C:\Users\Steve Williams\Documents\Classes\300

Total Page：16

File Type：pdf, Size：1020Kb

$C:\Users\Steve Williams\Documents\Classes\300$ MathEd/Math 300 Review for Final – Davis & Hersh Chapter 1 1. What are some reasons that mathematics cannot be carefully defined? 2. What is Ulam’s dilemma, and what does it say about the nature of modern mathematics? Chapter 2 3. What beliefs about mathematics are apparent from the Ideal Mathematician? To what extent are these ideas apparent in the mathematical culture you are familiar with? Is the Ideal Mathematician consistent in his beliefs? 4. Mathematics is commonly thought of as a way of getting at truth. Discuss this in light of Davis and Hersh's discussion on the role of models and modeling in mathematics. Chapter 3 5. One commonly held belief about mathematics is that "God is a mathematician" or that the universe expresses itself mathematically. Contrast this idea with what Davis and Hersh call mathematics "by fiat." Give some examples of how mathematics might be imposed on the universe by us (instead of the other way around). 6. Explain the differences between mathematical Hardyism and mathematical Maoism, and how this relates to Davis and Hersh’s discussion of “utility” in mathematics. 7. What does a pure mathematician mean when he says that a piece of mathematics is useful? What does an applied mathematician mean by this? Chapter 4 8. Davis and Hersh claim that mathematical objects exist in the same way that unicorns exist. Explain what they mean by that. 9. Read the quote by George Berkeley on p. 161, together with the two paragraphs before and after. What does this have to say about what is “intuitively obvious” in mathematics? 10. In what sense can mathematics be said to be beautiful? Chapter 7 11. We have said that the working mathematician is a “Platonist on weekdays and a formalist on Sundays.” Describe what this means. 12. Describe what mathematical Platonism is and outline the major arguments for and against it. 13. How does what we know about “Pascal’s Triangle” support a Platonic view of mathematics? 14. How would a mathematical humanist like Reuben Hersh respond to the above argument about Pascal’s Triangle? 15. What is the Euclid Myth and why is it significant? To what extent do people believe the Euclid Myth today? 16. To what extent do people believe the Euclid Myth today? 17. Explain what is meant by the "crisis in foundations" in mathematics. Include the major figures and schools of thought. How was the crisis resolved? 18. How did logicism attempt to resolve the crisis in foundations? Who was involved? Why was it not successful? 19. How did intuitionism attempt to resolve the crisis in foundations? Who was involved? Why was it not successful? 20. How did formalism attempt to resolve the crisis in foundations? Who was involved? Why was it not successful? 21. Formalists believe that they cannot assert that a theorem is true, even if it is proven. What do they assert instead of truth? 22. What was the purpose of Lakatos's Proof and Refutations? 23. Describe how each of the following would view the notion of proof in mathematics: a. The Ideal Mathematician b. A Formalist c. A Fallibilist Place the letter of the mathematician or philosopher in the list below into the category that best describes his approach to mathematics. Logicism Formalism Constructivism / Intuitionism Fallibilism Humanism A. David Hilbert E. L. E. J. Brouwer B. Alfred North Whitehead F. Ludwig Wittgenstein C. Imre Lakatos G. Philip J. Davis D. Bertrand Russell H. Reuben Hersh.

Copy Link

Recommended publications

Imre Lakatos's Philosophy of Mathematics
Imre Lakatos’s Philosophy of Mathematics Gábor Kutrovátz Eötvös Loránd University, Budapest e-mail: [email protected] Introduction Imre Lakatos (1922-1974) is known world-wide by at least two different groups of philosophers. For the philosophers of science, he was a great defender of scientific rationality: he gave up the idea of ‘instant’ methodological rationality and substituted it with the historiographical methodology based on the notion of ‘rational reconstruction’. For the philosophers of mathematics, he was a great attacker at the formalist school, and instead of the formal-axiomatic features of mathematics he put the emphasis on heuristics, the historical processes of theory building and conceptualisation. These two ‘incarnations’ of Lakatos are rarely discussed within a unique philosophical framework, and not without reason: Lakatos himself never really tried to harmonise these two fields of interest. There are signs showing that he intended to build a common ground for them, but perhaps this project could not even take a very definite shape before his early death. In this paper I shall concentrate on only one of these topics: his philosophy of mathematics; and my primary aim is to give a brief and coherent summary of his ideas concerning this field. It should be noted, however, that his philosophy of mathematics and that of science obviously have some characteristics in common, since they were contrived by the same man, and not at a great temporal distance in his academic life, when he was under very similar intellectual influences. My secondary purpose therefore is to build this summary in a way that is in accordance with his general philosophical theory of science, that is, to give a mild ‘rational reconstruction’ of his ideas.1 The context of Lakatos’s philosophy of mathematics In the Acknowledgements of his doctoral thesis wrote on the philosophy of mathematics, Lakatos mentions three authors as the most influential sources for his philosophy.
[Show full text]
Kuhn Versus Lakatos, Or Paradigms Ver Su S Research Programmes in the History of Economics
[HOPE Vol. 7 (1975) No. 41 Kuhn versus Lakatos, or paradigms ver su s research programmes in the history of economics Mark Blaug In the 1950’s and 1960’s economists learned their methodology from Popper. Not that many of them read Popper. Instead, they read Friedman, and perhaps few of them realized that Friedman is simply Popper-with-a-twist applied to economics. To be sure, Friedman was criticized, but the “Essay on the Methodology of Positive Economics” nevertheless survived to become the one arti- cle on methodology that virtually every economist has read at some stage in his career. The idea that unrealistic “assumptions” are nothing to worry about, provided that the theory deduced from them culminates in falsifiable predictions, carried conviction to economists long inclined by habit and tradition to take a purely instrumentalist view of their subject. All that is almost ancient history, however. The new wave is not Popper’s “falsifiability” but Kuhn’s “paradigms.” Again, it is un- likely that many economists read The Structure oj Scientific Revolu- tions (1962). Nevertheless, appeal to paradigmatic reasoning quickly became a regular feature of controversies in economics and “paradigm” is now the byword of every historian of economic thought.’ Recently, however, some commentators have expressed misgivings about Kuhnian methodology applied to economics, throw- ing doubt in particular on the view that “scientific revolutions” characterize the history of economic thought.* With these doubts I heartily concur. I will argue that the term “paradigm” ought to be banished from economic literature, unless surrounded by inverted commas. Suitably qualified, however, the term retains a function in the historical exposition of economic doctrines as a reminder of the Copyright 1976 by Mark Blaug MARKBLAUG is Head of the Research Unit in the Economics of Education at the University of London.
[Show full text]
PHI230 Philosophy of Science
Department of Philosophy Level 2 Module PHI230 Philosophy of Science LECTURER: George Botterill Office Hours Wed 12-1, Fri 1-2; external tel: (0114)2220580; email: [email protected] Lectures: Monday 2-3 in Hicks Building LT6 Friday 11-12 in Hicks Building LT5 Seminars: See MOLE unit to join: Monday 4-5 Jessop Building 215, Friday 12-1 Jessop Building 215 GENERAL OUTLINE This course will deal with major issues in the philosophy of science, with particular emphasis on the rationality of theory-change, explanation, the status of scientific laws, observation, and the structure of scientific theories. Most of the course will be devoted to the methodology of natural science, though we may also discuss some issues in the philosophy of the social sciences. Most of the issues presented revolve around two major debates in the philosophy of science: (1) the dispute between Kuhn and Popper about theory change (2) disagreements between positivist (or instrumentalist) and realist conceptions of science. Modern philosophy of science has become closely linked with the history of science. The course will reflect this by considering how well major developments in the history of science fit proposed methodological rules about how science ought to proceed. The Copernican Revolution in astronomy will be used as a case-study. 1 AIMS AND OBJECTIVES The main aims of this module are: • to introduce students to the debate about theory-change in science, deriving from the work of Popper, Kuhn, Feyerabend and Lakatos • to help students acquire an understanding of some important problems in the philosophy of science (concerning the topics of: explanation, observation, scientific realism, and the nature of laws) • to encourage students to enter into serious engagement with some of the major problems in the philosophy of science.
[Show full text]
Introduction
M889 - FULLER TEXT.qxd 4/4/07 11:21 am Page 1 Phil's G4 Phil's G4:Users:phil:Public: PHIL'S JO Introduction Science and Technology Studies (STS) is an interdisciplinary field usually defined as the confluence of three fields with distinct intellec- tual lineages and orientations: history of science, philosophy of science, and sociology of science. All three have been marginal to their named disciplines, because surprisingly few of the original practitioners of history, philosophy, or sociology of science were primarily trained in history, philosophy, or sociology. Rather, they were natural or exact scientists who came to be disenchanted with the social entanglements of their chosen fields of study. In effect, they fell victim to an intellec- tual bait-and-switch, whereby the reasons they entered science failed to explain science’s continued support in the wider society. This point often makes STS appear more critical than many of its practitioners intend it to be. STS researchers are virtually in agreement that people tend to like science for the wrong reasons (i.e. they are too easily taken in by its hype), but relatively few STS researchers would thereby con- clude that there are no good reasons to like science. There have been three generations of STS research, and each tells a different story of disenchantment with science. First, the logical pos- itivists, including Karl Popper, were keen to protect the theoretical base of science from the technological devastation that was wrought in its name during World War I. Next, Thomas Kuhn and his con- temporaries on both sides of the Atlantic – including Paul Feyerabend, Imre Lakatos, Stephen Toulmin, Derek de Solla Price – tried to do the same vis-à-vis World War II, though, more than the previous genera- tion, they relied on science’s past for normative guidance, largely out of a realization of technology’s contemporary role in scaling up the sci- entific enterprise and giving it forward momentum.
[Show full text]
“YOU MUST REMEMBER THIS” Abraham (“Abe”) Edel
MATERIAL FOR “YOU MUST REMEMBER THIS” Abraham (“Abe”) Edel (6 December 1908 – 22 June 2007) “Twenty-Seven Uses of Science in Ethics,” 7/2/67 Abraham Edel, In Memoriam, by Peter Hare and Guy Stroh Abraham Edel, 1908-2007 Abraham Edel was born in Pittsburgh, Pennsylvania on December 6, 1908. Raised in Yorkton, Canada with his older brother Leon who would become a biographer of Henry James, Edel studied Classics and Philosophy at McGill University, earning a BA in 1927 and an MA in 1928. He continued his education at Oxford where, as he recalled, “W.D. Ross and H.A. Prichard were lecturing in ethics, H.W.B. Joseph on Plato, and the influence of G. E. Moore and Bertrand Russell extended from Cambridge. Controversy on moral theory was high. The same was true of epistemology, where Prichard posed realistic epistemology against Harold Joachim who was defending Bradley and Bosanquet against the metaphysical realism of Cook Wilson.” He received a BA in Litterae Humaniores from Oxford in 1930. In that year he moved to New York City for doctoral studies at Columbia University, and in 1931 began teaching at City College, first as an assistant to Morris Raphael Cohen. F.J.E. Woodbridge directed his Columbia dissertation, Aristotle’s Theory of the Infinite (1934). This monograph and two subsequent books on Aristotle were influenced by Woodbridge’s interpretation of Aristotle as a philosophical naturalist. Although his dissertation concerned ancient Greek philosophy, he was much impressed by research in the social sciences at Columbia, and the teaching of Cohen at City College showed him how philosophical issues lay at the root of the disciplines of psychology, sociology, history, as well as the natural sciences.
[Show full text]
The Methodology of Scientific Research Programmes Philosophical Papers Volume I
The methodology of scientific research programmes Philosophical Papers Volume i IMRE LAKATOS EDITED BY JOHN WORRALL AND GREGORY CURRIE CAMBRIDGE UNIVERSITY PRESS Downloaded from https://www.cambridge.org/core. UB der LMU München, on 13 Apr 2020 at 02:49:26, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511621123 cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521280310 © Imre Lakatos Memorial Appeal fund and the Estate of Imre Lakatos 1978 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1978 First paperback edition 1980 Reprinted 1984, 1986, 1989, 1992, 1994, 1995, 1999 A catalogue record for this publication is available from the British Library isbn 978-0-521-21644-9 Hardback isbn 978-0-521-28031-0 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables, and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter.
[Show full text]
In-Between Science and Religion Dominic Balestra Fordham University, [email protected]
Fordham University Masthead Logo DigitalResearch@Fordham Hermeneutic and Phenomenological Philosophies Research Resources of Science 2002 In-Between Science and Religion Dominic Balestra Fordham University, [email protected] Follow this and additional works at: https://fordham.bepress.com/phil_research Part of the Philosophy of Science Commons Recommended Citation Balestra, Dominic, "In-Between Science and Religion" (2002). Research Resources. 44. https://fordham.bepress.com/phil_research/44 This Article is brought to you for free and open access by the Hermeneutic and Phenomenological Philosophies of Science at DigitalResearch@Fordham. It has been accepted for inclusion in Research Resources by an authorized administrator of DigitalResearch@Fordham. For more information, please contact [email protected]. DOMINIC BALESTRA IN-BETWEEN SCIENCE AND RELIGION 1. METHODOLOGICAL DEMARCATION: A WANING WALL Stripped of its seventeenth and eighteenth century Deistic, if not theistic, sensibility, science entered the twentieth century as agnostic and indifferent, if not outrightly antagonistic, toward religion. Religion was often viewed as the enemy of modern science; and its theology was presumed to be intellectually vacuous. “Real science” was clearly understood to be objective, empirical and rational. It bore no relationship to theology, none of any cognitive import – except perhaps to eliminate theology as contributing toward any understanding of the cosmos. This view of the relationship between science and religion was further reinforced by logical empiricism. In the latter’s view science aimed to establish empirically testable, generalized explanations of observable but problematic phenomena. Such phenomena might be freely falling bodies or the deviation of a planet from the expected path of its usual orbit. As the phrase “logical empiricism” suggests, in its view the objectivity of science rested with an evidential base of empirical facts, facts which were independent of any hypothesized theory under question and available to all through sense observation.
[Show full text]
International Journal of Science and Technology (STECH) Bahir Dar- Ethiopia Vol
IAARR 57 International Journal of Science and Technology Vol. 7 (1) February, 2018 International Journal of Science and Technology (STECH) Bahir Dar- Ethiopia Vol. 7 (1), S/No15, February, 2018: 57-71 ISSN: 2225-8590 (Print) ISSN 2227-5452 (Online) DOI: http://dx.doi.org/10.4314/stech.v7i1.5 SCIENTIFIC REVOLUTION, INCOMMENSURABILITY AND TRUTH IN THEORIES: OBJECTION TO KUHN’S PERSPECTIVE IHEJIRIKA, CARDINAL, PhD DEPARTMENT OF PHILOSOPHY UNIVERSITY OF PORT HARCOURT, NIGERIA PHONE: +2348028279646 E-mail: [email protected] ……………………………………………………………………….. EMEDOLU, CHRISTIAN C., PhD DEPARTMENT OF PHILOSOPHY UNIVERSITY OF PORT HARCOURT, NIGERIA PHONE: +2348035517505 E-mail: [email protected] …………………………………………………………………………………………… ABSTRACT This is to revaluate the truth-content of scientific theory in the event of scientific revolution. In this paper, we argued that the radical views on incommensurability offered, especially, by Thomas S. Kuhn (shared by Paul Feyerabend, and others) do not actually undermine either the realist or cumulative stature of science. The core of our discussion is, ultimately, to provide a clearer and broader picture of the general characteristics of scientific revolution or theory change. In doing this, the paper pinpoints the audacity behind this change and the nature of truth undergirding any emergent or overthrown scientific theory. The paper has some rebounding echoes of the realist and cumulative features of science while addressing the issue of the real character of theory change. Following some unique interpretations of scientific revolution, truth in science could still be redeemed in the face of theory change and Kuhnian Incommensurability Thesis. To this effect, the hermeneutical approach is considered most suitable in this paper.
[Show full text]
The Atheist Trap, Or the Argument from Design and Scientific Falsification
University of Rhode Island DigitalCommons@URI Open Access Master's Theses 1993 The Atheist Trap, or the Argument from Design and Scientific Falsification Joseph R. Mixie University of Rhode Island Follow this and additional works at: https://digitalcommons.uri.edu/theses Recommended Citation Mixie, Joseph R., "The Atheist Trap, or the Argument from Design and Scientific alsificationF " (1993). Open Access Master's Theses. Paper 1761. https://digitalcommons.uri.edu/theses/1761 This Thesis is brought to you for free and open access by DigitalCommons@URI. It has been accepted for inclusion in Open Access Master's Theses by an authorized administrator of DigitalCommons@URI. For more information, please contact [email protected]. BTI02 M589 1993 THE A THEIST TRAP, OR THE ARGUMENT FROM DESIGN AND SCIENTIFIC FALSIFICATION ISM BY JOSEPH R. MIXIE A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS IN PHILOSOPHY 2 I<;'�,,,,�,. -;; UNIVERSITY OF RHODE ISLAND ... 1993 THESIS ABSTRACT The argument from design is one of the most widely debated arguments for the existence of God. There has been much written in support of and in criticism of the argument's basic structure and conclusion. I shall attempt to clarify these positions, and to argue that the thei?ti c account provides a more rationally justified explanation of human life on earth than the atheistic account. Many philosophers think that any proof for the existence of God is mere "metaphysical speculation." Many times these philosophers use the criteria of scientific empiricism as the standard for an "acceptable" scientific theory, regardless of the subject matter.
[Show full text]
An Evaluation of Imre Lakatos' Attempt To
AN EVALUATION OF IMRE LAKATOS’ ATTEMPT TO RECONCILE NORMATIVE AND DESCRIPTIVE ACCOUNTS OF SCIENCE FRANCIS MWANGI NJENGA A THESIS SUBMITTED TO THE SCHOOL OF HUMANITIES AND SOCIAL SCIENCES IN THE PARTIAL FULFILLMENT FOR THE AWARD OF THE DEGREE OF MASTERS OF ARTS (PHILOSPHY) OF KENYATTA UNIVERSITY. JUNE 2017 ii DECLARATION This thesis is my original work and has not been presented for a degree in any other university or any other award. Signature…………………… Date………………………… FRANCIS MWANGI NJENGA (BA. Phil) (C50/23879/2013) Supervisors: I confirm that the work reported in this thesis was carried out by the student under my supervision. Signature…………………… Date………………………. DR. JOSPHAT OYIGO Department of Philosophy and Religious Studies Signature…………………… Date……………………….. DR. JACOB MAGERO Department of Philosophy and Religious Studies iii DEDICATION I dedicate this work to my Parents: Mr Wallace Njenga and Regina Njenga. Wishing them peace, health, happiness and longevity. iv ACKNOWLEDGMENT I wish to thank my supervisors Dr. Josphat Oyigo and Dr. Jacob Magero for their candid and insightful support in writing this thesis and Dr. Tom Namwamba for examining and shaping this work. I also thank the current and the former chairperson of the department Dr. Gecaga and Dr. Kahumbi plus the secretaries Jane and Cristobel for their whole hearted support in facilitating among other things, all the necessary arrangements for my defenses. I also recognize my classmates more so Nzioka John whom we have journeyed together up to this far, big brothers Ochieng’ Ojwang’ and Ndumpa F. for challenging, mentoring and supporting me diligently in the preparation and printing of my thesis and Massawa Charles for his editorial and moral support.
[Show full text]
The Pride of Losers: a Genealogy of the Philosophy of Science
History and Theory 41 (October 2002), 392-409 © Wesleyan University 2002 ISSN: 0018-2656 THE PRIDE OF LOSERS: A GENEALOGY OF THE PHILOSOPHY OF SCIENCE IMRE LAKATOS AND THE GUISES OF REASON. By John Kadvany. Durham, North Carolina: Duke University Press, 2001. Pp. xx, 378. In one respect, the history of the philosophy of science is striking: There seems to be an inverse relationship between the philosophical and the scientific signif- icance of the people talked about. Even the very greatest scientists, such as Galileo, Newton, Maxwell, and Einstein, tend to be treated as no more than pass- able philosophers of science. Sometimes others in that scientific league, most notably Charles Darwin, are relegated to polite philosophical silence. This curi- ous feature first becomes noticeable when examining periods when “natural philosopher” is used as an expression to cover people who, by our lights, are both scientists and philosophers. For example, seventeenth-century natural philoso- phers are nowadays divided into “scientists”—such as Galileo, Boyle, and Newton—and “philosophers”—such as Descartes, Hobbes, and Leibniz. The people we now call “philosophers” are basically the natural philosophers on the losing side of the battles that we now call “scientific.” This embarrassing tendency becomes more pronounced as we move into the nineteenth and early twentieth centuries. The major moments in the philosophy of science are defined by people like William Whewell, John Stuart Mill, Ernst Mach, Pierre Duhem, and Henri Poincaré. All were professional scientists, who are now represented as having engaged in major philosophical disputes with one another, other philosophers, and other scientists.
[Show full text]
A Classical Defense Against Mathematical Empiricism
The Crucial Role of Proof: A Classical Defense Against Mathematical Empiricism by Catherine Allen Womack Submitted to the Department of Linguistics and Philosophy in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 1993 @ Massachust•Os Institute of Technology 1993. All rights reserved. Author ... ................................... ............ ......... Department of Linguistics and Philosophy March 10, 1993 Certified by .............. .......................................... James Higginbotham Professor Thesis Supervisor Read by ...... .... , .... ....... ....................................... George Boolos Professor Thesis Reader Accepted by ..-. ,. ... ..................................... - . George Boolos Chairman, Departmental Committee on Graduate Students ARCHIVES MACSACHUSETTS INSTITUTE OF TECHNOLOGY rJUN 03 1993 The Crucial Role of Proof: A Classical Defense Against Mathematical Empiricism by Catherine Allen Womack Submitted to the Department of Linguistics and Philosophy on March 10, 1993, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Mathematical knowledge seems to enjoy special status not accorded to scientific knowledge: it is considered a priori and necessary. We attribute this status to math- ematics largely' because of the way we come to know it-through following proofs. Mathematics has come under attack from sceptics who reject the idea that mathe- matical knowledge is a priori. Many sceptics consider it to be a posteriori knowledge, subject to possible empirical refutation. In a series of three papers I defend the a priori status of mathematical knowledge by showing that rigorous methods of proof are sufficient to convey a priori knowledge of the theorem proved. My first paper addresses Philip Kitcher's argument in his book The Natuire of Mathematical Knowledge that mathematics is empirical. Kitcher develops a view of a priori knowledge according to which mathematics is not a priori.
[Show full text]