Speaking about importance of application of mathematics in natural sciences, we should not absolutize its role. Mathematical formulas in itself are abstract and deprived of the concrete contents. The mathematics is only the tool, or means, physical research. Only the physical researches coordinated with scientific supervision and experiment fill mathematical formulas with the concrete contents.
Using mathematical methods of research, involving them in informative search, sciences have to consider possibilities of mathematics, reckoning with limits of its applicability. That mathematical processing of the contents in itself, its translation into language of quantitative descriptions does not give information gain means.
In case of a constructivist method proceed from the mathematical constructs accepted intuitively obvious, on their basis build more difficult, than they, elements (but do not remove a formula), in the course of designing of these elements use sequence of steps, suitable for construction.
But this very plausible assumption fair not always it is moreover incorrect for wide intervals of time even in case of the normal (typical) course of natural processes. In it sense of the fascinating break which is carried out at research of dynamic systems.
However only negative conclusions do not follow from this circumstance. First, the mathematics differently cannot work, and secondly, in similar approach the advantage, it is interfaced, so to speak, to "purity" of the description: the accurate zadannost of research when it is necessary to track "behavior" of object on the basis of a certain property, to isolate the line of changes, a tendency of development and to transfer information in strict schedules, schemes, the equations is available.
Some methodological lessons inspired by mathematics are that. However, as the mathematical science is effective, and some shadows, and better to say are cast on it: these shadows - are continuation of its advantages (at inadequate use of the last).
In the bases of any mathematical discipline some mathematical elements and postiliruyemy distinctions between them by all means are found. Thus for creation of mathematical system two methods are used, as a rule: axiomatic and constructivist.