Here's an eclectic mix of examples from various perspectives across multiple domains. Understanding the representation of knowledge and the fundamental structure that underpins the connective essence of things is one of the prime ideas that drives my thinking, and as a consequence much of what I post about on here is related to that in some way. Below is a brief sampling of examples related to this idea, and for a more extensive list, browse through my postings (and if you can see how they all connect, let me know!).

In general, see the Classification of Finite Simple Groups [0]. In terms of specific examples, the minimal classifications of knots [1], trees [2], and braids [3] are 3 instances that popped to mind. From a number-theoretic perspective, the paper Enumerating the Rationals and its derivative works are interesting reads. In terms of spatial structure and spatial decomposition, look at the classification of crystal structures [4], lattices, polytopes (zonotopes are particularly interesting), periodic and aperiodic tilings, the classification of space-filling curves, and the classification of closed surfaces [5]. From a relational perspective, look at the classification of topological spatial relations [6] and Allen's interval algebra [7]. And the one my intuition finds most intriguing is the classification of Group Theory Single Axioms and its relation to my conjecture that division is primary.

[0] Classification of Finite Simple Groups https://en.wikipedia.org/wiki/Classification_of_finite_simpl...

[1] Prime Knots https://en.wikipedia.org/wiki/List_of_prime_knots

[2] Enumerating Trees https://www.cs.virginia.edu/~lat7h/blog/posts/434.html

[3] Braid Group https://en.wikipedia.org/wiki/Braid_group

[4] Crystal Structure https://en.wikipedia.org/wiki/Crystal_structure

[5] Classification of Closed Surfaces https://en.wikipedia.org/wiki/Surface_(topology)#Classificat...

[6] Topological Spatial Relations https://en.wikipedia.org/wiki/Spatial_relation

[7] Allen's Interval Algebra https://en.wikipedia.org/wiki/Allen%27s_interval_algebra

[8] "Yet Another Paper on Group Theory Single Axioms" https://www.cs.unm.edu/~mccune/projects/gtsax/

P.S. Clifford Algebra / Geometric Algebra [00] unifies much of mathematics, e.g. in GA, Maxwell's equations can be reduced to one equation and expressed on one line [01]:

[00] Geometric Algebra https://en.wikipedia.org/wiki/Geometric_algebra

[01] Maxwell's equations formulated in terms of Geometric Algebra https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...

Further: SQL is not composable. If I have a fragment of SQL in my hand, there is no reasonable way to insert that into a larger expression. You can further tell this is a problem by the way any nontrivial SQL function degenerates into an enormous hellstorm of SQL that any Haskell programmer will see as full of redundancy. This is because SQL is almost immune to abstraction. (I have to specify the "Haskell programmer" bit because of the characteristics of this sort of language is that working in it directly for a long time causes you to think natively in that language and thereby miss the opportunities for abstraction that you no longer see. FWIW, Erlang also bothers me this way.) Yes, the "bad programmer" issue is in play too, but as a good programmer who has tried to write good stored procedures and has experience in numerous other languages I can not help but notice how much SQL fights me on this front.

SQL is very difficult to debug in general, which combines rather poorly with the previous characteristic. Perhaps there are nonstandard extensions in Oracle that make it easier but there's nothing I know of in the core language.

SQL has fooled people into thinking it is equivalent to the Relational Model, when in fact as kunley points out elsewhere it is merely inspired by it.

SQL dates from the 1970s and it shows. There are syntax quirks around every corner. The behavior of NULL is sensible, but it is merely one of several sensible choices and there are reasons why the SQL choice is not the one that has "won". Why is "value IN ()" a syntax error? It's perfectly reasonable. What's with the syntactic drama around subselects, which ought to be a core part of the better language that we really ought to be using? (A language in which the equivalent is probably going to be so simple we won't even believe people had a special syntax for it.) What's with the rigid specification of the order of GROUP BY and ORDER BY and everything else, when those should simply be instances of some sort of combinator that applies in order, in a sensible manner. Why is SELECT used both for data queries and aggregate queries when in fact the two operations are so different they should have entirely different keywords? Why is it so hard to mix an aggregate and non-aggregate query together when it's obvious to me exactly what I want?

Why can't my columns contain tables themselves, recursively? The fact that there are databases that can do this proves it's not impossible.

SQL needs to be replaced by a combinator-based language that backs to a richer data store that is more sensible, both from a relational model point of view and from a recursive point of view. LINQ points the way, but still is hobbled by LINQ-to-SQL being the flagship LINQ product. (And I don't think it's complete as-is.)