Conservapedia - User contributions [en]

Science

2009-02-26T21:52:49Z

AMackenzie: Expand scope.

[[Image:Cassini-science-289.jpg|right]]
'''Science''' is a methodology for discovering and classifying [[knowledge]]. The scope of science is the portion of reality which is independent of [[religion|religious]], [[politics|political]], [[culture|cultural]], and [[philosophy|philosophical]] outlook; it includes all measurable [[phenomena]]. Science can be divided into two areas: [[natural science]], dealing with the [[physical]], [[natural]] world, and [[social science]], dealing with society and human nature.

Science differs from other methodologies of classifying knowledge in that a scientific theory is a description of the world which in principle is cabable of being disproved; this is known as [[falsifiability]]. It is this property which distinguishes science from other possible methods of discovering knowledge.

[[Epicurus]] is an important figure in the development of the [[scientific method]]. He insisted that nothing should be accepted except that which has been sufficiently tested through direct observation and logical deduction. [[Roger Bacon]] is hailed by many as the father of modern science. His focus on empirical approaches to science was influential. He wrote an encyclopedia, his ''Opus Majus''.

People who study science are called [[scientist]]s. Most of the early scientists who started many of the scientific fields, and some of history's greatest thinkers, such as [[Galileo Galilei]] and [[Isaac Newton]], believed in [[God]], or some other higher power, and many were [[creationists]].
In addition, [[Christianity]] [[Christianity and Science|played a pivotal role in the development of modern science]]. However, in recent years, American scientists have been much more [[atheism|atheistic]] as a group than the general public. <ref>http://www.atheists.org/flash.line/atheism1.htm</ref>

== Principles of science ==

The basis of modern science is observation and hypothesis. It involves constructing the best theory to explain an occurrence based on the evidence at the time. The generally accepted scientific procedure is:

* Observations of an unknown phenomenon are made
* A hypothesis is made to explain the observations
* A experiment or experiments are carried out to test the hypothesis.
** If multiple experiments support the hypothesis it is considered a theory
** If the experiment does not support the hypothesis it is either rewritten or discarded
* If at a later date evidence is produced which contradicts the theory, it is modified or discarded and a new hypothesis is developed

One of the fundamental tenets of science is that no theory is absolute; theories are constantly changing in response to the observation of new evidence. Hence, a scientific theory that begins with an immutable conclusion and attempts to "fit the facts" to that argument does not fall within the realm of true scientific research.<ref>Examples would include the claims that God created the world and that all living things evolved from a common ancestor.</ref>

On the other hand, in the absence of repeatable experiments being able to be done (past events such the very beginning of the universe), scientists believe in following the inference to the best explanation.

==Naturalism and science==
Since the beginning of modern science, scientists have worked under the assumption that their subjects of study have been controlled by consistent natural laws.
There is good evidence that this assumption was based on the [[Christianity|Christian]] view that the laws were created by a consistent creator Who didn't change those laws on a whim.<ref>See [[Natural science#Beginnings]]</ref>
This assumption is seen as a prerequisite for logical deduction to act on the observations made. Without the assumption that the universe is consistent we cannot apply the lessons drawn from an observation to any area other than the observations themselves. If a [[chemical reaction]] occurs in a given solution in a laboratory in one city it is assumed that the same reaction can occur in a different laboratory in a different city on a different day because the chemical [[solution]] and situations will be the same.

If a capricious supernatural force was to enter the equation they could not be controlled for and could not be studied.

The physical sciences largely concern themselves with questions involving the natural, not the supernatural, but this is not the same as assuming that the supernatural does not exist. In addition, there are also the social sciences like [[history]]. [[Christian apologetics|Christian apologists]] maintain that history testifies to the supernatural existing and that the physical sciences (such as [[Biblical archaeology]]) can aid in historical determinations and testify to the existence of God and the truth of biblical Christianity.

Three broad philosophies have developed in the scientific community.
* "[[Methodological naturalism]]" adheres to [[naturalism]] insofar as it concerns scientific experiments and observations, but does not rule out a personal deity. It does, however, ''a priori'' rule out the supernatural being an explanation for observations.
* "[[Philosophical naturalism]]" adheres to the belief that there is no beings or forces beyond what can be observed; this [[atheism|atheistic]] view rejects the supernatural, or is skeptical of such beliefs.
* The third approach is to follow the inference to the best explanation regarding whether or not a supernatural or natural cause best explains a past or present observation.<ref>http://creationontheweb.com/content/view/1315/</ref><ref>http://www.cgst.edu/publication/journal/43/J43_203_Forum04Abstract.pdf</ref> For example, this third approach is advocated by [[creation science|creation scientists]] and [[intelligent design]] theorists when it comes to the origins of the natural world. Creation scientists and intelligent design theorists rightfully maintain the falsity of the [[evolution|evolutionary]] position given the lack of evidence for evolutionary position and the many lines of evidence against the evolutionary position. Another example is that the [[First Law of Thermodynamics|first]] and [[Second law of thermodynamics|second]] laws of thermodynamics argue against an eternal [[universe]], and [[creation science|creation scientists]] claim that these laws point to the universe being supernaturally created.<ref>[http://godevidences.net/space/lawsofscience.php Evidences for God From Space—Laws of Science]</ref><ref>Thompson, Bert, [http://www.apologeticspress.org/articles/2329 So Long, Eternal Universe; Hello Beginning, Hello End!], 2001 (Apologetics Press)</ref><ref>http://www.creationscience.com/onlinebook/AstroPhysicalSciences14.html</ref> But in other respects, such as why [[Krakatoa]] exploded, a natural explanation would be considered the best explanation.

==Religious cultivation of early modern science==
''See also:'' [[Christianity and Science]]

According to the historian [[H. Floris Cohen]], there exists two distinct levels of argument along this line of historical scholarship. <ref> [http://books.google.com/books?id=wu8b2NAqnb0C The Scientific Revolution: A Historiographical Inquiry], [[H. Floris Cohen]], University of Chicago Press 1994, 680 pages, ISBN 0-2261-1280-2, pages 308-321 </ref> The first to be proposed was the [[Merton thesis]] in the late 1930's, which parallels the [[The Protestant Ethic and the Spirit of Capitalism|Weber thesis]] in suggesting that the rise of science was due, at first, to a [[protestant work ethic]] but later extended to a more general biblical ethic. The second to be proposed was that of [[Reijer Hooykaas]], who held the rise of early modern science was due to a unique combination of Greek and biblical thought. One of the main aspects of Hooykaas's argument was that the Greek disrepect for manual work prevented an experimental science from truly developing until the biblical view of honoring work with one's hands was socially sanctioned. Hooykaas reaches the conclusion that "Metaphorically speaking, whereas the bodily ingredients of science may have been greek, its vitamins and hormones were biblical." <ref> * [http://books.google.com/books?id=c6TEDHvAbXAC ''Religion and the Rise of Modern Science''], Regent College Publishing, 2000. ISBN 1-5738-3018-6 </ref>

Historian and professor of religion [[Eugene Marion Klaaren|Eugene M Klaaren]] holds that "a belief in divine creation" was central to an emergence of science in seventeenth-century England. The philosopher [[Michael B. Foster]] has published influential analytical philosophy connecting Christian doctrines of creation with empiricism. Historian William B. Ashworth has argued against the historical notion of distinctive mind-sets and the idea of Catholic and Protestant sciences in "Catholicism and early modern science."<ref> [http://books.google.com/books?id=hs2edDIGCqEC ''God and nature''], Lindberg and Numbers Ed., 1986, pp. 136-66; see also [http://cas.umkc.edu/history/faculty/AshworthW/pub.html William B. Ashworth Jr.'s publication list]; this is noted on page 366 of ''Science and Religion'', [[John Hedley Brooke]], 1991, [[Cambridge University Press]]</ref> Historians James R. Jacob and Margaret C. Jacob have published the paper "The Anglican Origins of Modern Science," which endeavors to show a linkage between seventeenth century [[Anglican]] intellectual transformations and influential English scientists (e.g., [[Robert Boyle]] and [[Isaac Newton]]).<ref> [http://www.compilerpress.atfreeweb.com/Anno%20Jacob%20&%20Jacob%20Anglican%20Fdn%20of%20Modern%20Science.htm The Anglican Origins of Modern Science], [[Isis (journal)|Isis]], Volume 71, Issue 2, June 1980, 251-267; this is also noted on page 366 of ''Science and Religion'', [[John Hedley Brooke]], 1991, [[Cambridge University Press]]</ref>

Two well-respected theological surveys, which also illustrate other historical interactions between religion and science occurring in the 18th, 19th, and 20th centuries, are [[John Dillenberger]]'s ''Protestant Thought and Natural Science'' ([[Doubleday]], 1960) and [[Christopher B. Kaiser]]'s ''Creation and the History of Science'' ([[Eerdmans]], 1991).

{{quotation|When natural philosophers referred to ''laws'' of nature, they were not glibly choosing that metaphor. Laws were the result of legislation by an intelligent deity. Thus the philosopher Rene Descartes (1596-1650) insisted that he was discovering the "laws that God has put into nature." Later Newton would declare that the regulation of the solar system presupposed the "counsel and dominion of an intelligent and powerful Being."<ref> [[John Hedley Brooke]], ''Science and Religion: Some Historical Perspectives'', 1991, [[Cambridge University Press]], ISBN 0-521-23961-3, page 19 </ref>
|Historian and [[Oxford University]] [[Science and Religion]] theologian [[John Hedley Brooke]]}}

[[University of California at Berkeley]]-educated historian [[Ronald L. Numbers]] has stated that this thesis "received a boost" from mathematician and philosopher[[Alfred North Whitehead]]'s ''[[Science and the Modern World]]'' (1925). Numbers has also claimed "Despite the manifest shortcomings of the claim that Christianity gave birth to science—most glaringly, it ignores or minimizes the contributions of ancient Greeks and medieval Muslims—it too, refuses to succumb to the death it deserves. The sociologist [[Rodney Stark]] at [[Baylor University]], a [[Southern Baptist]] institution, is only the latest in a long line of Christian apologists to insist that 'Christian theology was essential for the rise of science.'"<ref> ''Science and Christianity in pulpit and pew'', [[Oxford University Press]], 2007, [[Ronald L. Numbers]], p. 4, and p.138 n. 3 where Numbers specifically raises his concerns with regards to the works of [[Michael B. Foster]], [[Reijer Hooykaas]], [[Eugene M. Klaaren]], and [[Stanley L. Jaki]] </ref>

==Notes==

{{reflist}}

==See also==
* [[Scientific method]]
* [[Computing]]

[[category:science]]

Talk:Axiom of Choice

2008-08-02T21:45:10Z

AMackenzie: /* Not really controversial anymore */

== Not really controversial anymore ==

Any proof which uses the axiom of choice can be transformed into a proof that doesn't. Granted, it will be a somewhat more complicated proof, but it always works, and that's a fact. That is the reason that AC is much less controversial these days than it was, in the early 1900s.

There is a complete explanation of the process and the proof that it's reliable [http://web.unicam.it/matinf/aila/Scuola%20AILA/sulla%20varieta%20dei%20metodiAC_sem02.pdf here].

Also, the profoundly intuitive [http://en.wikipedia.org/wiki/Trichotomy_%28mathematics%29 trichotomy] is equivalent to AC, so be careful what you call controversial. [[User:BenjB|BenjB]] 20:29, 27 January 2008 (EST)

Actually, the Axiom of Choice has been proven '''independent''' of ZF, so there is no such transformation of a proof. Otherwise, "prove" AC as follows:
-----------------------------------------------
1. Axiom of Choice | Reason: Axiom of Choice

Then transform it to not need AC.
Result: AC proven in ZF,so ZFC=ZF.
But AC proved independent of ZF.
Therefore, no such transformation exists.
QED
[[User:SamSamson|SamSamson]] 21:50, 8 June 2008 (EDT)

This axiom actually can prove things which are (provably) unprovable without it; an example is Zorn's lemma. (Funnily enough, although Zorn is a guy's name, it's also a German word meaning "rage" ;-) --[[User:AMackenzie|AMackenzie]] 14:29, 2 August 2008 (EDT)

: The Axiom of Choice can also be used to "prove" absurdities, as explained by the entry here.--[[User:Aschlafly|Aschlafly]] 14:52, 2 August 2008 (EDT)
It is a wonderful thing in mathematics when an absurdity is proven - it expands the consciousness. Either the "absurdity" isn't as absurd as once thought (the Earth is a ball, not a plain; there are just as many whole numbers as there are fractions), or some axiom needs to be rethought, or you've made a mistake. Whatever, you can learn from it.--[[User:AMackenzie|AMackenzie]] 17:45, 2 August 2008 (EDT)

Talk:Axiom of Choice

2008-08-02T18:29:59Z

AMackenzie: Elucidate "unprovable without it".

Axiom of Choice

2008-08-02T18:26:16Z

AMackenzie: "unproven" back to "unprovable" (see talk)

The '''Axiom of Choice''' ('''AC''')<ref>There are numerous equivalent [[mathematical formula]]e which capture the idea of the Axiom of Choice. Here are two:
::<math>\forall x\;(\forall y\;y\in x\Rightarrow\exists z\; z\in y)\;\exists S\;(\forall z\;z\in x\Rightarrow\exists w\;w\in z\;\wedge\;w\in S)\;\wedge\;(\forall v\;v\in z\;\wedge\;v\in S\Rightarrow v=w)</math>

and
::<math>\forall x\neq\varnothing\;\exists S\;(\forall z\in x\;\exists_1w\in z\cap S).</math></ref> is a powerful [[axiom]] of [[Zermelo-Fraenkel]] set theory that is useful for proving otherwise unprovable problems, but which has also been used to prove statements that appear absurd.<ref>See, e.g., the [[Banach-Tarski Paradox]] discussed below.</ref> The '''Axiom of Choice''' holds that:<ref name="willard">Stephen Willard, "General Topology" 1.17, p. 9 (Dover 2004)</ref>

::given any collection of sets, however large, we can pick one element from each set in the collection.

In layman's terms, the Axiom of Choice is "For every nonempty set there is a choice function." Or, "We can choose one element from every element of a nonempty set of disjoint sets and this process by which we choose these sets will set up a new set." More precisely, the Axiom of Choice states that:

::For every collection of nonempty sets S, there exists a function f such that f(S) is a member of S for every possible S.

Mathworld explains the Axiom of Choice as follows:<ref>http://mathworld.wolfram.com/AxiomofChoice.html</ref>

::Given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets.

Yet another helpful explanation of the Axiom of Choice is this:<ref name="kuro5hin">http://www.kuro5hin.org/story/2003/5/23/134430/275</ref>

::If you have a collection of sets C (which may potentially contain an uncountably large number of sets), then there exists a set H, called the choice set, which contains precisely one element from each (non-empty) set in C. H is called the "choice set" because you are essentially going through each set in C and choosing one element from it. One feature of the Axiom of Choice is that H is simply assumed to exist; there is no algorithm given which might tell you how to construct an example of H.

It has been shown that the Axiom of Choice is consistent with and independent of the other axioms of set theory. Thus assuming the Axiom of Choice and assuming its negation each result in consistent set theories, differing from each other.

== Use of the Axiom of Choice ==

The Axiom of Choice has many equivalent statements, such as the [[Tychonoff theorem]], the [[Well-Ordering Theorem]], the existence of [[cardinal number]]s, the existence of a basis for every vector space, and the existence of subsets of the real line which do not have a well-defined [[Lebesgue measure]]. In [[algebra]] it is common to use [[Zorn's Lemma]] (also equivalent to the Axiom of Choice) to study [[ideal]]s in infinite [[Noetherian ring]]s.

Use of the Axiom of Choice has led to some seemingly absurd results. In the [[Banach-Tarski Paradox]], the Axiom of Choice is used to prove that a solid sphere of infinitely divisible parts may be chopped up and reconstructed as two new spheres of identical size, thereby creating 2 out of only 1. This paradox is proven only through use of the Axiom of Choice, and the authors of this proof did so to criticize this Axiom. One attempt to resolve this apparent contradiction is to show that physical spheres are not [[Lebesgue measurable]].<ref name="kuro5hin" />

The highly publicized proof of [[Fermat's Last Theorem]] relies on the Axiom of Choice, which was one of the points of criticism by [[Marilyn vos Savant]] in her column and book.<ref>Ask Marilyn ® by Marilyn vos Savant, Parade Magazine. November 21, 1993</ref><ref>''The World's Most Famous Math Problem: The Proof of Fermat's Last Theorem and Other Mathematical Mysteries'', Marilyn vos Savant. St. Martin's Griffin, 1993</ref>

== Controversy ==

Despite its usefulness, many mathematicians reject the Axiom of Choice. "It bothers some people because it asserts the existence of a set ... without giving enough information to determine that set uniquely (by applying a finite number of rules), and it is the ''only'' formal set-theoretical axiom which does this. For this reason it is customary to mention the axiom of choice whenever it is used. It need not be used if the number of sets is finite."<ref name="willard" />

The rejection of the Axiom of Choice reflects a preference for constructive [[Proof|mathematical proofs]]. AC, by its very nature, is nonconstructive, since it merely asserts that a choice function exists, but does not give an explicit method for its construction. Since the Axiom of Choice is independent of the other axioms, assuming this axiom results in a more constrained system than otherwise. Thus results provable with it might not be provable or might not even be true, without it.

The formulation of a constructive Axiom of Choice is one of three major problems which challenge 21st century logicians.<ref>Edna Ernestine Kramer, "The Nature and Growth of Modern Mathematics", Princeton University
Press, 1982. p. 595.</ref>

== Sources ==

*http://www.math.vanderbilt.edu/~schectex/ccc/choice.html

== References ==

<references/>

[[category:set theory]]
[[category:mathematics]]

Axiom of Choice

2008-08-02T11:48:49Z

AMackenzie: State that AC is independent of and consistent with the other ZF axioms.

The '''Axiom of Choice''' ('''AC''')<ref>There are numerous equivalent [[mathematical formula]]e which capture the idea of the Axiom of Choice. Here are two:
::<math>\forall x\;(\forall y\;y\in x\Rightarrow\exists z\; z\in y)\;\exists S\;(\forall z\;z\in x\Rightarrow\exists w\;w\in z\;\wedge\;w\in S)\;\wedge\;(\forall v\;v\in z\;\wedge\;v\in S\Rightarrow v=w)</math>

and
::<math>\forall x\neq\varnothing\;\exists S\;(\forall z\in x\;\exists_1w\in z\cap S).</math></ref> is an [[axiom]] of [[Zermelo-Fraenkel]] set theory holding that:<ref name="willard">Stephen Willard, "General Topology" 1.17, p. 9 (Dover 2004)</ref>

::given any collection of sets, however large, we can pick one element from each set in the collection.

In layman's terms, the Axiom of Choice is "For every nonempty set there is a choice function." Or, "We can choose one element from every element of a nonempty set of disjoint sets and this process by which we choose these sets will set up a new set." More precisely, the Axiom of Choice states that:

::For every collection of nonempty sets S, there exists a function f such that f(S) is a member of S for every possible S.

Mathworld explains the Axiom of Choice as follows:<ref>http://mathworld.wolfram.com/AxiomofChoice.html</ref>

::Given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets.

Yet another helpful explanation of the Axiom of Choice is this:<ref name="kuro5hin">http://www.kuro5hin.org/story/2003/5/23/134430/275</ref>

::If you have a collection of sets C (which may potentially contain an uncountably large number of sets), then there exists a set H, called the choice set, which contains precisely one element from each (non-empty) set in C. H is called the "choice set" because you are essentially going through each set in C and choosing one element from it. One feature of the Axiom of Choice is that H is simply assumed to exist; there is no algorithm given which might tell you how to construct an example of H.

It has been shown that the Axiom of Choice is consistent with and independent of the other axioms of set theory. Thus assuming the Axiom of Choice and assuming its negation each result in consistent set theories, differing from eachother.

== Use of the Axiom of Choice ==

The Axiom of Choice has many equivalent statements, such as the [[Tychonoff theorem]], the [[Well-Ordering Theorem]], the existence of [[cardinal number]]s, the existence of a basis for every vector space, and the existence of subsets of the real line which do not have a well-defined [[Lebesgue measure]]. In [[algebra]] it is common to use [[Zorn's Lemma]] (also equivalent to the Axiom of Choice) to study [[ideal]]s in infinite [[Noetherian ring]]s.

Use of the Axiom of Choice has led to some seemingly absurd results. In the [[Banach-Tarski Paradox]], the Axiom of Choice is used to prove that a solid sphere of infinitely divisible parts may be chopped up and reconstructed as two new spheres of identical size, thereby creating 2 out of only 1. This paradox is proven only through use of the Axiom of Choice, and the authors of this proof did so to criticize this Axiom. One attempt to resolve this apparent contradiction is to show that physical spheres are not [[Lebesgue measurable]].<ref name="kuro5hin" />

The highly publicized proof of [[Fermat's Last Theorem]] relies on the Axiom of Choice, which was one of the points of criticism by [[Marilyn vos Savant]] in her column and book.<ref>Ask Marilyn ® by Marilyn vos Savant, Parade Magazine. November 21, 1993</ref><ref>''The World's Most Famous Math Problem: The Proof of Fermat's Last Theorem and Other Mathematical Mysteries'', Marilyn vos Savant. St. Martin's Griffin, 1993</ref>

== Controversy ==

Despite its usefulness, many mathematicians reject the Axiom of Choice. "It bothers some people because it asserts the existence of a set ... without giving enough information to determine that set uniquely (by applying a finite number of rules), and it is the ''only'' formal set-theoretical axiom which does this. For this reason it is customary to mention the axiom of choice whenever it is used. It need not be used if the number of sets is finite."<ref name="willard" />

The rejection of the Axiom of Choice reflects a preference for constructive [[Proof|mathematical proofs]]. AC, by its very nature, is nonconstructive, since it merely asserts that a choice function exists, but does not give an explicit method for its construction. Since the Axiom of Choice is independent of the other axioms, assuming this axiom results in a more constrained system than otherwise. Thus results provable with it might not be provable or might not even be true, without it.

The formulation of a constructive Axiom of Choice is one of three major problems which challenge 21st century logicians.<ref>Edna Ernestine Kramer, "The Nature and Growth of Modern Mathematics", Princeton University
Press, 1982. p. 595.</ref>

== Sources ==

*http://www.math.vanderbilt.edu/~schectex/ccc/choice.html

== References ==

<references/>

[[category:set theory]]
[[category:mathematics]]