The non-mathematicians' common sense view, and why it is inadequate

There is a view of the nature of mathematics which is very common among non-mathematicians, and which, I think, by its very straightforwardness warrants the name 'the non-mathematicians' common sense view'. This view consists of the idea that mathematics is basically a language of abstractions of physical-world objects. The reasoning goes: First we counted fingers and pebbles, then we created an abstract idea of number that encompassed both, and from there mathematics developed. In this view, mathematics is no more than a language (albeit a formal one) to speak about abstractions of the real world, abstractions which are the products of human minds.

One advantage of this view, it seems at first glance, is that it gives an explanation to why mathematics is so successful in its applications to science. The common sense view would say that mathematics allows us to understand the physical world because it has been designed to do so. The claim is that mathematics is an organizational language constructed originally as a tool to deal with quantitative data and patterns observed empirically.
The first mistake that many non-mathematicians make is to think that mathematics is nothing but the language of the quantitative, or that which can be expressed numerically. Aside from the fact that this is more a description of statistics than of pure mathematics, this is a grosse misunderstanding of mathematics, but a misunderstanding that it takes a deep aquaintance of mathematics to spot. As a matter of fact, mathematics is much more 'qualitative' in its nature, dealing much more with concepts and structural relations, than the quantitative common sense view admits.
Practicing mathematicians who are generally well-aquainted with mathematics know that there is much more to mathematics than the language of quantity or abstractions of the physical-world. Why is this so? While it is no doubt true that the sequence of natural numbers 1, 2, 3, ... is an abstraction of physical phenomena such as counting pebbles or fingers, to say that all of mathematics is like this, is akin to saying that the music of Mozart is just a study in sound and a mental construction which comes directly from the natural sounds of insects, the wind, or other physical phenomena. Mathematics is, apart from a study of quantity, the science of precisely defined abstract structures. Some of the abstract structures which are fundamental to modern mathematics, such as groups, rings, or algebraic varieties, could (by a rather large stretch of imagination) be seen as abstractions and generalizations of the natural numbers or physical shapes such as curve-like objects. On the other hand, some of the most important objects in mathematics do not seem to be traceable back to counting fingers or to observing shapes of objects. Take for example group representations, infinite sets, sheaves, adeles, derived categories, etc. There is simply no reasonable way in which these objects can be reduced to simpler concepts or objects, until we reach the conceps and objects of our physical world.
But wait a minute, hasn't logicism (Russell et al) shown that all of mathematics is reducible to a few simple axioms of logic or set theory? Well, logicism has shown that mathematics is expressible in a formal language a posteriori, that is, after mathematics has already been constructed or discovered by non-formal means. The formal systems in question are almost completely non-conceptual at a higher level, that is, while they are capable of formulating our propositions and proofs, they would probably never have led to the discovery of the mathematical objects and concepts in the first place, and the meaning we give to the definitions in the formal system comes from the meaning of objects we have discovered previously. To claim that logicism proves mathematics to be an a priori study is thus outrageously unrealistic, because most of today's mathematics would never have seen the ligth of day if mathematicians had worked in the a priori setting of formal systems.
The reductionism of logicism is thus untenable, and the structure of the body of mathematics is better understood as a complex system with emerging properties. To be sure, the group representations could not exist without groups, and groups can be seen as vast generalisations of the set of integers, which itself is an abstraction and extension of finger counting. However, there is no immediate logically necessary path or physical metaphor between groups and group representations. The latter is simply an epi-entity of the former, something which could not exist on the simple level of the sequence of natural numbers itself, but that can be defined on a higher level, and indeed has profound implications for the objects on other levels as well as for physics, and even chemistry.
It is also a fact that many mathematical objects were discovered before (without any physical-world abstractions) their applications to physical phenomena. In higher science (i.e., way beyond finger counting), it is thus often a matter of concepts of mathematics informing empirical sciences, rather than the other way around.

The above discussion has made the point that mathematics cannot simply be an abstraction of physical objects and phenomena, as Aristotle thought. Aristotle would not allow infinite sets, and he would probably not have accepted such fundamental objects as negative numbers, not to mention complex numbers, which have no interpretation in the classical Greek geometry. Nevertheless, these mathematical objects are indispensable in today's science.

Now, if the objects of mathematics were just free creations of the mind, building on physical-world abstractions, then we would not be able to explain how these objects are so incredibly well-suited to understanding the physical world. Group representation theory was originally developped without any physical applications in mind. The physical study of atoms and elementary particles started independently of abstract algebra. Still, the two converged and came together in the application of representation theory to elementary particle theory. Moreover, this is not the only example of its kind. It is thus wholly unreasonable to assume that mathematics is a free game of the mind because we know that most abstract free games of the mind, such as many philosophical or political theories, card games, or board games, are useless for understanding the physical universe. Chess is an abstraction of objects in society (a kingdom), and it's played according to exact logical rules, much like mathematics. Still, chess is completely useless in explaining the universe, while mathematics is profoundly enlightening. In explaining the nature of mathematics one must therefore account for this difference in applictations to the understanding of the physical world. The success of mathematics in this area can, according to me, only be adequately explained if one understands the objects and results of mathematics as actually being facts of the world. But mathematics is not exactly facts about the physical world, because there are no perfect circles or infinite sets in the physical world. Rather, mathematics must be about general abstract facts of which the physical world is an instantiation, or particular manifestation.