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Topic: Rigorous proof

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	Gila Hanna: The Ongoing Value of Proof
		It maintains that proof deserves a prominent place in the curriculum because it continues to be a central feature of mathematics itself, as the preferred method of verification, and because it is a valuable tool for promoting mathematical understanding.
		The method of proof analysis is admittedly engaging, but the case for it as a general method rests upon a single sample, the study of polyhedra, an area in which it is relatively easy to suggest the counterexamples required.
		Certainly a proof offered by a very reputable mathematician would initially be given the benefit of the doubt, and in that sense the fact that this mathematician is considered an authority by other mathematicians would play some role in the eventual acceptance of the proof.
	fcis.oise.utoronto.ca /~ghanna/pme96prf.html (5483 words)

	Basel problem - Wikipedia, the free encyclopedia
		It is by far the simplest proof yet available; while most proofs utilise results from advanced mathematics, such as Fourier analysis, complex analysis, and multivariable calculus, the following does not even require single-variable calculus (although a single limit is taken at the end).
		Proof: This requires mathematical induction and some properties of the binomial coefficients.
		Proof: This is a consequence of the fundamental theorem of algebra.
	en.wikipedia.org /wiki/Basel_problem (1511 words)

	proof-writing 101 (Site not responding. Last check: )
		Proofs should always have a beginning (the statement of the problem you wish to solve), a middle (a course of reasoning, one statement following from the last), and an end (in which you arrive at the statement you intended).
		An elegant proof is the best proof of all; it is the proof that is clearly understood and to the point, sparing no unnecessary words.
		These are the parts of the proof that in some sense make the whole thing hang together, and really get you from point A to point B. In grading real Problems, there are often one (or more) such moves which you really have to figure out in order to get a solution.
	darkwing.uoregon.edu /~sdenton/proof.htm (1262 words)

	Aspaqlaria: The Kuzari Proof, part II
		In other words, if I want someone to accept my rigorous proof of G-d's existance, they must first accept all my givens, as well as the validity of each of my implications.
		The deeper faith is one in which the principles of Judaism are postulates, not theorems that require proving.
		Proofs have a role in deepening understanding -- after the basic principles have been accepted.
	www.aishdas.org /asp/2004/12/kuzari-proof-part-ii.shtml (664 words)

	[No title]
		Thus we claim that the role of rigorous proof in mathematics is functionally analogous to the role of experiment in the natural sciences.
		Complete proofs were achieved by Arnold in 1959 for the analytic case and by Moser in 1962 for the smooth case.
		Citing a theoretical paper for a structural ingredient of a supposedly rigorous proof must be handled with care, and a flag in the title would indicate when such care is needed.
	www.ams.org /bull/pre-1996-data/199329-1/Jaffe (6126 words)

	Interaction - Smart Board
		To produce a proof, we must somehow make use of the fact that the set is totally bounded to extract a Cauchy sequence from an arbitrary infinite sequence in the set.
		The key to the proof is in utilizing the finiteness of the cover; since the sequence is infinite, it must hit at least one of the e-balls in the cover "infinitely often," which is another common idiom of analysis that can be given a visual representation.
		Verifying this symbolically leads to a rigorous proof Another fact that the system might provide a visual hint of is that the construction does not ensure strictly nested s-balls, which may require special care in some constructions, although it causes no difficulty in this example.
	www.cs.brown.edu /stc/resea/interaction/research_I12a.html (974 words)

	[No title]
		It is unlikely that the proof of the general geometrization conjecture will consist of pushing the same proof further.Not all proofs have an identical role in the logical scaffolding we are building for mathematics.
		This particular proof has only temporary logical value, although it has a high motivational value in helping support a certain vision for the structure of 3-manifolds. When the general proof of the geometrisation conjecture is found, “proofs of special cases are likely to become obsolete.” At this point, Thurston’s conception of proof appears rather Lakatosian.
		The method of lemma-incorporation “upholds the proof but reduces the domain of the main conjecture to the very domain of the guilty lemma.” In this way, the lemma refuted by the counterexample is built into the conjecture.
	philsci-archive.pitt.edu /archive/00000286/00/lakps.doc (10116 words)

	Proof, Thought and Aesthetics
		Proofs of hundreds of pages, understandable to only a handful of mathematicians with expertise in highly specialized branches of mathematics, are now more typical of the work being done in mathematics.
		Proof is the method mathematicians use to communicate the truth or validity of a piece of knowledge.
		And the notion of proof transcends mathematics –; logical argument is at the heart of philosophy, the law, rhetoric and debate.
	www.chatham.edu /PTI/ProofinMathematics/proof_curriculum.htm (4646 words)

	Proof and Beauty
		This is the computer-assisted proof, and it is like a fast-food outlet that serves billions of dull, repetitive burgers.
		One of the earliest proofs to use this brute-force computer method was the proof of the four-colour theorem.
		The answer to Seifert's question is no. Her solution overcame what everyone considered to be the biggest obstacle-dealing with smoothness-with a brilliant trick in which she made the flow "eat its own tail", as a result of which smoothness ceased to be an issue.
	members.fortunecity.com /templarseries/proof.html (3115 words)

	Letters to the Editor
		Finding rigorous upper and lower bounds is a different skill, certainly a very demanding one, but dare I say it, a rather tedious one.
		A proof given by a working mathematician is a line of arguments which by certain standards of rigor leads from the unknown to the known, the new to the old, the difficult to the easy.
		The points about proof and correctness made by Joe Keller and Folkmar Bornemann are very valid ones; as computer scientists we know that we are always in danger of producing results that are wrong.
	www.siam.org /siamnews/01-03/letters.htm (952 words)

	Citations: The use of machines to assist in rigorous proof - Milner (ResearchIndex)
		The use of machines to assist in rigorous proof.
		Milner, R. The use of machines to assist in rigorous proof.
		As a result, the safe datatype is that of proofs, not (as here) that of tactics; tactics are valid if they are able to be validated by compositions of primitive rules.
	citeseer.ist.psu.edu /context/531570/0 (936 words)

	Formalizing the Proof of the Kepler Conjecture (Site not responding. Last check: )
		Although this statement has been regarded as obvious by chemists, a rigorous mathematical proof of this fact was not obtained until 1998.
		The proof involves methods of linear and non-linear optimization, and arguments from graph theory and discrete geometry.
		In view of the complexity of the proof and the difficulties that were encountered in refereeing the proof, it seems desirable to have a formal proof of this theorem.
	www.cs.utah.edu /tphols2004/hales.abstract.html (146 words)

	MAD Scientist: Fermat Theorem (Site not responding. Last check: )
		For the prerequisites, the mathematical concepts used in the proof, on which Wiles was able to build, had not even been developed in Fermat's time.
		Third, there are some shorter partial proofs, and I think even some short, seductive, but mistaken ones, that have been discovered over the years, some by good mathematicians.
		The inspired guess leads the way, motivates and guides the hard struggle to construct a rigorous proof, but no honest mathematician would ever knowingly say he had proved something that he had only guessed.
	spider.ipac.caltech.edu /staff/waw/mad/mad9.html (398 words)

	Syllabus for
		It is intended to be a rigorous course---both in terms of difficulty and mathematical precision.
		When I struggle with a proof in a book, I reach a point where I think, angrily, “this proof is just wrong or I am just stupid.” I reached that point several times in my first reading of the textbook.
		Proofs are social constructs in which the author convinces the reviewer that the author has indeed written a proof.
	lal.cs.byu.edu /~jones/cs611/syllabus.htm (2205 words)

	Outlining Proofs in Calculus. The Nature of Proof.
		Rigorous proof in the sense of Hilbert has an advantage, not shared by picture proofs, that proof outlines are suggested by the very language in which theorems are expressed.
		To those mathematicians that are satisfied only by rigorous mathematics, such a situation would merely represent proof by lack of imagination: we can't imagine any situation essentially different from the situation represented by the picture, and we conclude that because we can't imagine it that it doesn't exist.
		Rigorous and picture proofs are both necessary to a good course in calculus and both within grasp [5].
	germain.umemat.maine.edu /faculty/wohlgemuth/outintro.htm (1037 words)

	[FOM] Permanent value revisited
		But the further commentary will not constitute a "rigorous proof" of those axioms, on pain of having to be a proof, in which, of course, yet "deeper" axioms will be identifiable---contrary to our assumption that the regress of rigorous justification has been terminated.
		The point is that one ought to understand the logical structure of Cantor's proof, before adducing it in support of a philosophical or epistemological claim.
		Cantor's proof, construed in this minimalist fashion, makes one realize that IF one postulates that the set of all natural numbers exists AND postulates that every set has a powerset, THEN one is committed to an infinite sequence of infinite sets none of which stands in 1-1 correspondence with any of its predecessors.
	www.cs.nyu.edu /pipermail/fom/2004-May/008161.html (577 words)

	Math Forum: A Proof of the Pythagorean Theorem
		There are many different proofs of the theorem (even one supplied by President Garfield in 1876!), and we know that the Babylonians knew about the Pythagorean theorem about 1000 years before the time of Pythagoras (born in 572 B.C.).
		Nonetheless, a rigorous, general proof of the theorem requires the development of deductive geometry, and thus it is thought that Pythagoras probably supplied the first proof.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /isaac/problems/pythagthm.html (156 words)

	Key ideas in the context of a proof from collegiate calculus
		B is able to generate a proof without diagrams, he sees that picture (which presumably he could produce himself) provides a deeper sense of understanding, and that it is connected to the formal proof through the key idea.
		Key ideas are what connect the private and public aspects of proof, that is to say the informal, heuristic ideas that provide a sense that a claim ought to be true, and the procedural ideas that can help one produce a formal, rigorous proof.
		Often when one is trying to produce a proof or gain a deeper understanding of a proof, one goes back and forth between the informal, heuristic ideas and the formal, procedural ones.
	www.icme-organisers.dk /tsg19/Raman.htm (2046 words)

	ABSTRACT: "Moore’s Proof"
		The consequence is that the proof should not be understood as a dogmatic reply to the skeptic but as a criticism of his presuppositions.
		But given that the skeptic’s proof depends on a general premiss that leaves him unable to prove his conclusion: the truth of a general premiss is neither something that can be known immediately nor something that can be known by means of other propositions that themselves can be known immediately.
		The reason he has not given a proof of the premisses of his proof is that he thinks he knows the truth of these by immediate knowledge, and what we know by immediate knowledge we do not need a proof of.
	www.colorado.edu /StudentGroups/PhilosophyClub/Referees/Moore'sProof.htm (5036 words)

	Research Sampler 8: Students' difficulties with proof
		The traditional view is that “a mathematical proof is a formal and logical line of reasoning that begins with a set of axioms and moves through logical steps to a conclusion” [Griffiths, 2000, p.
		Fields Medalist William Thurston [1994] argues that it is important to distinguish between formal proofs and proofs that mathematicians actually construct.
		One reason that university students find proof so difficult is that their experience with constructing proofs is typically limited to high school geometry [Moore, 1994].
	www.maa.org /t_and_l/sampler/rs_8.html (4164 words)

	The Reference Frame: Ahmadinejad: a rigorous proof
		While Dr Ahmadinejad has one more proof than what is normally needed, some diplomats say that they have one fewer proof than what is needed to show that Iran is running 3000 centrifuges - an amount that needs to operate for one year to get one nuclear bomb.
		Apparently, the diplomats don't consider the self-confident words of Dr Ahmadinejad to be a proof.
		Bush stays peaceful and calm and consider the Anbar province to be a proof that progress is possible.
	motls.blogspot.com /2007/09/ahmadinejad-rigorous-proof.html (886 words)

	[No title] (Site not responding. Last check: )
		Since most 300 level math courses and all 400 level ones use a lot of proof and expect you to be able to find and write proofs it serves as one of the keystones of our curriculum.
		The argument in a proof in mathematics follows rules which have been codified in (usually) classical logic as forms of argument which will not introduce error.
		While the search for a proof may follow experimental, computational, or diagrammatic approaches and may make leaps of faith which leave holes to be filled in later, the exposition of a proof takes a deductive approach, starting from definitions and previously proved results and arguing to the result being proved.
	www.iwu.edu /~lstout/TechniquesOfProoff00.html (768 words)

	Today
		While the practitioners of mathematics differ in their view of what constitutes a rigorous proof, and there are fundamentalists who insist on even a more rigorous rigor than the one practiced by the mainstream, the belief in this principle could be taken as the defining property of mathematician.
		In this theory, it is possible to rigorously prove, or refute, any conjectured identity belonging to a wide class of identities, that includes most of the identities between the classical special functions of mathematical physics.
		Any such identity is proved by exhibiting a proof certificate, that reduces the proof of the given identity to that of a finite identity among rational functions, and hence, by clearing denominators, to that between specific polynomials.
	www.cecm.sfu.ca /organics/vault/theorem_price/html/node1.html (2071 words)

	The origins of proof III: Proof and puzzles through the ages (Site not responding. Last check: )
		New rules and operations were added and many of their proofs relied on the "evident" properties of integers and geometry.
		However, one of the most important trends was the return of rigorous proof.
		It became clear that many proofs previously thought to be rigorous were instead based purely on intuition.
	pass.maths.org.uk /issue9/features/proof3 (2348 words)

	Fund theorem of algebra
		In 1772 Lagrange raised objections to Euler's proof.
		In 1814 the Swiss accountant Jean Robert Argand published a proof of the FTA which may be the simplest of all the proofs.
		Euler gave the most algebraic of the proofs of the existence of the roots of an equation, the one which is based on the proposition that every real equation of odd degree has a real root.
	www-groups.dcs.st-and.ac.uk /history/HistTopics/Fund_theorem_of_algebra.html (1541 words)

	CPSC 513 Example Set 1
		When attempting to come up with a proof of a proposition, one sometimes-helpful method is to first "talk over" the solution, and try to come up with something phrased informally.
		With an inductive proof, you will often find that the base case can be demonstrated by a simple exhaustion of all possibilities, which is the case here.
		This is one of those potential "holes" than needs to be patched when you are turning your informal phrasing into a rigorous proof.
	pages.cpsc.ucalgary.ca /~nicholss/ta/w08_513/513ex01.php (1462 words)

	Writing Proofs
		For a constructive proof you proceed from the hypothesis, using their definitions or related axioms and write down a sequence of true statements (each justified by one of the definitions, axioms or theorems gathered in step 2).
		Nevertheless, it is often easier to do a proof by contradiction than a constructive proof because you need only lead to a contradictory statement and not towards a specific conclusive statement.
		A iff B must be proven first A implies B and then B implies A. Similarly a proof that two sets are equal requires a forwards and backwards proof, first proving a point in the first set must be in the second set and then the opposite.
	comet.lehman.cuny.edu /sormani/teaching/proof.html (602 words)

	Requirements for your final project (Site not responding. Last check: )
		If you are doing a proof project your final project page should contain a main page which states which project you have chosen to do, and a rigorous statement of what you have proved (something like "Theorem:The cop number of Q_n is blank").
		By "Rigorous Proof" we mean that it must look like something that you would see in a math book, and it must be correct.
		The problem is that if you don't have a correct proof, you may have missed the whole point of the assignment and you might get a very bad grade.
	math.ucsd.edu /~randerse/152/howto.html (891 words)

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