Mathematics as a Social Science
Most people (both mathematically versed and far from any math) would agree that mathematics is a science. Some of them, however, are apt to believe that this is a very special kind of science that lies in the basis of any science at all, providing the necessary pre-requisite for cognition as such. Finally, there are those who treat mathematics as an innate ability, the supreme knowledge embedded in the human animals in a mystical way, as a primordial touch of consciousness.
There are, indeed, serious reasons for all the above. Apparently, mathematicians behave like other scientists; at least, they talk the same way. On the other hand, it may be rather difficult to say what exactly they study in all those formal theories, whose practical significance finds recognition many decades (or even centuries) later, if ever. For an outer observer, mathematics is like a swarming ball of protoplasm that would suddenly sprout in one direction or another, to give birth to a new theoretical science, in the regular sense. To put it bluntly, a science must study something objective, that is, lying outside that very science and taken as an external source of facts and an application field. In this relative objectivity principle, we account for the whole hierarchy of indirect research, with higher-level sciences built upon a number of other sciences, however abstract, serving as a kind of empirical foundation. Like in any hierarchy, the "up" and "down" directions are readily interchangeable, and one could encounter the situations when two sciences lie in the empirical background of each other.
With mathematics, one cannot get rid of the impression of arbitrariness, inherent emptiness of any discussion, since one can easily develop a formally consistent theory starting from a collection of random assumptions, which are all equally acceptable, and there is no obvious reason for choice.
This was not that way in the ancient times, when mathematical knowledge came from immediate practical experience and satisfied people's everyday needs. Social differentiation and division of labor have detached the skill from its applications; however, up to the end of the XIX century, there was a hope for a "natural" foundation of mathematics, the common root of all the further abstractions. Multiple geometries, the algebraic revolution and computers have dispersed that illusion, and now, we are left face to face with our strange ability of constructing imaginary worlds that no one will ever inhabit.
But did the character of mathematical knowledge really change? Despite all the formal games, the basic feeling of a number and a shape remains intact, while the alternative notions of rigor manifestly continue the same line of causal arrangement that has always inspired technological progress, from the troglodyte magic to the modern robotized industry. If we do so and so, we are bound to finish as expected, unless some external circumstance (including the operator's blunders) abruptly modifies the operational environment.
This brings us back to the scientific status of mathematics. The object area of this peculiar science could be elicited using the hints from its early days, when its practical origin was yet evident enough.
Human activity is always aimed at producing some changes in the world. Each typical mode of such change gives impetus to developing a special science, with the object area related to the objective organization of the prototype activity. The hierarchy of sciences reflects the hierarchy of common activities. However, there is a fundamental distinction induced by the universal organization of any activity at all, which implies a conscious subject to take a portion of the world for an object and intentionally transform it into some product. As a result, each product can be characterized from two complementary aspects, as a kind of object (the material product) or as a representative of a certain way of action (the ideal product). This inner complexity of each product has eventually resulted in two complementary branches of science: the material aspect of activity is targeted by the so-called natural sciences, while knowledge about the modes of action is aggregated by social sciences. The latter name is appropriate since there is no individual that could exercise conscious activity outside any society at all, and the very idea of consciousness is only meaningful in the social context. That is, the subject of any activity is hierarchical, and any individuality belongs to that hierarchy along with the numerous forms of collective subject, from the family of two up to the humanity as a whole.
Now, does mathematical knowledge refer to any material things? No, it doesn't. There are very few people who would consider mathematical constructs as self-contained things, existing on themselves (this philosophical position, known as objective idealism, is usually associated with the name of Plato). Intuitively, mathematical knowledge is rather about some common properties of things; but, for us, the only relevant properties are those that are significant for using things in our activity. Mathematics is, therefore, to study certain modes of human activity, and hence it must belong to the class of social sciences. In other words, mathematics brings us knowledge about ourselves, just as any other humanitarian research. This perfectly explains the apparent arbitrariness of mathematical theories, since social sciences take the world under a subjective angle, including the freedom of choice. In our everyday life, we have to decide on the appropriate modes of action; this, in particular, is reflected in the versatility of mathematics. Philosophical materialism, however, holds that no choice can be entirely arbitrary, and that the variety of available options is always determined by the objective organization of the world, by the nature of things. This circumstance is responsible for the apparent rigidity of the mathematical method complementing the apparent arbitrariness of the premises.
One could expect that new modes of mathematical thought would come in response to significant cultural shifts; however, the humanity has not yet (at least on the memory of civilizations) experienced revolutions of that scope, and we are still quite comfortable within the existing paradigms. Nevertheless, some hints to the open possibilities might be drawn out of the several methodological turnovers known in the history of mathematics. In any case, no science can ever reach the state of permanent completion; though some sciences (including certain mathematical theories) seem to have exhausted their creative impetus, their abandonment is of an entirely local significance, as there is always a chance of running into an interesting feature that has earlier been irrelevant, or just overlooked.
During the periods of relative stability, natural sciences develop an inner organization that drives then away from nature, to resemble the humanities in the very occurrence. A typical mathematical (or physical) paper is packed up with metaphors and allusions, using the regular language (indispensable in any discourse, however formal) in a very loose manner, mentioning thousands of names (which is intended, but fails, to reference earlier introduced ideas), lacking conceptual and theoretical consistency, as a clear exposition of the matter is impossible without a range of assumptions and preliminaries beyond any tractability. Professionalism gets almost entirely reduced to mastering the parochial slang, while logical transparence is sacrificed to erudition. Everything is made to impress the public rather than educate it.
A novice will find modern science almost incomprehensible, since an individual life is not enough to get just acquainted with all the parental work, nothing to say about a critical examination. Considering the chaotic character and inevitable circularity of references, there is practically no way to check the logical consistency and factual substantiation of any special report. The validity of reasoning is no longer a matter of proof, but rather a kind of common consent, prejudice or academic fashion, so that the whole of science virtually develops from one level of belief to another, rather than from truth to more truth. In this context, credulity and good memory are much more important for a student than inquisitiveness and prehension.
As a result, the overall structure of mathematical knowledge remains utterly conventional, just like in social disciplines similar to law, or accounting. The absence of a natural organization makes it impossible to establish a standard reference frame, to make mathematics searchable. This is a dictionary with no alphabet, sorted by random criteria, like keys and the number of strokes in Chinese and Japanese hieroglyphic dictionaries (or abstract hash values in computers). Eventually, there are too many characters to learn, and the whole thing splits into a number of traditional areas poorly communicating with each other.
Well, every cloud has a silver lining. In its chaotic mass, mathematics just cannot come too restrictive in natural sciences, leaving more room for metaphorical usage and losing the aura of magic that led many scientists to overestimating the role of formal manipulation and starting to toss phantasies instead of studying nature. Given the limited accessibility of advanced mathematical methods, we have to organize knowledge according to the structure of the objective area rather than stretch observations to an arbitrary formalism; this may reveal uncommon structures that could eventually push forward our mathematical thought.
Typically, a working scientist (e.g. in physics) has an individual mathematical toolkit, a store of standard components to reuse in any new theoretical model. However, when it comes to a drastic conceptual change, the already available forms are no longer sufficient, and a mathematical description has to be invented from scratch, since it is almost impossible to find the relevant pieces in the body of modern mathematics (counting out the always-possible random encounters). In case of success, mathematicians would assimilate some of such handicraft to the earlier introduced constructs, or add yet another ad hoc theory to the rest, to keep on with piling up formal junk. The portions of mathematics that penetrate other sciences are nothing but the coming back of their own inventions reformulated and "refined", spiced up with a scent of "rigor". This, again, resembles the situation with the humanities: to gain an official status, a new teaching needs authoritative support, a formal assignment and right to compete; later on, each authorized discipline can play itself a role of official authority for the newcomers. The pretense of mathematics to the absolute dominance in science is a neat replica of the superpower image of the USA on the political and economic stage. The inner discrepancies of the American society leave enough room for the other nations to break the dictate and develop on their own, thus influencing the development of the USA as well. Someday, science will probably abandon the idea of scientific ranks and forget interdisciplinary competition, to grow a new hierarchy of knowledge that would not distinguish natural sciences from anything "unnatural" or "supernatural". This must obviously follow the overall democratization of the world order, annihilating the market economy, class society and any kind of competition, throughout both the human culture and the world.