Reading your article a radical thought came to my mind.
Being a mathematician or a physicist (like myself) we are used to treating mathematics as the ultimate tool in understanding nature. We develop mathematical formalisms that describe nature and then marvel at how well mathematics is suited to describe our subject of interest. But is mathematics really so universal as we make it out to be, or do we just see it in that light because we are so used to it?
The foundations of mathematics evolved in our ancestors for the need to count things in order to have to survive. Mathematics was used for centuries by the Babylonians, Egyptians, Greeks and other highly developed cultures. But it took the genius of a Newton or Leibniz to develop differential calculus and to be able to properly describe and object's motion when a variable force acts upon it. This formalism uses infinities and infinitely small intervals and is quite a long way from the simple act of counting.
So my answer is: The world can be understood mathematically, only because we develop the mathematical tools that allow us to understand it. Constructions, like differential calculus, do not follow inevitably from the basic principles of mathematics, but only from our need to describe nature. But this reverses the dependency. The universe is not inherently mathematical, but mathematics is constructed to be universal.