# It’s Just a Theory

# by D. G. Hoffman

“It’s just a theory”. You’ve probably heard someone say this, in an effort to debunk something they don’t agree with. But quite often, their opposition is unfounded, for two reasons.

The first is this. The word “theory” has different meanings, depending on the context in which it is used. Merriam-Webster’s online dictionary gives these two, and goes to some lengths contrasting them. One is “a scientifically acceptable body of principles offered to explain phenomena”. Another is “an unproved assumption, a conjecture”. The first meaning is the scientific one, e.g. “quantum theory”. The second one is the one often used in casual conversation, and typically suggests “fanciful balderdash”. The debunkers using the word “theory” in this sense are typically unaware of the first definition.

But here is the second reason. Scientists use a theory to unify and structure their observations. I maintain that in order to be useful in this sense, it doesn’t have to be true!

As an example, I have constructed a mythical universe. This universe consists of infinitely many one-mile square plots of land, lined up in a two-way infinite row. While the population and the area they occupy consistently grow, they are finite at any given time. The plots vary greatly in terms of their productivity as farmland, and the inhabitants have developed a ranking to describe this variation. The worst plots get assigned a ranking of 0. As it happens, every other plot, among those occupied so far, is ranked 0. Of the remaining plots, every other one is slightly better, but not as good as the rest. They get a ranking of 1. So every fourth plot is ranked 1. And this continues, so every eighth plot is ranked 2, every sixteenth plot is ranked 3, etcetera. As an example, perhaps the ranking of the occupied plots at some point in time looks like this:

020104010201030102010501020103010201040102 .

One of their scientists, Fred, has surmised that this pattern continues. He states this as follows:

Fred’s Theory: Every non-negative integer k is the rank of infinitely many plots, and every 2^{k+1}th plot has rank k.

Of course, the inhabitants of this universe have no way of verifying this theory, as they only occupy an ever-increasing, yet finite, number of plots. But I constructed this universe, and I can verify that Fred’s Theory is in fact true.

By Fred’s theory, all these rankings in the above examples are correct, with one notable exception; the plot ranked 5. Maybe its rank is 5. Or maybe it’s 6, or 47, or 509. They won’t know until more plots are explored.

Some years later, the brilliant scientist Mary, devised this extension of Fred’s Theory:

Mary’s Theory: This universe is the number line, and the plots are the integers. Odd numbers have a rank of 0. Even numbers that are not multiples of 4 are ranked 1, and so on. Specifically, every non-zero integer n can be written as n = 2^{k}x, where k and x are integers, with k non-negative, and x odd; this plot has rank k. That is, k is the greatest number of 2’s dividing n. But what about 0? The number 0 is divisible by every integer, and so by every power of 2. So its rank must be “infinity”! Mary dubbed this the “Paradise Plot”, it is far superior to every other plot.

But Mary’s theory is probably false! Let’s examine how you might go about constructing such a world. You have two choices for selecting which plots are ranked 0. After that, you have two choices of which plots to rank 1. And on and on. Constructing such a world consists of making an infinite sequence of binary choices. If you want to construct a world in which Mary’s theory is true, you could begin by selecting the plot to be the Paradise Plot, but then you have no more choices: in selecting one of the two possibilities for which plots to be ranked k, you must make the only choice that doesn’t rank the Paradise Plot. Thus there are infinitely many worlds for which Mary’s Theory is true, corresponding to the infinitely many choices for the Paradise Plot. The number of infinite sequences of binary choices, i.e. the number of possible world satisfying Fred’s Theory, is also infinite, but according to Cantor’s theory of cardinalities, it is a much, much larger infinity! So despite it being extremely useful, Mary’s Theory is almost certainly false.

All my references are online, I Googled “Theory definition”, and “Cantor’s cardinals”.

**Disclaimer:*** The views and opinions in this work are those of the author and do not necessarily reflect the views of Auburn University.*