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On the Origin of Numbers

The problem of what is more basic, the number concept of zero or the number concept of one, resembles the popular riddle of the chicken and the egg and which comes first.
You can only answer these questions if you know where and when the objects of your comparison came into being. And you must know what you are talking about, eg. what constitutes an egg, respectively what does a number stand for. Perhaps more a matter of taste than of definition…

Paleontology of the Chicken and the Egg

The Red Junglefowl, Gallus gallus, is a tropical member of the Pheasant family, and is often believed to be the direct ancestor of the domestic chicken. Recent evidence suggests that domestication of the chicken was under way in Vietnam over 10,000 years ago.
Fowl were the first neognath lineages to evolve. From the limited fossils that have to date been recovered, the conclusion that they were already widespread – indeed the predominant group of modern birds – by end of the Cretaceous is generally accepted nowadays.
The fossil record indicates that birds evolved from theropod dinosaurs during the Jurassic period, and the earliest known bird is the Late Jurassic Archaeopteryx, circa 155 Ma (million years ago).
Dinosaurs diverged from their archosaur ancestors during the Middle to Late Triassic period, roughly 20 million years after the Permian-Triassic extinction event wiped out an estimated 95% of all life on Earth.

The amniotes are a group of tetrapod vertebrates that have a terrestrially adapted egg. They include the Sauropsida (reptiles and dinosaurs, including birds) and later on in evolution the Mammals. Their eggs could survive out of the water, allowing amniotes to branch out into drier environments. The eggs could also "breathe" and cope with waste, allowing the eggs and the amniotes themselves to evolve into larger forms. Amniotes are defined by embryonic development that includes the formation of several membranes. The amnion is a membranous sac that surrounds and protects the embryo.
The first amniotes, such as Casineria kiddi, which lived about 340 million years ago, resembled small lizards. Their eggs were small and covered with a membrane, not a hard shell like most modern amniote eggs. The outer membrane, a soft shell, evolved as a protection against the harsher environments on land, as species evolved to lay their eggs on land where they were safer than in the water. In fish and amphibians there is only one inner membrane, also called an embryonic membrane.

From the above we may conclude that the egg as such is about 34,000 times older than the domesticated chicken Gallus gallus.
But what motivated the original question? And can any evolutionary perspective, with proper definition of both chicken and egg, supply the correct answer?
A chicken and a chicken's egg are mutually dependent, one of these objects cannot be formed without the other. Perhaps this children's riddle is meant to encourage the realization of this biological interdependence within the mental boundaries of a chicken run, not within the scientific framework of millions of years of evolution…

The Story of Counting

Long before man started counting, the brains of animals performed similar calculations. To distinguish between 2 and 3 bananas is not difficult for a monkey, and further down the ladder of life there's the sensory distinction of the presence or non-presence of chemical substances, light and heat as perceived by primitive living creatures in their immediate environment. Though sensory information can best be captured on a gliding or logarithmic scale, deep down in the nucleus of a cell sensory impulses must trigger some sort of yes/no switch, to change for example the outward behaviour of a bacteria.
A microbe already shows a response – depending on its inner state – to the quantity 1 (of something) and 0 (nothing). Of course less complex beings have less voluntary responses, resembling the automated output of a computer. In a simple non-random arithmetical program the same input always results in the same output number. In biology, as the incoming molecules more directly trigger the cellular reflex, cause and effect fall together in a single chemical process.
It is only when the children of man learn to reason and count, that the numbers 0 and 1 are dressed up with mathematical permanence, and become a philosophical problem in need of further clarification.

On the number line 0 is positioned before 1, but the origin of the number line lies in the counting of natural numbers.
What is counting? Counting is the repeated addition of 1. The most primitive form of counting (as embodied in a single binary switch) stops going upwards at the number 1. But for us, when we count natural numbers, first comes 1, second 2 which is 1+1, third is 3 = 2+1 = (1+1)+1, etcetera, until our ruler is filled to the edge (mathematicians may use a ruler of arbitrary or infinite length, without edges).
The concept of zero arises from the question: how much you have when you haven't started counting yet. Without a counter or ruler the number zero is meaningless on its own. It is only when the practise of counting is established that zero becomes a useful concept, namely as the basic natural number to count upwards from.
Counting is fundamental to all natural numbers, and we usually demonstrate the number 0 after we know how to count, that is how to move up and down a ruler. In this view the number 1 has primacy, because without the activity of "adding one" natural numbers cannot be defined.
Even so, in a situation where the counting number 0 still awaits definition (with the axioms 0+1 = 1+0 = 1, which presuppose the existence of 1), the number 0 arguably exists outside of counting as the primal concept "nothing" which lies before any (quantity of) "thing".

After all this talk about counting things it's time to address the question of the primacy of 0 or 1 using more advanced mathematics, drawing inspiration from the algorithms for Big numbers I developed recently.

Essential Operations

Addition and multiplication are the essential arithmetical operations to create natural numbers from entities 1.
As we write natural numbers in their singular form as series of ones 111... it is immediately clear that we can add two numbers just by placing them next to each other. Then there is no need for a special operator of addition "+".

1...[1#n] = n
ab = 1...1... [1#a 1#b] = a+b

In the above expressions the square brackets are there to explain what's going on – within these brackets we find a meta-expression which quantifies the length of the series of ones in the equation. The "#" here is a meta-operator for repetition, X#n repeats the characters X an n number of times.
In practice we won't need to write the meta-part, for example:

a = 11 & b = 1 => ab = 111

Introduce the concept of substitution of the entities 1 that make up a natural number n to define multiplication. As usual multiplication works out as repeated addition of a number a.

n [1:a] = 1...[1#n 1:a] = a...[a#n] = a*n = n×a

Super-exponentiation

Just like multiplication is repeated addition, so is exponentiation repeated multiplication.
We can generalize these types of recursive operations in a function bigE.

E(a,b,0) = E(a,b) = ab = a+b
E(a,b,1) = a*b = a...[a#b] = b×a
E(a,b,11) = a**b = a*... [a#b] = a^b
E(a,b,c1) = a*...b [*#c1] = a*...... [*#c a#b] = a^...b [^#c]

Here the meta-repetition *#c indicates that operators *...[*#c] are placed inbetween the repeated variables a. Therefore addition can be represented by *...[*#0], an operator with a series of * of length 0.

A complete definition for the first three parameters of super-exponentiation function bigE is given by:

E() =   = 0
E(a) = a
E(X,0) = E(X,) = E(X)
E(a,1) = E(a1) = a+1
E(a,b1) = E(E(a,1),b) = a+b+1
E(a,1,1) = E(a)
E(a,1,c1) = E(a,1,c) [c>0] = a
E(a,b1,c1) = E(a,E(a,b,c1),c)

Class Representation

A few introductions to the numbers zero and zero squared (which need not be totally equal):

01 = 10 = 1
0 = 1...[1#0]
0 = 0...[0#1] = 0*1 = E(0,1,1) = E(0,1,11) = 0^1
0...[0#0] = 0*0 = E(0,0,1) = E(0,E(0,1,11),1) = E(0,11,11) = 0^2

All numbers belong to certain classes which can be meta-numbered.
Operators and operator functions also can be sorted in classes. Addition is in the same class as multiplication and exponentiation, but series of ones are automatically added, which means addition is fundamentally not an operator (yet), so exponentiation also must have a class number zero.

It seems that last sentence produces something out of nothing. However, to come to this level one would have to have counted (meta levels) up to 2. Also, having "no operator class" doesn't exclude having "number classes" 1...

Zero Origination

With hindsight one can discern (and indeed meta-count) several starting points for the origination of the concept of nothing, which are levels of zero:

  1. Before the concept of a thing, 0 has no separate existence. All is one, everything is nothing, all is equal. Here true and false have no meaning. This state of 1=0 is the starting point for the axiomatisation of mathematics.
  2. When we allow a thing or quality to exist, it's not yet clear how we can count or measure its quantity. The fundamental 1 rests in its own, wholly different from its negation 0, now the sign for nothing. All we know about a number (of things), which is effectively described by the variable 1, is that it doesn't amount to 0.
    This state of the universe is essentially Boolean: any fact is either true or false. But there can be no grouping or listing or counting of facts, because facts do not yet have a separate existence and cannot be truly set apart. Things can be recognized, but each cognition starts anew in an eternal now.

Egg fell from the nest...

So far: Asankhyeya Shaktivedanta Okapi