Does a potential infinity actually exist?


It seems that there is a lot of confusion about the notion of a
"potential" infinity. Now I had always assumed that the people who
were confused by this concept didn't know how to use Google, but then
I tried myself to find a really clear explanation of the concept, and
I was unable to. So I guess it's up to me to explain it.

The idea of a potential infinity may be equated with the view of
infinity held by Gauss, Poincare, and Weyl. The following quotes
characterize this view:

Gauss (paraphrased): "The notion of a completed infinity doesn't
belong in mathematics; infinity is merely a figure of speech which
helps us talk about limits"

Poincare (quoted from Morris Klein): "Actual infinity does not exist.
What we call infinite is only the endless possibility of creating new
objects no matter how many exist already"

Hermann Weyl: "... classical logic was abstracted from the mathematics
of finite sets and their subsets .... Forgetful of this limited origin,
one afterwards mistook that logic for something above and prior to all
mathematics, and finally applied it, without justification, to the
mathematics of infinite sets. This is the Fall and original sin of
[Cantor's] set theory ...."

So the basic idea, as so eloquently stated by Weyl, is that infinite
sets do not exist in exactly the same sense as the sense in which
finite sets exist, and that mathematics should distinguish between the
two senses. The language of classical mathematics (in any sense the
reader may chose to interpret that phrase) does not make any
distinction whatsoever between different senses of existence, and
hence it impossible, or nearly so, to even talk about the potential
existence of infinity within the language of classical mathematics.
That seems to explain why there is so much confusion about the notion
of potential infinity.

The mathematical entities that have an "actual" existence are the
things that we can actually compute. Thus, for example, we can
actually compute an approximation of sqrt(2) accurate to seven digits,
and hence such an approximation actually exists. Likewise, with a
computer, we can compute a hundred digit approximation of sqrt(2), or
even a million digit approximation, and so such approximations
actually exist. In general, we can compute arbitrarily many digits of
sqrt(2), but we cannot actually compute infinitely many such digits.
That is, we cannot complete that task, and so the infinite string of
digits representing sqrt(2) exactly does not actually exist. Instead,
we say that it has only a potential existence, by which we mean that
we can only actually compute arbitrarily accurate approximations to
it.

What Gauss, Poincare and Weyl realized, but many mathematicians today
fail to realize, is that we don't actually need infinity to do
mathematics. That is certainly true about the mathematics of the real
world (i.e. the mathematics used in physics and computer science,
etc.) What we are actually interested in is the things that have an
actual existence, and the things with a potential existence should be
thought of as useful fictions or figures of speech which help us
reason about the things that actually exist. In other words, the
notion of infinity introduces no new theorems into mathematics;
everything that can be said using the notion of infinity could also be
said without it. (Or equivalently, those assertions that absolutely
require the axiom of infinity or the axiom of choice have nothing to
tell us about the things that actually exist)

To illustrate these points, consider the following two sentences:

1) sum (k=1..oo) 1/2^k = 1
2) Ap En Am (m>n) -> |sum (k=1..m) 1/2^k - 1| < 1/p

Here, 'A' and 'E' represent the universal and existential quantifiers,
and the variables p,n,m are natural numbers.

The two sentences say exactly the same thing, but the first sentence
invokes infinity, while the second doesn't. Most people would agree
that the first sentence is far more readily comprehended, and so we
see that we don't actually need infinity, but infinity is indeed a
very useful figure of speech.

So how would we develop a set theory if we insist that infinity has
only a potential existence? For starters, finite sets of integers are
no problem. They have an exact representation as data structures in a
computer, for example. The set of all integers is also not much of a
problem. We can define some sense in which the set of integers {1..N}
is an approximation to the set of all integers {1..oo}, and then we
can say that the set of all integers has a potential existence (i.e.
arbitrarily accurate approximations to it actually exist).

But what about the real numbers? The basic idea is that we have to
find a way to approximate the set of real numbers by entities that
actually exist. Let me outline one way in which the theory of real
numbers can be developed within the constraints imposed by maintaining
that infinity only has a potential existence.

Recall that with the usual topology, the set of real numbers has a
countable neighborhood base. The elements in that neighborhood base
could be taken to be the pairs of rational numbers, and we can
certainly say that that neighborhood base has a potential existence in
the same sense as the set of all integers does. Then, instead of
classical real analysis, we would develop a theory of interval
analysis (such a theory has indeed been developed). And it's a good
bet that interval analysis is capable of providing us with all the
tools we need for real world mathematics--after all, in the real
world, our measurements only yield real numbers within some interval;
they never give us infinite precision.

So in conclusion, when we say that infinity has only a potential
existence, we are making a distinction between different senses of the
word "existence". Classical mathematics fails to make the
distinction, and that failure pushes mathematics in the direction of
make-believe, and that's a deficiency of classical mathematics.

.

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