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Topic: Constructive proof

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	Constructivism (mathematics) - Wikipedia, the free encyclopedia
		Thus the proof of the existence of a mathematical object is tied to the possibility of its construction.
		In the constructive version, it is required that, for any given distance, it is possible to actually specify a point in the sequence where this happens (this required specification is often called the modulus of convergence).
		A proof which requires the axiom of choice is regarded as non-constructive, as it asserts the existence of a certain choice function or set without it being possible to say what it is. The Goodman-Myhill theorem further showed that the law of the excluded middle could even be derived from the full axiom of choice.
	en.wikipedia.org /wiki/Mathematical_constructivism (1331 words)

	Constructive proof - Wikipedia, the free encyclopedia
		In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object.
		This is in contrast to a nonconstructive proof (also known as an existence proof or pure existence theorem) which proves the existence of a mathematical object with certain properties, but does not provide a means of constructing an example.
		The contrast between a constructive proof and a nonconstructive proof is illustrated by the case of transcendental numbers (real or complex numbers that are not algebraic numbers).
	en.wikipedia.org /wiki/Constructive_proof (340 words)

	Nonconstructive proof - Wikipedia, the free encyclopedia
		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.
		According to the philosophical viewpoint of constructivism, nonconstructive proofs constitute a different kind of proof from constructive proofs.
		Nearly every proof which invokes the axiom of choice is nonconstructive in nature because this axiom is fundamentally nonconstructive.
	en.wikipedia.org /wiki/Nonconstructive_proof (401 words)

	Encyclopedia :: encyclopedia : Mathematical proof (Site not responding. Last check: 2007-10-31)
		The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.
		Proof by contradiction (also known as reductio ad absurdum): where it is shown that if some statement were false, a logical contradiction occurs, hence the statement must be true.
		Proof by construction: constructing a concrete example with a property to show that something having that property exists.
	www.hallencyclopedia.com /Mathematical_proof (615 words)

	Constructive Mathematics
		Constructive mathematicians believe that numbers, not sets, are the basis of mathematics and that since infinite sets do not seem to exist in nature, mathematicians can have no real, intuitive understanding on which to base judgments concerning them.
		Constructive mathematicians believe every problem in mathematics can be reduced to a numerical problem such as Goldbach's conjecture and every numerical question can be reduced to a problem of deciding whether or not a 1 appears in a sequence of O's and 1's.
		In Bishop's constructive revision of set theory, however, the axiom of choice appears to be noncontentious, because it is stipulated that to call a set nonempty one must show that the construction method specified in the definition of the set actually generates a member of the set.
	digitalphysics.org /Publications/Cal79/html/cmath.htm (8036 words)

	Methods of mathematics proof
		The only use of constructive proofs is in proving existence statements and sometimes it can be very difficult to find a constructive proof for a given existence statement.
		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.
		Although RAA proofs are often easier and more convenient, a direct proof is preferred for the reason that RAA depends for its validity on the assumption that the unprovability of the negation of p is tantamount to the provability of the negation of the negation of p.
	www.mathpath.org /proof/proof.methods.htm (2455 words)

	Constructive Mathematics
		It is important to realize that constructive algebra is algebra; in fact it is a generalization of classical algebra in that we do not assume the law of excluded middle, just as group theory is a generalization of abelian group theory in that the commutative law is not assumed.
		A constructive proof of a theorem is, in particular, a proof of that theorem.
		Ruitenburg, A course in constructive algebra, Springer-Verlag 1988.
	www.math.fau.edu /Richman/HTML/CONSTRUC.HTM (1293 words)

	Constructive Mathematical Truth
		In the early 20th century, Brouwer pioneered constructive mathematics, specifically intuitionism, as a distinct branch of mathematics with the rules of logic weakened to ensure that non-constructive results are not provable.
		The proof is by induction on n, and is analogous to the proof of Theorem 8.
		Also, in constructive arithmetic, computations should be based on numbers rather than on uncountable objects; so for constructive truth, we have to require that witnesses act based on finite segments rather than on the full witnesses, which leads to a kind of overspill exploited in Theorems 4 and 7.
	web.mit.edu /dmytro/www/ConstructiveTruth.htm (9105 words)

	Constructive Mathematics
		as “x is a proof of the proposition T”.
		He distinguishes carefully between proofs and derivations: a proof object is a witness to the fact that some proposition has been proved; whereas a derivation is the record of the construction of a proof object.
		Every constructive proof embodies an algorithm that, in principle, can be extracted and recast as a computer program; moreover, the constructive proof is itself a verification that the algorithm is correct — that is, meets its specification.
	plato.stanford.edu /entries/mathematics-constructive (6364 words)

	Logic Seminar Abstracts Spring 2002 (Site not responding. Last check: 2007-10-31)
		A new constructive proof uses the data already present in the formulation of the problem as well as special features of the problem _and_ the old proof.
		Proof mining is the process of logically analyzing proofs in mathematics with the aim of obtaining new information.
		This quantitative version, which was obtained by a logical analysis of the ineffective proof given by Borwein, Reich and Shafrir, could be used to obtain strong uniform bounds on the asymptotic regularity of such iterations in the case of bounded C and even weaker conditions.
	www-logic.stanford.edu /Abstracts/Seminar/Spring02.html (1060 words)

	GLEASON'S THEOREM HAS A CONSTRUCTIVE PROOF
		In [3] it is noted that Hellman's example leaves open the problem of finding a constructive substitute-a theorem with a constructive proof that is easily seen to be classically equivalent to Gleason's theorem.
		What precludes a constructive proof of Statement 1 is that determining the principal axes is, in general, an ill-posed problem : arbitrarily small changes in the bilinear form B can cause large changes in the principal axes.
		The constructive problem is with the former, not with the latter.
	www.math.fau.edu /Richman/Docs/glhasrev.html (2118 words)

	Investigating logical reflection, constructive proof, and explicit provability \| Decentralized Information Group (DIG) ...
		constructive proof is a much better way to think about proofs in
		And that version of the logic of proofs includes all of propositional calculus, including the law of the excluded middle.
		The Curry-Howard isomorphism converting intuitionistic proofs into typed lambda-terms is a simple instance of an internalization property of a our system lambda-infinity which unifies intuitionistic propositions (types) with lambda-calculus and which is capable of internalizing its own derivations as lambda-terms.
	dig.csail.mit.edu /breadcrumbs/node/89 (589 words)

	Proof Theory on the eve of Year 2000 (Site not responding. Last check: 2007-10-31)
		Add to this that it is closely connected with the proof theory of feasible arithmetic, and it seems clear to me that it is a classic problem of proof theory, though one that was quite unconsidered by the early pioneers.
		A precise representation of mathematical proofs as formal(izable) derivations is sought.
		Proof theorists, having failed in analysing proofs in mathematics, went on to apply their skills (somewhat opportunistically in my mind) in logical systems different from the two canonical ones, intuitionistic and classical.
	www-logic.stanford.edu /proofsurvey.html (19489 words)

	Constructive Versus Existential Proofs
		+ 1 is composite we constructed a factorization.
		Sometimes it is possible to prove the existence of something mathematical without actually constructing it.
		Well, it could be that you just cannot think of a constructive proof, or that a constructive proof is very long and tedious.
	zimmer.csufresno.edu /~larryc/proofs/proofs.construct.html (681 words)

	Omega-inconsistency in Gödel’s formal system: a constructive proof of the Entscheidungsproblem (Site not responding. Last check: 2007-10-31)
		However, we cannot construct a Turing machine T that is independent of n, and which will decide, for any given n as input, whether R(n, p) holds or not.
		Thus the omega-inconsistency of P can be seen as a constructive, and intuitionistically unobjectionable, negative proof of Hilbert’s Entscheidungsproblem under a constructive standard interpretation for P. References
		Assuming that r is a PROOF of a given FORMULA n, it seems to make the invalid assumption, “[r]=
	alixcomsi.com /CTG_00.htm (1528 words)

	A Constructive Completeness Proof for Intuitionistic Propositional Calculus, by Judith Underwood (Site not responding. Last check: 2007-10-31)
		This paper presents a constructive proof of completeness of Kripke models for the intuitionistic propositional calculus.
		The computational content of the proof is a form of the tableau decision procedure.
		If a formula is valid, the algorithm produces a proof of the formula in the form of an inhabitant of the corresponding type; if not, it produces a Kripke model and a state in the model such that the formula is not forced at that state in that model.
	www.nuprl.org /documents/Underwood/ConstructiveCompletenessProof.html (98 words)

	A Constructive Proof of Higman's Lemma, by Chetan Murthy and James R. Russell (Site not responding. Last check: 2007-10-31)
		A Constructive Proof of Higman's Lemma, by Chetan Murthy and James R. Russell
		Prior to this work, only classical (and impredicative) proofs of the Lemma were known.
		In this paper we present a direct constructive proof.
	www.nuprl.org /documents/Murthy/ConstructiveProofHigmansLemma.html (117 words)

	The law of the excluded muddle - useful books
		Worries about nonconstructive proofs began to surface in the 19th century, but the possibility of having a separate branch of mathematics restricted to constructive proofs really started in the 1920's with L. Brouwer's Intuitionism.
		Ray Mines deals with a specific area of constructive mathematics, but would be useful for those wanting to know more about the constructive version of the Hilbert basis theorem.
		Encyclopedia article on Constuctivism and proof theory (pdf file) and a Survey of the history of constructivism (ps file).
	www.chronon.org /books/excluded_muddle.html (449 words)

	Citebase - A Constructive Proof of Ky Fan's Generalization of Tucker's Lemma (Site not responding. Last check: 2007-10-31)
		A Constructive Proof of Ky Fan's Generalization of Tucker's Lemma
		We present a constructive proof of Ky Fan's combinatorial lemma concerning labellings of triangulated spheres.
		As a consequence, we produce a constructive proof of Tucker's lemma that holds for a larger class of triangulations than previous constructive proofs.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0310444 (113 words)

	Constructive Proof (Site not responding. Last check: 2007-10-31)
		In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by...
		A Constructive Proof of Vizing's Theorem (1990) (Make Corrections) (7 citations) J Misra, David Gries Home/Search Context Related View...
		In the FTA project in Nijmegen we have formalized a constructive proof of the Fundamental Theorem of Algebra.
	www.constructiveproof.info (299 words)

	Citebase - A short constructive proof of Jordan's decomposition theorem (Site not responding. Last check: 2007-10-31)
		A short constructive proof of Jordan's decomposition theorem
		Although there are many simple proofs of Jordan's decomposition theorem in the literature (see [1], the references mentioned there, and [2]), our proof seems to be even more elementary.
		In fact, all we need is the theorem on the dimensions of rang and kernel and the existence of eigenvalues of a linear transformation on a nontrivial finite dimensional complex vector space.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0312041 (132 words)

	constructive (Site not responding. Last check: 2007-10-31)
		Cantor's proof that the real numbers are uncountable can be thought of as a non-constructive proof that irrational numbers exist.
		(There are easy constructive proofs, too; but there are existence theorems with no known constructive proof).
		Obviously, all else being equal, constructive proofs are better than non-constructive proofs.
	burks.bton.ac.uk /burks/foldoc/65/24.htm (175 words)

	constructive from FOLDOC (Site not responding. Last check: 2007-10-31)
		A proof that something exists is "constructive" if it provides a method for actually constructing it.
		A few mathematicians actually reject all non-constructive arguments as invalid; this means, for instance, that the law of the excluded middle (either P or not-P must hold, whatever P is) has to go; this makes proof by contradiction invalid.
		Most mathematicians are perfectly happy with non-constructive proofs; however, the constructive approach is popular in theoretical computer science, both because computer scientists are less given to abstraction than mathematicians and because intuitionistic logic turns out to be the right theory for a theoretical treatment of the foundations of computer science.
	foldoc.org /?constructive (183 words)

	Contents (Site not responding. Last check: 2007-10-31)
		Proof: 3 is a prime number less than 10
		In the direct proof we use the representation of the object with property P
		The contrapositive proof is used when it is easier
	storm.simpson.edu /~sinapova/cmsc180a/L11-Proofs.htm (948 words)

	CONSTRUCTIVE PROOF OF EXISTENCE
		By finding an integer that is larger, I have shown how to construct that integer.
		In a constructive proof of existence, you not only prove that something exists, but you find it
		Because m is prime and m is greater than n, we have constructed a prime number
	nebula.deanza.fhda.edu /math/FT/bloom/Math22/ExistenceProofExamples.htm (411 words)

	A Short And Constructive Proof of Tarski's Fixed-Point Theorem
		I give short and constructive proofs of Tarski's fixed-point theorem, and of a much-used extension of Tarski's fixed-point theorem to set- valued maps.
		If you experience problems downloading a file, check if you have the proper application to view it first.
		"A short and constructive proof of Tarski’s fixed-point theorem," International Journal of Game Theory, Springer, vol.
	ideas.repec.org /p/wpa/wuwpge/0305001.html (286 words)

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