Millennia ago, primitive people were deeply impressed by the experience of how doing something a definite way would produce a quite expectable effect. They could not explain it, it was magic. They tried to do the same in different contexts. Sometimes it worked; this enhanced belief in the magical rite. Sometimes it failed; this was attributed to either an inaccurate reproduction of the right scheme, 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 can be no science without mathematics, and that, if anything has been mathematically proven, it must be true, unless an error has crept in the derivation.
In the infant centuries of the human consciousness, it was quite a miracle that a series of formal manipulations could produce a trustable result of a practical importance. This ability to predict (or prophesy) was considered as a mystical power granted to the select few. Today, the elements of mathematics have become a part of the general education standard; however, up to now, teachers of mathematics (stifled by the harsh competition for payable hours) tend to stick to the medieval dogmatic style, with the rules of operation presented as if they 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 gifted enough for math. Those few who like tossing abstract quantities still have no idea of how it works; so, they prefer to turn their ignorance into superiority and stay convinced that formal deduction is the highest form of rationality, its essence and law.
In science, the magical function of mathematics has lead to the distinction of the so called "exact" sciences from contemptible undersciences, which cannot be taken for serious until they grow up to the age when at least some mathematical slang gets in.
At a closer examination, one finds that the role of mathematics in science is immoderately 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 methods can pretend to a significant part is fundamental theory; but such theories constitute a very small (if not negligible) portion of science. Even there, in the realm of pure abstractions, the most important results usually 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 constructs by "physical conditions" and discourteously reject "unphysical" answers; in some other sciences we use mathematical labels as sheer metaphors, just because "it looks like that". Most often, as millennia ago, we just try our formal schemes in a range of object areas. Sometimes it works; this feeds our mystical belief in the power of mathematics. Sometimes it fails; this makes us seek for formal mistakes, or blame the experiment for insufficient purity. Like a capricious child, great theoreticians get sulk and say: you should not behave like that, I want you please me!
The fans of formal science forget a simple truth: before one can think formally, one is to acquire the very capacity of thinking. To shape something, you need something to shape. However vague and mutable, our tentative considerations lay the foundation of any superstructures, preceding any formal embellishment; in this sense, such science is truly fundamental. Deny that raw, 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 context, as well as on the idea of the required outcome. In many cases, a very general framework is quite enough, outlining a range of possibilities, without too much numeric detail. It would be unwise to employ a cumbersome (and expensive) computational technique just to get that gross estimate. Conversely, in applied engineering, we need a workable combination of anything at hand, right now; too much mathematical science would only hamper quick assembly of the product from the readymade blocks. That is, the right place for mathematical modelling is well in between, far from the creative frontier, on the level of mass consumption, when a wellknown thing is to be brought to the highest possible perfection; this has much in common with esthetical judgment, and that is why we appreciate the undeniable beauty of mathematics.
Science can be rigorous and predictive without exaggerated formalities. Simple logic (not necessarily formal) will often do. The attempts of philosophizing mathematicians to treat logic in general as a part of mathematics, a kind of calculus, cannot be but ridiculous. Anyway, in pure mathematics, all the new ideas come from outside; mathematical intuition does not obey formal prescriptions.
The success of mathematical methods in science can be explained by the relative rigidity of the forms of human activity, by their preservation in the course of cultural development. From time to time, this development requires a significant shift in the modes of action, and a new range of formal stability is to be established, to give birth to new mathematics and the new notions of mathematical rigor. The penetration of mathematical language and formal method in special sciences employs the same mechanism as any other boundary research: any interaction of earlier independent scientific disciplines is mutually advantageous, pouring in new blood in each of the original sciences; additionally, it may open new interdisciplinary domains.
Mathematics is a science like any other, and true scientists have nothing to compete for. Cultural distortions hinder universal cooperation; economic and social inequality is reflected in the dominance of one science over the others, the usurpation of power and formal autocracy. Still, no tyranny can last forever; mathematics is to join the free community of sciences some day, for the common benefit.
