
Applicability is the real strength of mathematics: its relationship to the scientific method is the fact that it is so successful in explaining the real world.
Mathematics began with the abstraction of properties from the real world around us; it proves its usefulness when the results obtained from this abstraction are turned back again to the real world. This is how science works; scientists abstract the properties they wish to study from experimental evidence (for example, the observations of planetary motion over many years were used to find the positions of the planets); this abstraction is then manipulated mathematically, possibly with other assumptions thrown in (for example, Newton assumed that the attraction between the planets varied according to the inverse of the square of the distance between them), to find a mathematical way of describing this data (in Newton\'s case, that the planets move in elliptical orbits around the Sun), which can then be verified by further experimentation (which in this example, had already been done a century before by Kepler). The whole of physics and much of many other sciences depends on this procedure. It has been used also in unexpected places in fine art in the early Renaissance: for example, the science of perspective was developed mathematically as a method for producing pictures that looked more realistic.
The method of applying mathematics outlined above has proved greatly successful. Today, if the experiment to verify the mathematical conclusion does not give results that match up with the predictions, the scientist does not decide that mathematics is of no use. On the contrary, he or she checks the other parts of the procedure first: that false results are not being obtained because of something which was not originally taken into account, or because there was an error (such as oversimplification) in the abstraction. Even if neither of these is the case, there is enough confidence in the reliability of the mathematics that the conclusion that follows is not that this is a case where mathematics is not applicable, but that the extra assumptions which were made (possibly unconsciously) were wrong. SMcL 
