Numbers can be Magical (Part 1)
Let’s critically examine our relationship with number.
This article will cover some preliminary terms and concepts. Future articles will concern themselves with broader examples and meanings attached to these numbers and their properties, such as numerological, religious, and mystical symbolisms.
Numbers themselves can have many different denotations, different connotations, and different expressions. We can understand, for the purposes of this discussion, that number is the thing that we measure when we count; it is quantity.
Some cultures do not actually have numbers above a certain “useful” limit. In his masterpiece Alex’s Adventures in Numberland, Alex Bellos begins with a chapter on the Munduruku of the Amazon. While I recommend reading Bellos’ work for this topic, as it is immensely enjoyable, I will mention here only a key fact: there is a limit to the number of numbers that they have. They have a system which I will call the pragmatic cardinals, for reasons that will make more sense as we move on. This system is distinguished by its diminuitive or minimalist nature: their language includes conceptions of 1, 2, 3, sometimes 4, and many. That is to say, any number greater than 4 is “many”. Now, I am certain that a Munduruku person would not equate 5 with the number of trees in the forest; but to them, number greater than 4 is practically equal.
Number truly began to arise with agriculture, though there is some evidence that some hunter gatherer tribes had broader mathematical concepts than the Munduruku.
Let us begin with some features of numbers.
Cardinality
So, for the newly initiated, cardinality is the measure of “size”. When we talk about the cardinality of a set S, we are talking about the number of things in the set. This goes like: 5 fingers per hand, 20 fingers and toes, 88 piano keys, ~365 days per year, and 6 protons in a carbon atom. We can visualize cardinality like this:
A = {a,b,c} → |A| = 3
So, we have a way to count? Well, we have one facet of the thing that is number: let’s look at another.
Note that ordinality is agnostic about the location of the 3rd element, but denotes generally that there are 3 elements.
Ordinality
Ordinality is the property of number that in English we list with “th”s: fifth, seventh, etc. When we order something, we assign ordinals to those things. Think about which finger (on either hand) is the third finger: this is always the same, but the second depends on which direction you count. This is ordinality. If we have an ordered set, which we will denote like this:
B = {b1, b2, b3}
B(2) = b2
Here, B(2) denotes the second element of B. This changes sometimes, depending on logical language used: for example, in a python “list”, which acts as a type of ordered set, you denote the second object of list L with L[2]. (We’ll talk about order itself next.)
Sometimes things come with a natural order, given by an ordering relation.
Order and Ordering Relations
Let’s think about two things: the number line and distance. For the number line, we can conceive of it as a ruler extending out to the horizon of our vision, with the beginning at zero.
We can see that 2 is farther from us than 1 is, and 3 farther than 2; so, what makes that true? If I take 10 steps, why is it farther than 5?
For what order is, we can see that (intuitively) 0 < 1 < 2 < 3 … gives us an ordering of the numbers themselves. In undergraduate Real Analysis, it is common to examine this more closely, for example proving that 0 < 1 or 0 is not equal to 1. The natural idea is just that, only “natural”; we need some way to get to the root of what an ordering relation is. From this, we get some basic axioms for an ordering relation.
It may not make sense to apply the idea of ordering relation to some things. For example, for the set of leaves on a tree, it is not very sensible to say that they are “ordered”. They are still collectible as a set, defined by attachment to the tree and the “act” of being a leaf, but they are not able to be ordered while still attached to the tree except for by artificial conceptual choices (mass, color density, distance from the observer of the farthest part of the leaf, etc).
For numbers, it often does make sense! Let us start from something we do daily: we count things. We need to know whether, how, why, or what we are doing when we count. This leads us to the following question: why does counting logically follow the same pattern? Why do you count the same that I do, aside from “learning” that set? This is answered by the idea of ordering relation.
Now, for some set S, where S is a collection of objects or numbers, there is an ordering relation on S only if it makes sense, or there exists a way, to apply the following properties:
For all a,b,c in S,
(i) a<b, b<a, or a=b.
(ii) if a<b, b<c, then a<b<c → a<c
(iii) a>b → a+c > b+c
(iv) if a>b, c>0, then ac>bc.
For our purposes, this is enough. Some sets may not include a “zero” element, so the last axiom may or may not be necessary. Other sets do not come equipped with multiplication, so it may not make sense to attach the last axiom at all.
Natural Numbers
The name, “natural”, again comes off the tongue: these are the numbers that “naturally” arise out of counting things we see around us. They are properly the positive integers (whole numbers) and zero; zero itself has a more complicated history, which we will not cover here. We need only to state that the natural numbers are the positive whole numbers and zero.
The natural numbers have the “natural” ordering; they are counted up (1, 2, 3…) where 0 < 1 < 2 … on as far as we like. (Those who have a math background will notice that we are also intentionally avoiding the idea of infinity; we will get there.) This is what we plot on the positive number line.
Now, we notice that we think of these numbers in several (spatial) ways. They are
(i) Amounts: the amount of apples in my kitchen is given number.
(ii) Distance: I can walk a number of steps.
(iii) Area/Volume: there are a number of tiles on my floor. (this can be extended to any number of directions, which we will discuss later [of course]).
(iv) Time: today is the fourth day of the week.
(v) Order: I am the third child (I am ‘number’ 3).
Conclusion (1)
This article was a terminology cover, but it gives us the beginning of what we need. We will be covering the other ways that number can be perceived, and relating it to this; number is much more than the sum of its parts, and humans seem to naturally understand this.
We can associate transcendent meaning to numbers, beyond their representations. This will be the subject of the next several essays.
Sources and Further Reading
https://www.amazon.com/Zero-Biography-Dangerous-Charles-Seife/dp/0140296476
