The Philosophy of Mathematics: a Study of Indispensability and Inconsistency
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Resonance in the Solar System
Resonance In the Solar System Steve Bache UNC Wilmington Dept. of Physics and Physical Oceanography Advisor : Dr. Russ Herman Spring 2012 Goal • numerically investigate the dynamics of the asteroid belt • relate old ideas to new methods • reproduce known results • the sky and heavenly bodies History The role of science: • make sense of the world • perceive order out of apparent randomness History The role of science: • make sense of the world • perceive order out of apparent randomness • the sky and heavenly bodies Anaximander (611-547 BC) • Greek philosopher, scientist • stars, moon, sun 1:2:3 Figure: Anaximander's Model Pythagoras (570-495 BC) • Mathematician, philosopher, started a religion • all heavenly bodies at whole number ratios • "Harmony of the spheres" Figure: Pythagorean Model Tycho Brahe (1546-1601) • Danish astronomer, alchemist • accurate astronomical observations, no telescope • importance of data collection • orbits are ellipses • equal area in equal time • T 2 / a3 Johannes Kepler (1571-1631) • Brahe's assistant • Used detailed data provided by Brahe • Observations led to Laws of Planetary Motion Johannes Kepler (1571-1631) • Brahe's assistant • Used detailed data provided by Brahe • Observations led to Laws of Planetary Motion • orbits are ellipses • equal area in equal time • T 2 / a3 Kepler's Model • Astrologer, Harmonices Mundi • Used empirical data to formulate laws Figure: Kepler's Model Isaac Newton (1642-1727) • religious, yet desired a physical mechanism to explain Kepler's laws • contributions to mathematics and science • Principia • almost entirety of an undergraduate physics degree • Law of Universal Gravitation ~ m1m2 F12 = −G 2 ^r12: jr12j • Commensurability The property of two orbiting objects, such as planets, satellites, or asteroids, whose orbital periods are in a rational proportion. -
Demystifying the Saint
DEMYSTIFYING THE SAINT: JAY L. GARFIELDʼS RATIONAL RECONSTRUCTION OF NĀGĀRJUNAʼS MĀDHYAMAKA AS THE EPITOME OF CONTEMPORARY CROSS-CULTURAL PHILOSOPHY TIINA ROSENQVIST Tampereen yliopisto Yhteiskunta- ja kulttuuritieteiden yksikkö Filosofian pro gradu -tutkielma Tammikuu 2011 ABSTRACT Cross-cultural philosophy approaches philosophical problems by setting into dialogue systems and perspectives from across cultures. I use the term more specifically to refer to the current stage in the history of comparative philosophy marked by the ethos of scholarly self-reflection and the production of rational reconstructions of foreign philosophies. These reconstructions lend a new kind of relevance to cross-cultural perspectives in mainstream philosophical discourses. I view Jay L. Garfieldʼs work as an example of this. I examine Garfieldʼs approach in the context of Nāgārjuna scholarship and cross-cultural hermeneutics. By situating it historically and discussing its background and implications, I wish to highlight its distinctive features. Even though Garfield has worked with Buddhist philosophy, I believe he has a lot to offer to the meta-level discussion of cross-cultural philosophy in general. I argue that the clarity of Garfieldʼs vision of the nature and function of cross-cultural philosophy can help alleviate the identity crisis that has plagued the enterprise: Garfield brings it closer to (mainstream) philosophy and helps it stand apart from Indology, Buddhology, area studies philosophy (etc). I side with Garfield in arguing that cross- cultural philosophy not only brings us better understanding of other philosophical traditions, but may enhance our self-understanding as well. I furthermore hold that his employment of Western conceptual frameworks (post-Wittgensteinian language philosophy, skepticism) and theoretical tools (paraconsistent logic, Wittgensteinian epistemology) together with the influence of Buddhist interpretative lineages creates a coherent, cogent, holistic and analytically precise reading of Nāgārjunaʼs Mādhyamaka philosophy. -
In Defence of Folk Psychology
FRANK JACKSON & PHILIP PETTIT IN DEFENCE OF FOLK PSYCHOLOGY (Received 14 October, 1988) It turned out that there was no phlogiston, no caloric fluid, and no luminiferous ether. Might it turn out that there are no beliefs and desires? Patricia and Paul Churchland say yes. ~ We say no. In part one we give our positive argument for the existence of beliefs and desires, and in part two we offer a diagnosis of what has misled the Church- lands into holding that it might very well turn out that there are no beliefs and desires. 1. THE EXISTENCE OF BELIEFS AND DESIRES 1.1. Our Strategy Eliminativists do not insist that it is certain as of now that there are no beliefs and desires. They insist that it might very well turn out that there are no beliefs and desires. Thus, in order to engage with their position, we need to provide a case for beliefs and desires which, in addition to being a strong one given what we now know, is one which is peculiarly unlikely to be undermined by future progress in neuroscience. Our first step towards providing such a case is to observe that the question of the existence of beliefs and desires as conceived in folk psychology can be divided into two questions. There exist beliefs and desires if there exist creatures with states truly describable as states of believing that such-and-such or desiring that so-and-so. Our question, then, can be divided into two questions. First, what is it for a state to be truly describable as a belief or as a desire; what, that is, needs to be the case according to our folk conception of belief and desire for a state to be a belief or a desire? And, second, is what needs to be the case in fact the case? Accordingly , if we accepted a certain, simple behaviourist account of, say, our folk Philosophical Studies 59:31--54, 1990. -
Mathematics 1
Mathematics 1 Mathematics Department Information • Department Chair: Friedrich Littmann, Ph.D. • Graduate Coordinator: Indranil Sengupta, Ph.D. • Department Location: 412 Minard Hall • Department Phone: (701) 231-8171 • Department Web Site: www.ndsu.edu/math (http://www.ndsu.edu/math/) • Application Deadline: March 1 to be considered for assistantships for fall. Openings may be very limited for spring. • Credential Offered: Ph.D., M.S. • English Proficiency Requirements: TOEFL iBT 71; IELTS 6 Program Description The Department of Mathematics offers graduate study leading to the degrees of Master of Science (M.S.) and Doctor of Philosophy (Ph.D.). Advanced work may be specialized among the following areas: • algebra, including algebraic number theory, commutative algebra, and homological algebra • analysis, including analytic number theory, approximation theory, ergodic theory, harmonic analysis, and operator algebras • applied mathematics, mathematical finance, mathematical biology, differential equations, dynamical systems, • combinatorics and graph theory • geometry/topology, including differential geometry, geometric group theory, and symplectic topology Beginning with their first year in residence, students are strongly urged to attend research seminars and discuss research opportunities with faculty members. By the end of their second semester, students select an advisory committee and develop a plan of study specifying how all degree requirements are to be met. One philosophical tenet of the Department of Mathematics graduate program is that each mathematics graduate student will be well grounded in at least two foundational areas of mathematics. To this end, each student's background will be assessed, and the student will be directed to the appropriate level of study. The Department of Mathematics graduate program is open to all qualified graduates of universities and colleges of recognized standing. -
Mathematics (MATH)
Course Descriptions MATH 1080 QL MATH 1210 QL Mathematics (MATH) Precalculus Calculus I 5 5 MATH 100R * Prerequisite(s): Within the past two years, * Prerequisite(s): One of the following within Math Leap one of the following: MAT 1000 or MAT 1010 the past two years: (MATH 1050 or MATH 1 with a grade of B or better or an appropriate 1055) and MATH 1060, each with a grade of Is part of UVU’s math placement process; for math placement score. C or higher; OR MATH 1080 with a grade of C students who desire to review math topics Is an accelerated version of MATH 1050 or higher; OR appropriate placement by math in order to improve placement level before and MATH 1060. Includes functions and placement test. beginning a math course. Addresses unique their graphs including polynomial, rational, Covers limits, continuity, differentiation, strengths and weaknesses of students, by exponential, logarithmic, trigonometric, and applications of differentiation, integration, providing group problem solving activities along inverse trigonometric functions. Covers and applications of integration, including with an individual assessment and study inequalities, systems of linear and derivatives and integrals of polynomial plan for mastering target material. Requires nonlinear equations, matrices, determinants, functions, rational functions, exponential mandatory class attendance and a minimum arithmetic and geometric sequences, the functions, logarithmic functions, trigonometric number of hours per week logged into a Binomial Theorem, the unit circle, right functions, inverse trigonometric functions, and preparation module, with progress monitored triangle trigonometry, trigonometric equations, hyperbolic functions. Is a prerequisite for by a mentor. May be repeated for a maximum trigonometric identities, the Law of Sines, the calculus-based sciences. -
Mathematics: the Science of Patterns
_RE The Science of Patterns LYNNARTHUR STEEN forcesoutside mathematics while contributingto humancivilization The rapid growth of computing and applicationshas a rich and ever-changingvariety of intellectualflora and fauna. helpedcross-fertilize the mathematicalsciences, yielding These differencesin perceptionare due primarilyto the steep and an unprecedentedabundance of new methods,theories, harshterrain of abstractlanguage that separatesthe mathematical andmodels. Examples from statisiicalscienceX core math- rainforest from the domainof ordinaryhuman activity. ematics,and applied mathematics illustrate these changes, The densejungle of mathematicshas beennourished for millennia which have both broadenedand enrichedthe relation by challengesof practicalapplications. In recentyears, computers between mathematicsand science. No longer just the have amplifiedthe impact of applications;together, computation study of number and space, mathematicalscience has and applicationshave swept like a cyclone across the terrainof become the science of patterlls, with theory built on mathematics.Forces -unleashed by the interactionof theseintellectu- relations among patterns and on applicationsderived al stormshave changed forever and for the better the morpholo- from the fit betweenpattern and observation. gy of mathematics.In their wake have emergednew openingsthat link diverseparts of the mathematicalforest, making possible cross- fertilizationof isolatedparts that has immeasurablystrengthened the whole. MODERN MATHEMATICSJUST MARKED ITS 300TH BIRTH- -
The Asteroidal Belt and Kirkwood Gaps—I. a Statistical Study
Pramfina, Vol. 8, No. 5, 1977, pp. 438-446i © Printed in India. The asteroidal belt and Kirkwood gaps--I. A statistical study R PRATAP Physical Research Laboratory, Ahmedabad 380009 MS received 2 February 1977 Abstract. In this paper we have made a spectral analysis study of matter distri- bution in the asteroidal belt. We have Fourier analysed this distribution and obtained the autocorrelation and power spectrum, and have identified the ratios from the resonance theory. We have shown that the Kirkwood gaps cannot be satisfactorily interpreted as due to mere resonance between the asteroid and Jupi(er orbital motions. We propose that they may be regarded as a consequence of density waves generated in the g~s disc in the ecliptic plane in the ncighbourhood of the Sun. We have also shown that the process is non-Marcovian and hence cannot be subjected to a hydrodynamical analysis. Keywords. D~nsity waves; Non-Marcovian; Kirkwood gaps. 1. Introduction It was observed by Daniel Kirkwood (1867) that the astetoidal belt had a ring struc- ture, and since then, resonance theory was invoked to explain the rings of minimal matter-gaps. The total mass distribution in these rings has been estimated to be about 0.1 M@. Recently Ovenden (1972) tried to estimate as to what would have b~n the mass in this region at the origin of the solar system. He invoked a "principle of least interaction action" and using this, he calaulated the semi- major axes of the various planets, trying to examine the physical contem of Titius-Bode's law. -
Dialectic, Rhetoric and Contrast the Infinite Middle of Meaning
Dialectic, Rhetoric and Contrast The Infinite Middle of Meaning Richard Boulton St George’s, University of London; Kingston University Series in Philosophy Copyright © 2021 Richard Boulton. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Vernon Art and Science Inc. www.vernonpress.com In the Americas: In the rest of the world: Vernon Press Vernon Press 1000 N West Street, Suite 1200 C/Sancti Espiritu 17, Wilmington, Delaware, 19801 Malaga, 29006 United States Spain Series in Philosophy Library of Congress Control Number: 2021931318 ISBN: 978-1-64889-149-6 Cover designed by Aurelien Thomas. Product and company names mentioned in this work are the trademarks of their respective owners. While every care has been taken in preparing this work, neither the authors nor Vernon Art and Science Inc. may be held responsible for any loss or damage caused or alleged to be caused directly or indirectly by the information contained in it. Every effort has been made to trace all copyright holders, but if any have been inadvertently overlooked the publisher will be pleased to include any necessary credits in any subsequent reprint or edition. Table of contents Introduction v Chapter I Method 1 Dialectic and Rhetoric 1 Validating a Concept Spectrum 12 Chapter II Sense 17 Emotion 17 Rationality 23 The Absolute 29 Truth 34 Value 39 Chapter III Essence 47 Meaning 47 Narrative 51 Patterns and Problems 57 Existence 61 Contrast 66 Chapter IV Consequence 73 Life 73 Mind and Body 77 Human 84 The Individual 91 Rules and Exceptions 97 Conclusion 107 References 117 Index 129 Introduction This book is the result of a thought experiment inspired by the methods of dialectic and rhetoric. -
History of Mathematics
History of Mathematics James Tattersall, Providence College (Chair) Janet Beery, University of Redlands Robert E. Bradley, Adelphi University V. Frederick Rickey, United States Military Academy Lawrence Shirley, Towson University Introduction. There are many excellent reasons to study the history of mathematics. It helps students develop a deeper understanding of the mathematics they have already studied by seeing how it was developed over time and in various places. It encourages creative and flexible thinking by allowing students to see historical evidence that there are different and perfectly valid ways to view concepts and to carry out computations. Ideally, a History of Mathematics course should be a part of every mathematics major program. A course taught at the sophomore-level allows mathematics students to see the great wealth of mathematics that lies before them and encourages them to continue studying the subject. A one- or two-semester course taught at the senior level can dig deeper into the history of mathematics, incorporating many ideas from the 19th and 20th centuries that could only be approached with difficulty by less prepared students. Such a senior-level course might be a capstone experience taught in a seminar format. It would be wonderful for students, especially those planning to become middle school or high school mathematics teachers, to have the opportunity to take advantage of both options. We also encourage History of Mathematics courses taught to entering students interested in mathematics, perhaps as First Year or Honors Seminars; to general education students at any level; and to junior and senior mathematics majors and minors. Ideally, mathematics history would be incorporated seamlessly into all courses in the undergraduate mathematics curriculum in addition to being addressed in a few courses of the type we have listed. -
214 Publications Of
214 Publications of the » On the other hand, how interesting it would be to possess a number of such drawings of the same object for all phases of illumination through a whole lunation, or for the same phase in the different degrees of libration ! The principle should always be, to sketch only when the atmo- sphere is transparent and steady, and then to reproduce everything that is seen within the specified limits with absolute truthfulness. Particular attention will, therefore, have to be paid to the moon in high declinations, and in case the observations are made on the meridian,—which, of course, is the most favorable point,—we must consider the convenience of the draughtsman; and we must either construct the pier of the instrument sufficiently high, or lower the seat of the observer below the floor. Unfortunately, such arrange- ments cannot be made at Prague. Prague, April, 1890. References to Professor Weinek's Drawings of the Moon. (see frontispiece.) No. i. Mare Crisium. 5. Columbus, Magellan. 2. Sinus Iridium. 6. Tycho Brahe. 3. Theopilus, Cyrillus. 7. Fracastor. 4. Gassendi. 8. Archimides. ON THE AGE OF PERIODIC COMETS. By Daniel Kirkwood, LL. D. Are periodic comets permanent members of the solar system? Is their relation to the sun co-terminous with that of the planets, or has their origin been more recent, and are they, at least in many in- stances, liable to dissolution? A consideration of certain facts in connection with these questions will not be without interest. In the brilliant discussions of Lagrange and Laplace, demon- strating the stability of the solar system, it was assumed (1), that the planets move in a perfect vacuum; and (2), that they are not subjected to disturbance from without. -
History and Pedagogy of Mathematics in Mathematics Education: History of the Field, the Potential of Current Examples, and Directions for the Future Kathleen Clark
History and pedagogy of mathematics in mathematics education: History of the field, the potential of current examples, and directions for the future Kathleen Clark To cite this version: Kathleen Clark. History and pedagogy of mathematics in mathematics education: History of the field, the potential of current examples, and directions for the future. Eleventh Congress of the European Society for Research in Mathematics Education, Utrecht University, Feb 2019, Utrecht, Netherlands. hal-02436281 HAL Id: hal-02436281 https://hal.archives-ouvertes.fr/hal-02436281 Submitted on 12 Jan 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. History and pedagogy of mathematics in mathematics education: History of the field, the potential of current examples, and directions for the future Kathleen M. Clark Florida State University, School of Teacher Education, Tallahassee, Florida USA; [email protected] The field of history of mathematics in mathematics education—often referred to as the history and pedagogy of mathematics domain (or, HPM domain)—can be characterized by an interesting and rich past and a vibrant and promising future. In this plenary, I describe highlights from the development of the field, and in doing so, I focus on several ways in which research in the field of history of mathematics in mathematics education offers important connections to frameworks and areas of long-standing interest within mathematics education research, with a particular emphasis on student learning. -
Lecture 1: Mathematical Roots
E-320: Teaching Math with a Historical Perspective O. Knill, 2010-2017 Lecture 1: Mathematical roots Similarly, as one has distinguished the canons of rhetorics: memory, invention, delivery, style, and arrangement, or combined the trivium: grammar, logic and rhetorics, with the quadrivium: arithmetic, geometry, music, and astronomy, to obtain the seven liberal arts and sciences, one has tried to organize all mathematical activities. counting and sorting arithmetic Historically, one has dis- spacing and distancing geometry tinguished eight ancient positioning and locating topology roots of mathematics. surveying and angulating trigonometry Each of these 8 activities in balancing and weighing statics turn suggest a key area in moving and hitting dynamics mathematics: guessing and judging probability collecting and ordering algorithms To morph these 8 roots to the 12 mathematical areas covered in this class, we complemented the ancient roots with calculus, numerics and computer science, merge trigonometry with geometry, separate arithmetic into number theory, algebra and arithmetic and turn statics into analysis. counting and sorting arithmetic spacing and distancing geometry positioning and locating topology Lets call this modern adap- dividing and comparing number theory tation the balancing and weighing analysis moving and hitting dynamics 12 modern roots of guessing and judging probability Mathematics: collecting and ordering algorithms slicing and stacking calculus operating and memorizing computer science optimizing and planning numerics manipulating and solving algebra Arithmetic numbers and number systems Geometry invariance, symmetries, measurement, maps While relating math- Number theory Diophantine equations, factorizations ematical areas with Algebra algebraic and discrete structures human activities is Calculus limits, derivatives, integrals useful, it makes sense Set Theory set theory, foundations and formalisms to select specific top- Probability combinatorics, measure theory and statistics ics in each of this area.