Philosophy of Natural Mathematics by Justin Coslor 2021-12-20
Rough Draft
     Small whole numbers get used far more than larger numbers in Natural Mathematics. It is a reassuring fact that there is a pair of even numbers around every odd number. Every even number is compatible and countable in every even base of counting. Every Natural Number treated as a base of counting contains the bases of its composite, and all of those bases are compatibly divisible and connectable by their contents.

     Every odd Natural Number contains a center object, including: on a numberline of whole number objects, including center objects of odd total counts of objects arranged into square plates and odd total counts of rectangular plates of rows or columns of numberlines of whole number objects, in stacks of such plates, as a line of stacks of such plates, in rows or columns of lines of stacks of those plates, in stacks of stacks of rows or columns of lines of stacks of plates of rows or columns of lines representably connectable as a numberline, as well as there being center objects in odd groups of objects in diameters and circles and spheres and hyperspheres of objects in space, and potentially in other ways. Numberlines of objects can be juxtaposed, such as in parallel or perpendicular, and connected or separated, placed next to one another for contrast, including when the objects are around circles and spheres and hyperspheres of objects in space. When the total count of objects is odd, at each scale of reference, those portions always have a center object. Individual groups of objects can be represented as one or more larger objects.

     Twin Primes . . . always have an even center object between them as odd even odd, and beyond the number four the Natural center object of twin prime pairs seems to always be divisible by six, because twin prime midpoints are even thus divisible by 2 and they are part of a three part sequence of ones as odd even odd. A pair of twin primes added together minus their center object number equals their center object number. Every three sequential number sequence in the Natural Numbers and Integers whose bounding numbers added minus its center number equals its center number. That concept is also true when a numbered line of objects has three objects symmetrically spaced any distance apart around the middle object between the other two objects. 

     Prime Number Neighborhoods as prime + numbers + center object + numbers + prime, beyond two, always have a center object and that center object is sometimes even and sometimes odd. The Prime Number Neighborhood PNN = prime + prime gap numbers with the Prime Number Neighborhood whole center object in the center + prime ... I like to explore this sometimes with sequential prime numbers and Prime Triangles On The Numberline with Prime Connection Stacks of exponent placeholders above primes of composite bases and this may also be useful with PNN's with one or more primes in the prime gap of the PNN but that would get complicated. While noting that 1 is not prime, here is a different process for comparison: 1 + 2 + [center object 4] + 2 + 1 = 7 and its center object is even, and with Prime Number Neighborhoods,[PNN 3 to 7 starting with 3 as the first 1] + 1 + [center object 5] + 1 + [outer PNN node 7] = 5 nodes and its center object is odd. PNN 3 to 13 has an even center number as [3 as the first 1] + 4 + [center object 8] + 4 + [outer PNN node 13] = 11, and the center object 8 is an even number in this 11 number sequence of whole numbers. [PNN 13 to 17] = 1 + 1 + [center object 15] + 1 + 1 and the center object in this five number whole number sequence i.e. 15, is odd as odd + even + odd center number + even + odd. [PNN 23 to 29] = 1 + 1 + 1 + even center object prime number neighborhood midpoint 26 + 1 + 1 + 1.

     Whole Numbers are Natural Numbers and Natural Numbers are Whole Numbers and they begin with the Whole Number 1. Natural Numbers that contain a two as one of its factors can be divided in half as a pair of blocks of objects, noting that even number zero is not an object though it is a position, and if the Natural Number contains more than one two in its group of factors it can be divided as a pair of pairs .. of blocks of numbers. If there are two 2's i.e. the group of non-even primes connected is multiplied by two to the exponent two. If there are three twos in the Natural Number's factors the number that represents the group of connected prime factor objects can be divided up into a pair of pairs of pairs of objects with its non-even prime factors existing on the numberline between exponents of the number 2. Natural Numbers (beginning with 1) are good for representing objects, and Integers (beginning with 0) are good for representing postions. All primes are Natural Numbers but not all Natural Numbers are prime because 1 is not considered to be a prime number though it is self-contained and continuous as the first object of the logic of mathematics.

     In Natural Number Systems of counting in odd number bases there is always a Prime Number Neighborhood containing a center object and bounding primes. All primes beyond two have a center object because they are odd Natural Numbers (whole objects that go odd, + number(s) + center object + number(s) + odd), and that center object is sometimes even and sometimes odd. Sometimes the "numbers" in this model are even and sometimes the "numbers" in this model are odd, when "numbers" = "numbers". With sequential odd primes in this beyond two model, a pairing symmetry of repetition "numbers" exists inside the Prime Number Neighborhood and when it has a center object, that center object is sometimes even and sometimes odd. When objects are sequential they can be thought of as a number line. When the PNN bounding primes are nonsequential there is not always a center object unless the PNN total count of nodes is an odd number, that center object is sometimes even and sometimes odd. When the total count of ones is even then in symmetrical hyperspheric object space there is no center object. Zero is not an object, it is just a position or node. Nodes are singular but Natural objects can be a prime or contain a composite of primes, think of space as zero and the First Object as One. Natural Number object numberlines begin with the first whole number object. On a Natural Number object numberline there is one more node than there is on a center-object-less circle or sphere or hypersphere that contains the same number of one-to-one transition lengths between nodes because a numberline has a beginning node or beginning object and an end node or end object. When a numberline is formed into a circle the beginning node overlaps the end node, and I call that a "Point Of Reference".

     In recursive rectangle block space, there is also a center number node when the total count at each perpendicular exponent dimensional juxtaposition is odd. One of the reasons why that is true is because there are an infinite number of ways to draw lines. An example of this is to imagine a never-ending story as a line of text or symbols.

     In the base ten natural number exponent equation 10^n / n! read as "ten to the nth power divided by n factorial" ... when n is at its 24th whole number step then 10^n read as "one octillion = 1E24 = 10^24" is the last step in which the numerator of the equation is larger than the denominator 24!, and since n! means there are n! unique objects, then only when n is at the 25th step and beyond, there are more unique objects than 10^24.

     A prime number is a number that does not fold perfectly with any whole number before it other than one, yet the first prime number is the number 2, and there is always a remainder when a prime number is divided by any whole number before it other than the number one. All primes except for the number 2 are odd and have a whole number center object because they are odd. It seems that very few odd numbers are prime. There are more non-prime numbers than prime numbers and the proof of that is the fact that beyond one there is a pair of even numbers around every odd number and many odd numbers are not prime, so the node sum of all of the even numbers plus the node sum of all of the non-prime odd numbers is far greater than one-half of all whole numbers.

     Primes can be organized as groups between powers of two.

Footnotes:   
Small numbers. Pairs. Even bases are compatible and countable. Even Countable. Odd numbers have a Center Object that makes their pairs around the center object odd. Twin Primes. Prime Number Neighborhoods as prime + number(s) + center object + number(s) + prime. Center objects of odd numbers are sometimes even and sometimes odd. Even Factors as pairs of pairs of pairs etc. All primes beyond two have a center object because all primes beyond two are odd in Natural Number Mathematics. In recursive rectangle block space, there is also a center number node when the total count at each perpendicular exponent dimensional juxtaposition is odd. 10^n / n! and n! unique objects. Primes can be organized as groups between powers of two.