Formal Logic and Mathematics

Formal Logic and Mathematics

Logic in general belongs to the synthetic level of reflection. No analytical study can cover all aspects of logicality. However, analysis is certainly useful for us to comprehend our logicality and seek for its possible improvements. In real life, logic is incorporated in all branches of conscious activity, and this is its only real mode of existence. Analytical approach mentally separates logic from its manifestations, splitting the whole into many partial aspects, each of them, in its turn, being a kind of integrity, so that the process could be continued as long as needed. Of course this is nothing but an expression of the hierarchical nature of human activity, with its movable distinction between activities, actions and operations. In this way, we eventually come to the art of logic, the science of logic, and the philosophy of logic. Basically, art provides us with the raw material to further formalize in science in the lines of some philosophical categorization. This "objective" treatment of logic is quite possible and justified, since logic is objectively present in any culture, on any level of development. The diversity of practical cases may result in numerous complementary views of logic tending to unfold into self-contained disciplines. Such abstract representations are objectively necessary and quite valid, provided we do not forget about logic as a universal aspect of any activity at all. In other words, discussing a particular feature of something, we should not forget about that very something behind the scene, without which our considerations would be pointless.

Leaving aside, for a while, the artistic visions and philosophical teachings, let us ponder a little on the scientific side of logic commonly known as formal logic. From the very beginning we turn down the pretense of some scientists to the exclusive right of dealing with the subject; even less logic can be restricted to an individual scientific theory. Formal logic as a science (taken in a however wide sense) is not identical with the logic of science (including the science of logic). And, stressing it once again, formal logic is not indeed logic; it is merely the similarity of the name that leads to confusion. Though a scheme of a science can be elevated to the level of synthetic reflection and thus become applicable to people’s activities as a part of their logic, this transformation is never inherent in the source science; it requires a special activity.

As science does not necessarily need to be institutionalized, formal logic too can penetrate all the levels of culture, from material production to the most sublime spirituality. We do not need a high school course to acquire outstanding skills in many everyday activities; though most of them admit formalized education, it is often an option, a matter of personal inclination, rather than an absolute necessity. Similarly, formal logic as a branch of science is a complement of the background usage, which does not render formal logic less scientific in character.

The full-fledged formal logic is a very peculiar branch of science, as it seems to have no real object and apply to anything at all. There is yet another science showing as wide universality, namely, mathematics. One is strongly tempted to identify the two, presenting them as a common foundation for scientific thought in general. Indeed, logical rules are readily representable by mathematical structures; and conversely, mathematics could be pictured as a reformulation of formal logic. Yet formal similarity does not mean identity. Who would serious contend that the smell of a rose is the same as electric current or market price levelling on the grounds of the standard diffusion law underlying these and many other phenomena? In any comparison, commonalities and distinctions are equally important. The great unification program somehow keeps glitchy, and there is yet no purely logical mathematics, and mathematical logic still does not (and will never) reflect all the aspects of formalization.

Intuitively, mathematics tells us how one thing is connected to another, in a most general sense. We admit that all the conceivable constructs co-exist in some huge formal domain, and all we need is to track the interdependencies and reveal formal similarities. However complex, this object area is deemed to be the same, so that the development of mathematics is to extend our acquaintance with its reign rather than the reign itself. Mathematical entities are essentially static; we can relate one entity to another, but we are not allowed to change them in the course of study. This vision can be expressed in a few words: mathematics is the science about structures. This readily explains the apparent ubiquity of mathematical methods, as structures are everywhere, and, in particular, any science is structured.

Formal logic is different. Its main purpose is to demonstrate how one structure can be obtained from another. What mathematics takes in simultaneity formal logic unfolds into a sequence. What mathematics pictures as mutually connected formal logic represents as a process of connecting one part to another through the work of some deductive machine. That is, we take some structure as input and produce another structure as the output following the rules incorporated in the derivation scheme. Obviously, this is a very general idea of a system. Formal logic can consequently be characterized as the most general science about systems. This, of course, assumes the presence of all the components of a regular science, including the empirical level, theory and general methodology; the traditional term "systems theory" currently tends to refer to any of these levels, depending on the context.

Now, the ubiquity of formal logic quite understandable, as systems are everywhere. Any science is a system too, including mathematics, whose deductive schemes are often explicit and intentional. Moreover, it is the prevalence of the systemic aspect that gives science its dedicated place in the modern culture. Indeed, knowledge (the principal product of science) is for learning. Early science just sorts out facts and skills, theoretical science allows sharing the principles of processing, but in any case we are given a formal tool for decision making, a collection of useful recipes to pass from one person to another.

The mutuality of formal logic and mathematics thus gets a simple explanation, since any structure can be "serialized" (albeit assuming a nontrivial underlying structure), as well as any system can be decomposed into a structure (albeit built of systemic elements). Structures and systems are the different aspects of the same; so are mathematics and formal logic. The opposites look much like one another, just because they are opposites. In the philosophical language, they are said to be mutually reflected. However, they can never be reduced to each other; again, just because they are opposites.

In particular, the area of mathematical research known as mathematical logic has nothing to do with formal logic, and even less with logic as such. The product of such theories remains structural, static; it does not imply any immediate application. To suppress terminological liberty, one could call it somewhat else, rather than logic; the result does not depend on the nomenclature. Such "logics" can be most intricate and peculiar; still, most of these mental toys will never refer to any real activity, and even mathematicians will hardly ever act like that. It is only in the framework of some other science that mathematical logic could develop systemic qualities, thus leaving the domain of mathematics and becoming a branch of formal logic. And it is only in the context of some practical activity (however reflexive) that both mathematics and formal logic transform into a kind of regulatory mechanism, logic as such. Thus applied science does not coincide with neither mathematical abstractions, nor formal derivation schemes; it will always include some elements foreign to scientific rigor.

Mathematical logic, just like any other branch of mathematics, was certainly inspired by some objective phenomena related to the organization of human activity (conventionally restricted to sheer reasoning). But a formal structural description is not enough for a realistic scientific model of (at least) formal reasoning; it must be complemented by systemic interpretations as well as certain pragmatic considerations relating the science to its object.

Of course, mathematics as science is never entirely structural; it incorporates the whole range of formal and informal components. The same hold for formal logic as a scientific discipline. It is this practical commonality that leads their numerous interdependencies.

From the logical viewpoint, the opposition and complementarity of the static (structural) and dynamic (systemic) modes of description means that there is yet another paradigm, a synthesis of the both, combining the features of a structure and a system. This idea could be conventionally called "hierarchy", dynamically layered complexity. Taken in any particular respect, a hierarchy shows up as a hierarchical structure (a number of levels) or a hierarchical system (transforming one hierarchical structure into another). When the levels of hierarchy are treated as structures, the inter-level transitions are systemic; conversely, considering the levels of hierarchy as systems, we get a structured collection of the possible scales.

Since any hierarchy can only manifest itself through hierarchical structures and systems, the idea of hierarchy as a self-contained paradigm may be different to grasp. To begin with, one could think about something that is left when we take the structural load from a hierarchical structure, and the systemic background from a hierarchical system. In other words, there is a clearly perceptible difference between hierarchical something and a "plain" thing. This distinction is neither structural nor systemic; that is why we need a special term to denote it, just to be able to proceed.

The irreducibility of hierarchy to mere structure or system means that there is no "natural" (or "inherent") structure or system in a hierarchy. We can start with any element of the hierarchy and unfold it into a number of hierarchical structures or systems, which are not arbitrary but widely flexible. Than we can fold the hierarchy in a different element and unfold it from scratch, in a different manner. Such hierarchical conversion makes the distinction between the levels relative, and the very notions of "up" and "down" depend on the particular representation. An unprejudiced mind will readily observe the same picture in any cultural sphere. Hierarchies are as ubiquitous as structures and systems. In particular, the hierarchical nature of science is quite obvious. There are all kinds of "vertical" relations (the structural aspect), as well as numerous examples of generalization or specification (systemic transitions). For each science there is a meta-science; thus, one could speak about meta-mathematics and meta-logic (meaning the reflexive form of formal logic). However, the principles of ordering may vary, and no science can be said to be above the others in an absolute sense.

As a complement to both mathematics and formal logic, one could also imagine some science of hierarchies, which would accumulate knowledge about the possible usage of structural and systemic models, treating the history of science as a manifestation of the its inner organization as well as revealing the cultural background in the foundations of science.

[Logic] [Hierarchies] [Unism]