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Synthesis of Modern Mathematics : Intuitive Category Theory (Philosophy)

On the first blog of the post, I wrote about the fact of around four theosophists in Grothendieck's list of mutants. Grothendieck is considered to be the greatest mathematicians of the last century informally by some of the eminent mathematicians. Although Samuel Eilenberg and Saunders Maclane gave the first definition of category, functor and natural transformations in their 1945 paper, the recognition and development of a separate mathematical discipline of category theory has been generally attributed to Grothendieck's influence by the mathematical community.

The following parallels between theosophy (as an occult philosophy) and category theory in contrast to current set-theoretic approach to science might give us a clue to the deeper thought patterns of this mathematician: 

1. The study of theosophy like that of geometry proceeds from the universal to particulars. This is also the approach of category theory which can be thought of universal mathematics. It is in distinct contrast to set-theoretic approach focusing on the study of separate objects as sets.

2. Set theory is particularly suited for the knowledge of the parts (or what David Bohm calls fragments) in contrast to category theory which emphasises knowledge of the whole (by connecting arrows across seemingly separate fragments) or the nature of universal in agreement with theosophy. 

3. Category theory is suited for global interactions based modelling while set-theory perfectly caters to local models.

4. From a physics perspective, category theory is a highly abstract theory much suited to understand the underlying cause or noumena as opposed to set-theory suitable for objective material phenomena or effect.

Next are the scans of the rough distinction as well as how the axioms of the set-theory can be deduced from pure category theory, both of which I had intuitively noted down last year. This is not precise and there might be changes in times to come.





Intuitive approach to defining a category (detailed axioms in another post)

Lets understand the intuition underlying the axioms of a category. For this I am taking the approach of Nathan Carter's book visual group theory.


Take a square card which we will call as object A (category consists of a collection of object) and label its 4 corners say 1,2,3 and 4. Start by placing this card as shown in top left corner. If we don't do any transformation on this object then we will call this as identity transformation(this transformation is the identity morphism of object in a category). If we rotate this object by 90 degrees clockwise (the axis of rotation is a line perpendicular to the page passing through its centre point) we get a transformation as shown as transformed card in the upper right corner (the transformation is nothing but a single morphism out of of a collection of morphisms in a category). Further rotation by 90 degrees clockwise yields right lower transformed card. This transformation can also be obtained directly from the card at the starting position in the top left corner and is considered as a composition of earlier two transformations one after other and is depicted as diagonal transformation which is clockwise rotation by 180 degrees (2 morphisms with a common codomain and domain can be composed in a category). No transformation composed by a rotation by 90 degrees clockwise is the same as first rotating by 90 degrees clockwise and then doing no transformation (unit law of a category). The transformations are associative and can be verified concretely by transforming the square (associativity law of morphisms in a category). This is a special example of category in which there is a single object and all the transformations are invertible (the counter-clockwise rotation by 90 degrees is the inverse transformation of clockwise rotation by 90 degrees) and is mathematically defined as a group (its a transformation group of the object A). In a general category there will be a collection of many objects and morphisms are not necessarily invertible.

We just saw a real world special easy example of category theory (or more specifically also group theory since group is a special case of category). 


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