Where Do the Natural Numbers Come From?

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Note: in this article, we assume that the natural numbers include 0.

The natural numbers are an unusually versatile construction. Perhaps you have not thought of it before, but they are lingustically overloaded. Consider the lingustic distinction between cardinal numbers and ordinal numbers, i.e. "five" versus "fifth", or the more extreme "one" versus "first." Yet, the same object works overtime, accounting for both cases. It was only in the 1800s that Georg Cantor showed that this is merely a coincidence, as his attempts at extending both notions to transfinite quantities could not continue conflating the two.

So, why is it that the natural numbers can take on a dual role of representing both (finite) cardinality and (finite) ordinality? Let's start with cardinality. Given a collection of objects, the base case is that a collection is, in fact, empty. However, we can always add an object to a collection. This is succession, the action that turns 0 into 1, 1 into 2, and so on. Notably, no quantity can be succeeded by a quantity that has already been constructed, and there are no quantities that cannot be obtained as such. More formally, we are describing an inductive type, meaning that the natural numbers are the initial such object equipped with a notion of succession. (More on this concept later...) We can also describe ordered sets in a similar manner. The base case of an ordered set is empty, which is trivially ordered. From this, we can take a given order and adjoin a new element that is greater than all prior elements. We make the same assumptions about how finite ordered sets are distinguished from each other as with cardinality, so we are again defining an inductive type. In fact, up to isomorphism, we are defining the same inductive type, hence the coincidence between finite cardinals and ordinals.

However, succession is far from the only thing one cares about with respect to natural numbers. For one, naturals can be added to each other. This can formally be defined as repetitive succession. Some basic arithmetic can illuminate essential properties of addition. For one, addition is both associative and commutative, meaning that the naturals form a commutative semigroup with respect to addition. Further, because 0 is an identity element with respect to addition, this refines the structure into that of a commutative monoid. In fact, the naturals form a very special kind of commutative monoid: they are the free commutative monoid generated by a single element. This means that we assume that this monoid contains a specified element (in this case, 1), and make no assumptions about equality not derived from the axioms. Again, this is an inductive definition. In fact, if we drop the assumption of commutativity, they are still the free monoid generated by a single element.

Going even further, we can add another piece of structure to the naturals: multiplication, formally defined as repetitive addition. Again, basic arithmetic shows important properties: multiplication is both associative and commutative, and 1 is an identity element for this operation. This means that the naturals are a commutative monoid in two different ways: with respect to addition and with respect to multiplication. Not only this, but these operations are compatible in an important way: multiplcation distributes over addition. This means that the naturals, with respect to addition and multiplication, form a commutative semiring. Further, they are the initial commutative semiring. Again, dropping the requirement of commutativity changes nothing: they are still the initial semiring.

Despite some relatively recent work, semirings are not well-studied objects in mathematics. However, rings are much more well-known. These are merely the semirings that have additive inverses for every element, but this assumption soothes the relatively pathological nature of semirings. The initial ring is also a well-known object: the integers play this role. In fact, we can take this development backwards, for the integers are also the free Abelian group generated by a single element. In both cases, removing the assumption of commutativity changes nothing.

The integers are well-known for being central to the study of algebraic geometry. This is, of course, something of an anachronism, as the origins of algebraic geometry lie in work done over algebraically closed fields, which was further extended to more general fields. Unfortunately, there is no such thing as an initial field, but for a given characteristic, there is an initial field of that characteristic. The closest one can get to an "initial" algebraic geometry is to "simultaneously work over" the rationals and prime modular arithmetic for every prime. The best possible interpretation of this is to work over the integers. (There are more formal justifications for this, but it is informally acceptable. Various attempts at defining a so-called "field with one element" are meant to replace the integers with a better candidate for an initial object. However, if one is only content with working over commutative rings, then the integers are the best possible base.)

This article is currently unfinished.