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