Order - Revision history

Lfox at 01:36, 21 April 2024

2024-04-21T01:36:26Z

← Older revision		Revision as of 01:36, 21 April 2024
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	A set <math>S</math> is said to possess an '''order''' (or, if necessary to disambiguate, a ''total'' order) if, for every pair of distinct elements <math>x,y \in S</math>, either <math>x</math> is ''before'' <math>y</math>, or <math>x</math> is ''after'' <math>y</math>, with respect to some [[Order#Order-relation\|order-relation]].		A set <math>S</math> is said to possess an '''order''' (or, if necessary to disambiguate, a ''total'' order) if, for every pair of distinct elements <math>x,y \in S</math>, either <math>x</math> is ''before'' <math>y</math>, or <math>x</math> is ''after'' <math>y</math>, with respect to some [[Order#Order-relation\|order-relation]].

	Besides what Objective Mathematics means by "order," there are two other significant uses of the word. One refers to a command (an order) given to other people, as in "he ordered a pizza." In the other usage, a set of things is said to possess an order if there is some pattern to which the things adhere; that pattern may or may not involve an order-relation. Both of these usages are related ''to'' the mathematical usage of the term; it's not a coincidence that they are all described by the same word. But getting into that would take us too far afield.		Besides what Objective Mathematics means by "order," there are two other significant uses of the word. One refers to a command (an order) given to other people, as in "he ordered a pizza." In the other usage, a set of things is said to possess an order if there is some pattern to which the things adhere; that pattern may or may not involve an order-relation. Both of these usages are ''related'' ''to'' the mathematical usage of the term; it's not a coincidence that they are all described by the same word. But getting into that would take us too far afield. I do not mean either of those things by "order."

	== Examples ==		== Examples ==

Lfox: /* The traditional concept */

2024-02-12T17:17:33Z

The traditional concept

← Older revision		Revision as of 17:17, 12 February 2024
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	# '''Transitivity''': If <math>R(x,y) = 1 </math> and <math>R(y,z) = 1 </math>, then <math>R(x,z) = 1 </math>.		# '''Transitivity''': If <math>R(x,y) = 1 </math> and <math>R(y,z) = 1 </math>, then <math>R(x,z) = 1 </math>.

	Although I accept that any order-relation will indeed satisfy properties (1) through (3), I do not like using them in the ''definition'' of an order-relation. I don't feel like it perfectly captures whatever essence that examples of order-relations share in common. For the time being, however, I don't have a better definition.		Although I accept that any order-relation will indeed satisfy properties (1) through (3), I do not like using them in the ''definition'' of an order-relation. I don't feel like it perfectly captures whatever essence it is that examples of order-relations share in common. For the time being, however, I don't have a better definition.

	== Partial order ==		== Partial order ==

Lfox: /* The traditional concept */

2024-02-12T00:23:57Z

The traditional concept

← Older revision		Revision as of 00:23, 12 February 2024
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	# '''Transitivity''': If <math>R(x,y) = 1 </math> and <math>R(y,z) = 1 </math>, then <math>R(x,z) = 1 </math>.		# '''Transitivity''': If <math>R(x,y) = 1 </math> and <math>R(y,z) = 1 </math>, then <math>R(x,z) = 1 </math>.

	Although I accept that any order-relation will indeed satisfy properties (1) through (3), I do not ~~think I accept~~ the ~~above as a~~ ''definition'' of an order-relation. I ~~think~~ it ~~is inconsistent with the principles~~ of ~~definition laid out~~ in ~~[[Concepts#Definition]]~~. For the time being, however, I don't have a better definition.		Although I accept that any order-relation will indeed satisfy properties (1) through (3), I do not like using them in the ''definition'' of an order-relation. I don't feel like it perfectly captures whatever essence that examples of order-relations share in common. For the time being, however, I don't have a better definition.

	== Partial order ==		== Partial order ==

Lfox: /* Before and after */

2024-02-12T00:21:24Z

Before and after

← Older revision		Revision as of 00:21, 12 February 2024
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	The concepts of "before" and "after," in their primary usage, are identifications of properties which are irreducible primaries, and thus they admit only ostensive definitions. Nevertheless, I will give a circular "definition" to indicate what is meant by them. The primary usage of these concepts, and the way that they are first arrived at by a conceptual consciousness, is as measurements of relative time-ordering. That is, we directly perceive that some events occur ''before'' other events, and that some events occur ''after'' other events. Then, by a process of measurement omission, we create the corresponding concepts.		The concepts of "before" and "after," in their primary usage, are identifications of properties which are irreducible primaries, and thus they admit only ostensive definitions. Nevertheless, I will give a circular "definition" to indicate what is meant by them. The primary usage of these concepts, and the way that they are first arrived at by a conceptual consciousness, is as measurements of relative time-ordering. That is, we directly perceive that some events occur ''before'' other events, and that some events occur ''after'' other events. Then, by a process of measurement omission, we create the corresponding concepts.

	Besides the primary usage, there are ~~man~~ derivative usages of "before" and "after." Consider the following example, wherein paragraphs are sorted from least abstract to most abstract.		Besides the primary usage, there are many derivative usages of "before" and "after." Consider the following example, wherein paragraphs are sorted from least abstract to most abstract.

	# In most contexts, the number 1 said to be ''before'' the number 2, because, when counting, the sound "one" is uttered ''before'' the sound "two." Here, the latter usage of "before" is primary; the former is derivative. Likewise, in most contexts, the letter X is said to be ''before'' the letter Y, because "X" is always pronounced ''before'' "Y" in the alphabet song.		# In most contexts, the number 1 said to be ''before'' the number 2, because, when counting, the sound "one" is uttered ''before'' the sound "two." Here, the latter usage of "before" is primary; the former is derivative. Likewise, in most contexts, the letter X is said to be ''before'' the letter Y, because "X" is always pronounced ''before'' "Y" in the alphabet song.

Lfox: /* Examples */

2024-02-12T00:15:10Z

Examples

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	== Partial order ==		== Partial order ==
	A ~~partial order~~ is ~~like~~ a ~~total~~ order, but not every pair of elements ~~of the set~~ are ~~required to be before or after others. [TODO define properly?]~~.		A set is said to possess a '''partial''' '''order''' if some, but perhaps not every, pair of elements are comparable using some given order-relation.

	=== Examples ===		=== Examples ===
	The leaves of a tree (a real, physical tree; not a mathematical tree) possess a partial order with respect to their height above the ground. For some ordered pairs of leaves, the first is higher than the second. For others, the first is lower than the second. For yet others, it cannot be said that the first leaf is higher than the second, nor can it be said that the first leaf is lower than the second; they are at approximately the same height.		The leaves of a tree (a real, physical tree; not a mathematical tree) possess a partial order with respect to their height above the ground. For some ordered pairs of leaves, the first is higher than the second. For others, the first is lower than the second. For yet others, it cannot be said that the first leaf is higher than the second, nor can it be said that the first leaf is lower than the second; they are at approximately the same height.

			A basket of blueberries could be partially ordered by their perceptual ''size''. Perceptually, we can see that some blueberries are larger than others, and some blueberries are smaller than others. But without introducing an explicit measure of volume, there are many incomparable pairs of blueberries, where we cannot tell which one is the smaller, and which one is the larger.

Lfox: /* Order-relation */

2024-02-11T03:00:21Z

Order-relation

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	== Order-relation ==		== Order-relation ==
	~~There~~ are ~~some~~ relations which are ''like'' "before" and "after" in some respect, but mean something else. The concept subsuming such relations is that of an ''order-relation''.		An '''order-relation''' is a binary relation satisfying reflexivity, antisymmetry, and transitivity (properties which are defined below). The reason for this concept is that there are many relations which are ''like'' "before" and "after" in some important respect, but mean something else. The concept subsuming such relations is that of an ''order-relation''.

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

Lfox: /* Before and after */

2024-02-11T02:55:42Z

Before and after

← Older revision		Revision as of 02:55, 11 February 2024
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	== Before and after ==		== Before and after ==
	The concepts of "before" and "after," in their primary usage, are irreducible primaries, and thus admit only ostensive definitions. Nevertheless, I will give a circular "definition" to indicate what is meant by them. The primary usage of these concepts, and the way that they are first arrived at by a conceptual consciousness, is as measurements of relative time-ordering. That is, we directly perceive that some events occur ''before'' other events, and that some events occur ''after'' other events. Then, by a process of measurement omission, we create the corresponding concepts.		The concepts of "before" and "after," in their primary usage, are identifications of properties which are irreducible primaries, and thus they admit only ostensive definitions. Nevertheless, I will give a circular "definition" to indicate what is meant by them. The primary usage of these concepts, and the way that they are first arrived at by a conceptual consciousness, is as measurements of relative time-ordering. That is, we directly perceive that some events occur ''before'' other events, and that some events occur ''after'' other events. Then, by a process of measurement omission, we create the corresponding concepts.

	Besides the primary usage, there are man derivative usages of "before" and "after." Consider the following example, wherein paragraphs are sorted from least abstract to most abstract.		Besides the primary usage, there are man derivative usages of "before" and "after." Consider the following example, wherein paragraphs are sorted from least abstract to most abstract.

Lfox at 22:15, 3 February 2024

2024-02-03T22:15:03Z

← Older revision		Revision as of 22:15, 3 February 2024
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	A set <math>S</math> is said to possess an '''order''' (or, if necessary to disambiguate, a ''total'' order) if, for every pair of distinct elements <math>x,y \in S</math>, either <math>x</math> is ''before'' <math>y</math>, or <math>x</math> is ''after'' <math>y</math>, with respect to some [[Order#Order-relation\|order-relation]].		A set <math>S</math> is said to possess an '''order''' (or, if necessary to disambiguate, a ''total'' order) if, for every pair of distinct elements <math>x,y \in S</math>, either <math>x</math> is ''before'' <math>y</math>, or <math>x</math> is ''after'' <math>y</math>, with respect to some [[Order#Order-relation\|order-relation]].

	Besides what Objective Mathematics means by "order," there are two other significant uses of the word. One refers to a command (an order) given to other people, as in "he ordered a pizza." In the other usage, a set of things is said to possess an order if there is some pattern to which the things adhere. Both of these usages are related ''to'' the mathematical usage of the term; it's not a coincidence that they are all described by the same word. But getting into that would take us too far afield.		Besides what Objective Mathematics means by "order," there are two other significant uses of the word. One refers to a command (an order) given to other people, as in "he ordered a pizza." In the other usage, a set of things is said to possess an order if there is some pattern to which the things adhere; that pattern may or may not involve an order-relation. Both of these usages are related ''to'' the mathematical usage of the term; it's not a coincidence that they are all described by the same word. But getting into that would take us too far afield.

	== Examples ==		== Examples ==

Lfox at 23:18, 2 February 2024

2024-02-02T23:18:25Z

← Older revision		Revision as of 23:18, 2 February 2024
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	A set <math>S</math> is said to possess an '''order''' (or, if necessary to disambiguate, a ''total'' order) if, for every pair of distinct elements <math>x,y \in S</math>, either <math>x</math> is ''before'' <math>y</math>, or <math>x</math> is ''after'' <math>y</math>, with respect to some order-relation.		A set <math>S</math> is said to possess an '''order''' (or, if necessary to disambiguate, a ''total'' order) if, for every pair of distinct elements <math>x,y \in S</math>, either <math>x</math> is ''before'' <math>y</math>, or <math>x</math> is ''after'' <math>y</math>, with respect to some [[Order#Order-relation\|order-relation]].

	Besides what Objective Mathematics means by "order," there are two other significant uses of the word. One refers to a command (an order) given to other people, as in "he ordered a pizza." In the other usage, a set of things is said to possess an order if there is some pattern to which the things adhere. Both of these usages are related ''to'' the mathematical usage of the term; it's not a coincidence that they are all described by the same word. But getting into that would take us too far afield.		Besides what Objective Mathematics means by "order," there are two other significant uses of the word. One refers to a command (an order) given to other people, as in "he ordered a pizza." In the other usage, a set of things is said to possess an order if there is some pattern to which the things adhere. Both of these usages are related ''to'' the mathematical usage of the term; it's not a coincidence that they are all described by the same word. But getting into that would take us too far afield.

Lfox at 23:18, 2 February 2024

2024-02-02T23:18:06Z

← Older revision		Revision as of 23:18, 2 February 2024
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	A set <math>S</math> is said to possess an '''order''' (or, if necessary to disambiguate, a ''total'' order) if, for every pair of distinct elements <math>x,y \in S</math>, either <math>x</math> is ''before'' <math>y</math>, or <math>x</math> is ''after'' <math>y</math>, with respect to some order relation.		A set <math>S</math> is said to possess an '''order''' (or, if necessary to disambiguate, a ''total'' order) if, for every pair of distinct elements <math>x,y \in S</math>, either <math>x</math> is ''before'' <math>y</math>, or <math>x</math> is ''after'' <math>y</math>, with respect to some order-relation.

	Besides what Objective Mathematics means by "order," there are two other significant uses of the word. One refers to a command (an order) given to other people, as in "he ordered a pizza." In the other usage, a set of things is said to possess an order if there is some pattern to which the things adhere. Both of these usages are related ''to'' the mathematical usage of the term; it's not a coincidence that they are all described by the same word. But getting into that would take us too far afield.		Besides what Objective Mathematics means by "order," there are two other significant uses of the word. One refers to a command (an order) given to other people, as in "he ordered a pizza." In the other usage, a set of things is said to possess an order if there is some pattern to which the things adhere. Both of these usages are related ''to'' the mathematical usage of the term; it's not a coincidence that they are all described by the same word. But getting into that would take us too far afield.
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	What is happening here is that we are using "before" and "after" to define an ''order-relation'' on things like the alphabet, the English language, and a set of people.		What is happening here is that we are using "before" and "after" to define an ''order-relation'' on things like the alphabet, the English language, and a set of people.

	== Order ~~relations~~ ==		== Order-relation ==
	There are some relations which are ''like'' "before" and "after" in some respect, but mean something else. The concept subsuming such relations is that of an ''order-relation''.		There are some relations which are ''like'' "before" and "after" in some respect, but mean something else. The concept subsuming such relations is that of an ''order-relation''.

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	=== The traditional concept ===		=== The traditional concept ===
	Standard mathematics says that, by definition, an order relation on a set <math>S</math> is a function <math>R : S \times S \rightarrow \{ 0, 1 \}</math> such that, for any <math>x,y, z \in S</math>, it satisfies		Standard mathematics says that, by definition, an order-relation on a set <math>S</math> is a function <math>R : S \times S \rightarrow \{ 0, 1 \}</math> such that, for any <math>x,y, z \in S</math>, it satisfies

	# '''Reflexivity''': <math>R(x,x) = 1</math>		# '''Reflexivity''': <math>R(x,x) = 1</math>
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	# '''Transitivity''': If <math>R(x,y) = 1 </math> and <math>R(y,z) = 1 </math>, then <math>R(x,z) = 1 </math>.		# '''Transitivity''': If <math>R(x,y) = 1 </math> and <math>R(y,z) = 1 </math>, then <math>R(x,z) = 1 </math>.

	Although I accept that any order relation will indeed satisfy properties (1) through (3), I do not think I accept the above as a ''definition'' of an order relation. I think it is inconsistent with the principles of definition laid out in [[Concepts#Definition]]. For the time being, however, I don't have a better definition.		Although I accept that any order-relation will indeed satisfy properties (1) through (3), I do not think I accept the above as a ''definition'' of an order-relation. I think it is inconsistent with the principles of definition laid out in [[Concepts#Definition]]. For the time being, however, I don't have a better definition.

	== Partial order ==		== Partial order ==