The general category theory

The general category theory studies the structures of scientific ideas in general, in disciplines such as physics, biology and mathematics, but also, philosophy, and human and social sciences. Furthermore it applies to the ideas of all research disciplines in general, in a unifying manner.

The study of idea categories has been driven by the existence of conceptual or mathematical structures which cut across some aspects of all disciplines, even though they are unaware of each other. These structures often take the form of exact analogies, including several mathematical analogies in physics, and also some conceptual analogies between the physical sciences, the biological sciences and the human and social sciences.

The definition of the general category theory consists primarily of setting out categories of scientific ideas. For example, the ideas of physics form one category, that is one algebraic structure whose objects, known as ideas, are concepts, theories, hypotheses or problems, etc. from physics. It is assumed that this definition is clear and accurate enough to allow researchers in general to obtain interesting results.

The abstract definition of ideometry is primarily modeled on the definition of mathematical object categories. A category of scientific ideas is made up of the data from a class of such ideas and the correspondence between these ideas, known as ideomorphisms. Therefore let us set out a function F: C → D of a category C to a category D, which associates a scientific idea F(X) of D to any scientific idea X of C so that any ideomorphism f:  X → Y of C, is associated with an ideomorphism F(f): F(x) → F(y) of D, which preserves the structure of C, i.e. which includes a neutral element F(IdA)= Id(Fa) and which preserves the composition: for all objects X, Y and Z and morphisms f: X → Y and g: Y → Z of C, F(g • f) = F(g) • F(f).

In ideometry, C and D are defined as two fields of scientific ideas, for example physics and biology, X and Y as two scientific ideas from a single field of ideas, for example <elementary particles> and the <atom>, an ideomorphism (ideometric morphism) f or g as a formal association of scientific ideas.

Natural transformations

Another essential aspect of the general category theory is what is known as natural transformations. Undoubtedly their importance stems from their usefulness in studying the frequent, fairly obvious, but poorly understood character of certain fundamental relationships of mathematical objects with each other. For the general category theory, this basically involves the different ways in which the functions of scientific ideas interconnect.

So let us define natural transformations in such a way that they might be useful for the development of ideometry.

For two categories of scientific ideas and ideomorphisms C and D, we first define Ideas(C), all the ideas of C, and ideo (D), all the ideomorphisms in D. Then we define two functions F and G of C in D; we call natural transformation of F in G, and we note m: F → G, an application m of Ideas (C) in Ideo (D) verifying:

1)         for any object A of C

m(A): F(A) in G(A);

2)         for any ideomorphism f: A → B of C,

we have

m(B)F(f)  =  G(f)m(A)

Example: It is possible to check that the two functions corresponding to the properties of being one constituent unit of an order of reality and being a basis for productive evolution are linked to the function applying a type of evolution in a type of higher complexification of evolution, so as to make up a natural transformation.

 An implicit structure in physics, biology and social sciences
Natural transformation of correspondences between two orders of reality:

<constituent unit of an order of reality>
<evolution specific to an order of reality>

The ideomorphisms of ideas from physics and those of biological ideas are once again the ideas of other categories of ideomorphisms, which represent structures of ideas. Ideometric natural transformations highlight the structures they preserve, which can be presented in sequences or in tables of ideometric sequences. It appears that structure L represents a set of features from several natural transformations.

Posted online on January 29, 2016 (first version)