Multitude - Revision history

Lfox at 01:34, 10 March 2025

2025-03-10T01:34:59Z

← Older revision		Revision as of 01:34, 10 March 2025
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	== Multiplication ==		== Multiplication ==
	In the context of natural numbers, '''multiplication''' is a process in which one identifies the multitude of a [[Sets#Cartesian product\|cartesian product]], based on his knowledge of the multitudes of its summands. (By the ''summands'' of a cartesian product <math>A \times B</math>, I mean the set <math>A </math> and the set <math>B</math>.) [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.]		In the context of natural numbers, '''multiplication''' is a process in which one identifies the multitude of a [[Sets#Cartesian product\|cartesian product]], based on his knowledge of the multitudes of its summands. (By the ''summands'' of a cartesian product <math>A \times B</math>, I mean the set <math>A </math> and the set <math>B</math>.) [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.]

			== References ==

Lfox at 00:52, 30 July 2024

2024-07-30T00:52:35Z

← Older revision		Revision as of 00:52, 30 July 2024
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	== Specific natural numbers ==		== 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."<ref>Harry Binswanger (in an answer to a question, in his 2024 class on philosophy of mathematics).</ref>

	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:		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:

Lfox: /* Specific natural numbers */

2024-06-23T18:00:07Z

Specific natural numbers

← Older revision		Revision as of 18:00, 23 June 2024
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	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:		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 [[Set#Empty set\|an empty set]].		'''0''' is the multitude of [[Set#Empty set\|an empty set]].

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

	2 is the [[Multitude#Successor\|successor]] to 1.		'''2''' is the [[Multitude#Successor\|successor]] to 1.

	3 is the successor to 2.		'''3''' is the successor to 2.

	4 is the successor to 3.		'''4''' is the successor to 3.

	Etc.		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 <math>\mathbb{N}</math> to denote nonzero natural numbers, and <math>\mathbb{N} \sqcup \{0\} </math> or <math>\mathbb{Z}_{\geq 0}</math> to denote natural numbers including zero.

	=== Examples ===		=== Examples ===

Lfox: /* Specific natural numbers */

2024-06-23T17:56:19Z

Specific natural numbers

← Older revision		Revision as of 17:56, 23 June 2024
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	== Specific natural numbers ==		== 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:		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 [[Set#Empty set\|an empty set]].

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

Lfox: /* Counting */

2024-06-23T17:47:07Z

Counting

← Older revision		Revision as of 17:47, 23 June 2024
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	The last sound spoken/thought is the perceptual symbol for the count.		The last sound spoken/thought is the perceptual symbol for the count.
			Oftentimes, to keep track of the set <math>A</math>, 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 <math>A</math>, and all the objects to the right of it as not belonging to <math>A</math>.

	=== Other examples of counting ===		=== Other examples of counting ===
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	* Counting using [[integers]], as in "3, 2, 1, 0, -1, -2, -3, … ."		* 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 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:		Borderline examples:
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	== Addition ==		== Addition ==
	'''~~Addition~~''' is a process in which one identifies the multitude of a [[Set#Disjoint union\|disjoint union]], based his knowledge of the multitudes of its summands. (By the ''summands'' of a disjoint union <math>A \sqcup B</math>, I mean the set <math>A </math> and the set <math>B</math>.) [TODO a problem with this definition is it doesn't include addition of integers, of fractions, of vectors, etc.]		In the context of natural numbers, '''addition''' is a process in which one identifies the multitude of a [[Set#Disjoint union\|disjoint union]], based his knowledge of the multitudes of its summands. (By the ''summands'' of a disjoint union <math>A \sqcup B</math>, I mean the set <math>A </math> and the set <math>B</math>.) [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 ==		== Multiplication ==
	'''~~Multiplication~~''' is a process in which one identifies the multitude of a [[Sets#Cartesian product\|cartesian product]], based on his knowledge of the multitudes of its summands. (By the ''summands'' of a cartesian product <math>A \times B</math>, I mean the set <math>A </math> and the set <math>B</math>.) [TODO a problem with this definition is it doesn't include multiplication of integers, of fractions, of matrices, etc.]		In the context of natural numbers, '''multiplication''' is a process in which one identifies the multitude of a [[Sets#Cartesian product\|cartesian product]], based on his knowledge of the multitudes of its summands. (By the ''summands'' of a cartesian product <math>A \times B</math>, I mean the set <math>A </math> and the set <math>B</math>.) [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.]

Lfox: /* Addition */

2024-06-23T17:35:52Z

Addition

← Older revision		Revision as of 17:35, 23 June 2024
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	\end{cases}</math>		\end{cases}</math>

	<math>\text{succ}(n)</math> is called "the successor to <math>n</math>."		<math>\text{succ}(N)</math> is called "the successor to <math>N</math>."

	=== Successor algorithm ===		=== Successor algorithm ===
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	== Addition ==		== Addition ==
	'''Addition''' is a process in which one identifies the multitude of a [[Set#Disjoint union\|disjoint union]], based his knowledge of the multitudes of its summands. (By the ''summands'' of a disjoint union <math>A \sqcup B</math>, I mean the set <math>A </math> and the set <math>B</math>.)		'''Addition''' is a process in which one identifies the multitude of a [[Set#Disjoint union\|disjoint union]], based his knowledge of the multitudes of its summands. (By the ''summands'' of a disjoint union <math>A \sqcup B</math>, I mean the set <math>A </math> and the set <math>B</math>.) [TODO a problem with this definition is it doesn't include addition of integers, of fractions, of vectors, etc.]

	== Multiplication ==		== Multiplication ==
	'''Multiplication''' is a process in which one identifies the multitude of a [[Sets#Cartesian product\|cartesian product]], based on his knowledge of the multitudes of its summands. (By the ''summands'' of a cartesian product <math>A \times B</math>, I mean the set <math>A </math> and the set <math>B</math>.)		'''Multiplication''' is a process in which one identifies the multitude of a [[Sets#Cartesian product\|cartesian product]], based on his knowledge of the multitudes of its summands. (By the ''summands'' of a cartesian product <math>A \times B</math>, I mean the set <math>A </math> and the set <math>B</math>.) [TODO a problem with this definition is it doesn't include multiplication of integers, of fractions, of matrices, etc.]

Lfox at 01:37, 23 June 2024

2024-06-23T01:37:38Z

@@ Line 6: / Line 6: @@
 == Counting ==
 '''Counting''' is a process in which one identifies the multitude of a set, through iteratively identifying multitudes of its subsets.
 == Specific natural numbers ==
-Multitude can be divided up into sub-concepts such as "1," "12," and "73."
+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:
-is the multitude containing only a unit. [ostensive definition? is 1 a synonym for unit?]
+is the multitude of a set containing only a unit.
-is the multitude succeeding 1 in the counting algorithm.
+is the [[Multitude#Successor|successor]] to 1.
-is the multitude succeeding 2 in the counting algorithm.
+is the successor to 2.
-is the multitude succeeding 3 in the counting algorithm.
+is the successor to 3.
 Etc.
 == Addition ==

Lfox at 16:20, 21 June 2024

2024-06-21T16:20:55Z

← Older revision		Revision as of 16:20, 21 June 2024
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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''', '''discrete~~ quantity''', '''cardinality''', and the symbol <math>\mathbb{N}</math>; though all those concepts have different shades of meaning, Objective Mathematics uses them interchangeably.		'''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 <math>\mathbb{N}</math>; though all those concepts have different shades of meaning, Objective Mathematics uses them interchangeably.

	== Comparison ==		== Comparison ==
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	== Specific natural numbers ==		== Specific natural numbers ==
	Multitude can be divided up into sub-concepts such as "1," "12," and "73."		Multitude can be divided up into sub-concepts such as "1," "12," and "73."

			1 is the multitude containing only a unit. [ostensive definition? is 1 a synonym for unit?]

			2 is the multitude succeeding 1 in the counting algorithm.

			3 is the multitude succeeding 2 in the counting algorithm.

			4 is the multitude succeeding 3 in the counting algorithm.

			Etc.

	== Addition ==		== Addition ==

Lfox at 03:17, 21 June 2024

2024-06-21T03:17:49Z

← Older revision		Revision as of 03:17, 21 June 2024
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	'''Multitude''' is the [[quantity]] of a finite [[set]]. Synonyms for multitude are '''natural number''', '''whole quantity''', '''discrete quantity''', '''cardinality''', and the symbol <math>\mathbb{N}</math>; though all those concepts have different shades of meaning, Objective Mathematics uses them interchangeably.		'''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''', '''discrete quantity''', '''cardinality''', and the symbol <math>\mathbb{N}</math>; though all those concepts have different shades of meaning, Objective Mathematics uses them interchangeably.

	== Comparison ==		== Comparison ==
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	== Counting ==		== Counting ==
	Counting is a process by which one identifies the ~~isomorphism class~~ of a set. ~~[TODO lol]~~		'''Counting''' is a process in which one identifies the multitude of a set, through iteratively identifying multitudes of its subsets.

	== Specific natural numbers ==		== Specific natural numbers ==
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	== Addition ==		== Addition ==
	[~~TODO~~]		'''Addition''' is a process in which one identifies the multitude of a [[Set#Disjoint union\|disjoint union]], based his knowledge of the multitudes of its summands. (By the ''summands'' of a disjoint union <math>A \sqcup B</math>, I mean the set <math>A </math> and the set <math>B</math>.)

			== Multiplication ==
			'''Multiplication''' is a process in which one identifies the multitude of a [[Sets#Cartesian product\|cartesian product]], based on his knowledge of the multitudes of its summands. (By the ''summands'' of a cartesian product <math>A \times B</math>, I mean the set <math>A </math> and the set <math>B</math>.)

Lfox: Created page with "'''Multitude''' is the quantity of a finite set. Synonyms for multitude are '''natural number''', '''whole quantity''', '''discrete 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 by which one identifies the isomorphism class of..."

2024-06-19T18:20:00Z

Created page with "'''Multitude''' is the quantity of a finite set. Synonyms for multitude are '''natural number''', '''whole quantity''', '''discrete quantity''', '''cardinality''', and the symbol <math>\mathbb{N}</math>; 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 by which one identifies the isomorphism class of..."

New page

'''Multitude''' is the [[quantity]] of a finite [[set]]. Synonyms for multitude are '''natural number''', '''whole quantity''', '''discrete quantity''', '''cardinality''', and the symbol <math>\mathbb{N}</math>; 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 by which one identifies the isomorphism class of a set. [TODO lol]

== Specific natural numbers ==
Multitude can be divided up into sub-concepts such as "1," "12," and "73."

== Addition ==
[TODO]