Theorem

A theorem is essentially a statement to the effect that some domain is structured in a particular way. If the theorem is interesting, the structure characterized by the theorem is hidden and perhaps surprising.

Why are there theorems?

Why do so many structures have hidden internal structures?

Take the natural numbers: 0, 1, 2, 3, 4, ... It seems so simple: just one thing following another. Yet we have number theory, which is about the structures hidden within the naturals. So the naturals aren't just one thing following another. Why not? Why should there be any hidden structure?

If something as simple as the naturals has inevitable hidden structure, is there anything that doesn't? Is everything more complex than it seems on its surface? If so, why is that? If not, what's a good example of something that isn't.

Connections and Shortcuts

The Princeton companion to mathematics lists 35 major theorems, from the ABC conjecture and the Atiyah-Singer Index Theorem to the Weil Conjectures. Theorems are based on connections in the structure of mathematical systems: to find a new theorem is like revealing a hidden structure. Some connections are shortcuts between different points, others are bridges between different areas.

Why do so many structures have hidden internal structures? It is the reason why we do Mathematics, otherwise it would be boring. Perhaps because there are systems where simple elements and rules can produce complex structures.

The basic mathematical elements and axioms allow a whole universe of combinations and connections which is consistent and complex at the same time. Algebraic and geometric systems seem to contain an infinite number of complex structures. The integers 0,1,2,3,4,.. may be simple, but there is an infinite number of them. If we consider only the numbers of the finite Group with 4 elements, then Number Theory becomes less interesting.

In general the patterns and structures which can emerge in a system depend on the basic axioms, elements and operations, and on the size of the system. A kind of emergence again, perhaps..

Hidden structures

Another reason for "hidden" structures is our limited capacity for instant in-depth analysis. They only appear to be "hidden" for us. Look at this XKCD Cartoon: There seems to be nothing but flat empty water as far as the eye can see, but there is a large number of complex structures below the surface.

In this sense, there is a large number of hidden structures in some systems because we have cognitive limitations. There is a limit in our cognitive abilities to perceive complex structures. We can recognize certain patterns and superficial structures at once, but we are not able to make an instant in-depth analysis of a complex system.