What exactly are the numbers we use everyday?

Question

Pi can be defined as diameter / circunference of a circle. But what is a circle? You can't tell a computer: "build a circle and divide its diameter by its circumference". You need to define what a circle is. Wikipedia defines a circle as: It is the set of all points in a plane that are at a given distance from a given point, the centre. But that is just a statement about a property that circles have that allows us to uniquely identify them among the space of existing mathematical ideas. How do you actually build that circle? And how do actually find pi from that definition? There are no instructions on this definition. This definition doesn't explain how to build a circle, just identifies what it is.

As an example, lets talk about trees. Trees are well-understood structures with an obvious computational representation. There are even many kind of trees and I can actually write down all of them how they truly are. Using the λ-calculus, for example, I can represent a complete, untagged binary tree of depth 2 using the lambda-church encoding:

tree : (∀ (t : *) -> (t -> t -> t) -> t -> t)
tree = (λ branch tip . branch (branch tip tip) (branch tip tip))

That is a tree, on its true form. It doesn't just "identify" the essence of a tree by listing an unique property it has. I actually defines it. The type above perfectly captures the set of untagged binary trees, and the element I wrote is a member of that set. You can see it, you can inspect its structure and you can use it to do meaningful computation. If you study that type, you are studying untagged binary trees; it has all the properties you could expect from an untagged binary tree, because it is the actual definition of an untagged binary tree.

Now, when it comes to numbers, we don't have that. If we ask a mathematician what is a natural number, or a real number, or a fractiona lnumber, he will keep proposing "wordy" statements that uniquely identify those sets, but he won't show you how to actually build those things. If you ask him to write down a number on paper, he will write 5, or 6. But that isn't a number. That is a decimal representation of a number that humans happen to like. Similarly, if you ask a computer scientist, he will talk about properties numbers have. If you ask him to show you how a number actually looks like, he might show you a 64-bit vectors. But those vectors aren't numbers too. Those only represent numbers.

If we wanted to study bit vectors, we could, too:

bits : (∀ (t : *) -> (t -> t) -> (t -> t) -> t -> t)
bits = (λ o z e . (o (z (o (o (z (z (z (z e)))))))))

That is as good a definition of a bitvector as the definition above is of a tree. But it says nothing about numbers. It is, at most, isomorphic to the set of natural numbers. But that's where it ends. It doesn't say anything about natural numbers because it doesn't capture its essence in any meaningful way. Studying numbers through that definition would be like studying Japanese grammar in english. It is a layer of indirection.

So, what is a number? What is its structure like? How do you represent, computationally, syntactically, the phenomena of a real number, of a complex number, a tensor? What algebraic structure has all the properties that a quaternion has, without layers of indirections?

Answer 1

Real numbers can be approximated by rational numbers. Rational numbers can be "constructed" explicitly once we've constructed the integers. We can construct the integers once we have the natural numbers.

And there's a perfectly simple definition of what the natural numbers are, entirely analogous to the way people specify grammars in computer science. A grammar is specified by an alphabet and a set of production rules, allowing you to build valid formulas from shorter valid formulas. The corresponding definition of a natural number is this:

1. $0$ is a natural number.

2. If $n$ is a natural number then $Sn$ is a natural number.

So the first few natural numbers are $0$, $S0$, $SS0$, etc. More commonly written as $0$, $1$, $2$, etc. See here for more details.

You can even calculate $2+2$:

class Int:
  def __init__(self, pred=None):
    self.pred = pred

  def __str__(self):
    if self.pred:
      return 'S' + str(self.pred)
    else:
      return '0'

  def __add__(n,m):
    if m.pred:
      return S(n) + m.pred
    else:
      return n

def S(n):
   return Int(n)

Zero = Int()

print S(S(Zero)) + S(S(Zero))

Answer 2

Circle's actually do have representation in Lambda Calculus.

circle : distance -> point -> (point -> point -> distance) -> *
circle = (λ (r : distance). λ (c : point) . λ (d : point -> point -> distance). forall (x : *) . (forall (p : point). d c p = r -> x))

Where point is a data type for points, distance is a data type for lengths, d is a metric, and = is a type for equality (such as defined here.) circle r c d is the type for points that lie on a circle with center c and radius r with metric d.

Answer 3

The question "what is a number" is not an easy one to answer, even in reference to the natural numbers as you mention in your question.

Unlike a finite tree, dealing with the collection of natural numbers involves infinitary issues. Within the usual set-theoretic foundations of mathematics the numbers are defined by declaring $0$ to be the empty set $\emptyset$, $1$ to be the set $\{\emptyset\}$, $2$ to be the set $\{\emptyset,\{\emptyset\}\}$, etcetera. Soon enough in a set-theoretic context you obtain the set $\mathbb{N}$.

Most mathematicians adhere to the view that $\mathbb{N}$ corresponds to the intuitive counting numbers. Such an identification is sometimes referred to as the Intended Interpretation (as opposed to the usual meaning of the term interpretation when certain syntactic structures are mapped onto semantic ones, usually in a set-theoretic context). An extensive discussion of this issue can be found here.