Many millennia ago, primitive people wondered how doing something a definite way would produce a definite effect. They could not explain it, it was magic. They tried to do the same in different contexts. Sometimes it worked; this enhanced the belief in the magical rite. Sometimes it failed; this was attributed to either an inaccurate reproduction of the rite, or the interference of some supernatural forces. This magic attitude to the world is reproduced today in the traditional belief of many people that there is no science without mathematics, and anything mathematically proven must be true, unless an error has crept in the derivation.
For the ancient consciousness, it was quite impressive, that a series of formal manipulations could produce a trustable result of practical importance. The ability to predict was considered as a mystical power granted to the select few. Then the elements of mathematics have become a part of common education; however, up to now, mathematical education mainly preserves the dogmatic medieval style, with the rules of operation presented as if they have descended to us from heaven. That is why many people beware of coming too close to these sacred truths in school, pretending to be not enough gifted for math. And even those, who like playing with abstract quantities, remain convinced that this is the highest form of rationality.
In science, the magical function of mathematics has lead to the distinction of "exact" sciences from the rest, which is not commonly considered as science until at least some mathematical slang gets in.
At a closer consideration, one finds that the role of mathematics in science is greatly exaggerated. Thus, in experimental science, success is by 99% due to the instrumental skills of the observer and the eclectic mentality of the interpreter. Applied science is entirely dependent on the ability to adapt any formal results to the real needs. The only domain where mathematical derivations can play a significant role is fundamental theory, which is only a very small portion of science. But even there, most important results come from the considerations far from mathematical reasoning, like the sense of completeness, love for beauty, taste for unification, personal predispositions etc. In physics, we justify the choice of mathematical constructions by "physical considerations"; in some other sciences we use mathematical labels just because "it looks like that". Then, as millennia ago, we apply our formal schemes to different areas. Sometimes it works; this enhanced our belief in the power of mathematics. Sometimes it fails; this makes us seek for mathematical mistakes or blame the experiment for insufficient purity.
The adepts of formal science forget a simple truth: before one can think formally, one needs to be able to think at all. Deny this syncretic thought—and you will annihilate any thought at all. As any other human activity, science combines different levels of reasoning, including formal derivation and formal construction. But the weight of the latter largely depends on the practical circumstances and the required outcome. In many cases, we only need a general framework, a range of possibilities, without too much numeric detail. It would be unwise to employ a cumbrous computational machine just for that. Conversely, in engineering, too much mathematical science will only hamper quick assembly of the target device from ready-made blocks.
Science can be rigorous and predictive without too much mathematics. Simple logic (not necessarily formal) can often be enough. The attempts of mathematicians to declare logic in general a part of mathematics cannot be but ridiculous. Even within mathematics, all the new ideas come from outside and mathematical intuition does not obey any formal prescriptions.
The success of mathematical methods in science can be explained by the relative rigidity of the forms of human activity, by their conservation in the course of cultural development. Eventually, this development requires a significant shift in the modes of action, and a new range of formal stability is established to give birth to new mathematics. The mechanism of mathematization in science is the same as in any other boundary research: the interaction of different scientific disciplines is often fruitful, infusing new blood in the original sciences and opening new interdisciplinary domains.
Mathematics is a science like any other. The usurpation of the reign in the land of science by one of them and mathematical tyranny can only be temporary, and the universal equality and partnership will be restored some day.