What are natural numbers?

We use them as one of the most convenient tools in our everyday lives. We use them for counting any tangible items. With their theoretical applications like arithmetic, the concept of natural numbers is one of the primary grids that we see the world.

There are, however, two things we tend to take for granted.

First, natural numbers don’t necessarily show the world as is. A set of them is merely one of the many other grids through which we perceive the world, while it is one of the most basic and intuitive grids that we use consciously and unconsciously.

Second, because of this, we tend to forget the fact that the concept of natural numbers requires a rigorous definition. We have to define it to use them more consciously and to see the limitations as a grid that what we can see and can not the world and the universe.

It reminds me of this story. Mathematically, proving 1 + 1 = 2 is a bit complicated. It requires lines of formulas to conclude it. What intrigued me is the initial portion of this mathematical inquiry. We have to start with the definition of natural numbers. Usually, we use Peano five axioms.

In 1889, Italian mathematician, Giuseppe Peano introduced them. Like the axioms for geometry devised by Greek mathematician Euclid (c. 300 BCE), the Peano axioms define a rigorous foundation for natural numbers.

These are as follows:

- Zero is a natural number.
- Every natural number has a successor in the natural numbers.
- Zero is not the successor of any natural number.
- If the successor of two natural numbers is the same, then the two original numbers are the same.
- If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.

What surprised me are the first and third axioms. Zero is part of natural numbers; at the same time, it posses an exceptional position.

As mathematicians point out, these two are arbitrary. And yet, this arbitrariness implies a very nature of the fact that all grids we use to see the world and the universe are arbitrary in one way or another.

Let me rephrase these five axioms in my words.

In natural numbers, there is a beginning. Such is “zero,” which means nothingness. There was nothingness, out of which, “one” came out. It sounds like this for me.

And God said, Let there be light: and there was light.

Genesis 1:3

The second axiom is apparent. It is self-evidently axiomatic. All natural numbers have their successors. 1 has 2. 6 has 7. 124 has 125, and the like. And even 0 has 1, like “And God said, Let there be light: and there was light.”

As above-mentioned, however, the third axiom sounds tricky. Among all natural numbers, 0 is indeed exceptional. As it is the ultimate beginning, 0 is not the successor of any natural numbers. It is the starter and the creator of the natural world and the universe, as far as this particular grid is concerned.

The fourth axiom explains a sort of inductive functionality of natural numbers as a tool. Because of this, we can use them for counting any items on various comparisons. If I have 5 items and you also have 5 items of something else, then at least we can agree on the fact that the number of items is the same. It is one of the primary bases for institutional knowledge such as money.

Lastly, the fifth axiom is also intriguing. All-natural numbers have their successors, including zero. All of them are under the control of causality. 1 is after 0. 2 is after 1. 8 is after 7. These can never be like 13 is after 7 (skipping) or 3 is after 8 (looping). In other words, we can never have a time machine in the world of natural numbers. Time travel is impossible.

However, there are exceptions. There is nothing before 0. And, 0 also means nothing; nevertheless, 0 “exists” as one of the natural numbers. So, it looks like:

Nothingness that does not exist, 0 (nothingness) that do exist, and 1, 2, 3, 4, 5, 6…

Like God, as we can not see Him, we can never see anything before zero through the grid of natural numbers. Such absolute nothingness or emptiness did create zero, just like “And God said, Let there be light: and there was light.” .

If so, could this light be zero? Both are equally elusive. They don’t exist; at the same time, they do exist. How? Perhaps they are the connectors between eternal and temporal. We can’t see the things outside the grid of natural numbers through the same grid of natural numbers.

That was the true Light, which lighteth every man that cometh into the world. He was in the world, and the world was made by him, and the world knew him not.

John 1:9-10

While we look not at the things which are seen, but at the things which are not seen: for the things which are seen are temporal; but the things which are not seen are eternal.2 Corinthians 4:18

Image by Gerd Altmann