Friday, February 15, 2013

Down the rabbit hole--infinity, paradoxes, and weirdness, Part 4

In this segment of the paradoxes related to infinity, I want to develop a concept that I've actually already talked about somewhat. That concept plays at the heart of the problems with attempting to assign a uniform probability density function (uniform PDF) to a space that is infinitely big.

Some definitions, quickly: A probability density function is a rule that assigns probabilities to various events or ranges of events in a given space known as the sample space. A requirement of probability density functions is that they must integrate to the value one over the entire space. A PDF is said to be uniform if it assigns the same probability to every element in the sample space and to every equal-sized interval within the space. See the second post in this series for more information (Link).

This theme has recurred because it's actually central to the construction I put forth in God Doesn't; We Do. Further, since I already explored this topic and developed it in Part 2 of this series, I suggest you go read that. Particularly, I developed it there to discuss the psychology behind why it seems so intuitive to be able to put a uniform distribution on an infinite space.

Here, I want to force the situation to show how odd and absurd things get if we attempt to force the impossible. Doing so is cheating, but it's sort of justified cheating because it's a goal a lot of mathematicians chase after and that many use (including me when I misused it, as is typically the case). Somewhat sophisticated, if fringe, nonstandard methods of dealing with analysis, the relevant branch of mathematics, have developed a mathematics of infinitesimals that deals with it... kind of. Because of this Holy Grail-esque quality, forcing the situation isn't too out of bounds for a philosophical exploration.

Here's the problem at the heart of doing so, even if we use infinitesimals to do it, and this is as good as any an example of how weird and paradoxical infinity can get.

Suppose we use the example of the natural numbers, for relative ease and frequency of occurrence. The natural numbers, again, is the set of numbers {1, 2, 3,...}. Suppose we decide that we're going to force the situation, using infinitesimals if needed, and argue that a uniform PDF is placed upon the natural numbers, i.e. that we are able to pick any natural number, from amongst all of them, at random with an equal likelihood of picking literally any of them. What can we conclude.

As I've discussed before, including in this series, since every number is smaller than most (that is, only finitely many are smaller and still positive while infinitely many are bigger than any number we consider), we can ask the question: "What is the probability that our randomly selected number will appear in the interval (n,), meaning the set of every number larger than some arbitrary number n?"

However we measure it, the probability that we will pick a number within that interval is one since if any number divided by has any meaning, it either means zero (standard mathematics) or an infinitesimal (nonstandard mathematics). Thus, the probability that the chosen number will occur in the interval (n,) is one minus that, either zero or an infinitesimal. One minus an infinitesimal is sometimes said to be 0.999..., but it's generally recognized to be indistinguishable from one except formally.

Let me rephrase that in plainer English, and in bold: The probability that a randomly chosen natural number, assuming uniform likelihood of any natural number, will be larger than some arbitrary number n is one (or one minus an infinitesimal). This is true for every natural number n.

Now consider what happens when we start making n bigger.

In plain language consider the following series of questions exploring the matter: Will our selected number be lower than 100? Never. Lower than 1000? Never. Lower than one million? Never. Lower than one trillion? Never. Lower than a googolplex to the googolplex power? Never. It will always be bigger than those in any trial. Pick any number you want. The random selection will definitely be bigger than that. Now note that "for any" also means "every."

That means that our random selection isn't going to be a number. More formally, if we let n tend toward infinity, we arrive at the fact that the number we will select, with probability one, which means absolute (or all but absolute) certainty, is going to be in the "interval" (,), which is better known as the empty set, the set that contains nothing.

What does this mean? It means if we try to force the impossible situation of picking a number from amongst all natural numbers, uniformly distributed, we will actually fail to pick a number at all.

What if we use infinitesimals to get around this problem? It doesn't work. Why? The probability that we pick any number at all is infinitesimal, while the probability that we are unable to pick a number is one, short that infinitesimal (which means essentially short nothing at all).

What if we force it again by looking at a conditional probability? Suppose we assume that we pick a number and ask the same probability question we asked before. Note that this requirement means that with probability one, we will pick a number. Under this condition that we do pick a number (even though it's infinitely unlikely to happen), this entire construction merely repeats itself, and it is still infinitesimally likely that any number will be picked. Thus, we find ourselves in the position that the conditional probability of picking any number given that we pick a number is simultaneously infinitesimal (essentially zero) and one at the same time. This is a contradiction.

What gives?

Simple: We cannot put a uniform probability distribution on a sample space that is infinitely big (for the pedants: formally this means on a space with infinite measure). We can't even force it.




This post is the fourth in an ongoing series discussing the topic of the weirdness of infinity. To read more, you can read...
  • Part 1 (Link to Part 1) to see the interesting paradox that the (old) axioms of number theory predict infinity but don't allow it. 
  • Part 2 (Link to Part 2) develops the idea at the center of this post and suggests at the psychology of why infinity is so counterintuitive.
  • Part 3 (Link to Part 3) develops the ideas in Parts 1&2 more fully and discusses looking at the problem from the other side.
EDIT (21 Feb. 2013): Part 5 (Link to Part 5) to see how the question of infinity forces us to choose various foundations for mathematics and the philosophy that comes from it.


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