Over at
Backreaction, Max Tegmark's
Mathematical Universe Hypothesis (MUH) was evaluated. What is the MUH exactly?
The Mathematical Universe Hypothesis (MUH): Our external physical reality is a mathematical structure.
What kind of structure could this be? We have general relativity and quantum field theory, and these are based on (pseudo) Riemannian geometry and Lie algebraic fiber bundle theory, respectively. Tegmark has asserted the MUH implies a so-called "Theory of Everything" (TOE) will be a purely mathematical theory. This seems reasonable, and the devil is always in the details. So what are the details?
One can approach the problem by uniting general relativity with quantum field theory, and this is usually dubbed quantum gravity. Being that general relativity does not inherently contain, say, SU(3) symmetry, which is central to our understanding of quarks and gluons and baryonic matter in general, it is logical to seek a mathematical formalism that does include it, and attempt to derive general relativity at large scales.
Some remark that general relativity is akin to a hydrodynamic theory which cannot discern individual H2O molecules and their bonding properties. In the case of spacetime, the "molecules" would analogously be
gravitons. Garrett Lisi's
E8 model takes the Lie algebraic structure as an axiom and studies a unified model in which spacetime and the standard model arise from gradings of the largest exceptional Lie algebra. Alain Connes refines fiber bundle theory with noncommutative geometry and takes a certain C*-algebra that acts on finite points as
encoding the standard model over a 4D base space. In string theory, there are
D-brane models that encode standard model structure in worldvolumes, and this is also an example of noncommutative geometry. Once the graviton is identified, it is possible, such as in string theory, to show large curvature in general relativity is equivalent to a
coherent state of gravitons.
Surely progress is being made in our search for a TOE. And the closer we get, Max Tegmark's hypothesis seems much more likely. However, the MUH does not give the mathematical structure we seek. To find such a structure, researchers must journey past the boundaries of known mathematics and physics. And no single road will dominate all searches, and one must be versed in a myriad of approaches to arrive at the coveted TOE.