I recently discovered this image
by René Magritte, a Belgian surrealist artist. The caption translates as "This is not a pipe," which makes us think about the abstract representation of concrete or abstract ideas.
Before this discussion continues, however, it's important to know how we accumulate knowledge, willingly or not. I'd say there are three ways, two of which can play tricks on us, whether we realize or not (this is where the third one takes over): intuition and senses. With intuition, we take things as real without arguments for its existence; we take them without thinking about whether what is being presented is logical or not. This is very clearly shown in mythology, which used to be accepted in whole without even the slight ponder of opposition. It's just as easy for our senses to trick us. Look up to the moon one of these nights. Your eyes just tricked you, as the moon seems to be about the size of your thumb.
You won't accept this as true. Of course you won't; you know for a fact that the moon is not as large as your thumb. You know it's in fact a relatively large object floating around a few thousand miles away. This is where reason comes in. It is by reason and only by reason that we can acquire consistent elements of observation. It is only by reason that we may question things we find untrue. (On the other hand, we can't completely abandon intuition and senses, because it is by them that we can use reason to logically sort out what we can assume true or not. The three are very closely linked together.)
This concept of how intuition can trick us can be easily seen in the following example. Take a look at the following:
1 2 3
A B C
Writing systems and numbers exist solely by reason. They cannot exist concretely.
Let's take a look at the idea of numeric values. Numbers don't exist. Have you ever run into the number 2 on the street? It's impossible. Even a statue or a model of 2 isn't the number. It's only a statue. It's only a clay model. Any sort of measurement that can be represented by number is nonexistent. It is simply the result of the quantification, abstraction, and materialization of a completely cognitive idea. Taking systems of methods into consideration, a tree can be five feet away from a person. This conceptual measurement is put in numeric form by humans, and doesn't exist in its complete essense. The tree will still be away from the person be there a system of measurement or not.
Taking a turn into the world of writing systems, I'll use the English language written in the Latin alphabet for examples. H O U S E. That word is a group of non-existent symbols that are, in this case, represented by means of electric phenomena. The same word can easily be represented by the production of ink agglomeration - a process known as writing. Either way, it isn't a structure built for lodging; it is and will continue to be a representation of such.
After all, the word "representation" is self-defining: re
(again, newly) present
(to show, make known). We are taking something concrete and making it known again, but this time in an abstract manner.
When we accept that "2 HOUSES" that really is a group of two lodgings of a certain kind without even thinking about it, we are using our intuition. When we see a photograph of two houses and say "these are two houses," we are being tricked by our senses. Once we finally take the time to think about how these are simply cognitive representations of something concrete, then only then are we using our reason.