Theorem

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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. In mathematics, a theorem is a statement which has been proved on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms.

Contents

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.

Structures and Relationships

The key to the solution has to do with relations -- and that this is related to emergence. There are theorems because systems have relationships as well as elements, from which arise emergent properties. Systems are more than just components, or elements. A system must also have relationships among its elements before they it is worthy being called a system. But, when you take these component relationships into account, the possibilities for what characteristics, or properties, a system may exhibit begins to ramify into a potentially large and surprising number, due to combinatorics. With so many possible component relationships, it often becomes non-intuitive as to which potential properties (true statements) of the system are true. Thus the need for theorems arises due to a system having relationships among its components. And we haven't even mentioned emergent properties yet! This is simple, of course, because it is elemental, foundational to systemics.

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.

Reducibility

Some systems are reducible, explainable and understandable because they can be decomposed and divided in sub-domains and there are sometimes similarities in different sub-domains. These systems can often be explained and understood by theorems. Some systems like cellular automata are irreducible, because they are based on irreducible computations, see Computational Irreducibility

An example for a reducible and explainable system is the natural world itself. If we consider the system of natural language to describe it, then every valid sentence or statement is a "theorem" about the state of the natural world. The interesting theorems correspond to metaphors.

Words can be combined to sentence, and sentences to paragraphs and pages, and pages to books. Sentences describe actions, paragraphs episodes, pages short stories, and books long stories. But there are also metaphors, which describe cross domain relationships. Metaphors describe or reveal a hidden structure.

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