Warning: this article can make you feel dizzy.

My son is almost four years old, that age when children push the limits all the time. He asks for a candy, and we say "ok, just one, after the lunch!". Then he thinks, looks at his hands' fingers, and replies "just three" or "just four". Sometimes, he smiles and cheerfuly says "just ten!".

Then I grew curious: how does he grasp numbers? For example, the number three; what does he think when he asks "three" candies? A sweet aftertaste that is longer than usual? Three fingers? Or perhaps one candy in each hand and another in mouth?

But, thinking over it, what is the real meaning of "three"? Three oranges, three apples. Why do we say that there are three fruits in the basket, instead of four, five, or two?

What is this important for? What's the difference, anyway? Why count things? Animals don't count and do well, and may be happy in spite of that. Is "three" something that really exists in Nature, or is it a product of human imagination, like fairy tales?

Another wrinkle in this subject: if the set of natural numbers is infinite, how does God handle this? Does He know every integer number? Would that be possible, even for Him? (Now that's something that Christians should spend time on, instead of debating if the sexual position XYZ is sinful.)

We learn that 2 + 3 = 5. Is that really true? If we can't define precisely what "two" or "three" mean, the sum operation does not make sense either. The teacher said that we can't add oranges and apples. But, perhaps, "two" and "three" could be unique and incompatible entities as well, and they can't be summed.

Ok, before I go forward on this way, I will challenge another thing: the way we employ to represent numbers. For example, why "three" is 3? Or, more to the point, why "twenty" is 20?

Twenty fingers I can visualize. If a hammer hits any finger of my hand or of my foot, I know which one it is, even if I was blind. But does "2" and "0" put together mean the same thing?

Well, it is just the decimal system, or more precisely the decimal positional system. It is "positional" because the digit position changes its meaning. A "2" left of "0" means "two hands full of fingers". It is a practical way to represent big quantities, like the public deficit.

Computer programmers learn other systems, all positional: hexadecimal, binary, elder hands have learned octal. People exposed to pure computer science might have had contact with the ternary system.

Even the binary system has two digits: 1 and 0. Would it be possible a system with just one digit? Common sense says no, because then we have only the 0, and any string of zeros is worth zero, no matter how long.

But the unary system does exist, and the only digit is "1". Common sense is wrong because every numeric system has the digit 0, but unary is the exception. This is because unary is not positional.

Well, how do unary numbers look like? Three is 111, five is 11111, one is 1, ten is 1111111111. Unary really exists and people do use it (without knowing) when they e.g. use tally marks.

Remember the prisioner stereotype? Scratching tally marks on cell walls to mark time served. When you were a child, you learnt to count in unary, using the fingers.

The zero, or nil quantity, is represented in unary by the absence of symbols. This may be a problem because there is no distinction between "zero" and "no value written in that wall". A cell with no tally marks in walls may have been recently painted, or it is a cell where prisoners never spent a whole day.

An "antidote" for the lack of zero is to add a bias. The prisoner scratches a tally mark as soon as it is thrown in the jail, effectively adding a bias of 1. Even if it goes away five minutes later, The next prisoner will know that someone spent a fraction of one day in there.

Roman numerals are essentialy unary. The only "improvement" is some degree of compression. For example, you write IV instead of IIII, and X instead of IIIIIIIIII. But it is not wrong to write four as IIII; some old clocks actually use IIII.

As a unary system, roman numerals don't have a symbol for zero, and are terrible for pencil-and-paper arithmetic as well.

Anyway, we saw that the unary system exists, it is pervasive and it is a perfectly valid device for some cases. More: it is easier to see the real meaning of a number like "three" in unary.

More: unary numbers have the interesting property of being palatable to regular expressions. This is more of scientific/theoretical value than practical (in the current state of technology). You can do arithmetic operations and find if a number is prime just using regular expressions, provided they are feeded with unaries.

Back to our existential angst: what is "three"? Why two plus two equals four? Perhaps these are all things that we are force-fed in school to distract us and kill time, while our parents go to work.

At the same time, we "smell" some actual, practical meaning in numbers. It is even hardwired in our brains, even more than reading and writing. I've seen many illiterate persons, and everyone of them could count money!

Very well, a mathematician called Giuseppe Peano sweated on these questions, and managed to find a small set of definitions, or axioms, that perfectly define what is a number, what is a sum etc.

Peano begins by defining the successor function (which I will express as S(x)). Every number has a successor; this is the essence of a number.

On top of this, we can define what is "three": it is the successor of "two". Nothing more! Sure, we now have to define what is "two". But that's easier: "two" is the successor of "one", and "one" is the successor of zero.

So, the grounds-up definition of "three" is S(S(S(0))). The label "three" is just a label that we stamped on the third successor of zero.

Peano axioms are a formalization of natural numbers. There are no negative numbers. Zero is special because it is not the sucessor of any number ("there is no natural number x so S(x) = 0").

Zero is also the neutral element in addition, which by the way, is defined by Peano like this:

x + 0 = x
x + S(y) = S(x + y)

# Using these definitions for 2 + 3
# Applied recursively
2 + 3
2 + S(2)
S(2 + 2)
S(2 + S(1))
S(S(2 + 1))
S(S(2 + S(0)))
S(S(S(2 + 0)))
S(S(S(2)))
5

Likewise, multiplication is defined as:

x . 0 = 0
x . S(y) = x + (x . y)

The curious thing in defintion of multiplication is: the neutral number (1) emerges naturally from the definition, without explicit mention and without "special status" in Peano world.

From these basic rules we can define all the rest, including complicated things like the definition of a prime number.

And, of course, it is a bit clearer (for me, at lest) how God handles the set of natural numbers. He just laid out those simple rules... or created Peano for that purpose :)

Now, the realization that fries our brains: we can define some kind of "numeric system" that is completely arbitrary, without any connection to reality or intrinsic meaning, yet it follows Peano rules. (Well, "reality" and "intrinsic meaning" are abstract concepts themselves.)

For example, I will put together a new numeric system based on fancy Unicode characters. Its name is "Mariju":

  ⥁
S(⥁) = ⤨      S(⤨) = ✺     S(✺) = ∇
S(∇) = ⧯      S(⧯) = ℗     S(℗) = ♣
S(♣) = ♠      S(♠) = ♕     S(♕) = ☹
S(☹) = ✈      S(✈) = ⚑     ...

Mariju has a neutral element and admits additions, multiplications, prime numbers, etc. but each Mariju "number" does not mean ANYTHING, and does NOT HAVE TO.

For instance, ♕ is simply ♕, nothing more. The only thing that we know about ♕ for sure, is that it is the successor of ♠. We are not allowed to say that ♕ is worth 9. Free your mind: "nine" and ♕ are equally abstract concepts, both divorced from physical world.

But it's true that N (the set of natural numbers) and Mariju have the same cardinality because both follow Peano axioms. Any other Peano-compliant system will share the same cardinality with them.

Because of that, we can actually establish a relation between the two sets. Again, we are not forced to relate ♕ com 9, eventhough the simplest relation between N and Mariju (justapose the neutral elements, then the immediate successors and so on) leads to this very relation.

Someone could tell that Mariju is not equivalent to naturals because we can't make Mariju infinite. We would have to attribute a new symbol for each successor, and time or creativity would eventually dry up.

Well, this is not a problem. Nobody wrote down every natural number as well; they are composed on demand, as the need for bigger numbers arises, and we can do the same for Mariju.

It's true that we just found a big advantage of "normal" numeric systems: they are recursively defined and use justa a handful of symbols (ten in decimal system, two in binary, only one in unary).

Still, it is a practical advantage, not a theoretical limitation that affects Mariju. We can patch this issue in another way: make Mariju finite, that is, a field. It just takes this:

S(⚑) = ⥁

That is, the successor of "flag" (related to number 11) is the neutral element (related to 0). All elements now have successors in Mariju and we don't need more symbols, either.

Ok, that makes Mariju a finite arithmetic system, with slightly different characteristics, but by no means illegitmate or worthless. It is analogous to a integer field with numbers 0 to 11.

Since the successor of 11 is 0, summing two numbers may deliver a surprisingly small result. For example, 10 + 3 = 1 (in the 0..11 field). Just like a clock: three hours after ten o'clock, it will be one o'clock. (I admit that I made Mariju with twelve symbols just to be able to compare with the clock.)