Mathematical Relativity

For many centuries, the power of mathematics has been associated with its deductive structure, so that many properties of a mathematical object could be formally derived from just a few fundamental statements. This circumstance tempts mathematicians into a common belief that the whole mathematics could be constructed as a deductive system, as soon as we fixed the most basic ideas. The search for the universal foundations of mathematics has never ended; a few general platforms have been largely explored, but neither of them came out to be satisfactory in all respects, and the question remains open up to now. Moreover, there are different schools that do not get along well enough; they seem to be irreducible to each other, leading to the kinds of mathematics that can intersect in most practically important domains, but do not coincide in full, with each theory producing a range of statements impossible in another approach. That is, instead of an all-embracing consistent formalism, we get a number of options, equally rigorous in their own sense. Once believed to give the ultimate truth, mathematics tends to dissolve in the numerous alternatives, and no mathematical truth is absolute nowadays. This makes mathematics a regular science like any other, rather than a common arbitrator and decisive authority. It may be bad news for those who came there for a bit of soul comfort in the uncomfortable world, but, no doubt, getting rid of blinding exceptionality will be useful for the development of the mathematical science proper.

In particular, we find that the paradigms of other sciences readily penetrate mathematical thought opening new promising directions of research. In fact, this is the principal way of development in mathematics, since any insider interests do not get beyond minor specifications and postponed proofs, while inventing a really valuable mathematical construct is impossible without a recourse to the current cultural demands (both in the material sphere and in all kinds of reflection). The job of a working mathematician is to attentively look at the world and seek for the ways of action common enough to admit of a kind of formalization; given the bulk of the already existing knowledge, such formalizations mostly adapt earlier theories, but sometimes, they give birth to an entirely new idea, and the lure of discovery is an essential complement to personal curiosity.

When it comes to the foundations of science, that is, the science's self-reflection, comparison with the others is of crucial importance. One cannot look at oneself but with the others' eyes. Here is a wide field for attractive metaphors, which gradually take the form of an intentional framework, and finally a unification platform.

On the topmost level, we are to explain why we need all those different paradigms together and outline the conditions for their interoperability. That is where the idea of relativity naturally enters. The possible unification platforms become the obviously analogs of the physical frames of reference, while the standard proofs of equivalence play the role of coordinate transforms. Just like in physics, there are families of dynamically different frames that cannot be reduced to each other without additional assumptions (inertial forces); however, within each family, all the relevant science is the same regardless of the choice of a particular formalism. Just like in physics, we are facing the problem of objectivity, since distinguishing real topics to discuss from spurious issues due to an inappropriate model is a non-trivial task requiring a wider context to embrace the whole hierarchy of such distinctions, allowing for hierarchical conversion and hierarchical development. Each individual layered structure can become a mathematical theory; but there is no universal mathematics equally applicable to any conceptual frame at all.

The impressive efficiency of science is in its power of abstraction, and hence lack of universality. Still, there is no abstraction without something to abstract from, and hence any science (including mathematics) can only approach the integrity of its object from one side or another, never capable of any exhaustive description; even combining all the existing and possible sciences together, we still remain abstract and approximate, since, to be truly concrete, one needs to advance beyond science, into the practical sphere. And here is the explanation for the frame-of-reference mystery. We are never interested in the particulars of any individual science; we only need schemes of action (prescriptions, recipes) applicable within a range of common situations. Of course, such schemes cannot be of an absolute value, they essentially depend on the modes of activity. As long as different people do something in a similar manner, they share a collection of formal implements, differently arranged in each individual reflective household. This objective structure of activity underlies the notion of a reference frame (or a conceptual platform).

The universality of reason means, in particular, that any aspect of activity can become a separate activity, and any separate activity can be included in another activity as its specific aspect. This is how scientific reflection becomes institutionalized science, and one science can be used as a paradigm for another. However reflexive development of mathematics does not mean that we could seek for the grounds of mathematics in mathematics itself, just laying out the foundations of science in terms of that very science. Mathematics is hierarchical; it cannot be reduced to a trivial flat structure. There are different mathematical theories, the abstract models of the corresponding objects; a mathematical theory of another theory is essentially different from its target, they belong to the different levels of hierarchy. Within a definite hierarchical structure, we can find that many theories are basically isomorphic to each other; still, isomorphism does not mean identity, it could rather be pictured as identity in a relative way, within a higher-level integrity and in this particular respect. This closely resembles physical frames of reference related to each other by inertial transforms (isomorphism) which are considered to be "physically equivalent". There is an obvious logical circularity, as we have to refer to physical equivalence to define inertial systems, while the notion of inertial motion refers to a class of physical commonalities; this circularity can only be broken from a higher-level perspective (the practical context). Similarly, seeking for the foundations of mathematics, we only establish a common practical context, allowing for formal transitions from one theory to another within the same higher-level paradigm. But we never invent this paradigm from nothing; it must reflect the common ways of action, and any change of paradigm will always come as a result of a cultural shift, virtually due to the development of the mode of material production.