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Topic: Mathematical proof

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Mathematical induction

List of mathematical proofs

Direct proof

Foundations of mathematics

Axiom

Reductio ad absurdum

Logic

Proof of mathematical induction

Axiomatic set theory

Louis de Branges de Bourcia

Axiom of choice

Proof that e is irrational

Applied mathematics

Riemann hypothesis

Scientific evidence

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	Metamath Site Selection
		...let's look at why mathematical proofs are so difficult to understand for most people...any realistic mathematical proof will leave out a great many steps, which are considered to be the "required background knowledge" for anyone who wants to understand the proof.
		By the way, a very interesting project called the Metamath project is trying to create an online archive of mathematical proofs which are specified all the way to the bottom, starting from set theory.
		But this is a very rare exception to the general rule.
	www1.shore.net /~ndm/java/mm.html (984 words)

	Mathematical proof - Wikipedia, the free encyclopedia
		In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true.
		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.
	en.wikipedia.org /wiki/Mathematical_proof (581 words)

	Mathematical practice - Wikipedia, the free encyclopedia
		This distinction is considered especially important by adherents of quasi-empiricism in mathematics, which denies the possibility of foundations of mathematics and attempts to refocus attention on the ways in which mathematicians arrive at mathematical statements.
		One motivation to study mathematical practice is that, despite much work in the 20th century, some still feel that the foundations of mathematics remain unclear and ambiguous.
		If mathematics has been "informally" used throughout history, in numerous cultures and continents, then it could be argued that "mathematical practice" is the practice, or use, of mathematics in everyday life.
	en.wikipedia.org /wiki/Mathematical_practice (449 words)

	"The Mathematical Experience" by Philip J Davis & Reuben Hersh
		Proof serves many purposes simultaneously.In being exposed to the scrutiny and judgment of a new audience,the proof is subject to a constant process of criticism and revalidation.
		Mathematics is expressible in language that employs a finite number of symbols strung together in sentences of finite length.
		Mathematical axioms have the reputation of being self-evident,but it might seem that the axioms of infinity and that of God have the same character as far as self-evidence is concerned.
	www.fortunecity.com /emachines/e11/86/mathex4.html (6667 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.
		Mathematics educators and mathematicians believe that establishing the veracity of a statement is only one of many reasons for constructing or presenting a proof.
		Many mathematics educators argue that explanation should be the primary purpose of proof in the mathematics classroom [e.g., Hanna, 1990; Hersh, 1993].
	www.maa.org /t_and_l/sampler/rs_8.html (4175 words)

	When is a proof?
		The left wing answer (fuzzy, democratic, and human centered) is that a proof is an argument that convinces a typical mathematician of the truth of a given statement.
		In the case of the Goldston-Yildirim result, they and the rest of the mathematical community were still sipping their celebratory champagne when Andrew Granville of the University of Montreal and Kannan Soundarajan of the University of Michigan discovered a flaw in the new proof, a flaw that is almost certainly fatal.
		Proofs, especially in topology and geometry, rely on intuitive arguments in situations where a trained mathematician would be capable of translating those intuitive arguments into a more rigorous argument.
	www.maa.org /devlin/devlin_06_03.html (2581 words)

	Peter Suber, "Mathematical Induction"
		Mathematical induction is deductive, however, because the sample plus a rule about the unexamined cases actually gives us information about every member of the class.
		The induction step is the proof of a conditional statement, namely, "if the theorem is true of the ancestor case, then it is true of the descendant cases." The if-clause of this conditional statement, asserting that the theorem is true of the ancestor case, is called the induction hypothesis.
		It is assumed for the sake of a conditional proof; we don't have to prove it.
	www.earlham.edu /~peters/courses/logsys/math-ind.htm (1191 words)

	Mathematical Proof of Intelligent Design in Nature
		Actually, something which has a totally formal "proof" is based on deductive reasoning which is absolute and cannot possibly have any other answer, such as a "proof" in geometry or algebra which is based on definitions, assumptions and theorems as the basis for the inescapable conclusions of logical deduction.
		Thus, in our proof, we move on to the possibility of the random assembly of proteins: To look most simply at the probability of the random assembly of a protein, note that proteins are made of 20 amino acids, which are linked together into strings or "chains" (polymers).
		Although the scientific proof of intelligent design may be demonstrated quite rigorously, the personal identity of the designer(s) is not a scientific issue, and is outside the realm of scientific inquiry.
	www.geocities.com /Athens/Aegean/8830/mathproofcreat.html (6885 words)

	Proposal: The Role of Inquiry-based Instruction in Undergraduate Mathematics Students' Development of Proof
		The development of an understanding of mathematical proof is one of the benchmarks of a major in mathematics.
		It is in such courses that most students develop their understandings of formal mathematical proof; hence these courses provide an opportunity for researchers to study students' conceptions of proof in mathematics.
		As a brand new mathematics education researcher at the University of Texas at Austin in the fall of 2002, I was invited by an experienced professor of mathematics to help design an evaluation of an MMM course.
	teachnet.edb.utexas.edu /~jenn_smith/spencer.html (2061 words)

	[No title]
		The abstract nature of proof seems to be such a mystery to the majority of students that it often is disguised as something less intimidating, or even passed over in a general curriculum.
		Either the user may be given a random mathematical statement to prove, with the user choosing the method of proof, or the user may ask to practice a particular method of proof, in which case appropriate example statements will be displayed.
		For some methods of proof, such as mathematical induction, part of the process of establishing method is to lead the user into articulating both the starting point and the desired final statement, so that the success of each intermediate step may be evaluated.
	www.cs.unb.ca /profs/fritz/edmed_97.htm (820 words)

	Royal Society \| Events diary \| The nature of mathematical proof
		The increasing use of computers both within mathematics and to automate mathematical reasoning has raised new questions about the nature of mathematical proof.
		Mere length of the conventional proof is not the issue.The essential problem is how to establish, at anywhere near the standards prevailing in pure mathematics, the correctness of the large-scale computational components of the proof.
		Given that similar examples are likely to appear, the concept of mathematical proof is bound to change.
	www.royalsoc.ac.uk /event.asp?id=1334 (389 words)

	Proof by Mathematical Induction
		Proof by Mathematical Induction is a method of proof which Computer Scientists encounter on a regular basis.
		Intuitively, this method of proof uses the validity of one instance of an algorithm to prove the validity of another instance.
		After such a proof is constructed we are left with the statement - IF there is an instance of the algorithms that is true THEN there is this second instance which we have proved to be true.
	www.neiu.edu /~css/archive/induction.html (648 words)

	Randomness and Mathematical Proof
		It is in the realm of mathematical proof that Gödel's incompleteness theorem is such a conspicuous landmark; my version of the theorem predicts that the required proof of randomness cannot be found.
		The question at issue was: ``What constitutes a valid proof in mathematics and how is such a proof to be recognized?'' David Hilbert had attempted to resolve the controversy by devising an artificial language in which valid proofs could be found mechanically, without any need for human insight or judgement.
		He represented a scientist's observations as a series of binary digits; the observations are to be explained and new ones are to be predicted by theories, which are regarded as algorithms instructing a computer to reproduce the observations.
	www.umcs.maine.edu /~chaitin/sciamer.html (5230 words)

	Short History of the Mathematical Miracle of Qur'an-PROOF THAT THE QURAN IS THE WORD OF GOD-Mathematical Miracle of the ... (Site not responding. Last check: )
		The discovery of mathematically coded scripture assures us that the verses, words, letters and all parameters of the original scripture were written down in accordance with an intricate pattern that is clearly superhuman.
		Mathematical composition of a literary work is a totally new concept, though we now realize it has existed for centuries in sacred writings.
		Mathematically composed liturgies were reported by Rabbi Judah the Pious in the 11th century.
	www.submission.org /miracle-history.html (4331 words)

	Proof - David Auburn
		"Proof is David Auburn's first major production; and if it is not exactly the brilliant debut that some have been claiming, it certainly represents the work of a writer with a fairly decent grasp on his not terribly fanciful material.
		Auburn isn't to concerned with specifics: it is a proof of "a mathematical theorem about prime numbers, something mathematicians have been trying to prove since...
		The puzzle of who came up with the proof isn't the most intriguing of issues, nor is how easily Hal and Claire can (or could) write off Catherine as a nutcase (or how easily she could write herself off).
	www.complete-review.com /reviews/usplays/auburnd1.htm (1451 words)

	Relativity?? $50,000(US) Awards for mathematical Proof
		MATHEMATICAL DEMONSTRATION ON HUBBLE'S LAW This paper differs itself from the belief that is preached by the Big Bang Theory and is able to unify all major astronomical observations in one systematic explanation.
		By the same exact mathematical demonstration, however, relativity must lead us to conclude that any object is destined to move at the speed of light, and at the speed of light only, regardless.
		Paradox four: Relativity claims that light in vacuum space possesses the maximum speed in nature, then it uses this speed to claim, with equation, that speed of light at the mass center of a gravity body to be the highest in nature.
	members.aol.com /crebigsol/awards.htm (1038 words)

	The Flyspeck Project Fact Sheet (Site not responding. Last check: )
		A formal proof is understood in the sense of the QED manifesto
		The Annals of Mathematics solicited the paper for publication in 1998 and hosted a conference in January 1999 that was devoted to understanding the proof.
		Tom Hales has produced a formal proof of the Jordan Curve theorem in HOL Light (included in version 2.0 of HOL Light), which is one step of the proof of the Kepler conjecture.
	www.math.pitt.edu /~thales/flyspeck (2100 words)

	Cryptography FAQ (04/10: Mathematical Cryptology) (Site not responding. Last check: )
		Message-ID: X-Last-Updated: 1994/07/05 Newsgroups: sci.crypt, talk.politics.crypto Subject: Cryptography FAQ (04/10: Mathematical Cryptology) From: crypt-comments@math.ncsu.edu Reply-To: crypt-comments@math.ncsu.edu Date: 19 Mar 2003 10:52:36 GMT Archive-name: cryptography-faq/part04 Last-modified: 93/10/10 This is the fourth of ten parts of the sci.crypt FAQ.
		Read part 3: we keep saying ``a strong cryptosystem must have this property, but having this property is no guarantee that a cryptosystem is strong!'' In contrast, the purpose of mathematical cryptology is to precisely formulate and, if possible, prove the statement that a cryptosystem is strong.
		Often he can try to construct a proof of security for a system, see where the proof fails, and use these failures as the starting points for his analysis.
	www.faqs.org /faqs/cryptography-faq/part04 (1368 words)

	Proof and Mathematical Reasoning - ASL ARG on NCTM 2C - The Association of Symbolic Logic and the NCTM Standards (Site not responding. Last check: )
		Answer: Mathematical language is the foundation of mathematical reasoning, so we would like to see it emphasized across all grades.
		The inclusion of logic puzzles in the video Labyrinth and in video games attests to their appeal.Anecdotal evidence suggests that playful mathematics like logic puzzles that test one's power of mathematical reasoningmay encourage some individuals to remain interested in mathematics.
		See [Smith, 1996] for examples of puzzles based on a variety of mathematical skills and field tested with 6th graders.See [Herr and Johnson, 1994] for examples of playful problems like the problem of the title, crossing the river with dogs.
	www.phil.ucalgary.ca /asl-cle/nctm/Q2C.html (749 words)

	MathML Conference 2002: Presentations
		When developing formal mathematical proofs on the computer, dissemination of the proofs is an important part of their life cycle.
		The next challenge is to display Arabic proof explanations, where the text follows the Arabic direction (from right-to-left), and the mathematical formulas are written from left-to-right (as is done in Morocco) or from right-to-left (as is done in Egypt).
		In the last case, several mathematical symbols or characters are mirrored and some mathematical elements have a non-standard behavior (for example, matrix terms are oriented from right to left).
	www.mathmlconference.org /2002/presentations/naciri (812 words)

	42 Methods of Mathematical Proof
		If the proof of a theorem is not immediately apparent, it may be because you are trying the wrong approach.
		Proof by Necessity: "It had better be true or the whole structure of mathematics would crumble to the ground."
		Proof by Postponement: "The proof for this is so long and arduous, so it is given in the appendix."
	www.pen.k12.va.us /Div/Winchester/jhhs/math/humor/proof.html (491 words)

	Egyptian Science, the Greeks, and Mathematical PROOF (Site not responding. Last check: )
		That is what mathematical PROOF ultimately boils down to: stating premises which are hopefully self-evident and therefore not themselves requiring of PROOF, then applying syllogistic reasoning based on the premises to obtain the result (theorem) that is sought.
		That their axiomatization has not survived is not proof that there was not one; rather, the converse seems more reasonable, namely that if they could implement a result, they must have got to that point by some syllogistic reasoning process.
		If arrived at by mathematical intuition alone, this would be even more remarkable than if arrived at by the imperfect axiomatic method to which we are heirs today, and for which we credit the Greeks.
	www.theafrican.com /Magazine/Athena/1.htm (2182 words)

	Invalid techniques of proof (Site not responding. Last check: )
		An issue or two of a journal devoted to your proof is useful.
		We were asked in an exercise to proof this theorem.
		This method of proof is one of the two pillars of modern cryptography.
	www.maths.uwa.edu.au /~berwin/humour/invalid.proofs.html (848 words)

	fantastic planet » Mathematical proof of God! (Site not responding. Last check: )
		The best proofs of the existence of Divinity are the example of how you live your own life, the transformative power of prayer/meditation, and of course, good ‘ol Gnosis.
		The fourth case includes a simple mathematical error, in the forth line what should be a minus is replaced with a plus.
		I’ll say again also; logical or mathematical proofs are useless, because it’s not the logical mind’s job to “prove” the existence of god.
	www.snant.com /fp/archives/mathematical-proof-of-god (2402 words)

	Drama in Numbers: Science News Online, Dec. 21, 2002
		The plot centers on the authorship of a potentially outstanding mathematical proof in number theory, which was found among notebooks filled with Robert's less-than-lucid scribbles.
		In "Proof," Catherine's spiritual mentor is the 19th-century mathematician Sophie Germain, who sent her own highly original mathematical results to Carl Friedrich Gauss under a man's name because women then had no credibility in mathematics.
		Like "Proof," the play "Arcadia" features a very clever young woman who has remarkable mathematical insights yet faces the skepticism of well-trained scholars who are less original in their thinking.
	www.sciencenews.org /20021221/bob8.asp (2245 words)

	36 Methods of Mathematical Proof (Site not responding. Last check: )
		"The proof for this is long and arduous, so it is given in the appendix."
		Limit of proof by postponement as it approaches infinity
		If it's not true in today's math, invent a new system in which it is.
	www.bluemoon.net /~watson/proof.htm (194 words)

	Amazon.com: 100% Mathematical Proof: Books: Rowan Garnier,John Taylor (Site not responding. Last check: )
		Proof has been and remains one of the concepts which characterises mathematics.
		Covering basic propositional and predicate logic as well as discussing axiom systems and formal proofs, the book seeks to explain what mathematicians understand by proofs and how they are communicated.
		The authors explore the principle techniques of direct and indirect proof including induction, existence and uniqueness proofs, proof by contradiction, constructive and non-constructive proofs, etc. Many examples from analysis and modern algebra are included.
	www.amazon.com /exec/obidos/tg/detail/-/047196199X?v=glance (511 words)

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