Monadic Mathematics researches the symmetry concept itself, where symmetry is represented as an association between continuum (represented by a smooth line) and discreteness (represented by a collection).

These associations are ordered by quantity where each quantity is ordered by several internal symmetrical degrees, where redundancy AND uncertainty are their first-order properties.

This ordering method can also be defined as a systematic transition between totally parallel state to totally serial state.

The ordered building-blocks are like a Mendeleyev periodic table of symmetries, that can help us to define the deep relation between them and also to systematically define new symmetries.

Let us examine a very famous and important symmetry that can be found in the basis of many natural phenomena, the Fibonacci series.

By standard Math, Fibonacci series is the sequence 1,1,2,3,5,8,13,21,… where the next number is the sum of the two previous numbers, where the numbers are quantitative information forms.

By this approach, there is one and only one Fibonacci series.

By Monadic Mathematics there are many Fibonacci series because each quantitative value has several internal structures that are ordered by their internal symmetrical degrees.

For example:

http://www.geocities.com/complementarytheory/fibon.j...

By this method we have a very powerful and systematic way to research many already known and unknown symmetries, and also define the deep relations between them.

This is just a one powerful example of Monadic Mathematics in its first-order level, even before we used any arithmetic.

Symmetry:

Let x be a general notation for a singleton.

When a finite collection of singletons has the same level, it means that all singletons are identical, or have the maximum symmetrical-degree.

When each singleton has its own unique level, it means that each singleton in the finite collection is unique, or the collection has the minimum symmetrical-degree.

Multiplication can be operated only among identical singletons, where addition is operated among unique singletons.

Each natural number is used as some given quantity, where in this given quantity we can order several different sets, that have the same quantity of singletons, but they are different by their symmetrical degrees.

In a more formal way, within the same quantity we can define all possible degrees, which exist between a multiset and a "normal" set, where the complete multiset and the complete "normal" set are included too.

As this example of transformations between multisets and "normal" sets shows, the internal structure of n+1 > 1 ordered forms, constructed by using all previous n >= 1 forms:

1
(+1).=.{x}

2
(1*2)......=.{x,x}
((+1)+1).=.{{x},x}

3
(1*3).............=.{x,x,x}
((1*2)+1)…...=.{{x,x},x}
(((+1)+1)+1).=.{{{x},x},x}

4
(1*4)........................=.{x,x,x,x}.<---------- Maximum symmetrical-degree,
((1*2)+1*2).............=.{{x,x},x,x}.............Minimum information's
(((+1)+1)+1*2)........=.{{{x},x},x,x}..........clarity-degree
((1*2)+(1*2))...........=.{{x,x},{x,x}}.........(no uniqueness)
(((+1)+1)+(1*2))…..=.{{{x},x},{x,x}}
(((+1)+1)+((+1)+1)).=.{{{x},x},{{x},x}}
((1*3)+1).................=.{{x,x,x},x}
(((1*2)+1)+1)..........=.{{{x,x},x},x}
((((+1)+1)+1)+1).....=.{{{{x},x},x},x}.<---- Minimum symmetrical-degree,
.......................................................................Maximum information's
5.....................................................................clarity-degree
... ...................................................................(uniqueness)