Greetings Matt
I enjoyed your essay. A network view of evolving mathematical knowledge does indeed exist. Even now, we are connecting more nodes, and physics seems to be giving a "garden of insights", fueling this process.
What is interesting about our current era, is how advanced physics has become. In an attempt formulate a theory of quantum gravity, it is clear Riemannian geometry must give way to a new geometrical structure. What this new structure is, will become one of the ultimate examples of the unity of mathematics and physics. For at present it seems current mathematics, in all its complexity, is not sufficient to describe the microscopic structure of spacetime.
There a different approaches to quantum gravity, such as loop quantum gravity, Connes' noncommutative geometry, and string/M-theory. And although there has been a backlash in public opinion towards string/M-theory (our most promising theory of unified physics), one must remember it has its origins in hadronic physics, not quantum gravity. Moreover, pure bosonic string theory is only consistent in 26-dimensions, a fact which is very mysterious from a mathematical viewpoint.
If string/M-theory, in its complete nonperturbative form, proves to be the unique theory of quantum gravity, not only for our universe, but for all other universe in a multiverse system, it is a testament to the usefulness of exceptional mathematics, instead of general mathematical structures. General mathematical structures, hold no preference for a given n-dimensional space, with fixed value of n. Special values of n occur in the study of sporadic groups, exceptional Lie groups, and other "odd ball" mathematical constructions.
What seems to be occuring, at present time, is the unifying of these exceptional mathematical structures, using insights from string/M-theory, loop quantum gravity and noncommutative geometry. For example, it has been argued by Horowitz and Susskind that it is difficult to relate bosonic string theory in 26-dimensions to M-theory in 11-dimensions because there should be a 27-dimensional bosonic M-theory that reduces to bosonic string theory, just as M-theory reduces to the 10-dimensional superstring theories by compactification. In other words, at strong coupling, the extra dimension "opens up" and there should be a stable ground state as the full extra dimension is restored.
Smolin (a father of loop quantum gravity), years ago, introduced a Chern-Simons type matrix model in 27-dimensions as a candidate for bosonic M-theory, that effectively unites loop quantum gravity with M-theory. The matrix model makes use of an algebra of observables, that was discovered as an exceptional case by Jordan and Von Neumann in 1934. The algebra makes of 8-dimensional variables called the octonions (found in 1843), which themselves are a maximal case of the so-called (finite dimensional) division algebras. The real, complex and quaternion algebras are the only three algebras of this type, existing in 1, 2 and 4-dimensions.
There is mounting evidence that bosonic M-theory may indeed be formulated as a matrix model in 27-dimensions. If this is the case, the dimension with value n=27, is a result of the maximality of construction of an algebra of observables over the octonions. Such an algebra, permits only 3x3 matrices, at most, which together close into a hermitian matrix algebra that leads to a sensible quantum mechanics. This hermitian matrix algebra, the exceptional Jordan algebra, is the self-adjoint part of a unique Jordan C*-algebra. This C*-algebra allows a noncommutative geometrical interpretation.
So here, a historical series of accidental mathematical and physics discoveries, could lead to a theory of unified physics, that unites all promising approaches to quantum gravity. Moreover, it seems likely sporadic groups, exceptional Lie groups, L-functions, Shimura varieties and noncommutative geometry can all be connected in the process. This is quite an embarrassment of riches considering the haphazard process by which it stemmed. This is not to say, ultimate reality is necessarily mathematical. For as Hawking once asked, "What is it that breathes fire into the equations and makes a universe for them to describe?" What is the structure of the ancient Pythagorean's apeiron? Is there a pre-mathematical description of reality? Is Nature, at its core, acausal? Hopefully, in the coming years, we can begin to address such questions in a sensible manner.