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3.2 How accurately are the existing mathematical theories predicting the natural phenomena?
All the existing mathematical theories have limited applications. For example, classical mechanics cannot describe the behavior of electrons and theory of gravitation cannot describe the beginning of universe (singularity). But, how accurately are the theories (for example Newton's laws of motion) describing the physical processes which are within their domain of applicability (for example the falling of a stone)?Let us first consider the laws of motion given by Isaac Newton. For simple situations like free falling of a stone or swinging of a pendulum, Newton's laws of motion can mathematically describe the phenomena quite accurately. But for most of the real life situations, which are quite complex, the predictions are at the best approximate. For instance, let us consider the movement of a rock rolling down the mountain. We can very well understand that Newton's laws of motion are applicable to this situation. But what about describing the geometries of the rock and the mountain surface? Thus it is extremely cumbersome to solve the motion precisely.
Let us turn our attention to predictions based on quantum mechanics. How well does quantum mechanics describe the electronic state in real solid having all sorts of defects like grain boundaries, dislocations and vacancies as shown in Figure 2? Precise description is not possible even with the best computers and the state of the art density functional theories (based on quantum mechanics). In most situations, only an approximate description of a real life phenomena is possible by a mathematical theory although it is perfectly applicable. Roger Penrose  cautions us -"It (Plato's world of mathematics) tells us to be careful to distinguish the precise mathematical entities from the approximation that we see around us in the world of physical things."
One can readily see two reasons for this viz. the problem of geometrical complexity in a natural phenomenon and the different length and time scales in nature.