The mathematical model is used in modern mathematics which uses axioms to develop each theory. The language and logic used is almost always the classical language and logic investigated first by Aristotle. In English, this language, at the very least, includes such language-forms as "and", "or", "not," "If .... then ....," "... if and only if....," "for each," "there exists" and all equivalent forms. In addition, the language includes forms that are classified as "predicates" as well as the notion of the logical variable. For example, the sentence "God is good" can be replaced with a one-place predicate "God is --- => G(-)." Then the --- is replaced with either a constant "good" (G(good)) or a variable "x" (G(x)). Consider the two-place predicate "The boy went to --- with --- => B(-,-)." Again each --- can be replaced with a constant "term" or a variable. For mathematics, consider the three place predicate "--- + --- = --- => A(-,-,-)." For number theory, A(2,3,5) holds. For a mathematical language, additional notions such as functions and formal terms might also be included.
Modern mathematical theories
A modern mathematical theory is the set of all conclusions deduced from a set of axioms. The axioms are written in these classical forms and, usually, classical logic, along with other acceptable processes, is used for the deductions. Independent from the motivation that determines a set of axioms, the actual pure symbols used within a mathematical theory are not supposed to carry any meaningful content.
Constructing a mathematical model
There are, in general, two types of mathematical models. On this page, only one type that is of the most interest is presented. Suppose that statements from another discipline, not classified as pure mathematics, can, at least, be partially expressed using the classical language-forms. If the terms used within these statements are associated consistently with the pure abstract symbols used to develop a pure mathematical theory, then the most basic requirements for a mathematical model have been met.
Often the associated symbols are given names that are exactly the same as the discipline terms. For example, the physical theory of "dynamics" can be associated with the pure mathematical theory "linear algebra." Symbols called "vectors" in the mathematical theory can take on the names "force, velocity, acceleration." Even special symbols F, v, a, are used in order to represent these physical notions. There are cases where the names used indicate a collection of terms from the discipline. For example, in the discipline that investigates logical discourse, an entire collection of processes that leads to a deduction from a set of hypotheses has been termed as a "consequence operation." Then this consequence operation generates a specific "consequence operator" which is a member of a mathematical model. In all cases, this constructed association is called an interpretation.
The mathematical model
Either mathematical theorems have already been deduced that contain these interpretations or new theorems are mathematically obtained that contain these interpretations. If the original discipline uses a language restricted to a defined physical language, then, often, the mathematical model mimics physical behavior or physical processes. Thus, any new discipline statements are deduced in the most accurate logical manner known to humankind.
The mathematical model itself is, at the least, the collection of all of the mathematical theory statements that contain the original interpretations as well as others that have been specially named.
The mathematical statements can contain other terms that are not part of the original interpretation of the disciple statements being modeled but these terms do have meaning for the original discipline. For example, these meanings can be relative to comparative statements or related to various modes of measurement and counting. For the General Grand Unification (mathematical) model, terms such as ultralogics, ultranatural laws, ultranatural events, and subparticles denote mathematical objects generated automatically by the pure mathematical theory. They have been given these names since they have many characterizing properties that are similar to the properties that characterize the objects obtained when the "ultra" prefix is removed. Further, they have new properties that are different from the modeled original objects and these new properties can be meaningfully compared with the interpreted original objects.
The General Grand Unification Model (GGU-model) is an example of a reinterpretation. The original mathematical model is used to model general logical discourse. Then the objects used for this purpose are reinterpreted as models for collections of physical processes that produce or alter the behavior of physical-systems. The reason this is possible, in this case, is that physical laws and scientific theories can be modeled by general logic-systems and, hence, corresponding consequence operators. The other standard physical processes represented by GGU-model objects are similar to the three significant human activities of finite choice, finite ordered choice and using finite combinations of defined "simple" objects to produce a more complex ensemble.