Multitude

Multitude is the quantity of a finite set. [TODO maybe circular because what do I mean by finite set?] Synonyms for multitude are natural number, whole quantity, cardinality, and the symbol $\mathbb {N}$; though all those concepts have different shades of meaning, Objective Mathematics uses them interchangeably.

Comparison

"Few" and "many" describe relative multitudes.

Counting

Counting is a process in which one identifies the multitude of a set, through iteratively identifying multitudes of its subsets.

Successor function

The successor function is a function which takes a numerical symbol, and returns the next one. [TODO this definition is circular, because what I mean by "the next one" is the one that successor returns.] For example, it takes a symbol such as "thirteen" (spoken or thought) or "13" (written) or "1101" (computer code), and returns "fourteen" or "14" or "1110." I will give a full specification of the successor function for numerical symbols. First we define it manually for some one-digit numerical symbols,

$n$	${\text{succ}}(n)$
0	1
1	2
2	3
3	4
4	5
5	6
6	7
7	8
8	9

then we may define it in general for a tuple $(n_{1},n_{2},\cdots ,n_{k})$ of numerical symbols:

$${\text{succ}}(n_{1},n_{2},\cdots ,n_{k}):={\begin{cases}(n_{1},n_{2},\cdots ,{\text{succ}}(n_{k})),&n_{k}<9\\(n_{1},n_{2},\cdots ,{\text{succ}}(n_{k-1}),0),&n_{k}=9{\text{ and }}n_{k-1}<9\\\cdots &\cdots \\\underbrace {(1,0,0,\cdots ,0)} _{k+1{\text{ entries }}},&n_{1}=n_{2}=\cdots =n_{k}=9\end{cases}}$$

${\text{succ}}(N)$ is called "the successor to $N$."

Successor algorithm

Successor counting, or counting by increments of 1, is the counting process where the first subset contains 1 element, and each subsequent subset in the process contains the previous subset, and also 1 more element. Traditionally, this is the first counting method that a child learns; it is the one which, when spoken aloud, sounds like "one, two, three, four, five, … ."

We may now specify, in more detail, how the successor counting algorithm works.

focus on a set $S$, and keep it in mind
focus on a subset $A\subset S$ which initially contains a single element
nextSound := "one"
do:
  think or say the sound nextSound 
  nextSound := succ(nextSound)
  add one extra element to $A$, if there are any left
while $A\subsetneq S$.

The last sound spoken/thought is the perceptual symbol for the count.

Oftentimes, to keep track of the set $A$, one uses extra tools. For example, at each step of counting, one might point to a different object, and regard that object and all the objects to the left of it as belonging to the set $A$, and all the objects to the right of it as not belonging to $A$.

Other examples of counting

• Counting by 5s, as in "zero, five, ten, fifteen, twenty, … ."
• Doing the successor counting algorithm, but instead of counting the whole set, you stop after "five."
• Counting backwards, as in "ten, nine, eight, seven, six, five, four, three, two, one, blast-off!"
• Counting using fractions, as in "1/3, 2/3, 1, 4/3, 5/3, 2, … ."
• Counting using integers, as in "3, 2, 1, 0, -1, -2, -3, … ."
• Counting using the hexadecimal numbers, as in "1c, 1d, 1e, 1f, 20, 21, … ."
• Counting by keeping track of the count using one's fingers rather than auditory symbols.

Borderline examples:

• "Counting" by using tally marks.

Non-examples

• "Counting" the candies in a rectangular jar by counting the number of candies along its height, width, and length, then multiplying the three quantities together.

Specific natural numbers

A natural number is "an identification of a quantity by means of a symbol whose position in a fixed sequence of those symbols is the same amount as what you are identifying."

Multitude can be divided up into sub-concepts such as "1," "12," and "73." These subconcepts are known as natural numbers. The definitions of the natural numbers are as follows:

0 is the multitude of an empty set.

1 is the multitude of a set containing only a unit.

2 is the successor to 1.

3 is the successor to 2.

4 is the successor to 3.

Etc.

Whether or not 0 is to be considered a natural number is optional. In most contexts, I will not consider 0 to be a natural number. Symbolically, I will use the symbol $\mathbb {N}$ to denote nonzero natural numbers, and $\mathbb {N} \sqcup \{0\}$ or $\mathbb {Z} _{\geq 0}$ to denote natural numbers including zero.

Examples

• 

  1 cat.

• 

  4 engines.

Addition

In the context of natural numbers, addition is a process in which one identifies the multitude of a disjoint union, based his knowledge of the multitudes of its summands. (By the summands of a disjoint union $A\sqcup B$, I mean the set $A$ and the set $B$.) [TODO a problem with this definition is it doesn't work in all contexts. Doesn't include addition of integers, of fractions, of vectors, etc.]

Multiplication

In the context of natural numbers, multiplication is a process in which one identifies the multitude of a cartesian product, based on his knowledge of the multitudes of its summands. (By the summands of a cartesian product $A\times B$, I mean the set $A$ and the set $B$.) [TODO a problem with this definition is it doesn't work in all contexts. Doesn't include multiplication of integers, of fractions, of matrices, etc.]