Date: Feb 12, 2013 12:07 PM Author: fom Subject: Re: infinity can't exist 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.