fom
Posts:
1,038
Registered:
12/4/12
|
|
Re: infinity can't exist
Posted:
Feb 12, 2013 12:07 PM
|
|
On 2/12/2013 9:19 AM, Craig Feinstein wrote:
> Let's say I have a drawer of an infinite number of identical socks at time zero. I take out one of the socks at time one. Then the contents of the drawer at time zero is identical to the contents of the drawer at time one, since all of the socks are identical and there are still an infinite number of them in the drawer at both times. But the contents of the drawer at time zero is also identical to the contents of the drawer at time one plus the sock that was taken out, since they are exactly the same material. So we have the equations:
>
> Contents of drawer at time 0 = Contents of drawer at time 1
> Contents of drawer at time 0 = (Contents of drawer at time 1) plus (sock taken out of drawer).
>
> Subtracting the equations, we get
>
> Nothing = sock taken out of drawer.
>
> This is false, so infinity cannot exist.
>
> How does modern mathematics resolve this paradox?
>
Consider your use of the word "identical".
In Aristotle, one finds the discussion that one can
never define x=y. To be precise, he says that one
can never prove a definition, but one can destroy
a definition. But, definitions rely on the notion
that some word is "the same" as the object toward
which its language act of referring is directed.
Now, what does Aristotle mean by this? He means
precisely the kind of thing that Einstein said about
relativity. It cannot be proved true, but it can
be proved false by a single experiment.
In finite-state automata -- a finite mathematical
discipline -- the definition of distinguishability
is categorized in terms of experiments of length
k, where k is the length of a given input string.
Two automaton states are k-distinguishable if
their is an experiment of length k for which the
automaton is started in each of the candidate
states and the two output strings differ. They
are k-equivalent if their is no m<=k for which
they are m-distinguishable.
Thus, two states are distinguishable if they
are k-distinguishable for some particular k.
Two states are equivalent if they are not
distinguishable.
How many experiments does it take to prove
that they are equivalent?
==============================
Suppose now, that you do not like this explanation
because you are talking about socks, and, socks
are material objects.
How do we understand material objects? We can
think of them as bodies in the sense of impenetrability.
We can think of them as bodies in terms of our
visual field.
Both senses of body are correlated with certain
forms of logic. In general, one thinks of deontic
logic in terms of social norms of what is and
what is not permitted. But, the general sense
of deontic logic is simply the logic of lawfulness.
Clearly, the notion of a physical law like gravity
arises from a "law" like "birds can fly, but man
cannot." Or, returning to the impenetrability
of bodies mentioned above, "one cannot walk
through walls."
The logic associated with the visual field is
somewhat harder. I will argue here that it is
temporal logic. I probably cannot do that
successfully because of the complexities. But,
our visual field delineates objects on the basis
of color distinctions. Our science has invested
a great deal of effort to explain color in our
visual field, and, that has led to optics and
quantum mechanics. It has also led to special
relativity in the sense that the color of
light is presumed to be different to observers
in different inertial reference frames.
But, if one grants me this position, then what
one has are two distinct logics whose common
elements form what is called propositional
logic. Typically, what is true and what is
false in propositional logic is based on
a set of functions called basic Boolean functions,
or truth tables.
Now, the basic Boolean functions are part of
a class of functions called switching functions.
These switching functions can have a property
called linear separability. Not all switching
functions are linearly separable. But, what
makes linear separability important is that
it has a geometric analogue.
The sense by which you cannot walk through
walls is the sense by which one represents
linear separability. In your typical introductory
mathematics classes, a line divide a plane into
two halves and a plane divides space into two
halves.
And, now we can return to the question of
your socks. For how do we think of our
material world as consisting of a plurality
of objects if we do not first divide the
field of sensory experience into parts?
When we divide the world into parts, we
are applying the logics mentioned above
in the sense that we are explaining boundaries.
The origin of the mathematical theory of sets
arose in combination with the mathematical theory
of point set topology. The two arise together
because they are attempts to address the problem
of Xeno's paradox. Calculus and the numerical
methods of approximation arising from the solution
of its equations address Xeno's paradox by
quantizing the last step to the finish line.
That is, they get informationally "close enough"
and then treat the error as a discrete quanta of
noise. Point set topology addresses the question
of Xeno's paradox as a lawlike limitation related
to the boundaries of material objects.
Once again, we arrive at the problem of
identity. In particular, the issue here is
Leibniz' principle of identity of indiscernibles.
Since the resources available to you will
not explain this as Leibniz actually wrote
it, I shall give you the quote:
"What St. Thomas affirms on this
point about angels or intelligences
('that here every object is a lowest
species') is true of all substances,
provided one takes the specific difference
in the way that geometers take it with
regard to their figures."
In point set topology, this is expressed
for topologies that have a way to measure
specific differences by Cantor's intersection
theorem. This theorem relates a sequence
of nested non-empty closed sets having
vanishing diameters.
It is in the definition of a closed set
where one is confronted with the boundary
where Xeno's paradox comes into play once
more.
So, now if we return to the discussion of
linear separability, it turns out that the
planes by which we divide our sensory
experience into parts cannot be represented
by linearly separable switching functions.
The basic Boolean function that has the
same properties as the sign of identity
is not linearly separable, and to synthetically
represent linearly separability by some
other means is an infinitary process.
So, we are back to Aristotle's explanation
and the question of how many experiments of
length k are need to prove that two input
strings are equivalent.
The theory of infinity in mathematics arises
because it is necessary if one wants to
say
This sock is this sock.
or:
"...I take out one of the socks"
and have it mean anything substantive.
|
|
