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."
Abstract In this article, the writer discusses how Eric Carle's 'The Grouchy Lady Bug' may be used as a first grade math tool. The writer notes that although a number of printed and Internet sources have already expressed how to adapt this book for student exercises in mathematics and literature, this book shows itself amenable to other lessons a teacher devises, directly from the book in relation to what the curriculum must cover. The writer concludes that in its seeming lack of limitation for grade one learners, and others, the book can be strongly recommended to teachers accustomed to using literary and visual sources in the teaching of elementary mathematics.
Outline:
Introduction
Class Activities
Examining the Text
Concluding Remarks
Works Cited
From the Paper "To generate interest in a book that will be used for a number of lessons, learners can be helped to talk about the ladybug in general. Some Grade One students will say that they have seen one, and others can state words they would use to describe a ladybug to someone who had never seen one. Other students will answer questions as to how large a ladybug is in relation to other things in the room, reinforcing ideas of larger than and smaller than, the teacher framing questions that can be answered in simple responses of "Yes" or "No". Grade One students will giggle when asked if a ladybug is larger than the teacher's chair, or smaller than a speck on the ceiling, if it would fit in the teacher's pocket or handbag, or if a ladybug is larger than a cat? If the teacher had a pet ladybug, would he need to take it for walks?"
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."
Abstract This paper explains that charting, in its most basic forms, is used to put fundamental measurements from an observation into a rational way of thinking ,thus bringing clarity to confusion. The author points out that charting primarily is dependent upon what data is being analyzed and who is doing the analysis. The paper stresses that charting can often become confusing because people make charts that display too much data within a single chart. Five charting techniques are illustrated in this paper: bar chart, candlestick charting, line charts, point and figure charts and three line break charts.
Table of Contents:
Introduction
Charting Rationale
Charting Techniques
Charting Types
The Bar chart
Candlestick Charting
Line Charts
Point & Figure chart
Three Line Break Chart
Conclusion
From the Paper "This type of charting shown below is very similar to that of the bar chart. Except during the period between the open of trading and the close of trading a solid thick line is drawn in during the time-period in question. The same line appears in the bar chart but is not as defined and is the section between the open and last trade. Often this type of charting is used to analyze the short term forecasts of the stock. In addition to this the basic solid square represents a day which closes with a low and the open square in the chart represents a day where closing is on a high note/price."
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."
This paper discuses that mathematics and human language are very similar in structure and form because they can both be broken down into ever smaller functional units.
Abstract This paper explains that regressions are preformed all the time in mathematics, which involve the division of numbers into innate and precise formal units; however, this is not a common practice in human language other than by theorists of deconstruction techniques. The author points out that the deconstruction of language, both verbal and non-verbal, has been a practice of linguists, philosophers and critical theorists for many years. The paper relates that verbal and non-verbal human communication is comprised of both signs and symbols,which together form a recognized code, or what laymen commonly refer to as a language. The author underscores that there is a significant problem in reaching some consensus on what constitutes a verbal sign or symbol because of significant confusion regarding both meaning and intent.
From the Paper "The solution to developing a better understanding of the relationship between sign and symbol in order to make the case for a deep similarity between human language and mathematics is to develop a more pragmatic framework within which to develop a more complete paradigm of the communicative process of verbal and non-verbal communication. Devlin does this when he speaks of the grammar generated, deep structure strings in the text of the "Language in the Mind". Some theorists say this need is a distinction that must be better developed between components of a sign to define as the signified and the signifier."
Abstract This paper explains that industrial relations within the context of the British economy and the character of its workforce have long been dominated by the power and presence of its unions. The author points out that, because of the stakes involved in the collective bargaining negotiations, game theory (GT) and coalition theory, which is a subset of GT, is relied upon to achieve fractional improvements in contract negotiations. The paper relates that game theory (GT) is most often associated with a zero-sum scenario; however, it also encompasses positive-sum and negative-sum scenarios where a party may gain or win without the necessity of an equivalent loser. The author relates that, because of the necessity to form alliances in order to reach consensus among diverse stakeholders, industrial relations often employ a type of GT known as coalition theory,which examines the nature, reasons and underlying dynamics of these coalitions that form in all the various settings. The paper includes graphs.
Table of Contents
Introduction
Game Theory
Industrial Relations and Game Theory
Conclusion
From the Paper "Of particular value has been research integrating sender-receiver frameworks that analyze how knowledge is transferred, both symmetrically and asymmetrically, with GT whereby advantages gained through asymmetrical knowledge transfer creates zero-sum advantages for one player or the other in an industrial relations setting such as the collective bargaining platform. This concept is explained in terms of being a signal that one side uses to inform the other of a possible solution, such as concessions that can be made on benefits."
Abstract This paper explains that the solution to Stepran's infinity puzzle
is not so difficult and has nothing to do with infinity, although the calculus of this equation may in fact be infinite. The author underscores that the puzzle is not a puzzle at all and is not indicative of infinity but rather is purely an exercise in the limitations of physics. The paper agrees with Rucker's concept of infinity as simply a natural element of the universe or of being one of the basic functional elements of mathematical device. The author concludes that the useful concept of infinity is that it does naturally occupy points in both physical and mathematical space ,which truly cements it within the context of a tangible mathematical and physics principle rather than some far-off rationale construct created and identifiable only by mathematical theorists.
Table of Contents:
The Puzzle
The Solution
Response Page to Postings
Discussion
From the Paper "Stepran's states that a person is tasked with turning a light switch off and on starting with on at 2 minutes and then in increments by half of the time remaining flipping the switch to the opposite position. On the surface the outcome appears as if it will be a simple persuasion of the ineluctable quality of time; that, time is unavoidable and all things must come to an end. Yet, as one begins the calculations it becomes apparent that the half increments are, apparently, infinite starting with two in terms of seconds: 120, 60, 30, 15, 7.5, 3.75, 1.875, .93, .46, .23, .117, .058, .029, ad infinitum, at least to the extent that a common calculator is capable of dividing."
Abstract In this article, the writer explains that manipulatives are defined as materials that are physically handled by students in order to help them see actual examples of mathematical principles at work. The writer notes that manipulatives are incorporated into curriculum with the aim of helping the student understand mathematics, rather than increasing efficiency in calculation. The writer maintains that manipulatives are very useful especially in early education. The writer notes that there is a wide array of math manipulatives on the Internet. Some may be bought while others can be enjoyed for free on the web. The writer provides examples and pictures and discusses how it would be possible to use them in teaching children.
Outline:
What are Manipulatives?
References
From the Paper "Manipulatives are incorporated into curriculum with the aim of helping the student understand mathematics, rather than increasing efficiency in calculation. Manipulatives are very useful especially in early education. Moreover, its use is not exclusive to teachers and schools, parents who would choose to help their children with school lessons can also employ them to help their children understand math concepts. Most students dislike math because they think it is very complicated. This prejudice towards this subject result to poor performance of students in math subjects. The development of this negative mind set on the subject may have started when in their childhood. Traditional ways of teaching may have bored them and cause them to dislike the subject, which they will carry to adulthood. That is why it is important that at a young age, kids should learn to enjoy math. And the use of manipulatives can help them enjoy and appreciate it. Manipulatives come in colorful packages that attract children, their interactive design also allows children to play with them as they learn. There is a wide array of math manipulatives in the Internet."
Abstract This paper explains that John Napier, in 1614, took an algebraic approach by defining logarithms as a ratio of two distances within a geometric pattern, base 1/e, which substituted for his lack of a base as in the currently used common logarithmic base 10. The author points out that the real benefit of these logarithms was that they simplified mathematical calculations by providing a shortcut for exponential factors just as exponents are a shortcut for multiplication. The paper relates that, since Napier's original logarithms lacked a common base, they were more consistently accurate but not as easy to manipulate as the common logarithms employed today. The author states that the common logarithms are much easier to calculate but only sufficiently accurate as compared to Napier's original logarithms. The paper includes graphs.
Table of Contents:
Introduction
Historical Background
Napier's Logarithms
Base 10 Logarithms
Conclusion
From the Paper "While initially Napier's logarithms did not employ a base in the traditional sense he eventually adjusted his logarithms to account for a consistent base in much the same way they are currently employed today. Napier worked with another mathematician, a man by the name of Henry Briggs, to change his logarithmic forms to the form now currently common which is the L/e equation. Thus, Naperian logarithms now are described as points that are moving along a straight trajectory and indicated by units prescribed in length."
Tags: astronomers, napier's bones, exponential functions, base 1/e, accurate
Abstract This paper discusses the significant contributions of Galileo Galilei to the field of mathematics. It provides a brief history of his life and then focuses on some examples of the contributions that he made to mathematics. The paper also discusses his misinterpreted-battle between science and religion and how it overshadows many of the other contributions that Galileo made during his lifetime as a scientist and mathematician.
From the Paper "We often hear of a Copernican revolution in science, but Galileo was the instigator of a much more fundamental revolution that influenced both science and mathematics. The worldview that Galileo created to replace the Aristotelian worldview that dominated at the time contended that the world was made up only of matter whose properties and motions could be described in terms of mathematics (Machamer). In other words, Galileo advanced the now-obvious notion that mathematics was nothing short of the language of the universe. Using mathematics, Galileo was able to describe and understand the mechanics of the universe, effectively gaining a deeper understanding of the way that the world is put together. This is Galileo's most significant contribution to mathematics. He removed the idle, superstitious philosophy from the study of the natural world and pushed mathematics to the forefront of natural inquiries, demonstrating again and again that it could be used to understand the way the world works."
Abstract The paper looks at the extensive use of geometric and symmetrical patterns in Islamic art. The paper provides a definition of geometry and looks at translations, rotations and reflections in Islamic art. The paper then examines the mathematics of symmetry and how symmetry, as manifested in Islamic art, can be utilized to teach geometry in the contemporary classroom.
Outline:
Introduction
Symmetry in Islamic art, Part I
Symmetry in Islamic Art, Part II
Symmetry in Islamic Art, Part III
From the Paper "Many civilizations have long used artistic designs for a variety of purposes. For instance, some civilizations have used artistic designs for emblematic purposes, while some have used artistic designs for ornamental and/or architectural purposes; still others, perhaps unsurprisingly, have used artistic designs for spiritual symbolism. Another thing that is not at all a surprise is that artistic designs almost invariably utilize mathematical concepts. Specifically, within the Islamic art tradition, there has long been the extensive use of geometric and symmetrical patterns - so much so that it may be put forward that one of the defining features of Islamic art is its ability to incorporate mathematical concepts and ideas in ways that are rich, vibrant and aesthetically pleasing."
Abstract This paper examines the origins and significance of hyperbolic geometry. Specifically,it briefly discusses the men who conceived of it, as well as how hyperbolic geometry differs from Euclidean geometry. Finally, and most importantly, the paper looks at the significance of hyperbolic geometry when it comes to exploring the universe around us.
From the Paper "Delving deeper, the contemporary significance of non-Euclidean geometry grows more and more unavoidable - even to those disinclined to give it its "due". For one thing, it is well-known that hyperbolic geometry has shed some light on the immersion and curvature of spaces. More importantly, Einstein's theory of relativity is, at least in part, indebted to non-Euclidean geometry - though it is admittedly not clear from the available literature the precise extent to which hyperbolic geometry made his revolutionary findings vis-a-vis relativity possible. In any event, this writer - drawing upon course work completed in previous introductory classes that dealt with geometry and its relationship to modern cosmology - would be remiss if he did not also point out the fact that the "empty" regions in outer space where no matter exists can really only be described adequately using a hyperbolic model. In effect, understanding the Hubble Constant involves understanding and appreciating non-Euclidean, hyperbolic geometry."
Abstract The paper is an in depth explanatory narrative on the subject of linear transformation. After an analytical definition of the term, the paper gives examples of the many applications of linear transformation and explains that linear transformation is a method of altering geometric figures into another form. The paper also explains the basic requirements for this and quotes examples. The paper also provides descriptive explanations of linear transformation (also referred to as the algebra of matrices )and interprets how the process occurs. The paper further relates an extensive explanation on conic sections and how they are determined. Throughout the paper the various terms are fully explained together with examples and methods of application.
From the Paper "For then transformation (of the plane), let S be the set of points in the plane. A transformation of the plane is then a one-to-one mapping from S to S. The most important transformations of the plane are the linear transformations, meaning those that can be represented by linear equations. For a linear transformation T, there are constants a, b, c, d, h and k such that T maps the point P with coordinates (x, y) to the point P' with coordinates (x', y') where h = k = 0. The origin O is a fixed point, since T maps O to itself, at which point the transformation can be written x' = Ax. Such transformations include rotations about O, reflections in lines through O, and dilatations from O. Translations are examples of linear transformations in which O is not a fixed point."
Abstract The paper outlines the paradigms within the programming arena that have a close link to mathematical logic and provides an explanation for that link. The paper points out that it is difficult to separate the mathematical logic from the programming paradigms, which highlights how connected each programming logic is to the mathematical concepts, functions, methods and logic.
From the Paper "The procedural paradigm refers to the programming language that specifies steps it takes to reach a desired state. Within the language the operations contain a series of steps that are completed to finalize a desired action. The object-oriented paradigm is the programming language where each object is considered a separate entity that translates processes, receives and sends data throughout the process, (Hudak 360). The objects are collectively responsible for operations, but each object has its clearly defined role within the system. Functional programming paradigm uses mathematical functions for processing and evaluating data, and focuses on the application of functions as the avenue for programming languages. The logic paradigm is the mathematical concepts for computer programming, with the programming language utilizing logic for the problem solving and model development process."
