From Alice in Wonderland:

–snip–
What can you see on the road?

Nothing.

What great eyesight! What does nothing look like?
–snip–

In these cases, we have not revealed some mystical truth or problem with the universe, we have simply made a semantic error with the usage and application of our nouns. We misinterpret or mis-state what we denote. We ask, what color is 42, what is the next real number, or we speak of an “infinite quantity” all of which are simply undefined. The reason we have trouble seeing that these latter examples are undefined is because we have obfuscated the facts of what we are denoting..

Here is how we get there:

First we are shown a picture of 2 apples (a map of a concrete thing) then 3 more apples and now we see that we have 5 apples. Then we are given another “map” that corresponds to the map (picture) of the apples. Now we see numerals next to the pictures of our apples. 1, 2, 3, 4, 5… then we get rid of the pictures of apples and deal only with our numerals (our map of a map), and with this image of an image of something concrete we slowly start to think of these numerals in the same concrete way we think of apples. We think now of the “number 5″ in a very concrete way.

How thoroughly we have tricked ourselves! Eventually, we are unable to understand the difference between “map-ness” and “concrete-ness” altogether. Or, if we can understand it, we yet do not “see” this difference when we think of “a number.”

Furthermore, it is always in this way that we move from the concrete, “up” to the abstract level, and then bring the abstract back “down” to the concrete. We act “as if” the abstract number 5 denotes something in the same (concrete) way the picture of the apple denotes something.

A good example of how a map can distort or “flatten” the concrete is to think of the map of the world. When we first compare a flattened map of the world to the globe we find that the sizes of the continents are distorted, even though we have made a one-to-one correspondence between the two. Nevertheless, straight lines have been transformed to curves in the correspondence of these two representations. This is what happens when we “flatten our maps.”

And this is what happens when we bring these abstractions of an abstraction “down” from the realm of our minds into the physical, concrete world.

Thus we have completely forgotten that a decimal is not a number, but a special notation that represents a number. It is a shortcut way of representing an abstraction; it is an abstraction on top of an abstraction. It is only because we have become so used to treating all these abstractions “as if” they were concrete apples that we have trouble understanding what 0.999… = 1 could possibly “mean.” We look at this as some odd distortion as if we are looking at some flattened map of the world and complaining about the apparent size of one of the continents. It must be wrong, we say.

But we are too clever by half. We no longer even recognize our own maps as maps. When we ask, what is the color of 42, or what is the next real number, we must remind ourselves that we are talking nonsense. It is not some big mystery, it is simply undefined. We are just confused because of all of the semantic shortcuts we have taken so as to facilitate communication. The opaqueness of what we are really saying is why the mathematical rigor came into being – as a way of avoiding all this semantic sloppiness.

So when we speak of an “infinite quantity” we must realize that we speak nonsense; that when we do this, we speak of the “abstract-concrete” and make a kind of category mistake. By definition, quantity cannot be infinite. Quantity is discreet by nature; infinity is by nature non-discreet. Infinity doubled is the same “size” because “doubling” has “meaning” only when applied to the discreet. Just because we can denote “the infinite” in the same semantic form that we can denote “a quantity” does not mean that we can give it the same meaning. When we apply this discreetness to the idea of the infinite, we make a category mistake. In other words, we fall into the rabbit hole with Alice, asking her what nothing looks like.

The analogy of having a static representation for 0.999… is very helpful in understanding why our intuition gets confused. We think this infinite process stops and measured, when it can never stop. Now I’m starting to appreciate the 0.999… = 1 argument a lot more from an intuitive side (and not just an appeal to limits and differences that diminish below any given error margin). Awesome stuff, keep on writing!