Hyperoperations Wiki - Revision history

MphLee at 12:37, 19 May 2025

2025-05-19T12:37:51Z

MphLee at 10:09, 10 May 2025

2025-05-10T10:09:44Z

MphLee at 09:30, 10 May 2025

2025-05-10T09:30:03Z

MphLee at 09:25, 10 May 2025

2025-05-10T09:25:50Z

MphLee at 09:23, 10 May 2025

2025-05-10T09:23:22Z

MphLee at 07:59, 10 May 2025

2025-05-10T07:59:22Z

MphLee at 06:57, 10 May 2025

2025-05-10T06:57:13Z

MphLee at 06:16, 10 May 2025

2025-05-10T06:16:26Z

MphLee at 06:09, 10 May 2025

2025-05-10T06:09:21Z

MphLee at 22:41, 9 May 2025

2025-05-09T22:41:15Z

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-The [[Hyperoperations Wiki]] is intended to complement [http://math.eretrandre.org/tetrationforum/ the Tetrationforum]. By creating a ''cartographic tool'' for the exploration done at the [[Tetrationforum]], keeping track of of definitions, terms, ideas, and bibliographic references; and by formally re-expressing all modern mathematical concepts from a preparatory viewpoint that is compatible with ''hyperoperative'' inquiry.
+The [[Hyperoperations Wiki]] is intended to complement [http://math.eretrandre.org/tetrationforum/ the Tetrationforum]. By keeping track of the definitions, notations, ideas, and bibliographic references, it aims to create a ''cartographic tool'' for the exploration done at the [[Tetrationforum]]. Additionally, by formally re-expressing all modern mathematical concepts from a preparatory viewpoint that is compatible with ''hyperoperative'' inquiry, the wiki could help bridge the historical gap to contemporary mathematics.
 [[File:hw.png|250px|center]]
@@ Line 6: / Line 6: @@
 [[File:hw_2.png|thumb|350px|right|The [[hyperoperations]] emerge from the investigation of the nature of [[arithmetical operations]].]]
-The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperations]], thereby situating them within the landscape of modern and contemporary [[mathematics]]. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to structures derived from hyperoperations, presenting them from what could be termed a '''''hyperoperational perspective ([[hPOV]]).'''''
+Organizing the knowledge and folklore surrounding the [[tetration]] function, as well as the broader circle of ideas encompassed by the term "hyperoperation," presents two major challenges:
 ===The hPOV and the concept of iteration===
@@ Line 13: / Line 20: @@
 [[File:hw_3.png|thumb|600px|center|Intuition about extending the conceptual relationship between sum and product to the whole space of operations.]]
-There is a circular relation between the hPOV and the iterative point of view a kind of duality.
+There is a circular relation between the hPOV and the iterative point of view, a kind of duality.
-* At first sight it may seem that iteration supplies the ambient theory where to investigate hyperoperations. Hyperoperations seems to be reduced to an '''iteration''' of the link between addition and multiplication that goes beyond them.
+* At first sight, it may seem that iteration supplies the natural ambient theory where to investigate hyperoperations. Once a process linking addition to multiplication is chosen, the study of Hyperoperations seems to be reduced to '''iterating''' that chosen process beyond them.
 *Yet also the opposite is true. Iteration emerges naturally just as one of the many possible ways to understand the connection between addition and multiplication.
-This reveals the possibility of an intersection of the two points of view by definition of a deeper and fundamental concept, an intersection where the ''iteration of iteration itself'' happens.
+This reveals the possibility of an intersection of the two points of view by definition of a deeper and fundamental concept, an intersection where the ''iteration of iteration itself'' could happen.
 ===Useful pages===
 [[Special:AllPages]] brings you to a list of all so far described terms.

@@ Line 3: / Line 3: @@
 [[File:hw.png|250px|center]]
-=Idea=
+==Idea==
 [[File:hw_2.png|thumb|350px|right|The [[hyperoperations]] emerge from the investigation of the nature of [[arithmetical operations]].]]
 The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperations]], thereby situating them within the landscape of modern and contemporary [[mathematics]]. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to structures derived from hyperoperations, presenting them from what could be termed a '''''hyperoperational perspective ([[hPOV]]).'''''
-Just like the concept of a [[ring]], the concept of hyperoperation serves as a potential avenue for exploring the conceptual nature of [[arithmetical operations]] through [[generalization]] and [[abstraction]]. The approach taken by the [[hPOV]] lies on the assumption that some relationships between addition and multiplication have a hierarchical nature. Certain relationships indicate a unidirectional increase in complexity and reveal pathways to explore further in those directions, '''''above and beyond''''', i.e. ''ὑπέρ ([[hyper]])'', '''''sums and products.'''''
+===The hPOV and the concept of iteration===
 [[File:hw_3.png|thumb|600px|center|Intuition about extending the conceptual relationship between sum and product to the whole space of operations.]]
 ===Useful pages===
 [[Special:AllPages]] brings you to a list of all so far described terms.

@@ Line 4: / Line 4: @@
 =Idea=
-[[File:hw_2.png|thumb|350px|right|The hyperoperations emerge from the investigation of the nature of [[arithmetical operations]].]]
+[[File:hw_2.png|thumb|350px|right|The [[hyperoperations]] emerge from the investigation of the nature of [[arithmetical operations]].]]
-The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperation]], thereby situating them within the landscape of modern and contemporary [[mathematics]]. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to structures derived from hyperoperations, presenting them from what could be termed a '''''hyperoperational perspective ([[hPOV]]).'''''
+The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperations]], thereby situating them within the landscape of modern and contemporary [[mathematics]]. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to structures derived from hyperoperations, presenting them from what could be termed a '''''hyperoperational perspective ([[hPOV]]).'''''
-Just like the concept of a [[ring]], the concept of hyperoperation serves as a potential avenue for exploring the conceptual nature of [[arithmetic operations]] through [[generalization]] and [[abstraction]]. The approach taken by the [[hPOV]] lies on the assumption that some relationships between addition and multiplication have a hierarchical nature. Certain relationships indicate a unidirectional increase in complexity and reveal pathways to explore further in those directions, '''''above and beyond''''', i.e. ''ὑπέρ ([[hyper]])'', '''''sums and products.'''''
+Just like the concept of a [[ring]], the concept of hyperoperation serves as a potential avenue for exploring the conceptual nature of [[arithmetical operations]] through [[generalization]] and [[abstraction]]. The approach taken by the [[hPOV]] lies on the assumption that some relationships between addition and multiplication have a hierarchical nature. Certain relationships indicate a unidirectional increase in complexity and reveal pathways to explore further in those directions, '''''above and beyond''''', i.e. ''ὑπέρ ([[hyper]])'', '''''sums and products.'''''
 [[File:hw_3.png|thumb|600px|center|Intuition about extending the conceptual relationship between sum and product to the whole space of operations.]]

← Older revision		Revision as of 09:25, 10 May 2025
Line 1:		Line 1:
−	The [[Hyperoperations Wiki]] is intended to complement [http://math.eretrandre.org/tetrationforum/ the Tetrationforum]. By creating a ''cartographic tool'' for the exploration done at the [[Tetrationforum]], keeping track of of definitions, terms, ideas, and bibliographic references; and to formally re-~~express~~ all modern mathematical concepts from a preparatory viewpoint that is compatible with ''hyperoperative'' inquiry.	+	The [[Hyperoperations Wiki]] is intended to complement [http://math.eretrandre.org/tetrationforum/ the Tetrationforum]. By creating a ''cartographic tool'' for the exploration done at the [[Tetrationforum]], keeping track of of definitions, terms, ideas, and bibliographic references; and by formally re-expressing all modern mathematical concepts from a preparatory viewpoint that is compatible with ''hyperoperative'' inquiry.

	[[File:hw.png\|250px\|center]]		[[File:hw.png\|250px\|center]]

@@ Line 1: / Line 1: @@
-The [[Hyperoperations Wiki]] is intended to complement [http://math.eretrandre.org/tetrationforum/ the Tetrationforum]. By creating a "cartographic" tool for the exploration done at the [[Tetrationforum]], keeping track of of definitions, terms, ideas, and bibliographic references; and to formally re-express all modern mathematical concepts from a preparatory viewpoint that is compatible with "hyperoperative" inquiry.
+The [[Hyperoperations Wiki]] is intended to complement [http://math.eretrandre.org/tetrationforum/ the Tetrationforum]. By creating a ''cartographic tool'' for the exploration done at the [[Tetrationforum]], keeping track of of definitions, terms, ideas, and bibliographic references; and to formally re-express all modern mathematical concepts from a preparatory viewpoint that is compatible with ''hyperoperative'' inquiry.
 [[File:hw.png|250px|center]]
@@ Line 6: / Line 6: @@
 [[File:hw_2.png|thumb|350px|right|The hyperoperations emerge from the investigation of the nature of [[arithmetical operations]].]]
-The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperation]], thereby situating them within the landscape of modern and contemporary [[mathematics]]. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to "hyperoperations-derived" structures, presenting them from what could be termed a "hyperoperational perspective ([[hPOV]])."
+The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperation]], thereby situating them within the landscape of modern and contemporary [[mathematics]]. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to structures derived from hyperoperations, presenting them from what could be termed a '''''hyperoperational perspective ([[hPOV]]).'''''
-Just like the concept of [[ring]], the concept of hyperoperation is just a possible road to investigate the conceptual nature of the [[arithmetical operations]] by means of [[generalization]] and [[abstraction]].
+Just like the concept of a [[ring]], the concept of hyperoperation serves as a potential avenue for exploring the conceptual nature of [[arithmetic operations]] through [[generalization]] and [[abstraction]]. The approach taken by the [[hPOV]] lies on the assumption that some relationships between addition and multiplication have a hierarchical nature. Certain relationships indicate a unidirectional increase in complexity and reveal pathways to explore further in those directions, '''''above and beyond''''', i.e. ''ὑπέρ ([[hyper]])'', '''''sums and products.'''''
 ===Useful pages===
 [[Special:AllPages]] brings you to a list of all so far described terms.

← Older revision		Revision as of 06:57, 10 May 2025
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		+	===Useful pages===
	[[Special:AllPages]] brings you to a list of all so far described terms.		[[Special:AllPages]] brings you to a list of all so far described terms.

@@ Line 1: / Line 1: @@
 The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperation]], thereby situating them within the landscape of modern and contemporary [[mathematics]]. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to hyperoperative structures, presenting them from what could be termed a "hyperoperational perspective ([[hPOV]])."
-Thus, we have a twofold objective: to create a "cartographic" tool for the exploration and cataloging of definitions, ideas, and bibliographic references; and to formally re-express all modern mathematical concepts from a preparatory viewpoint that is compatible with "hyperoperative" inquiry.
+Thus, we have a twofold objective: to create a "cartographic" tool for the exploration done at the [[Tetrationforum]] by keeping track of of definitions, ideas, and bibliographic references; and to formally re-express all modern mathematical concepts from a preparatory viewpoint that is compatible with "hyperoperative" inquiry.
 [[Special:AllPages]] brings you to a list of all so far described terms.

← Older revision		Revision as of 22:41, 9 May 2025
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−	~~Hello~~, ~~Welcome~~ to the ~~Hyperoperations Wiki~~.	+	The Hyperoperations Wiki serves as an online collaborative [[encyclopedia]] with a dual purpose. On one hand, it aims to map the [[definitions]] and [[relationships]] among the mathematical structures that emerge from and are connected to the [[abstract]] [[concept]] of [[hyperoperation]], thereby situating them within the landscape of modern and contemporary [[mathematics]]. On the other hand, to maintain a self-contained nature, the wiki seeks to provide all fundamental definitions—logical, set-theoretical, algebraic, and geometric— in a manner that makes them directly applicable to hyperoperative structures, presenting them from what could be termed a "hyperoperational perspective ([[hPOV]])."
		+
		+	Thus, we have a twofold objective: to create a "cartographic" tool for the exploration and cataloging of definitions, ideas, and bibliographic references; and to formally re-express all modern mathematical concepts from a preparatory viewpoint that is compatible with "hyperoperative" inquiry.

	[[Special:AllPages]] brings you to a list of all so far described terms.		[[Special:AllPages]] brings you to a list of all so far described terms.