A model is nothing more or less than a mental picture of a reality, whose description of that reality satisfies the full capacity of our reason. The real thing is, in fact, far above the capacity of our reason to grasp. This is why the models we produce must continually be expanded, or else entirely remade, as we uncover more and more realities along with all their properties and interactions. No sooner have we opened the door of understanding to one facet of a reality than we are dumbfounded by a whole new world on the other side. There seems to be no end to the depths of the riches of reality.
Mathematics are the most basic description of physical reality we have. They are not a picture, however; they are a true and unimaginable description. Models are invented; mathematics are found. They can be analogized, and they can even be applied, but they cannot be fully and entirely translated.
Our models are based on mathematics; they are not mathematics themselves. When we speak of a “mathematical model,” we are speaking of a collage made from mathematics which appear to govern the system we are modeling — a selection of mathematical rules, revised and edited, transcribed onto a canvas that our mind’s eye can read. We are squeezing mathematics into something that our imaginations can picture, at which point they cease to be true mathematics. Like a three dimensional landscape flattened into a two dimensional photograph. Where you happen to be standing, and at what angle you happen to be holding your camera, has everything to do with how the picture comes out.
Models, therefore, do not and cannot explain reality. They can be useful tools insofar as they allow us to capitalize on some aspect of reality, but they are not to be confused with reality itself. They serve only to give us a measure of predictive power over it. A negligible measure, I would add, in comparison with the infinity of the real.
All things proceed from the Nothing, and are borne towards the Infinite. Who will follow these marvellous processes? The Author of these wonders understands them. None other can do so.
~ Blaise Pascal, Pensées