| Papers [1-16] of 34 :: [Page 1 of 3] | | Go to page : 1 2 3 —> | Search results on "GODEL THEOREM": |
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Godel's Theorem, 2007.
A review of Godel's theorem and the limitations of an allegory in trying to understand it.
1,495 words (approx. 6.0 pages), 5 sources, MLA, AU$ 71.95
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Abstract
This paper discusses Godel's theorem and how it is sometimes used to imply that all machine logic can eventually become self-aware. The paper also discusses the criticisms of the theorem and its limitations. The paper then provides an allegory to explain Godel's theorem and discusses the advantages of this explanation, as well as the limitations in using an allegory to try to understand the theorem.
Table of Contents:
Allegory and Godel: Oil and Water
From the Paper
"Godel recognizes that his theory in fact could not be fully described in human language and concepts and this is a fact that Hofstadter completely misses. When Godel is quoted as saying the epistemological descriptions in a given language cannot be restated in that same language, he directly disallows the use of allegory in retelling his theory. The unfortunate aspect of Hofstadter's allegory is that most readers get lost in trying to decide what the various characters represent, what is meant by the way the dialogue is spoken and, ultimately, what the Omega record player looks like. None of which, of course, has anything to do with Godel's Theorem."
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Free Will in Godel's Theorem, 2006.
This paper discusses the concept and assertions of free will in Godel's Theorem.
1,125 words (approx. 4.5 pages), 4 sources, AU$ 64.95
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Abstract
This document discusses the Smullyan text, "Is God a Taoist?". The writer examines how its concept of free will is substantiated or invalidated by Godel's Theorem. The conclusion provided in this article is that this dialogue by Smullyan erroneously relies on Godel's Theorem to support his argument in that the reader is required to make assumptions regarding which characters are still undecided and at play.
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Godel's Theorem, 2008.
An analysis of the advantages of Godel's theorem within mathematics.
1,596 words (approx. 6.4 pages), 5 sources, MLA, AU$ 75.95
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Abstract
This paper explains Godel's theorem and its application to the machine mind. It describes the advantages of Godel's theorem in mathematics and how it is used in practice by mathematicians who lack understanding of a specific principle. The paper also provides the writer's opinion of the use of the theorem and suggests that it is almost commonsensical in nature.
Table of Contents:
Response to Postings
Discussion
From the Paper
"This could in fact be yet another referral to Cherniak's Riddle but that fact would only be left to the literary critic to decide and because human language is a series of referential signs and symbols that always refer to something else this could never be known absolutely. Here is the key difference in the two languages in question. When a mathematical principle is discovered and proven it is self evident to all and taken as fact. When a literary concept is created it is, conversely, always up for debate and its meaning always at play. Thus, Godel's theorem is both an apologetic and a principle best left explained in the language it was conceived in--mathematics."
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Godel's Theorem, 2008.
A discussion as the to the proof or lack thereof in support of Godel's theorem of the self-awareness of machines.
1,358 words (approx. 5.4 pages), 3 sources, MLA, AU$ 65.95
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Abstract
This paper discusses Godel's theorem and its lack of proof, absolute or otherwise, that machines do or may in the future experience self-awareness of one type or another. It discusses the assertions of the theory and the problems with it. The paper then provides a personal response, by the writer, to the issue of the present and future self-consciousness of machines.
Table of Contents:
Discussion
Response
From the Paper
"Free will is a concept that cannot be even remotely defined with any degree of consensus. Talking about free will with religious groups results in completely different concepts of free will than when talking with political groups or academic groups or any number of different types of groups. Conversely, arithmetic calculations are easy to quantify and easy to define within the confines of the overall system. Somehow Smullyan would like his readers to believe that defining free will is as self-apparent as 2 plus 2 or similar arithmetic equation. Some researchers have described Godel's Theorem as being some type of alternate description of a value system: "The system of values could be part of the program the computer followed in making its choices. The computer system would then appear to have those values, and be guided by them (Machina 3). Thus Smullyan's entire argument regarding free will is based on a number of unfounded and unproven assertions that have no basis except in extreme positives or negatives. These equate to a world that is either black or white and all decisions are, ultimately, yes or no questions."
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Godel and the "Theorem on Incompleteness", 2002.
Review of R. Rucker's discussion of the concept of infinity and how it relates to Godel's "Theorem on Incompleteness".
650 words (approx. 2.6 pages), 4 sources, AU$ 38.95
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Abstract
R. Rucker helps us better understand Godel's "Theorem on Incompleteness" by discussing infinity and whether it can be seen as a real entity. In his view, infinity can be seen as a tangible reality. He argues that it is quite possible that time may actually continue forever - and that is precisely what infinity is. Rucker also sees the possibility of the potential infinite divisibility of space into smaller and smaller pieces.
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Godel's Theorem, 2005.
An analysis of the implications of Kurt Godel's theorem on mathematics.
900 words (approx. 3.6 pages), 3 sources, AU$ 51.95
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Abstract
This paper discusses Godel's theory on mathematical truths as being that they cannot be found in any set of axioms or rules and ultimate truth cannot be achieved. The paper suggests that Kurt Godel's incompleteness theorem encompasses the fact that all formal systems turn out to be incomplete by their very nature and it discusses the implications of this theory.
From the Paper
"Godel stated that there can be no proof of any statement (P). If P is true, there is no proof of it. If P is false, there is a proof that P is true. This is a contradiction. It cannot be decided whether P is true in a symbolic system."
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Freewill and Incompleteness, 2002.
A comparison of Smullyan's dialogue to Godel's incompleteness theorem.
650 words (approx. 2.6 pages), 2 sources, AU$ 38.95
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Abstract
This essay will explore some of the dynamics of Smullyan's dialogue and compare them to Godel's Incompleteness Theorem, which was a break through mathematical theorem that showed that axioms of a rational system cannot be proved within the system.
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Mathematics: Fermat's Last Theorem, 2002.
This paper describes the interesting phenomenon called Fermat's Last Theorem, written in layman's terms.
613 words (approx. 2.5 pages), 5 sources, MLA, AU$ 31.95
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Abstract
Fermat's Last Theorem (FLT) has been one of the most fascinating theorems in mathematics. This paper looks at the conformities and the disparities of this statement. It assesses the theorem and the problem it comes to solve as well as the theorem's proof. It gives a detailed mathematical exercise which he solves using the theorem.
From the Paper
"Pierre de Fermat was born near Montauban in 1601. He was born in a family reared by a leather-merchant who was his father and was educated at home. He was essentially a lawyer and was an amateur mathematician. Throughout his life, Fermat published only one mathematical paper, which was written anonymously and appeared as an appendix to a book. He died in 1655."
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Existence and Uniqueness Theorem, 2002.
A study of the relationship and accuracy of the existence and uniqueness theorems.
1,835 words (approx. 7.3 pages), 3 sources, MLA, AU$ 84.95
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Abstract
This paper aims to provide information on generalization of existence and uniqueness theorem for ordinary differential equations (ODE). This contains mathematical definitions, notations, symbolisms, formulas, graphics, and equations of the theorems mentioned. A few examples were also provided to illustrate and prove the definitions of the theorems.
From the Paper
The fundamental theorem of Existence and Uniqueness answer the questions does a solution exists and is it unique. If at least one solution can be determined for a given problem, a solution to that problem is said to exist. Frequently, mathematicians seek to prove the existence of solutions (by means of a so-called existence theorem) and then investigate their uniqueness (by means of a so-called uniqueness theorem). The solutions to an ordinary differential equation (frequently abbreviated as ODE) satisfy the existence and uniqueness properties.
The foundation of the theory of differential equations is the theorem on the existence of solutions for the initial value problem
x = f ( x ), x ( t o ) = x o
for a function x: R - Rn and f: Rn - Rn The main tool that we will use in developing our theory is the reformation of the differential equation as an integral equation. Suppose that both x(t) and f(x) are continuous functions. Then we can formally integrate both sides with respect to t to obtain:
x ( t ) = x o + ....
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Fermat?s Last Theorem, 2001.
This paper takes a look at this mathematical theorem and how it has fascinated mathematicians for hundreds of years.
650 words (approx. 2.6 pages), 5 sources, MLA, AU$ 34.95
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Abstract
This paper briefly gives a background of Pierre de Fermat and states this famous theorem - FLT. It looks at a few working examples of problems related to the theorem and how mathematicians think that they have finally solved them.
From the Paper
"Pierre de Fermat was born near Montauban in 1601. He was born in a family reared by a leather-merchant who was his father and was educated at home. He was essentially a lawyer and was an amateur mathematician. Throughout his life, Fermat published only one mathematical paper, which was written anonymously and appeared as an appendix to a book. He died in 1655. (Ball) Fermat?s Last Theorem (FLT) has been one of the most fascinating theorems in mathematics. This theorem has been one the great, unsolved problems in this field for three hundred and fifty some years. Some experts believe, however, that the problem has been solved."
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Pythagoras and the Pythagorean Theorem, 2002.
This paper discusses the ancient Greek philosopher Pythagoras of Samos and the Pythagorean School
650 words (approx. 2.6 pages), 3 sources, AU$ 38.95
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Abstract
.The author examines the influence Pythagoras had on ancient learning, the Pythagorean Theorem, and the Pythagorean School, and notes that that the Pythagorean School was inspired by Pythagoras's genius. It was half religious and half scientific, and followed a code of secrecy which served its purpose in ancient times, but which has prevented historians from obtaining much information about Pythagoras other than through later second-hand sources.
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Smullyan's Incomplete Dialogue, 2002.
Examines the philosophical text of Smullyan and compares its ideas and style to Godel's Incompleteness Theorum.
900 words (approx. 3.6 pages), 3 sources, AU$ 51.95
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Abstract
In this essay, I will examine Smullyan's work in the explicit context of Godel's Incompleteness Theorem and argue that not only does Smullyan show statements that are intimately linked to some of Godel's notions, but that the overall structure of his work is that of incompleteness thus mirroring Godel's model in a fundamental manner. In addition to this, the important notion of self-reference will be inserted into the argument.
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Language and Mathematics, 2005.
This paper explores the similarities that exist between language and mathematics.
675 words (approx. 2.7 pages), 4 sources, AU$ 38.95
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Abstract
This paper explains that obvious similarities conclude that human language may be reducible to mathematical formulation. The author points out that that mathematics consists of sets of axioms in which statements can be either true or not. The paper relates, while this does not necessarily seem very much like language, Godel's Incompleteness Theorem relates that meaning can exist outside of axiomatic sets, providing a new basis for similarity.
From the Paper
"It should not be surprising that mathematicians and linguists have drawn parallels between these two disciplines. There are obvious similarities that have made many believe that human language may be reducible to mathematical formulation. Some have even attempted to use the assumption to teach machines how to speak, constructing complex utterances based on a limited number of syntactical rules. However, these efforts and others to fully connect mathematics and language have proved largely unsuccessful. The following paper will briefly examine some of the similarities between language and mathematics. By its nature, language has a combinational structure, known as syntax or grammar, that permits the communication of complex ideas (Devlin "Born")."
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The Man Who Was Pythagoras, 2001.
This paper is a biography of the mathematician, his work outside mathematics, a description of the Pythagorean Theorem, and the Pythagorean Society.
815 words (approx. 3.3 pages), 2 sources, AU$ 42.95
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Abstract
This paper offers a biographical look at Pythagoras. The author discusses the many mysteries surrounding this man, in addition to his many contributions to mankind. Included are some explanations of some Pythagorean theorems, with pictures to highlight textual information.
From the Paper
"Numbers play a large part in our everyday lives, from the time we get up, how long we cook our food, the distances we travel, and other such aspects, many of which we take for granted. A scholar who played a large part in the way we view certain numbers and objects people use regularly is Pythagoras. Pythagoras was a philosopher, medical practitioner, astronomer, and mathematician. Although he contributed many thoughts and ideas to society, such as those of the Pythagorean Society, the Pythagorean Theory is by far the most practiced and well-known."
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Bell?s Inequality, 2002.
An insight into Bell?s Theorem (Bell?s Inequality) of quantum theory.
2,115 words (approx. 8.5 pages), 3 sources, MLA, AU$ 96.95
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Abstract
This paper discusses the work of the physicist John Bell, who's great recognized achievement occurred during the 1960s when he brought new life into the foundations of quantum theory. It examines how Bell demonstrated that discussion of such concepts as 'realism', 'determinism' and 'locality' could be formed into a rigorous mathematical statement, 'Bell's Theorem?, which is capable of experimental test. It looks at how his work has become a point of interest for scientists throughout the world who have found applications not only in quantum theory, but in investigations of the physical universe as well and how current applications of Bell?s Inequality have been found in the development of quantum computing and quantum cryptography.
From the Paper
"Quantum mechanics, however, fails to satisfy Bell's Inequality. He predicts correlations that cross over boundaries that are delineated by the structure of that inequality. Quantum mechanics predicts ?odd? correlations that seem to defy a common, classical conception of reality, and in fact it is for basically this type of reason that Einstein, Podolsky, and Rosen argued in 1935 that it was incomplete. The formulation of Bell's Inequality, however, allowed the possibility for determining, through experiment, which was right: quantum mechanics or a local reality theory of the sort Einstein postulated, because it stated what must be the case in our experiments if a locally real theory is correct."
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Pi, 1996.
Examines mathematical theorem that ratio of circle's circumference to diameter is 3.14159. History, impact on science, search for extraterrestrial life, symbols.
2,250 words (approx. 9.0 pages), 8 sources, AU$ 114.95
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From the Paper
"This research examines the mathematical theorem that the ratio of the circumference of any circle to its diameter is 3.14159 . . ., an irrational number, describing a relationship that is irreducable to a whole number, and which is referred to as ?, hereinafter referred to as [Pi]. The primary focus of this research is on the wider societal impact of [Pi], as opposed to the theorem's mathematical properties.
History
Motz and Weaver (1993, p. 4) stated that no evidence is known that supports a contention that Archimedes deduced the value of [Pi] in geometric terms to support technology. Nevertheless, they ..."
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