**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 [3] 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.