Theorem - Revision history

Jfromm: /* Reducibility */

2011-02-16T22:37:28Z

Reducibility

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	Some systems are '''reducible''', explainable		Some systems are '''reducible''', explainable
	and understandable because they can be		and understandable because they can be decomposed
	divided in sub-domains and there are		and divided in sub-domains and there are
	similarities in different sub-domains.		sometimes similarities in different sub-domains.
	These systems can often be explained		These systems can often be explained
	and understood by theorems.		and understood by theorems.

Jfromm: /* Reducibility */

2011-02-16T22:26:36Z

Reducibility

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	=== Reducibility ===		=== Reducibility ===

	Some systems are reducible, explainable		Some systems are '''reducible''', explainable
	and understandable because they can be		and understandable because they can be
	divided in sub-domains and there are		divided in sub-domains and there are
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	These systems can often be explained		These systems can often be explained
	and understood by theorems.		and understood by theorems.

	Some systems like cellular automata are		Some systems like cellular automata are
	irreducible, because they are based on		'''irreducible''', because they are based on
	irreducible computations, see		irreducible computations, see
	[http://mathworld.wolfram.com/ComputationalIrreducibility.html Computational Irreducibility]		[http://mathworld.wolfram.com/ComputationalIrreducibility.html Computational Irreducibility]

			An example for a reducible and explainable
			system is the natural world itself.
			If we consider the system of natural language to
			describe it, then every valid sentence or statement
			is a "theorem" about the state of the natural world.
			The interesting theorems correspond to
			metaphors.

			Words can be combined to sentence, and sentences to paragraphs and pages,
			and pages to books. Sentences describe actions, paragraphs
			episodes, pages short stories, and books long stories.
			But there are also metaphors, which describe cross domain
			relationships. Metaphors describe or reveal a hidden structure.

Jfromm at 22:19, 16 February 2011

2011-02-16T22:19:35Z

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	This is simple, of course, because it is elemental, foundational to		This is simple, of course, because it is elemental, foundational to
	systemics.		systemics.


	=== Connections and Shortcuts ===		=== Connections and Shortcuts ===
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	basic axioms, elements and operations,		basic axioms, elements and operations,
	and on the size of the system. A kind		and on the size of the system. A kind
	of emergence again, perhaps..		of [[Emergence\|emergence]] again, perhaps..

	=== Hidden structures ===		=== Hidden structures ===

Jfromm at 22:18, 16 February 2011

2011-02-16T22:18:02Z

← Older revision		Revision as of 17:18, 16 February 2011
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	we are not able to make an instant		we are not able to make an instant
	in-depth analysis of a complex system.		in-depth analysis of a complex system.

			=== Reducibility ===

			Some systems are reducible, explainable
			and understandable because they can be
			divided in sub-domains and there are
			similarities in different sub-domains.
			These systems can often be explained
			and understood by theorems.

			Some systems like cellular automata are
			irreducible, because they are based on
			irreducible computations, see
			[http://mathworld.wolfram.com/ComputationalIrreducibility.html Computational Irreducibility]

Jfromm at 22:00, 16 February 2011

2011-02-16T22:00:33Z

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	structured in a particular way. If the theorem is interesting, the structure		structured in a particular way. If the theorem is interesting, the structure
	characterized by the theorem is hidden and perhaps surprising.		characterized by the theorem is hidden and perhaps surprising.
			In mathematics, a theorem is a statement which has been proved on the basis of
			previously established statements, such as other theorems, and previously
			accepted statements, such as axioms.

	== Why are there theorems? ==		== Why are there theorems? ==

Jfromm: /* Why are there theorems? */

2011-02-16T22:00:13Z

Why are there theorems?

← Older revision		Revision as of 17:00, 16 February 2011
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	isn't.		isn't.

	=== ~~Components~~ and Relationships ===		=== Structures and Relationships ===

			The key to the solution has to do with relations -- and that
			this is related to [[Emergence\|emergence]].
	There are theorems because systems have relationships as well as		There are theorems because systems have relationships as well as
	elements, from which arise emergent properties.		elements, from which arise emergent properties.

Jfromm: /* Why are there theorems? */

2011-02-16T21:56:46Z

Why are there theorems?

@@ Line 16: / Line 16: @@
 surface? If so, why is that? If not, what's a good example of something that
 isn't.
 === Connections and Shortcuts ===

Jfromm at 21:53, 16 February 2011

2011-02-16T21:53:41Z

@@ Line 16: / Line 16: @@
 surface? If so, why is that? If not, what's a good example of something that
 isn't.
 The Princeton companion to mathematics
@@ Line 21: / Line 23: @@
 conjecture and the Atiyah-Singer Index
 Theorem to the Weil Conjectures.
 Theorems are based on connections in
 the structure of mathematical systems:
@@ Line 43: / Line 44: @@
 geometric systems seem to contain an
 infinite number of complex structures.
 The integers 0,1,2,3,4,.. may be simple,
 but there is an infinite number of them.
@@ Line 55: / Line 55: @@
 and on the size of the system. A kind
 of emergence again, perhaps..

Jfromm: Created page with "A '''theorem''' is essentially a statement to the effect that some domain is structured in a particular way. If the theorem is interesting, the structure characterized by the t..."

2011-02-16T21:49:24Z

Created page with "A '''theorem''' is essentially a statement to the effect that some domain is structured in a particular way. If the theorem is interesting, the structure characterized by the t..."

New page

A '''theorem''' is essentially a statement to the effect that some domain is
structured in a particular way. If the theorem is interesting, the structure
characterized by the theorem is hidden and perhaps surprising.

== Why are there theorems? ==

Why do so many structures have hidden internal structures?

Take the natural numbers: 0, 1, 2, 3, 4, ... It seems so simple: just one
thing following another. Yet we have number theory, which is about the
structures hidden within the naturals. So the naturals aren't just one thing
following another. Why not? Why should there be any hidden structure?

If something as simple as the naturals has inevitable hidden structure, is
there anything that doesn't? Is everything more complex than it seems on its
surface? If so, why is that? If not, what's a good example of something that
isn't.

The Princeton companion to mathematics
lists 35 major theorems, from the ABC
conjecture and the Atiyah-Singer Index
Theorem to the Weil Conjectures.

Theorems are based on connections in
the structure of mathematical systems:
to find a new theorem is like revealing
a hidden structure. Some connections are
shortcuts between different points, others
are bridges between different areas.

Why do so many structures have hidden
internal structures?
It is the reason why we do Mathematics,
otherwise it would be boring. Perhaps
because there are systems where simple
elements and rules can produce complex
structures.

The basic mathematical elements and axioms
allow a whole universe of combinations
and connections which is consistent and
complex at the same time. Algebraic and
geometric systems seem to contain an
infinite number of complex structures.

The integers 0,1,2,3,4,.. may be simple,
but there is an infinite number of them.
If we consider only the numbers of the
finite Group with 4 elements, then Number
Theory becomes less interesting.

In general the patterns and structures
which can emerge in a system depend on the
basic axioms, elements and operations,
and on the size of the system. A kind
of emergence again, perhaps..