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Topic: Nonconstructive proof

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	Mathematical proof - Encyclopedia, History, Geography and Biography
		A proof is a logical argument, not an empirical one.
		The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.
		For example, the first proof of the four colour theorem was a proof by exhaustion with 1,936 cases.
	www.arikah.com /encyclopedia/Mathematical_proof (1325 words)

	The law of the excluded muddle
		An example of the sort of nonconstructive proof which became acceptable in this way is that of the Hilbert basis theorem (If a commutative ring R is Noetherian then so is the ring of polynomials over R), of which Paul Gordan said:"This isn't mathematics, it is theology!".
		Cantor's proof of the existence of transcendental numbers is sometimes thought to be non-constructive, but if you accept that an algorithm for generating a sequence of decimal digits defines a real number then Cantor's proof is certainly constructive.
		Proofs which use the axiom of choice are prime examples of nonconstructive proofs (as they don't give a description of how to calculate f).
	www.chronon.org /articles/excluded_muddle.html (1035 words)

	Mathematical Proof Encyclopedia Article @ Proves.org (Site not responding. Last check: 2007-11-05)
		A proof is a logical argument, not an empirical one.
		The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.
		Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists.
	www.proves.org /encyclopedia/Mathematical_proof (1430 words)

	Encyclopedia :: encyclopedia : Nonconstructive proof (Site not responding. Last check: 2007-11-05)
		In mathematics, a nonconstructive proof, is a mathematical proof that purports to demonstrate the existence of something, but which does not say how to construct it.
		Many nonconstructive proofs assume the non-existence of the thing whose existence is required to be proven, and deduce a contradiction.
		Nearly every proof which invokes the axiom of choice is nonconstructive in nature because this axiom is fundamentally nonconstructive.
	www.hallencyclopedia.com /topic/Nonconstructive_proof.html (349 words)

	Home > Port Jefferson, New York, NY, 11777, Port Jefferson Real Estate, Port Jefferson Yellow Pages, Port Jefferson ... (Site not responding. Last check: 2007-11-05)
		Proof by induction is where a "base case" is proved, and an "induction rule" used to prove an (often infinite) series of other cases.
		Proof by contradiction (also known as reductio ad absurdum, Latin for "reduction into the absurd") is where it is shown that if some statement were false, a logical contradiction occurs, hence the statement must be true.
		Proof by exhaustion is where the conclusion is established by dividing it into a finite number of cases and proving each one separately.
	www.portjeffersonnyus.com /topic/Mathematical_proof (1315 words)

	Elementary Number Theory and Methods of Proof
		Start the proof by supposing that x is a particular but arbitrarily chosen element of D for which the hypothesis P(x) is true.
		Writing proofs is similar to writing a computer program based on a set of specifications: organize your thoughts, declare your variables, document thoroughly [italicized brackets here], and follow a logical progression.
		Understanding the ideas of generalizing from the generic particular and the method of direct proof, allows one to write the beginnings of a proof even for a theorem not well understood.
	people.uncw.edu /norris/133_sp04/proofs/proo.htm (1437 words)

	Constructive Mathematics
		Cantor's proof that the set of transcendental numbers was infinite failed rather spectacularly to meet these requirements, however, in that it did not give rise to one example of a transcendental number, let alone an infinity of them.
		A proof by reductio ad absurdum that establishes the existence of an object in a finite set is perfectly acceptable to any mathematician; one can always in principle produce the object by checking through all the members of the set.
		For example, in Cantor's proof that there are transcendental numbers the assumption that all numbers are algebraic is shown to lead to a contradiction; therefore by the law of the excluded middle there must be transcendental, or nonalgebraic, numbers.
	digitalphysics.org /Publications/Cal79/html/cmath.htm (8036 words)

	Nonconstructive proof (Site not responding. Last check: 2007-11-05)
		Another example of a nonconstructive theorem is John Nash's proof that the game of Hex is a first-player win.
		Practically every proof which invokes the axiom of choice is nonconstructive in nature because this axiom is nonconstructive at heart.
		According to the philosophical viewpoint of mathematical constructivism, nonconstructive proofs are invalid.
	www.black-science.org /wikipedia/n/no/nonconstructive_proof.html (244 words)

	Methods of mathematics proof
		Nonconstructive proofs are used in proving statements other than existence statements; they are also used sometimes to prove existence statements.
		We recommend that a Proof by Contradiction be one that begins with p and ~q and ends up obtaining the negation of the premise, and that a Reductio Ad Absurdum Proof be one that ends up obtaining any contradiction of a known truth.
		Since both RAA and proof by contrapositive are indirect proofs, it is clearer to the reader of the proof not to mention RAA as just an indirect proof.
	www.mathpath.org /proof/proof.methods.htm (2459 words)

	Proofs
		A conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion.
		In mathematics, a proof by infinite descent is a particular kind of proof by mathematical induction.
		Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence.
	www.shortopedia.com /P/R/Proofs (1298 words)

	A transcendental number (Site not responding. Last check: 2007-11-05)
		Some people claim this as an example of a nonconstructive proof - the existence of something is shown without giving any idea of how to construct it.
		However I find that most proofs that are supposed to be nonconstructive are nothing of the kind.
		Certainly in this case it is possible to use Cantor's proof to construct a non-algebraic real number.
	www.chronon.org /applets/transcendental.html (352 words)

	PlanetMath: Gödel's incompleteness theorems
		This second form of the theorem is the one usually proved, although the theorem is usually stated in a form for which the nonconstructive proof based on Tarski's result would suffice.
		The proof for this stronger version is based on a similar idea as Tarski's result.
		The first version of the proof can be used to show that also many non-axiomatisable theories are incomplete.
	planetmath.org /encyclopedia/GodelsIncompletenessTheorems.html (723 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Proofs are essential in the theoretical underpinnings of computer science, and will be used extensively in the rest of this class.
		The most important things for you to get from the section at this time are what constitutes a valid proof, what some of the proof techniques are, and what some of the most common fallacies are.
		There is a guide to proof writing in the 'Student Center' in the textbook web site, but much of the notation and many of the examples in that guide are from later sections of the text.
	www-users.itlabs.umn.edu /classes/Spring-2004/csci2011/notes/n15.txt (339 words)

	Nonconstructive proof
		In mathematics, a nonconstructive proof, as opposed to a constructive proof, is a mathematical proof that purports to demonstrate the existence of something, but does not reveal how to construct it.
		The term "pure existence proof" is often used as a synonym for "nonconstructive proof", where "pure" means that the proof just shows existence and yields nothing else.
		The statement "Either q is rational or it is irrational", from the above proof, is an instance of the law of excluded middle, which is not valid within a constructive proof.
	www.danceage.com /biography/sdmc_Nonconstructive (391 words)

	Mathematical proof - Wikipedia, the free encyclopedia
		These were once the primary study of philosophers of mathematics.
		A famous example of a proof by contradiction shows that
		What are mathematical proofs and why they are important?
	en.wikipedia.org /wiki/Mathematical_proof (1296 words)

	Proofs Encyclopedia Article @ PSAMathe.net (PSA Mathe) (Site not responding. Last check: 2007-11-05)
		Proof of the Euler product formula for the Riemann zeta function
		Proof that the sum of the reciprocals of the primes diverges
		Proofs of Fermat's theorem on sums of two squares
	www.psamathe.net /encyclopedia/Category:Proofs (209 words)

	[No title]
		A proof of a thesis under these labels is invoked whenever the thesis cannot be proved constructively by deriving what it is supposed to prove.
		In factor language, "proof by contradicction" (double negation) means that a number is equivalent to the complement of its complement.
		Elsewhere I argue that only mathematics by constructive proof is "First Class Mathematics"; that math by nonconstructive proof is "Second Class Mathematics".
	members.fortunecity.com /jonhays/proofby.htm (718 words)

	Kids.Net.Au - Encyclopedia > Talk:Complexity classes P and NP (Site not responding. Last check: 2007-11-05)
		If they explicitly exhibit a polynomial algorithm for an NP complete problem, then we would know polynomial algorithms for all NP problems, and if the degrees of the involved polynomials are reasonable, that would be huge news in many fields, but is considered extremely unlikely.
		However, it isn't really possible for there to be a nonconstructive proof of P=NP with no known algorithm.
		But then my original statement stands: if we find a non-constructive proof of P=NP, then we still wouldn't have a polynomial algorithm for NP problems, since a polynomial algorithm has to stop after p(n) steps.
	encyclopedia.kids.net.au /page/ta/Talk:Complexity_classes_P_and_NP (1151 words)

	MATH220 Formal Methods -- L. E. Rogers, Pepperdine University ... (Site not responding. Last check: 2007-11-05)
		Proofs of such theorems are called existence proofs.
		Sometimes these proofs will actually exhibit the desired value (a constructive proof), but other times the proof may simply verify that such a value does exist without actually finding it (a nonconstructive proof).
		This new conjecture would then be subject to investigation -- perhaps being shown true (by a proof) or false (by a counterexample).
	faculty.pepperdine.edu /lrogers/ma220/ch1/1-5.htm (868 words)

	Citations: Systems of explicit mathematics with nonconstructive ¯-operator and join - Glass, Strahm (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		for instance [10] theorem 18.4) The proof of b) is by induction on #.
		for instance [23] theorem 18.4) The proof of b) is by induction on #.
		In this paper the logical relationship between three weak forms of induction on the natural numbers in BON is investigated: set induction, operation induction, and N induction.
	citeseer.ist.psu.edu /context/79187/0 (1666 words)

	Research at UVSC (Site not responding. Last check: 2007-11-05)
		One of my areas of research interest is called “constructive mathematics”; where mathematical proofs must be done “constructively”.
		A nonconstructive existence proof would merely show the impossibility that such a function could fail to exist.
		A constructive proof is also a valid classical proof, but not vice versa.
	research.uvsc.edu /default.asp?Dept=MATH (112 words)

	NONCONSTRUCTIVE
		When mathematicians cannot find a constructive proof of a theorem, resort is made to a nonconsructive proof, such as reductio ad absurdum (a.k.a.
		For example, here is Euclid's Proof of the Theorem: The square root of two is not a rational number.
		But I also accept a nonconstructive proof as second class mathematics for another very good reason: history shows that existence of a nonconsructive proof often motivates discovery of a constructive proof of the same mathematical statement, advancing mathematics.
	members.fortunecity.com /jonhays/nonconstruct.htm (919 words)

	Constructive Mathematics
		Constructive mathematics is based on the idea that the logical connectives and the existential quantifier are to be interpreted as instructions on how to construct a proof of the statement involving these logical expressions.
		This suspicion is doubtless reinforced by recollection of the furore that arose after Hilbert's original, highly nonconstructive proof of that theorem: witness Gordan's famous remark, `Das ist nicht Mathematik.
		The basic idea in these applications is that from each proof in BISH we can extract a programme whose correctness is provided by the proof from which it is extracted.
	www.seop.leeds.ac.uk /archives/fall1998/entries/mathematics-constructive (1652 words)

	Proof Theory At Work: Program Development In The Minlog System (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		Abstract: INTRODUCTION The old idea that proofs are in some sense functions, has been made precise by the Curry-Howard-correspondence between proofs in natural deduction and terms in typed-calculus.
		Since the latter can be viewed as an idealized functional programming language, this amounts to an interpretation of proofs as functional programs.
		Proof Theory At Work: Program Development In The..
	citeseer.ist.psu.edu /316017.html (442 words)

	Proof -- from Wolfram MathWorld
		I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof....
		There is some debate among mathematicians as to just what constitutes a proof.
		The four-color theorem is an example of this debate, since its "proof" relies on an exhaustive computer testing of many individual cases which cannot be verified "by hand." While many mathematicians regard computer-assisted proofs as valid, some purists do not.
	mathworld.wolfram.com /Proof.html (394 words)

	ICS141 Lecture Notes #7 (02/02/99) (Site not responding. Last check: 2007-11-05)
		All proof techniques in propositional logic + the following 5 additional proof techniques (1) Proof for Non-Existence Show that p(x) is false for every x.
		We show that x is odd, using proof by contradiction.
		Review examples of proof techniques for propositional logic in Proof Techniques (Postscript 57K).
	www2.ics.hawaii.edu /~sugihara/course/ics141s99/note/02-02n07 (348 words)

	Constructive Mathematics
		This suspicion is doubtless reinforced by recollection of the furore that arose after Hilbert's original, highly nonconstructive proof of that theorem: witness Gordan's famous remark, `Das ist nicht Mathematik.
		In fact, the results and proofs in BISH can be interpreted, with at most minor amendments, in any reasonable model of computable mathematics, such as, for example, Weihrauch's TTE (Weihrauch [1987], [1996]).
		The basic idea in these applications is that from each proof in BISH we can extract a programme whose correctness is provided by the proof from which it is extracted.
	seop.leeds.ac.uk /archives/fall1998/entries/mathematics-constructive (1652 words)

	Nonconstructive Proof (Site not responding. Last check: 2007-11-05)
		Please look for Nonconstructive Proof, Trigonometric and Trigonometric Functions to find more Nonconstructive Proof information.
		In mathematics, a nonconstructive proof, as opposed to a constructive proof, is a mathematical proof that purports to demonstrate the...
		For infinite societies, Fishburn Kirman and Sondermann and Armstrong gave a nonconstructive proof of the existence of a social welfare function...
	www.nonconstructiveproof.info (323 words)

	[No title]
		Mathematical proofs can themselves be represented formally as discrete structures.
		The fundamental activity of mathematics is the discovery and elucidation of proofs of interesting new theorems.
		Rules of inference - Patterns of logically valid deductions from hypotheses to conclusions.¡´`0``` `` `` `V``I``ª óÏ.¨More Proof Terminology¡`ª ¨Lemma - A minor theorem used as a stepping-stone to proving a major theorem.
	bi.snu.ac.kr /Courses/dm2005spring/Module-10-ProofStrategy.ppt (236 words)

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