Klein bottle and Talk:Belief: Difference between pages

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[[Image:KleinBottle3d.PNG|thumb|300px|right|Two views of a Klein bottle immersed in three-dimensional space.]]

{{philosophy|class=start|importance=}}

In [[mathematics]], the '''Klein bottle''' is a certain [[genus (mathematics)|genus]]-1 non-[[orientability|orientable]] [[surface]], ''i.e.'' a surface (a two-dimensional [[topological space]]), for which there is no distinction between the "inside" and the "outside" of the surface. The Klein bottle was first described in [[1882]] by the [[Germany|German]] mathematician [[Felix Klein]]. It is closely related to the [[Möbius strip]] and embeddings of the [[real projective plane]] such as [[Boy's surface]].

{{WikiProject Psychology|class=start|importance=Mid}}

{{WP1.0|class=Start|category=category|VA=yes}}

Hmm. Im wondering, how come Hume and Kant seem to be quoted so often here in WP.

Certainly they are [[pillar]]s of western thought, but they do have some [[holes]] in their ideas, and besides, I thought we had long ago begun the process of [[weening]] ourselves off of our [[sacred cow]]s of [[westernism]].

----

"Westernism"? What's that? If you mean Western culture generally, um, no, I'm not aware that anyone other than some "postmodern" and extremely politically correct types are making a move to "weaning ourselves" off of this material. We've got to have a huge amount of such material on Wikipedia if it's going to be complete. But this doesn't stop you from adding as much "non-Western" (whatever that means) type material as you like. --[[User:Larry Sanger|Larry Sanger]]

----

Not again... More silly resentment towards "postmodernism" and "politically correct types".... they're not out to kill you. So you disagree with them, get over it. I'd be willing to bet that you (yes, you, Larry Sanger) will be dwelling on this absurd cynicism for a very long time. Postmodernism is just a catch-all phrase for something easy to criticise; the fact is that there is no such thing as a postmodern "movement" or "school of thought" or "belief system"... The obsession with postmodernism is simply a phenomenon among critics who are desperate for a board to throw darts at.

----

Picture a bottle with a hole in the bottom. Now extend the neck. Curve the neck back on itself, insert it through the side of the bottle (a true Klein bottle in four dimensions would not require this step, but it is necessary when representing it in three-dimensional [[Euclidean space]]), and connect it to the hole in the bottom.

Would it be relevant (or interesting) to mention the logical convolutions of [[Raymond Smullyan]], eg characters who believe one thing, but consistently lie, so say the opposite, etc?

----

I'm not sure--why would it (on this page)? Wouldn't that belong on [[lying]] or something like that? --[[User:Larry Sanger|Larry Sanger]]

Unlike a drinking glass, this object has no "rim" where the surface stops abruptly. Unlike a balloon, a fly can go from the outside to the inside without passing through the surface (so there isn't really an "outside" and "inside").

: Just a thought (I'll crib what I've typed here to pad out the stub on Smulllyan, at any rate). At one point he introduces characters who only believe only false things, yet lie: hence all their statements are true. -- [[User:Tarquin|Tarquin]]

==Properties==

Topologically, the Klein bottle can be defined as the [[square (geometry)|square]] [0,1] × [0,1] with sides identified by the relations (0,''y'') ~ (1,''y'') for 0 ≤ ''y'' ≤ 1 and (''x'',0) ~ (1-''x'',1) for 0 ≤ ''x'' ≤ 1, as in the following diagram:

---->

^ ^

| |

<----

I wonder what point he was making with that. Sounds interesting...

Like the [[Möbius strip]], the Klein bottle is a two-dimensional differentiable [[manifold]] which is not [[orientability|orientable]]. Unlike the Möbius strip, the Klein bottle is a ''closed'' manifold, meaning it is a [[compact]] manifold without boundary. While the Möbius strip can be embedded in three-dimensional [[Euclidean space]] '''R'''³, the Klein bottle cannot. It can be embedded in '''R'''⁴, however.

==Is belief voluntary?==

Actually, there is something interestingly relevant we could add from the literature in epistemology: it's widely held that most people have no control over most of what they believe... --[[User:Larry Sanger|Larry Sanger]]

: I made a stub section on this matter. Please expand and improve. [[User:Andries|Andries]] 11:03, 17 Apr 2004 (UTC)

If I may add my own experience (and I am quite sure many people would recognize a pattern here)...

I have a firm belief that reincarnation exists because instinctively I can't imagine I could stop being conscious after death, but I also admit I can't live forever. But by rational thinking I also know that nothing to my knowledge can justify reincarnation. This is only one example among others where belief seems to oppose knowledge. I think there are many other such examples, essentially about concepts difficult or impossible to prove, for example involving the existence or non-existance of God.

[[User:Fafner|Fafner]] 09:47, 3 Sep 2004 (UTC)

---

If I find the time... I'll try to add sometime here. Hume (amongst others) noted that we acquire beliefs passively, that the aquisition of them is not subject to the will. Bernard Williams' paper 'Deciding to Believe' investigated this and tries to show that the coneptual relations between belief, truth and evidence rule out voluntary believing. While some have shown that his argument for the incoherence of 'believing at will' is not quite right, most philsophers do believe that decision and belief can't be linked in the same way as, for instance, decision and imagination : I can successfully decide to imagine a scene, but I can't successfullly decide to belief that scene represents truely. However, as Williams noted, this doesn't rule out deciding and influencing our belief by more "roundabout routes". One could embark on a course of action, hypnosis or drugs were his suggestions, such that afterwards you would have brought it about that you belive some proposition or other. Williams remarks that this would make the person "deeply irrational". Some have questioned this but it reamins to be seen whether any convincing account of belief at will can be found. ([[User:Fabulist|Fabulist]] 18:58, 14 February 2006 (UTC))

The Klein bottle can be constructed (in a mathematical sense) by joining the edges of two Möbius strips together, as described in the following [[anonymity|anonymous]] [[limerick (poetry)|limerick]]:

==Degree of certainty==

: A mathematician named Klein

: Thought the Möbius band was divine.

: Said he: "If you glue

: The edges of two,

: You'll get a weird bottle like mine."

Why is there no mention of degree of certainty? If I believe something then it means that I think that the chance that something is true is >50%. I can believe something with 51% or 99% certainty. Quite a big difference [[User:Andries|Andries]] 20:35, 17 Mar 2004 (UTC)

==Dissecting the bottle==

---

If a Klein bottle is dissected into halves along its [[plane of symmetry]], the result is the surface shown in the following figures.

{|

| [[Image:KleinBottleCrossSection.PNG|thumb|205px|Figure A: Dissection of a Klein bottle along a symmetry plane.]]

| [[Image:Dissection of Klein bottle ROT13.PNG|thumb|250px|Figure B: ROT13 cipher inscribed along perimeter of dissection.]]

|}

''Attempted anwer'': Certainty looks like an absolute, and it may be hard to see how something can be 'a bit certain', or 'fairly certain'. Perhaps it can only be 'absolutely certain'. Sceptics seem to have a similar problem over ‘knowledge’ and conclude, rigorously, that it cannot be truly achieved. Anyway, if belief is accepted as ‘a strong feeling’ this confusion as to whether it must entail any particular degree of certainty seems to go away[[User:Yanx|Yanx]] 19:48, 15 July 2007 (UTC)

In Figure B, twenty-six points on the dissection's [[perimeter]] (the blue curve) have been labeled with the twenty six letters of the [[alphabet]]. But the dissection is a surface, not a curve. The red lines show how the surface is subtended by the perimeter.

==Belief system==

[[Image:Moebius strip ROT13.PNG|thumb|300px|right|A [[Möbius strip]], a surface with only one side. Glueing together two Möbius strips, one obtains the Klein bottle.]]

Please help with the [[belief system]] entry at [[Talk:belief system]]. Thanks. [[User:Adraeus|Adraeus]] 02:06, 7 Sep 2004 (UTC)

The Möbius strip is a surface: its perimeter is shown as a blue curve, and the red lines show how the surface is subtended by the perimeter.

:Because that article is on VfD and looks to be deleted due to no content, I am moving the associated talk page, which does have content to here:

=== Moved content from [[Talk:Belief system]], currently on [[WP:VfD|VfD]] ===

In both the dissected Klein bottle and Möbius strip, the red lines connect letters which are related mutually in the [[ROT13]] [[cipher]]. This helps to illustrate that half a Klein bottle is [[homeomorphism|homeomorphic]] to a Möbius strip.

'''Note:''' This entry needs work. [[User:Adraeus|Adraeus]] 02:10, 7 Sep 2004 (UTC)<br>

A '''belief system''' (also ''system of beliefs'') is...<br>

Here is my small contribution. It will probably need lots of works,

but after all we have to start from somewhere ;-)

I don't know if the comparison has been used somewhere, but a belief

system really looks like a mathematical logical system with a set of

axioms (unproved beliefs) and inferring rules (reasonnings).

Axioms (beliefs) are very debatable since it usually involves beliefs

in God(s), supernatural, or even science after all (how many people

among you has ever ''seen'' and ''verified'' an experiment in quantum

mechanics? probably not the majority, certainly not my case but I

''believe'' in quantum mechanics) ;-)

Inferring rules (reasonnings) are usually common to most people.

Deduction is the most reliable, induction is used to assert probable

conclusions (although I met someone acknowledging ''only'' induction

as reliable and rejecting deduction).

[[User:Fafner|Fafner]] 08:05, 7 Sep 2004 (UTC)<br>

See also

[[belief]],

[[worldview]],

[[paradigm]],

[[model]]<br>

External links

[http://www.general-semantics.org/library/conf-papers/eddy.pdf On Belief and Belief Systems] by the late [[Bob Eddy]] (Institute of [[General Semantics]])<br />

[http://www.cognitivebehavior.com/theory/beliefsystems.html Belief Systems] by [http://www.cognitivebehavior.com/ CognitiveBehavior.com]

[[User:Eric Herboso|Eric]] [[User_talk:Eric_Herboso|Herboso ]] 04:16, 23 Feb 2005 (UTC)

== Self-consistent sets of beliefs ==

It is also possible to perceive directly that Figure A is a Möbius strip, by imagining that the narrower, re-entrant part of the bottle no longer intersects line segment ''DB'' after the dissection is performed, but that it becomes loose from dissecting plane and Figure A is actually three-dimensional, with line segments ''VW'' and ''IJ'' hovering above line segment ''DB''. Then, suddenly, Figure A looks like a [[roller coaster]], and by imagining the motion of a rail car along the blue rails of this roller coaster, one perceives that this roller coaster is non-orientable.

I seem to recall something about the application of G&ouml;del's proof to beliefs, to demonstrate that one's beliefs cannot, taken as a whole, be logically self-consistent. It seemed very interesting at the time, but I can't pull up a cite -- can anyone help? (Yes, I know that G&ouml;del's proof actually demonstrates "incomplete or inconsistent", but the argument did something plausible at this point...) -- [[User:Karada|Karada]] 07:57, 13 Apr 2005 (UTC)

==Parametrizing the Klein bottle==

Since the Klein bottle is a topological surface, there can be several ways of describing it parametrically. One way is the following:

[[Gödel's incompleteness theorem#Misconceptions about Gödel's theorems]]: "The theorem only applies to systems that are used as their own proof systems"; it follows that the theorem might imply that you can't be consistent if you justify your beliefs with other beliefs; on the other hand if, as most people, you justify your beliefs from one or several external referrents, the theorem does not apply. [[User:Jules.lt|Jules LT]] 19:36, 2 November 2005 (UTC)

:<math> R(\phi) = {2 \over \pi} (1 + \sin \phi) + {1 \over 2} (1 - \sin \phi) \left\{ {4 \over \pi} - \sin \left( {\pi \over 4} - {\pi \over 4} \cos \phi \right) \cdot {\exp \left( {\pi \over 4} - {\pi \over 4} \cos \phi \right) - 1 \over \exp \left( {\pi \over 2} \right) - 1} \right\} \qquad \qquad (1) </math>

== belief is assigning probability greater than 50% ??? ==

:<math> R'(\phi) = {dR \over d\phi} </math>

::<math> = {2 \over \pi} \cos \phi - {1 \over 2} \cos \phi \left\{ {4 \over \pi} - \sin \left( {\pi \over 4} - {\pi \over 4} \cos \phi \right) \cdot { \exp \left( {\pi \over 4} - {\pi \over 4} \cos \phi \right) - 1 \over \exp \left( {\pi \over 2} \right) - 1 } \right\} -</math>

::<math> -{\pi \over 8} \cdot { \sin \phi (1 - \sin \phi) \over \exp \left( {\pi \over 2} \right) - 1 } \left\{ \sin \left( {\pi \over 4} - {\pi \over 4} \cos \phi \right) \exp \left( {\pi \over 4} - {\pi \over 4} \cos \phi \right) + \cos \left( {\pi \over 4} - {\pi \over 4} \cos \phi \right) \left( \exp \left( {\pi \over 4} - {\pi \over 4} \cos \phi \right) - 1 \right) \right\} \qquad \qquad (2) </math>

Removed from the article: "To believe something can be interpreted as assigning a [[probability]] of more than 50% that something is true."

:<math> x(\phi) = -{4 \over \pi} + R(\phi) \cos \left( {\pi \over 4} - {\pi \over 4} \cos \phi \right) \qquad \qquad (3) </math>

:<math> x'(\phi) = {dx \over d\phi} = R'(\phi) \cos \left( {\pi \over 4} - {\pi \over 4} \cos \phi \right) - {\pi \over 4} R(\phi) \sin \left( {\pi \over 4} - {\pi \over 4} \cos \phi \right) \sin \phi \qquad \qquad (4) </math>

:<math> y(\phi) = R(\phi) \sin \left( {\pi \over 4} - {\pi \over 4} \cos \phi \right) \qquad \qquad (5) </math>

:<math> y'(\phi) = {dy \over d\phi} = R'(\phi) \sin \left( {\pi \over 4} - {\pi \over 4} \cos \phi \right) + {\pi \over 4} R(\phi) \cos \left( {\pi \over 4} - {\pi \over 4} \cos \phi \right) \sin \phi \qquad \qquad (6) </math>

(also removed "The rule of the thumb from a school of [[epistemology]] that says that certainty should be as big as the corresponding evidence is called [[evidentialism]].", which is useless without the preceding "definition")

:<math> x_N(\phi) = { -y'(\phi) + \left( \cos {\phi \over 2} \right) \tanh \left( \delta \left( \sin {\phi \over 2} \right) \right) \over \sqrt{x'^2(\phi) + y'^2(\phi) + \tanh \left( \delta \left( \sin {\phi \over 2} \right) \right)}} \qquad \qquad (7) </math>

:<math> y_N(\phi) = {x'(\phi) \over \sqrt{x'^2(\phi) + y'^2(\phi) + \tanh \left( \delta \left( \sin {\phi \over 2} \right) \right)}} \qquad \qquad (8) </math>

This has little to do with [[evidentialism]], which is a [[theory of justification]], in any case.

:<math> \alpha = {1 \over 5} \qquad \qquad (9) </math>

:<math> w(\phi) = {4 \over 3} + \tanh \left( \sqrt[3]{\sin(\phi + \sin \phi)} - {3 \over 4} \right) \qquad \qquad (10) </math>

Who said that? In what book? Is it so widely accepted among scholars that it deserves mentionning so high in the article? This is not only unsourced, it also looks pretty preposterous to me. When you say "X has a probability of more than 50%", you don't believe that "X", you believe that "X is more probable than not"; this is entirely different. [[User:Jules.lt|Jules LT]] 19:13, 2 November 2005 (UTC)

:<math> X(\phi,\psi) = x(\phi) + \alpha \, x_N(\phi) \left[w(\phi) \cos(\psi) - {1 \over 4} + {1 \over 4} \cos \phi \right] \qquad \qquad (11) </math>

:<math> Y(\phi,\psi) = y(\phi) + \alpha \, y_N(\phi) \left[x(\phi) \cos (\psi) - {1 \over 4} + {1 \over 4} \cos \phi \right] \qquad \qquad (12) </math>

:<math> Z(\phi,\psi) = \alpha \, w(\phi) \sin (\psi) \qquad \qquad (13) </math>

where parameters ''&psi;'' ranges from 0 to ''2&pi;'' and ''&phi;'' ranges from 0 to ''2&pi;''; ''&delta;'' is the [[Dirac delta function]] and ''tanh'' is the [[hyperbolic tangent]] function. Variables ''X'', ''Y'' and ''Z'' in the last three equations are the actual coordinates in <math>\mathbb{R}^3</math> of a point on the Klein bottle whose parameters are ''&phi;'' and ''&psi;''.

== Definition of Belief ==

[[Image:KleinBottleParametricVersion2.PNG|thumb|300px|right|A Klein bottle which is defined by the set of parametric equations above.]]

Equations (2), (4) and (6) are [[derivative]]s with respect to ''&phi;'' of equations (1), (3) and (5) &mdash; respectively. Functions ''x(&phi;)'' and ''y(&phi;)'' are the components of a position [[vector]] function which defines the [[locus (mathematics)|locus]] of points of a plane curve which has one [[cusp]] at the origin. This curve might be called the "spine" of the Klein bottle, but this spine is not itself a subset of the Klein bottle.

A [http://www.yesselman.com/glosindx.htm#ReligiousBelief belief], in its varying degrees, can be a guess, a dogma, a hope, an intuition, a leap-of-faith. Belief is to make an hypothesis which then must pass the test of [http://en.wikipedia.org/wiki/Talk:Religion#Cash_Value Cash Value]—bringing Peace of Mind. [[User:Yesselman|Yesselman]] 20:35, 24 December 2005 (UTC)

Equations (3) and (5) not only describe a plane curve, they also describe the movement of a point along that plane curve. The cusp is a special point where the curve abruptly reverses direction by 180°. The movement of the point along the "spine" has been defined in such a way that the point slows down as it gets closer to the cusp, stops when it reaches the cusp, and then begins to accelerate in the opposite direction.

Functions ''x'(&phi;)'' and ''y'(&phi;)'' are the components of a vector function which is the tangent vector of spine. Functions ''x_N(&phi;)'' and ''y_N(&phi;)'' are the components of a vector function which is the unit normal vector of the spine (see [[Frenet-Serret formulas]]). The binormal unit vector always points in the ''+z'' direction &mdash; since it is a curve on the ''x-y'' plane &mdash; but is not necessary for parametrizing the Klein bottle. The unit normal was obtained from the tangent by [[coordinate rotation|rotating]] 90° &mdash; that is, by changing (''x&prime;'', ''y&prime;'') to (''&minus;y&prime;'', ''x&prime;'') &mdash; and then normalizing. The Dirac delta terms were added to Equations (7) and (8) in order to fix discontinuities at the pinch point.

(edited to correct it in a way)

Equations (9) and (10) describe the "amplitude constant" ''&alpha;'' and the "amplitude function" ''w(&phi;)''. Both of these are used to modulate the "[[amplitude modulation|amplitude]]" &mdash; i.e. [[magnitude]] &mdash; of the normal vector in Equations (11), (12) and (13). The points on the Klein bottle are defined by rotating the amplified normal vector 360° around the position vector's point on the spine (the tail of the normal vector is at the head of the position vector; the point on the head of the normal vector belongs to the Klein bottle; the position vector moves along the spine). Parameter ''&phi;'' specifies a point on the spine, and parameter ''&psi;'' specifies the degree of rotation of the normal around the spine.

->

==See also==

To belief is diffrent from the word believe, believe is to trust and see something in another person.

*[[Topology]]

But belief is like to imagen to trust and have faith into a higher being.

*[[Algebraic topology]]

Belief can't just be put out in words it comes from you and is within you.

I think what you ment was believe and even there is a mistake in that.

== External links ==

If you believe in a person you either do it or not you can not just believe have trust and faith in them her him or what ever just 50% else what kind of person would you be?

*[http://www.kleinbottle.com/ Acme Klein Bottles] - Actual Klein bottles! (Or at least 3D projections of them, sold by [[Clifford Stoll]]).

* [http://www.cut-the-knot.org/do_you_know/paper_strip.shtml Paper Strip Activities]

* [http://www.cut-the-knot.org/Curriculum/Combinatorics/Sliders.shtml Slider puzzles] (Slider puzzles on the plane, cylinder, torus, Klein's bottle and Projectie plane)

* [http://www.cut-the-knot.org/shortcut.shtml Klein bottle construction] (an avi movie)

*[http://www.lipsons.pwp.blueyonder.co.uk/mathlego.htm Andrew Lipson's Mathematical LEGO Sculptures] - [[Lego]] constructions of [[Möbius strip]] and Klein bottle structures. This site also shows the dissection of the Klein bottle.

*[http://www.math.uiuc.edu/~jms/Images/klein.html Klein Bottle Images by John Sullivan]

*[http://www.geom.uiuc.edu/zoo/toptype/klein/ The Klein bottle]

*[http://hektor.umcs.lublin.pl/~mikosmul/origami/misc.html A modular origami model of the Klein bottle]

== Reasoning?? ==

[[Category:Surfaces]]

''Beliefs can be acquired through perception, reasoning, contemplation or communication''

[[Category:Geometric topology]]

[[Category:Articles with ASCII art]]

This statement is plain incorrect, How on Earth can resoning be related to 'belief' . Infact they have completely opposite meanings. Obviously if you can reason(or if there is a logical explanation) to something, then there won't be any 'need' to believe because that 'thing' would be undeniable fact(like a maths equation). The point of belief only arises if there is an absence of resoning!!

[[de:Kleinsche Flasche]] [[es:Botella de Klein]] [[ja:&#12463;&#12521;&#12452;&#12531;&#12398;&#22775;]]

The only possibility here is if 'resoning' is being referred to as 'bias' dependent on culture/surroundings etc. [[User:Reasonit|Reasonit]] 00:26, 24 February 2006 (UTC)

I think this results from a confusion between belief as an unproven fact and belief as a conviction adopted after a reasonning (for example a political position). The difference between the two of them might be thin in some cases. Just a thought... [[User:Fafner|Fafner]] 08:01, 24 February 2006 (UTC)

Yes. A belief can be adopted based on a number of criteria:

- authority

- experience

- perceived phenomena

- reasoning

- discussion (e.g. clarification/debate)

"Beliefs" don't necessarily have any relation to reason. Especially those induced by authority figures. An associated topic might be rigidity of belief systems and conflicts arising therefrom..

== "Is Belief Voluntary?" section ==

"''Most philosophers hold the view that belief formation is to some extent spontaneous and involuntary.''

Most philosophers!? That's a bold and sweeping statement. I'm not sure if to just suggest that is radically POV or ask for some kind of verification. For now I've added a "citeation needed" tag and left it.

Maybe "many philosophers" would be a better choice of words, and easier to add a few references for. The word "most" suggests that nearly all philosophers past-and-present agree about this - somehow, I seriously doubt that... -[[User:Neural|Neural]] 03:42, 17 July 2006 (UTC)

== Introduction ==

The introduction:

<blockquote>Belief is usually defined as a conviction of the truth of a proposition without its verification; therefore a belief is a subjective mental interpretation derived from perceptions, contemplation(reasoning), or communication.</blockquote>

is simply wrong. At least, there is no such definition in my SOD, and if it were the case, one would not be able to believe a verified proposition. Nor is "1+1=2" a "subjective mental interpretation" (Can you think of something that is subjective and yet not mental? Interpretation of what?), yet it is something one might believe.

What is it about introductions to philosophical articles that attracts such stuff? [[User:Banno|Banno]] 07:31, 19 January 2007 (UTC)

==Religion==

The paragraph:

<blockquote>In the religious sense, "belief" refers to a part of a wider spiritual or moral foundation — generally called faith. Historically, faiths were generated by groups seeking a functionally valid foundation to sustain them. The generally accepted faiths usually note that, when the exercise of faith leads to oppression, clarification or further revelation is called for.</blockquote>

has been removed. I can;t see a reason to give prominence to religious belief. Someone may wish to insert it into a new section within the article. [[User:Banno|Banno]] 07:38, 19 January 2007 (UTC)

== Deductive vs. Inductive ==

It seems that the epistimology section contradicts itself, saying that belief is a deductive process, but the building of the belief system is an inductive one. Am I missing something? I'm in favor of stating all belief systems are inherently inductive, and that all deductive processes used in the belief system are based off of premises that require induction.

[[User:140.233.44.55|140.233.44.55]]AME 2/21/07

:I'd say rather that the whole section is OR,and should be removed. [[User:Banno|Banno]] 04:40, 22 February 2007 (UTC)

Done[[User:Peterdjones|1Z]] 17:38, 14 March 2007 (UTC)

== Belief necessarily True ==

I disagree with the lead sentence "Belief is the psychological state in which an individual is convinced of the truth of a proposition." This is easily refuted, I and many others believe in God and would agree with a proposition such as "God exists" but would not necessarily argue that it can be proven as "True". In other words you can recognize that you have a belief, such as religion, or race or sexuality, and know that it not necessarily "True" but that you believe it anyway.[[User:Tstrobaugh|Tstrobaugh]] 14:32, 16 April 2007 (UTC)

And does that apply to "2+2=4" or "the sky is blue"? Or is there a difference between mere belief,

and Belief with a capital B?

[[User:Peterdjones|1Z]] 18:04, 16 April 2007 (UTC)

: Actually I'm not sure what you consider to be Beliefs and/or beliefs, perhaps you could provide some more examples, which category is the "2+2" in? or the sky? The "2+2" one is obviously incorrect as others have stated above "Gödel had shown that mathematics is both incomplete and inconsistent. Mathematics must be incomplete because there will always exist mathematical truths that can’t be demonstrated. Truths exist in mathematics that do not follow from any axiom or theorem."[[User:Tstrobaugh|Tstrobaugh]] 20:16, 16 April 2007 (UTC)

::GIT doesn't have the slightest impact on the necessary truth of 2+2=4.

[[User:Peterdjones|1Z]] 21:40, 16 April 2007 (UTC)

::: Really? Explain how GIT has no influence on elementary math. Here's my rebuttal when you're done. (and thanks for answering all my questions, I can see this will be productive) "Gödel showed that "it is impossible to establish the internal logical consistency of a very large class of deductive systems--elementary arithmetic, for example--unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the systems themselves."(10) In short, we can have no certitude that our most cherished systems of math are free from internal contradiction." from [http://www.rae.org/godel.html].[[User:Tstrobaugh|Tstrobaugh]] 14:00, 17 April 2007 (UTC)

rems.[[User:Peterdjones|1Z]] 19:03, 17 April 2007 (UTC)

[http://www.sm.luth.se/~torkel/eget/godel/prove.html GIT does not stop you being able to prove individual theorems] [[User:Peterdjones|1Z]] 19:03, 17 April 2007 (UTC)

::Exactly my point about beliefs to begin with. Just as belief in God is accepted without proof and those that accept it know it can't be proved. From the page you cited:"So suppose we accept the axioms and methods of proof formalized in T as valid without proof."[[User:Tstrobaugh|Tstrobaugh]] 13:46, 18 April 2007 (UTC)

:::But '''that''' point has nothing to do with Godel. We don't need GIT to tell us we can't prove every axiom. (And we can adopt the formalist's approach of defining truth only within an axiomatic system). [[User:Peterdjones|1Z]] 14:34, 18 April 2007 (UTC)

:::If you think "god exists" is not necessarily true, you presumably think there is some evidence or argument which could disprove it. Would you continue to believe in God if the disproof were presented to you? if not, doesn't that show there is ''some'' connection between truth and belief? [[User:Peterdjones|1Z]] 14:48, 18 April 2007 (UTC)

:::::That is not true. I do not believe that there is any evidence or argument to disprove it, also no evidence or argument to prove it. Where prove means using empirical, objective evidence and Popperian hypo-thetico-deductive logic. The connection, as you say, between proof and belief is in mine and other believers minds and beyond the reach of scientific inquiry and objective "Truth".[[User:Tstrobaugh|Tstrobaugh]] 16:38, 18 April 2007 (UTC)

Point 1: You can think what you like, Tstrobaugh, but if you can't find your ideas in the literature, then it can't go in the Wiki. [[User:Banno|Banno]] 22:00, 15 July 2007 (UTC)

Point 2: The implication of your opening statement is that one can believe something while holding it not to be true; for example, that one could coherently say "I believe god exists , but it is not true that god exists". See [[Moore's paradox]]. You seem simply to have confused truth with proof of truth. [[User:Banno|Banno]] 22:00, 15 July 2007 (UTC)

== Removed Paragraph, For Now... ==

"If one has an external inducement to belief, such as a prospective marriage partner, he may be unable to drastically change his true belief in order to obtain the desired reward. The best he might do would be to pretend at belief. There is a possibility that with study, he would come to change his belief, depending on his earlier sources and his confidence in the validity of new ones."

I believe this paragraph needs rewritten, because the example is unclear. What I mean is the relevence to the example given in connection with the topic. (Yes, I know the connection is implied. Yet an encyclopedia is meant to give [[information]] and describe, not [[imply]].) The paragraph also did not seem consistent with the section it was previously in and probably needs moved. If no one else does, I hope to rewrite this, but I'll have to research how beliefs play roles in marital relationships (and since I am not married, well, I'll have to trust sources that are plausibly verifiable.) [[User:69.245.172.44|69.245.172.44]] 18:19, 22 July 2007 (UTC)