TITLE: The enigmatic complexity of number theory
QUESTION [81 upvotes]: One of the most salient aspects of the discipline of number theory is that from a very small number of definitions, entities and axioms one is led to an extraordinary wealth and diversity of theorems, relations and problems--some of which can be easily stated yet are so complex that they take centuries of concerted efforts by the best mathematicians to find a proof (Fermat's Last Theorem, ...), or have resisted such efforts (Goldbach's Conjecture, distribution of primes, ...), or lead to mathematical entities of astounding complexity, or required extraordinary collective effort, or have been characterized by Paul Erdös as "Mathematics is not ready for such problems" (Collatz Conjecture).
(Of course all branches of mathematics have this property to some extent, but number theory seems the prototypical case because it has attracted some of the most thorough efforts at axiomization (Russell and Whitehead), that Gödel's work is most relevant to the foundations of number theory (more than, say, topology), and that there has been a great deal of work on quantifying complexity for number theory--more so than differential equations, say.)
Related questions, such as one exploring the relation between Gödel's Theorem and the complexity of mathematics, have been highly reviewed and somehow avoided any efforts to close.  This current problem seems even more focused on specific references, theorems, and such.
How do professional mathematicians best understand the foundational source of the complexity in number theory?  I don't think answers such as "once relations are non-linear things get complicated" or its ilk are intellectually satisfying.  One can refer to Gödel's Theorem to "explain" that number theory is so complicated that no finite axiomitization will capture all its theorems, but should we consider this theorem as in some sense the source of such complexity?
This is not an "opinion-based" question (though admittedly it may be more appropriate for metamathematics or philosophy of mathematics):  I'm seeking theorems and concepts that professional mathematicians (particularly number theorists) recognize as being central to our understanding the source of the breadth and complexity of number theory.  Why isn't number theory trivial?  
What references, especially books, have been devoted to specifically addressing the source of the deep roots of the diversity and complexity of number theory?

By contrast, I think physicists can point to a number of sources of the extraordinary wealth and variety of physical phenomena:  It is because certain forces (somehow) act on fundamentally different length and time scales.  The rules of quantum mechanics govern the interaction of a small number of subatomic particles.  At sufficiently larger scales, though, quantum mechanics effective "shuts off" and classical mechanics dominates, including in most statistical mechanics, where it is the large number of particles is relevant.  At yet larger scales (e.g., celestial dynamics), we ignore quantum effects.  Yes, physicists are trying to unify the laws so that even the quantum mechanics that describes the interactions of quarks is somehow unified with gravitation, which governs at the largest scales... but the fact that these forces have different natural scales leads to qualitatively different classes of phenomena at the different scales, and hence the complexity and variety of physical phenomena.

REPLY [39 votes]: What references, especially books, have been devoted to specifically addressing the source of the deep roots of the diversity and complexity of number theory?

To a first approximation, I would say that the answer to this question is, there are none.
You have emphasized that you are interested in the point of view of professional number theorists.  I would say that for most number theorists, a term such as the "diversity and complexity of number theory" brings to mind central problems in modern number theory, such as the generalized Riemann hypothesis, the parity problem in sieve theory, the Langlands conjectures, the structure of Gal(${\overline{\mathbb{Q}}}/\mathbb{Q})$, etc.  These are the sorts of things that professional number theorists might cite as the "source" of the diversity and complexity of number theory.  Note in particular that things such as Hilbert's Tenth Problem are interesting to relatively few professional number theorists.  The kind of "complexity" that logicians and theoretical computer scientists are interested in is not what interests most number theorists.  Roughly speaking, it is because undecidability questions are regarded as signs of chaos whereas number theorists are interested in finding structure.
If we want an explanation of why there is so much diversity and complexity in number theory even when we focus on the structures that occupy the attention of number theorists, then I do not think that looking towards undecidability will give us the answer.  Generalized chess, for example, exhibits that kind of "complexity" but the deeper one studies chess, the more it seems to exhibit seemingly "random" behavior that defies elegant description (just take a look at the record-holders in the endgame tablebases for example).  In generalized chess we find no sign of anything with the beautiful and deep structure of, say, class field theory.
Seeking an explanation of what number theorists regard as the diversity and complexity of their subject is instead likely to elicit essays with "unreasonable effectiveness" or some such in the title, and the discussion will likely follow the same path that the discussion of Wigner's article has taken.  For example it can be pointed out that there is a natural selection process taking place, with number theorists deliberately gravitating towards the areas of diversity and complexity and abandoning areas that are sterile.