Mathematical
logic is an attempt to fomalise
reason. Statements and their meaning become mathematical objects; reasoning becomes a computational process. The purpose of mathematical logic is to make the assumptions and the reasoning steps in logical arguments
explicit.
The foremost example of this approach is Euclid's work on the laws of (Euclidean) geometry. Its basic idea is to define the world of geometry in terms of a few logical statements about the primary geometrical concepts. All geometrical reasoning can then be performed by applying universal logical reasoning to these logical statements.
This approach is very appealing for its elegance: it creates an autonomous formal universe of logical statements about geometrical concepts, and all geometry can be done in terms of such statements. You can define additional concepts (in terms of logical statements about them) and hypothesize properties (more logical statements) that can then be investigated for their validity. Logical statements and universal laws of logical reasoning are all you need. This makes a very clean, objective, powerful, and elegant tool for problem solving.
Within this approach, definitions and properties look exactly the same: they are logical propositions about concepts. When reasoning about a part of the world in mathematical logic (i.e. modeling that world in terms of logic), you take a basic set of axioms: propositions that are assumed to hold in that world. This establishes a set of basic concepts and defines properties. You can then define other concepts in terms of the given ones, and study arbitrary propositions about the defined concepts. A proposition may turn out to be either
- valid (it logically follows from the axioms)
- invalid (it logically contradicts the axioms)
- contingent on the axioms: given the assumptions it may be true or false
This method only makes sense if some of these propositions are
interesting - if they represent real and relevant questions about the piece of the world being modelled.
The method provides a very clean abstraction of the deductive reasoning process. Every statement used in the reasoning is modelled explicitly, and so is the reasoning itself, which operates entirely on the logical statements, without drawing on any additional knowledge of the world. So when a problem arises, it can be pinpointed exactly: the reasoning may be incorrect, or the model may be incorrect, with assumptions that do not hold, or incomplete in which case too few assumptions are used to capture the subject in sufficient detail.
If the subject matter itself is described mathematically, we can apply model theory, the branch of mathematical logic that studies the relationship between the truth of statements and their provability. Kurt Gödel proved that reasoning systems have a fundamental limitation in this regard: in general, not all valid propositions (i.e. those that must be true given the assumptions) are provable (i.e. can be derived from the assumptions using the logic's deduction rules).