Re: Metaphysics of Potential Infinity
From: Mitch Harris (harrisq_at_tcs.inf.tu-dresden.de)
Date: 01/27/05
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Date: Thu, 27 Jan 2005 10:16:24 +0100
examachine@gmail.com wrote:
> Mitch Harris wrote:
>>examachine@gmail.com wrote:
>>
>>>Here are my answers.
>>
>>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
...
> 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.
I don't see that analogy at all. It looks like a typing problem (also
the technical meanings of series and sequence might be confused here).
Z is a set (of all naturals), R is a set (of all reals). A sequence of
naturals can always be extended, likewise a sequence of reals. Both
sets (either the whole thing, or just an infinite sequence) can be
completed.
>>>>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,
So I take it that you take it that 2 is immaterial.
>>>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.
So do you take 2 to be immaterial but a potential infinity to be material?
...
>>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.
I don't think so. Whatever you think of our friend 2 (as a platonic
form, or just an appropriate chemical configuration in most people's
brains), I think the phrase applies.
...
>>>>>[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?
...
> 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).
OK, then. The Hilbert curve is questionable. How about, on a physical
sphere, a physical path that is actually infinitely long? just keep
going around? If it is nly potentially infinite to you, then the
difference between potential and actual for you seem to come from
which world it lies in, the physical or ... the other world which is
not physical (whether it exists or not).
>>>>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.)
\sum_{n=0..infinity} 1/2^n = 2
and such a statement -absolutely requires- the concept of infinity. Of
course it could be easily argued over what kind of infinity that is.
but it won't change the result.
-- Mitch Harris (remove q to reply)
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