Bye taxonomy...
jk. There is not enough development in ontology yet to completely discard taxonomy. Even if ontology had more development, taxonomy will still dominate because it's simplicity.
I'm going to use Taxonomy for taxonomy schemes below unless I specifically address it.

I'm interested in classifying data due to my need of creating a system that can find a particular math problem fast when all math problem are provided with description. Mathematics has concrete definitions of each of it's individual elements. It's the ideal model to test any classification system.

My previous views on classification are in fact restricted on the taxonomy system. It's simple, but can get really complicated. There is something extremely difficult to address. For instance. The problem between how to exactly describe a term.
My confusion with taxonomy is: Does terms intended to be the only object describe terms?
In the most common taxonomies, only terms can be used to describe terms. Because it only have terms and parent-child relationships. There is nothing else, it's easy to implement and get the job done.
I tried to divide term into quality and quantity, which is like creating attributes to extend the taxonomy. I'm not a genius, there are shortcomings in the model. Include using taxonomy to describe attributes, which I think it's a problem in all models ever created by man.

An example:
Triangle is a child term of Polygon.
Triangle have 3 sides but Polygon can have [positive integer between and include 3 to infinity] sides.
So sides should be an attribute. The different quantities in attribute created the difference of terms.
In fact, even without manually associate Polygon as the parent of Triangle. After examine all the possible attributes of the two term, the computer can see that Triangle is a child of Polygon.
So side is an attribute. But side(synonym of edge) is just a 1-face. So side can have the attribute of n-face set to 1. So an attribute is getting described by another attribute. Replace the word "attribute" with "term" in my last sentence. It would be the standard model of taxonomy. The attribute+term model of taxonomy seems like the normal taxonomy system forced to create a unnatural border between completely same concepts. I get confused and I really want to understand if attribute and term can be used interchangeably or not.

But worry no more... I start to focus my attention on ontology.
All the quotes are from wikipedia

Definition of Ontology:

An ontology is a representation of a set of concepts within a domain and the relationships between those concepts.

Review what Taxonomy means.

Taxonomy is the practice and science of classification.

We can't compare ontology with taxonomy. The definition shows compare them are like comparing water with kittens1.
We can compare ontology model and taxonomic schemes. They both can do the essential thing I want: Show the relationships between items and their properties.

If there is unlimited system resource. A taxonomy can really in fact classify everything by exhaustively convert all attribute and associated values into terms. Like the following one:

With unlimited computational and storage resource. I would tell you right now that the current multi-inheritance taxonomy scheme is perfect. There is no need of develop a specific ontology model for anything. We can all sleep at night knowing another great challenge is defeated.

But obviously no one is going to represent every single number as a separate term in taxonomy. You can, if you have relatively few attributes and value combinations. Each combination requires 3 slot for storage. One refer to the attribute, one refer to the value and one refer to the term that's the combination of those two. It construct a huge web, and need huge computational power.

A ontology model basically solved that problem. Ontology is almost like the math we see everyday! It has three significant advancement compare to normal taxonomy--Attributes, axioms and restrictions. Refer to Wikipedia on these, because I'm not as expert as Wikipedia in this particular topic. Wikipedia is like the living proof of a very loose ontology.

I'm looking forward to OWL become part of Drupal one day. Then, when it start to have service APIs for distributed Drupal and data, Drupal will be the perfect CMS for everything.

What is beyond the current ontology?
Humans start to talk in constructed language that's syntactically unambiguous(The only one known to me is Lojban) so machines can now, seriously, understand what we are talking about. Then all the ontology and taxonomy structures are built by machines automatically.

1. 1. No, they are not both eatable