of abstractions, e.g. number, addition,
etc.
> > There is no _fundamental_ difference between abstract common sense
> > concepts like "past" and "the number two" from this perspective.
>
> My question was rhetorical in the sense that it seemed that one main
> distinction that you see between actual and potential infinity is
that
> one, actual infinity, is immaterial (without spatial extent (there
> -is- a secondary meaning to immaterial which is "irrelevant" which
> neither of us intends at this point)), but the other, potential
> infinity, is not immaterial. But, (the rhetorical point is that) if
> something like "2" is immaterial, then certainly "potential infinity"
> is also.
The other senses would be irrelevant as you say. Following is some
boring possible-worlds talk, if you wouldn't mind (e.g. metaphysics).
The actual infinity is not necessarily immaterial. We can conceive of
an actually infinite space-time. However, the potential infinity is
always a material concept. That's I think what I argue for. It is a
material concept *either* in an infinite or finite universe.
The physicalist stance would bring into question how sensible it is to
try to talk of a completely metaphysical "theory". You know, we have a
lot of old idealist philosophy and theology just like that. While as
seen above I don't reject entire metaphysics like Wittgenstein, I do
think it's worth trying to "fix" our language and concepts. What do we
really mean by infinite? If infinity meant something like "angel" in
Christianity, surely it would be no better than a fiction. It would
then have the same _ontological status_ as an angel: a word with no
referent. No existence. What exists about angelhood is merely fiction.
According to many philosophers, sentences involving things with no
referents like angels could not be considered true... So, what's the
difference between the status of angel and infinity?
I'm trying to give you a flavor of what kind of a reasoning lies behind
physicalism. Either you commit to physicalism or reject it. There is no
in-between. If you reject it, there you go, you have souls, gods,
angels and all sorts of fantastic objects. If you accept it, you have
to give physical explanations.
One physical explanation is Turing mechanics. Some programs would never
halt if it were not for the physical constraints of the universe. But
it seems in our universe they would eventually have to halt. Never
mind. There is nothing that is free of physical constraints in a
physical world. (This part is in fact important). So we see that we
mean being able to *give* practical bounds on things other than the
trivially huge bounds of the ultimate physical bounds. That is
potential infinity for us.
If our universe is finite, then there are such trivially huge bounds
for any given physical process. If you can give substantially better
upper bounds for things (spatial or temporal) for a process, then it is
bounded, otherwise it is *practically* unbounded. The physicalist would
take this practical unbounded-ness to be also the correct sense of
being unbounded or potentially infinite.
If our universe is infinite, then similar arguments follow, either
there are hard bounds or not. Only if the universe is infinite *and*
there are no hard bounds, then the concepts of potential and actual
infinity come close. (This is not too easy to explain I think, so I'll
just omit it, because I think the universe is finite...)
> Maybe the term you want to use is "physically realizable"?
I am not sure if that is a term that I would like.
The term "physically realizable" carries the Platonist tendencies in
our language. If something is physically realizable, then perhaps there
is an extra-physical antecendent, e.g. a Platonic form, to its
existence? You see, it is as if the presupposition is built into our
language. I would like to avoid such sense distortion.
If there is such a property as physically realizable, is there such a
property as immaterially realizable? But nothing exists beyond our
space-time Only concepts are realized, and concepts, as _models_ of
things in the world, are perfectly physical. That would be okay, as it
is consistent with a physicalist theory of _mind_.
> > If you say on the other hand that immaterial things do exist, this
is a
> > direct admission of substance dualism, which is generally seen as
an
> > untenable position in philosophy.
>
> I am not so deep or studied in philosophy to be able to claim to hold
> even tenable positions.
Nobody is. Common sense and curisoity is sufficient to be make
philosophy.
> > Instead, let me ask you: what do you mean by immaterial? Do you
have
> > something else in your mind other than the well-established
> > philosophical definition?
>
> I couldn't say. I think I agree with your definition above
(immaterial
> = not having physical extent), but a lot of philosphy is teasing out
> exactly what the connotations are and what they should be.
That is also the definition Descartes used in formulating his
metaphysics. But of course, in his philosophy there was a place for
God.
> >>>[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.]
> >>
> >>Couldn't one say that there are -actual- infinite length paths on
the
> >>sphere?
> >
> > No, as far as I can tell.
>
> Really? How about potentially infinite paths?
> How about the area filling Hilbert curve?
It sounds like a word play to say that "X exists" where X is a
potentially infinite construct. If we accept the validity of such a
sentence, yes, there are potentially infinite paths.
I cannot say if a completed fractal could exist. As a metaphysical
concept, yes, in a strange possible world where such things could exist
and different laws of physics hold, it could be actually infinite. It's
better to keep in mind that we are talking about limits when we
theorize such fractals, not completed infinities, at least in worlds
like our own.
By sphere, I meant a physical sphere, not the concept of a n-sphere in
geometry which assumes that it's okay to talk of actually infinite
division (e.g. the set R).
On the abstraction quality of positing actual infinity: yes, it can cut
down on the length of mathematical statements, trading reality for
conciseness. Not always a good deal.
> >>I sitll don't see how one can
> >>prove the nonexistence of something else, like Hegel's mistake
about
> >>the number of planets). So what? all these concepts that involve
> >>actual infinity are still useful ones.
> >
> > Useful in what sense? That is the question.
>
> er, lots of useful math depends on it?
Good point. By depends, I assume you mean that such useful math could
not exist without the concept of actual infinity. e.g. the conception
of actual infinity is a _necessary_ condition. I know it's most
unpleasant to do this, but may I ask you an example for the most useful
kind of mathematics that absolutely requires the concept of actual
infinity? (Because frankly, I cannot think of one.)
Regards,
-- Eray Ozkural
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