Multitude

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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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n_1, n_2, \cdots, n_k) } of numerical symbols: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{succ}(N)} is called "the successor to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S}
, and keep it in mind
focus on a subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}
, if there are any left
while Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \subsetneq S}
.

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

Oftentimes, to keep track of the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , and all the objects to the right of it as not belonging to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

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.

Examples

  • 1 cat.

    1 cat.

  • 4 engines.

    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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \sqcup B} , I mean the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A } and the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \times B} , I mean the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A } and the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.]

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