Number - Revision history

Lfox: /* Counting */

2025-03-12T21:54:09Z

Counting

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	The '''successor''' to a numeral is the numeral which comes after it in the fixed sequence of numerals.		The '''successor''' to a numeral is the numeral which comes after it in the fixed sequence of numerals.

	There is an algorithm computing successorship, which is learned by schoolchildren. [TODO copy paste the part where I wrote it?]		Successorship is the [[function]] computing successor. There is a very simple [[algorithm]] computing successorship, which is learned by schoolchildren. [TODO copy paste the part where I wrote it?]

	=== Counting ===		=== Counting ===
	'''Counting''' is ~~an algorithm~~ which produces a number by means of repeated application of the ~~successor operation~~.		'''Counting''' is the process which produces a number by means of repeated application of the successorship function.

	=== Addition ===		=== Addition ===
	The '''sum''' of two quantities of existents is the quantity of those existents considered together as a single group. (See also the "disjoint ~~set~~" thing.)		The '''sum''' or '''disjoint union''' of two quantities of existents is the quantity of those existents considered together as a single group. (See also the "disjoint union" thing.)

	'''Addition''' is the function which takes two numbers as input, and outputs the number identifying the sum of their corresponding quantities.		'''Addition''' is the function which takes two numbers as input, and outputs the number identifying the sum of their corresponding quantities.
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	=== Multiplication ===		=== Multiplication ===
			see page [[multiplication]].

	Before we can give a definition of multiplication, we must say a bit more about how numbers work.		Before we can give a definition of multiplication, we must say a bit more about how numbers work.

Lfox: /* Subtraction */

2025-03-12T21:31:36Z

Subtraction

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	=== Subtraction ===		=== Subtraction ===
	The '''difference''' of two quantities is the quantity which would have to be added to the lesser in order to make it equal to the greater. (Note: this sense of the word is not to be confused with ~~the~~ axiomatic concept of "difference," which stands in opposition to the axiomatic concept of "similarity.")		The '''difference''' of two quantities is the quantity which would have to be added to the lesser in order to make it equal to the greater. (Note: this sense of the word is used derivatively. It should not be confused with that primary and axiomatic concept of "difference," which stands in opposition to the axiomatic concept of "similarity.")

	'''Subtraction''' is the function which takes two numbers as input, and produces the number identifying the difference between their corresponding quantities.		'''Subtraction''' is the function which takes two numbers as input, and produces the number identifying the difference between their corresponding quantities.

Lfox: /* Subtraction */

2025-03-12T20:50:40Z

Subtraction

← Older revision		Revision as of 20:50, 12 March 2025
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	=== Addition ===		=== Addition ===
	The '''sum''' of two quantities is ~~their~~ quantity ~~when~~ considered together as a single group. (See also the "disjoint set" thing.)		The '''sum''' of two quantities of existents is the quantity of those existents considered together as a single group. (See also the "disjoint set" thing.)

	'''Addition''' ~~(of~~ two numbers~~) is an algorithm which produces~~ the number identifying the sum of their corresponding quantities.		'''Addition''' is the function which takes two numbers as input, and outputs the number identifying the sum of their corresponding quantities.

	=== Subtraction ===		=== Subtraction ===
	The '''difference''' of two quantities is the quantity which would have to be added to the lesser in order to make it equal to the greater. (Note: this sense of the word is not to be confused with the axiomatic concept of "difference," which stands in opposition to the axiomatic concept of "similarity.")		The '''difference''' of two quantities is the quantity which would have to be added to the lesser in order to make it equal to the greater. (Note: this sense of the word is not to be confused with the axiomatic concept of "difference," which stands in opposition to the axiomatic concept of "similarity.")

	'''Subtraction''' ~~(of~~ two numbers~~) is an algorithm which~~ produces the number identifying the difference between their corresponding quantities.		'''Subtraction''' is the function which takes two numbers as input, and produces the number identifying the difference between their corresponding quantities.

	=== Multiplication ===		=== Multiplication ===
	Before we can give a definition of multiplication, we must say a bit more about how numbers work.		Before we can give a definition of multiplication, we must say a bit more about how numbers work.

	In the identification of a quantity ~~with~~ a number, a unit is implicit. Strictly speaking, one cannot just say "there are x," where x is a number. One must say "there are x ''meters''" or "there are x ''cows''" or "there are x ''Hertz''" etc.		In the identification of a quantity as a number (in the "numbering" of a quantity), a unit is implicit. Strictly speaking, one cannot just say "there are x," where x is a number. One must say "there are x ''meters''" or "there are x ''cows''" or "there are x ''Hertz''" etc.

	[TODO wait is that really true? what about ratios? I guess one could say that there's a unit involved with ratios, but it doesn't matter which unit you pick. Are ratios not numbers? Actually I am thinking now that I will probably bite the bullet and say that a ratio is not a number. Hmm I guess I'll come back to this later. OK it's later now. Ratios ARE numbers. And ratios actually do come with units. Like I am driving at 30 MPH, that's expressing a ratio but it has units. Even if I take something with the same units, like the ratio of two measurements of length, maybe I shouldn't omit that fact from the notation. Like maybe we should write 3.14 m/m instead of just 3.14. The former retains some information about where it came from, the latter doesn't; it's just a numeral and that's it.]		[TODO wait is that really true? what about ratios? I guess one could say that there's a unit involved with ratios, but it doesn't matter which unit you pick. Are ratios not numbers? Actually I am thinking now that I will probably bite the bullet and say that a ratio is not a number. Hmm I guess I'll come back to this later. OK it's later now. Ratios ARE numbers. And ratios actually do come with units. Like I am driving at 30 MPH, that's expressing a ratio but it has units. Even if I take something with the same units, like the ratio of two measurements of length, maybe I shouldn't omit that fact from the notation. Like maybe we should write 3.14 m/m instead of just 3.14. The former retains some information about where it came from, the latter doesn't; it's just a numeral and that's it.]

Lfox: /* Multiplication */

2025-03-12T03:54:02Z

Multiplication

← Older revision		Revision as of 03:54, 12 March 2025
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	In the identification of a quantity with a number, a unit is implicit. Strictly speaking, one cannot just say "there are x," where x is a number. One must say "there are x ''meters''" or "there are x ''cows''" or "there are x ''Hertz''" etc.		In the identification of a quantity with a number, a unit is implicit. Strictly speaking, one cannot just say "there are x," where x is a number. One must say "there are x ''meters''" or "there are x ''cows''" or "there are x ''Hertz''" etc.

	[TODO wait is that really true? what about ratios? I guess one could say that there's a unit involved with ratios, but it doesn't matter which unit you pick. Are ratios not numbers? Actually I am thinking now that I will probably bite the bullet and say that a ratio is not a number. Hmm I guess I'll come back to this later.]		[TODO wait is that really true? what about ratios? I guess one could say that there's a unit involved with ratios, but it doesn't matter which unit you pick. Are ratios not numbers? Actually I am thinking now that I will probably bite the bullet and say that a ratio is not a number. Hmm I guess I'll come back to this later. OK it's later now. Ratios ARE numbers. And ratios actually do come with units. Like I am driving at 30 MPH, that's expressing a ratio but it has units. Even if I take something with the same units, like the ratio of two measurements of length, maybe I shouldn't omit that fact from the notation. Like maybe we should write 3.14 m/m instead of just 3.14. The former retains some information about where it came from, the latter doesn't; it's just a numeral and that's it.]

	'''Multiplication''' (of A and B) is an algorithm which produces the number identifying a quantity in units of ''v'', where B identifies the number of ''v''s making up one ''u'', and ''A'' is the quantity of ''u''s.		'''Multiplication''' (of A and B) is an algorithm which produces the number identifying a quantity in units of ''v'', where B identifies the number of ''v''s making up one ''u'', and ''A'' is the quantity of ''u''s.

Lfox at 03:10, 10 March 2025

2025-03-10T03:10:47Z

← Older revision		Revision as of 03:10, 10 March 2025
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	In the identification of a quantity with a number, a unit is implicit. Strictly speaking, one cannot just say "there are x," where x is a number. One must say "there are x ''meters''" or "there are x ''cows''" or "there are x ''Hertz''" etc.		In the identification of a quantity with a number, a unit is implicit. Strictly speaking, one cannot just say "there are x," where x is a number. One must say "there are x ''meters''" or "there are x ''cows''" or "there are x ''Hertz''" etc.

	[TODO wait is that really true? what about ratios? I guess one could say that there's a unit involved with ratios, but it doesn't matter which unit you pick. Are ratios not numbers? Hmm I guess I'll come back to this later.]		[TODO wait is that really true? what about ratios? I guess one could say that there's a unit involved with ratios, but it doesn't matter which unit you pick. Are ratios not numbers? Actually I am thinking now that I will probably bite the bullet and say that a ratio is not a number. Hmm I guess I'll come back to this later.]

	'''Multiplication''' (of A and B) is an algorithm which produces the number identifying a quantity in units of ''v'', where B identifies the number of ''v''s making up one ''u'', and ''A'' is the quantity of ''u''s.		'''Multiplication''' (of A and B) is an algorithm which produces the number identifying a quantity in units of ''v'', where B identifies the number of ''v''s making up one ''u'', and ''A'' is the quantity of ''u''s.

Lfox at 03:05, 10 March 2025

2025-03-10T03:05:40Z

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 A '''number''' is an identification of a [[quantity]], by means of a symbol (a "numeral") whose position in a fixed sequence of those symbols is the same amount as that which is being identified.<ref>I took this definition from Harry Binswanger, which he stated in his course ''Philosophy of Mathematics'', 2024. </ref>
 This is also basically the definition of "numeral" and "counting."
 Numbers work by identifying a quantity of things with a quantity of symbols. That is, the quantity of these dots {*,*} is ''the same as'' the quantity of these symbols {"1", "2"}.
 == Definitions of specific numbers, like "2" ==
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 The definition of the ''adjective'' "2" is: the quantity of numerals up to and including "2," in the counting algorithm.
 I use "two" and "2" interchangeably.
 The two definitions above can be generalized to arbitrary natural numbers. Indeed, we define "3" as the numeral succeeding "2" in the counting algorithm, "844" as the numeral succeeding "843" in the counting algorithm, etc.
 == Is a fraction a number? ==

Lfox at 01:31, 10 March 2025

2025-03-10T01:31:59Z

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	A '''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 that which is being identified.<ref>I took this definition from Harry Binswanger, which he stated in his course ''Philosophy of Mathematics'', 2024. </ref>		A '''number''' is an identification of a [[quantity]], by means of a symbol (a "numeral") whose position in a fixed sequence of those symbols is the same amount as that which is being identified.<ref>I took this definition from Harry Binswanger, which he stated in his course ''Philosophy of Mathematics'', 2024. </ref>

			This is also basically the definition of "numeral" and "counting."

			Numbers work by identifying a quantity of things with a quantity of symbols. That is, the quantity of these dots {,} is ''the same as'' the quantity of these symbols {"1", "2"}.

	== Definitions of specific numbers, like "2" ==		== Definitions of specific numbers, like "2" ==
	[TODO unify this page with the natural number page.]		[TODO unify this page with the natural number page.]

	I use "two" and "2" ~~completely~~ interchangeably.		The definition of the ''noun'' "2" is: the numeral succeeding "1" in the counting algorithm.

			The definition of the ''adjective'' "2" is: the quantity of numerals up to and including "2," in the counting algorithm.

			I use "two" and "2" interchangeably.

	~~It is tempting~~ to ~~say something like "the definition of~~ "2" ~~is:~~ the ~~symbol~~ succeeding "1" in the counting algorithm~~." That is not correct, because when I say something like "those are two cows~~," ~~I do not mean~~ "~~those are (~~the ~~symbol~~ succeeding "1" in the counting algorithm~~) cows." Rather~~, ~~I am identifying the ''quantity'' of cows with a ''quantity'' of symbols~~.		The two definitions above can be generalized to arbitrary natural numbers. Indeed, we define "3" as the numeral succeeding "2" in the counting algorithm, "844" as the numeral succeeding "843" in the counting algorithm, etc.

	== Is a fraction a number? ==		== Is a fraction a number? ==

Lfox at 20:48, 9 March 2025

2025-03-09T20:48:58Z

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	A '''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 that which is being identified.<ref>I took this definition from Harry Binswanger, which he stated in his course ''Philosophy of Mathematics'', 2024. </ref>		A '''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 that which is being identified.<ref>I took this definition from Harry Binswanger, which he stated in his course ''Philosophy of Mathematics'', 2024. </ref>

			== Definitions of specific numbers, like "2" ==
			[TODO unify this page with the natural number page.]

			I use "two" and "2" completely interchangeably.

			It is tempting to say something like "the definition of "2" is: the symbol succeeding "1" in the counting algorithm." That is not correct, because when I say something like "those are two cows," I do not mean "those are (the symbol succeeding "1" in the counting algorithm) cows." Rather, I am identifying the ''quantity'' of cows with a ''quantity'' of symbols.

	== Is a fraction a number? ==		== Is a fraction a number? ==

Lfox: /* References */

2025-02-11T08:55:00Z

References

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-A '''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 that which is being identified.<ref>I took this definition from Harry Binswanger, in his course ''Philosophy of Mathematics'', 2024. </ref>
+A '''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 that which is being identified.<ref>I took this definition from Harry Binswanger, which he stated in his course ''Philosophy of Mathematics'', 2024. </ref>
 == Is <math>\pi</math> a number? ==
 [[π]] is the [[ratio]] between the circumference and the diameter of a [[circle]]. Unlike standard mathematics, by "circle," OM means real, finite, ''physical'' circles, with thickness and height and bumps and other non-uniformities. Observe the following facts about OM's concept of π, which are not true of the standard concept:
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 To treat something as a circle is to say like "all these ways in which I might measure its circumference, they're the same."
 == References ==

Lfox at 23:43, 7 February 2025

2025-02-07T23:43:58Z

← Older revision		Revision as of 23:43, 7 February 2025
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	A '''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 that which is being identified.<ref>I took this definition from Harry Binswanger, in his course ''Philosophy of Mathematics'', 2024. </ref>		A '''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 that which is being identified.<ref>I took this definition from Harry Binswanger, in his course ''Philosophy of Mathematics'', 2024. </ref>

			== Is <math>\pi</math> a number? ==
			[[π]] is the [[ratio]] between the circumference and the diameter of a [[circle]]. Unlike standard mathematics, by "circle," OM means real, finite, ''physical'' circles, with thickness and height and bumps and other non-uniformities. Observe the following facts about OM's concept of π, which are not true of the standard concept:

			# Since π is the result of a continuous measurement, it is not a specific value, but is rather a range of possible values. For example, one circle might have a circumference and diameter whose ratio is found to be 3.1 ± 0.1.
			# π is not the same range for every circle. For one circle, we might measure π = 3.1 ± 0.1; for another we might measure π = 3.15 ± 0.02; for yet another, we might measure π = 3.14159 ± 0.00005.

			OK I'm failing to differentiate between two things. On the one hand, there's the actual ratio of the actual circumference and diameter of the actual circle. On the other hand, there's the thing we measure....

			To treat something as a circle is to say like "all these ways in which I might measure its circumference, they're the same."

	== References ==		== References ==