The final equation

It is often said that, in the end, all of physics, when it will have been properly unified, will be contained in one simple, very elegant, mathematical equation. While I believe that the current trend in physics (including my own work on gravitation theory) shows that this is probably true, I do not think that it means that physics is trivial or simple. Indeed the most general equations of physics are based on the most general and therefore the most advanced mathematics and although the basic relationships it describes can be embodied in a simple equation, the mathematics itself can be quite complicated. This is where the complexity of a final equation would be hidden. Also, in physics, a fundamental equation is the mathematics that applies to the most simple phenomena. But not all phenomena are simple and the real difficulty is always to show how it is possible to explain even the most complex processes using the very simple fundamental laws which are assumed to be valid under all circumstances. This is, of course, the main task of the working physicist, the bread and butter of physics.

The most general mathematical frameworks contain as particular instances many distinct, less general theories which apply in a variety of particular circumstances, even though they do not apply in general. For instance, even though we know that the equations of the general theory of relativity that describe curved spacetime always apply, it is still true that, in the most common situations, space is flat and is describable by the less general equations of special relativity theory. This is also a source of complexity, because there is a near infinite number of particular cases in which physics can be most appropriately described using equations which are not universally valid, but which are still derivable from a more general mathematical framework.

Thus, reality itself is very complex and in practice is most appropriately described by equations that are not very general and not very simple. Yet, what is truly amazing is that it can be expected that, ultimately, the most simple processes underlying all of physical reality will be described by a universal theory embodied in one simple mathematical relationship, the final equation of fundamental theoretical physics. It is simply the fact that we always learn about the multiple particular cases before we are allowed to assimilate the more general mathematical theory from which they all arise that explains our natural tendency to believe that it is unlikely that reality could ever be described by one simple mathematical relationship, while from a historical perspective the exact opposite conclusion appears to be the most reasonable.