DOCSLIB.ORG

On What There Is in Philosophy of Mathematics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On What There Is in Philosophy of Mathematics ON WHAT THERE IS IN PHILOSOPHY OF MATHEMATICS SOBRE LO QUE HAY EN FILOSOFÍA DE LAS MATEMÁTICAS ICE B. RISTESKI 7RURQWR&DQDGD (PDLOLFH#VFLHQWLVWFRP CARLOS E. GARCÍA D. 8QLYHUVLGDGGH&DOGDV8QLYHUVLGDGGH0DQL]DOHV (PDLOFDUORVJDUFLDBG#XFDOGDVHGXFR RECIBIDO EL 30 DE NOVIEMBRE DE 2007 Y APROBADO EL 14 DE MARZO DE 2008 ABSTRACT RESUMEN ,QWKLVSDSHUZHVNHWFKURXJKO\VHYHUDO (QHVWHDUWtFXORSUHVHQWDPRVXQERVTXHMR PRPHQWVWKDWKDYHDFRQVLGHUDEOHLQÁXHQFH GHGLYHUVRVPRPHQWRVTXHKDQHMHUFLGR RQWKHGHYHORSPHQWRIWKHSKLORVRSK\RI una influencia considerable sobre el PDWKHPDWLFV:HLOOXVWUDWHWKHPDLQ GHVDUUROORGHODÀORVRItDGHODVPDWHPiWLFDV problems of this discipline and discuss Ilustramos los principales problemas de WKHLPSRUWRIWKHVRFDOOHG¶SRVLWLYLQHVV HVWDGLVFLSOLQD\GLVFXWLPRVHOVHQWLGR\ RIPDWKHPDWLFV·%\H[DPLQLQJVHYHUDO DOFDQFHGHODGHQRPLQDGD¶SRVLWLYLGDG DSSURDFKHVWRWKLVSUREOHPZHDQDO\]H GHODPDWHPiWLFD·$GHPiVPHGLDQWHHO RQWRORJLFDODQGHSLVWHPRORJLFDOTXHVWLRQV examen de diversos enfoques sobre este ZKLFKFDQKHOSWRFODULI\WKLVFRPSOLFDWHG WySLFRDQDOL]DPRVDOJXQDVFXHVWLRQHV area. HSLVWHPROyJLFDV\RQWROyJLFDVTXHSXHGHQ DUURMDUDOJRGHOX]VREUHHVWDFRPSOLFDGD iUHDGHODPHWDÀORVRItD KEY WORDS PALABRAS CLAVE 3KLORVRSK\ RI PDWKHPDWLFV ORJLF )LORVRItDGHODVPDWHPiWLFDVOyJLFD DEVWUDFWPDWKHPDWLFVSXUHPDWKHPDWLFV PDWHPiWLFDDEVWUDFWDPDWHPiWLFDSXUD application of mathematics. aplicación de las matemáticas. 'LVFXVLRQHV)LORVyÀFDV$xR1(QHUR²-XQLRSS Ice B. Risteski & Carlos E. García D. 3KLORVRSK\RIPDWKHPDWLFVLVDYHU\FRPSOLFDWHGGLVFLSOLQHDQGLVIXOORI GLYHUJHQWLGHDV3KLORVRSK\RIPDWKHPDWLFVGRHVWKHSUREOHPDWL]DWLRQRI PDWKHPDWLFVi.e.LWEULQJVWKHODWWHULQWRTXHVWLRQ,WLVDFULWLFDOVXUYH\ RQPDWKHPDWLFVDVXUYH\WKDWUHTXLUHVGLVWDQFHDQGDEDQGRQPHQW RIPDWKHPDWLFDOWKLQNLQJ+HQFHLWPXVWEHGLVWLQJXLVKHGIURPWKH XQFULWLFDOSHQHWUDWLRQLQWKHIUDPHZRUNRIWKHPDWKHPDWLFDOWKLQNLQJ LQZKLFKPDWKHPDWLFVLWVHOILVGHYHORSHG +RZHYHUWKLVDSSURDFKFDQEHPLVLQWHUSUHWHGE\TXDVLSKLORVRSKHUV ZKREHOLHYHWKDWWKHLUYLVLRQRIDVFLHQWLVWZKRFDQQRWXQGHUVWDQG WKHHVVHQFHRIVFLHQFHLVFRQÀUPHGRQFHPRUHi.e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the present discussion as do the so called pure philosophers. :HGLVFXVVKHUHDSUREOHPDWL]DWLRQRIPDWKHPDWLFVDVDVFLHQFHRU PRUHSUHFLVHO\DVDSDUDGLJPDWLFH[DPSOHRIDVFLHQFHDFFRUGLQJWR WZRGLUHFWLRQV 0DWKHPDWLFVLVDSDUDGLJPDWLFH[DPSOHRIDSRVLWLYHVFLHQFHDQGWKXV LWEHFRPHVDÀHOGRQZKLFKDQ\HSLVWHPRORJ\FDQEHWHVWHG 0DWKHPDWLFV LV D SDUDGLJPDWLF H[DPSOH RI D VFLHQFH ZKRVH SRVLWLYHQHVVPXVWEHMXVWLÀHGE\DJHQHUDOHSLVWHPRORJ\+HUHWKH positiveness itself is questioned. ,Q WKH ÀUVW DSSURDFK WKH HSLVWHPRORJ\ SDVVHV RU IDLOV GHSHQGLQJ RI ZKHWKHU WKH SRVLWLYHQHVV RI PDWKHPDWLFV i.e. WKH SRVLWLYHQHVV RI mathematical science can be explained or not. In the second approach LWLVPDWKHPDWLFVWKDWSDVVHVRUIDLOVGHSHQGLQJRIZKHWKHULWFDQEH HSLVWHPRORJLFDOO\MXVWLÀHG 7KH ÀUVW DSSURDFK LV FKDUDFWHULVWLF RI WKH SKLORVRSKHUV ZKR WKLQN DERXWPDWKHPDWLFVZKLOHWKHVHFRQGDSSURDFKLVFKDUDFWHULVWLFRIWKH PDWKHPDWLFLDQVZKRWKLQNSKLORVRSKLFDOO\ 152 'LVFXVLRQHV)LORVyÀFDV$xR1(QHUR²-XQLRSS ON WHAT THERE IS IN PHILOSOPHY OF MATHEMATICS 7KHÀUVWDSSURDFKHQFDSVXODWHVWKHold PHDQLQJRIWKHSKLORVRSKLFDO DWWLWXGHWRZDUGVWKHSRVLWLYHPDWKHPDWLFVZKLOHWKHVHFRQGDSSURDFK represents the newPHDQLQJRIWKHSKLORVRSKLFDODWWLWXGHWRZDUGVWKH positiveness of mathematics. /HWXVH[DPLQHQRZWKHIROORZLQJTXHVWLRQ:KDWLVPDWKHPDWLFDOO\ EURXJKWLQWRTXHVWLRQLQWKHold philosophy of mathematics"&HUWDLQO\QRW LWVSRVLWLYHQHVV:KDWHYHUFRXOGFRQWUDGLFWWKLVSRVLWLYHQHVVLVDOUHDG\ GRXEWIXOGXHWRWKLVWHQVLRQ%ULHÁ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abstract truths. The positiveness of mathematics lies in its abstractness ZKLFKHQDEOHVSURRI 7KDOHV VWDUWHG DQG 3\WKDJRUDV ZHQW RQ GHYHORSLQJ PDWKHPDWLFV DV D SRVLWLYH DEVWUDFW VFLHQFH :LWK 3\WKDJRUDV WKLV DEVWUDFW VFLHQFH LV QRWRQO\DEVWUDFWFRQVFLRXVQHVV:LWKKLPPDWKHPDWLFDOFRQFHSWLVD VWUXFWXUDODQGFRQVWLWXWLYHHOHPHQWRIWKHFRQFUHWHRUDV3\WKDJRUDV ZRXOGVD\nature is made up of numbers7KRXJKWKLVSKUDVHLVRIWHQ FRQVLGHUHGDVPDWKHPDWLFDOP\VWLFLVP\HWLWVHHPVWKDWWKLVLVMXVWD JHQHUDOHVWLPDWHZKLFKVD\VQRWKLQJDQGH[SODLQVVWLOOOHVV%XWZKDW LV WKLV DOO DERXW" ,W LV DERXW WKH UHODWLRQVKLS EHWZHHQ WKH SRVLWLYH DEVWUDFW PDWKHPDWLFV DQG WKH FRQFUHWH QDWXUH +HUH ZH ÀQG IRU WKH ÀUVWWLPHWKHSUREOHPDWL]DWLRQRIWKLVUHODWLRQVKLSDVZHOODVWKHÀUVW VROXWLRQRIWKHSUREOHPQRWLFHGZKLFKWKURXJKWKHLGHQWLÀFDWLRQRI FRQFUHWHQDWXUHZLWKDEVWUDFWPDWKHPDWLFVGHVFULEHVWKHDSSOLFDELOLW\ of abstract mathematics to the comprehension of concrete nature. A JUHDW QXPEHU RI FRQWHPSRUDU\ DQG IRUPHUO\ OLYLQJ PHWDOOXUJLVWV SK\VLFLVWV DQG FKHPLVWV ZKR EHOLHYH LQ WKH DSSOLFDELOLW\ RI DEVWUDFW PDWKHPDWLFVDUHLQIDFWDVPXFKRIP\VWLFVDV3\WKDJRUDVZDVVLQFH WKHLUEHOLHILVEDVHGPDLQO\RQDQLGHQWLÀFDWLRQRI3\WKDJRUHDQW\SH 7KHSUREOHPDWL]DWLRQRIWKHUHODWLRQVKLSEHWZHHQDEVWUDFWPDWKHPDWLFV DQGFRQFUHWHQDWXUHRUDVLWZDVFDOOHGDERYHWKHSUREOHPDWL]DWLRQRI 153 Ice B. Risteski & Carlos E. García D. WKHDSSOLFDELOLW\RIDEVWUDFWPDWKHPDWLFVWRFRQFUHWHQDWXUHLQQRZD\ FDQEHFDOOHGP\VWLFLVP7KHSRLQWLVDERXWWKHPDLQSUREOHPRIWKHROG SKLORVRSK\RIPDWKHPDWLFVZKLFKGRHVQRWTXHVWLRQWKHSRVLWLYHQHVV RIPDWKHPDWLFVWUHDWLQJLWDVZHOOIRXQGHGLQLWVDEVWUDFWQHVVEXWLW EULQJVLQWRTXHVWLRQLWVDSSOLFDELOLW\ZLWK3\WKDJRUDVDVLWVIRXQGHU +HUH DQ DGGLWLRQDO H[SODQDWLRQ LV QHFHVVDU\ 7KH DERYH PHQWLRQHG LGHQWLÀFDWLRQE\3\WKDJRUDVWDNHVSODFHLQWZRVWHSV,QWKHÀUVWVWHS WKHJHRPHWULFDODEVWUDFWLRQVDUHFRQVLGHUHGDVFRQVWUXFWLYHHOHPHQWV RI WKH FRQFUHWH QDWXUH DQG LQ WKH VHFRQG VWHS DV D ÀQDO UHGXFWLRQ of these abstractions the number is discovered. The second step RI WKHVH LGHQWLÀFDWLRQV LV VKDWWHUHG E\ WKH GLVFRYHU\ RI WKH SRVVLEOH LQFRPPHQVXUDEOH OHQJWKV ,W DOVR VKDWWHUV WKH VSHFLÀF 3\WKDJRUHDQ UHGXFWLRQRIWKHFRQFUHWHQDWXUHRIWKHQXPEHUEXWQRWWKH3\WKDJRUHDQ YLHZWKDWWKHDEVWUDFWIRXQGDWLRQRIPDWKHPDWLFVEULQJVLQWRTXHVWLRQ LWV DSSOLFDELOLW\ WR FRQFUHWHQHVV i.e. WKH YLHZ ZKLFK UHSUHVHQWV D IXQGDPHQWDOSUREOHPRIWKHROGSKLORVRSK\RIPDWKHPDWLFV +HUHZHVKDOOXQGHUOLQHWKDWWKLVSUREOHPUHDOO\DQGEDVLFDOO\UHPDLQV DSUREOHPRIWKHROGSKLORVRSK\RIPDWKHPDWLFV 7RPDQ\SHRSOHLWLVGLIÀFXOWWRXQGHUVWDQGWKHJHQHVLVRI'HPRFULWXV· PDWKHPDWLFDO DWRPLVP >@ ,W FDQ EH HDVLO\ XQGHUVWRRG LI ZH VHH LQ LW D FRQFRUGDQFH RI WKH DEVWUDFW PDWKHPDWLFV ZLWK WKH FRQFUHWH DWRPLVWLFDOO\ FRPSUHKHQGHG QDWXUH D FRQFRUGDQFH ZKLFK VKRXOG MXVWLI\WKHDSSOLFDELOLW\RIPDWKHPDWLFV 6LPLODU SUREOHPV DUH HQFRXQWHUHG E\ PDQ\ LQYHVWLJDWRUV RI =HQR·V DSRULD:KLOHVRPHVHHLQWKHVHMXVWDQLGOHVRSKLVWLFDOWDONRWKHUVÀQG PDWKHPDWLFDOHUURUVLQ=HQR·VDUJXPHQWV7KHZHOONQRZQVRSKLVPV FDQQRWFRQIXVHWKRVHZKRORRNSULPLWLYHO\DQGFRQFUHWHO\DWWKHPRWLRQ RIWKHDUURZDQG$FKLOOHV·VYLFWRU\RYHUWKHWRUWRLVH7KH\FDQQRWEH GHFHLYHGDQGOHGDORQJDZURQJZD\E\DEVWUDFWDUJXPHQWV2QWKH RWKHUKDQGWKHLVRODWHGDEVWUDFWDUJXPHQWLVEDVHGRQWKHHUURQHRXV PDWKHPDWLFDOSUHPLVHWKDWWKHVXPRILQÀQLWHO\PDQ\ÀQLWHTXDQWLWLHV PXVWEHLQÀQLWHO\JUHDW,VQRW=HQR·VDUJXPHQWMXVWDVRSKLVWLFDOJDPH ZKLFKEULQJVLQWRTXHVWLRQWKHHOHPHQWDU\XQGHUVWDQGLQJRIWKHFRQFUHWH QDWXUH E\ XVLQJ DQ DEVWUDFW PDWKHPDWLFDO DUJXPHQW ZKLFK UHPDLQV VLOHQWDERXWDIDOVHPDWKHPDWLFDOSUHPLVH"0D\EHEXWRQO\XQGHUWKH FRQGLWLRQWKDWWKHDSSOLFDELOLW\RIWKHDEVWUDFWPDWKHPDWLFDODUJXPHQW WRWKHFRQFUHWH$FKLOOHV·VVLWXDWLRQLVFRQVLGHUHGVHOIHYLGHQWRQO\LI 154 'LVFXVLRQHV)LORVyÀFDV$xR1(QHUR²-XQLRSS ON WHAT THERE IS IN PHILOSOPHY OF MATHEMATICS ZHLGHQWLI\WKHFRQFUHWHSDWKRI$FKLOOHVZLWKDQDEVWUDFWPDWKHPDWLFDO OHQJWK RQO\ LI ZH WKLQN WKDW WKH LQÀQLWH GLYLVLELOLW\ RI WKH DEVWUDFW PDWKHPDWLFDOOHQJWKLVWKHVDPHWKLQJDVWKHLQÀQLWHGLYLVLELOLW\RIWKH FRQFUHWHSDWKRI$FKLOOHV7KXV=HQR·VHIIHFWFDQEHFRQVLGHUHGDKROORZ VRSKLVPRQO\LIWKHDSSOLFDELOLW\RIWKHDEVWUDFWPDWKHPDWLFDODUJXPHQW WR WKH FRQFUHWH $FKLOOHV·V VLWXDWLRQ LV FRQVLGHUHG VHOIHYLGHQW %XW LV QRWMXVWWKLVVHOIHYLGHQFHRIWKHDSSOLFDELOLW\RIDEVWUDFWPDWKHPDWLFV EURXJKWLQWRTXHVWLRQE\=HQR·VDUJXPHQW",VQRWWKHSRLQWKHUHDERXW WKHIXQGDPHQWDOSUREOHPRIWKHROGSKLORVRSK\RIPDWKHPDWLFV"2I FRXUVHWKLVLVH[DFWO\WKHSRLQW ,QWKHPRGHUQSKLORVRSK\RIPDWKHPDWLFVZHFDQWUDFHWKHVKLIWIURP WKH REMHFWLYH QRUPDWLYH UHODWLRQVKLS RI WKH DEVWUDFW WR WKH FRQFUHWH WRZDUGV WKH WKRXJKW DERXW WKH DEVWUDFW ZKLFK VWDQGDUGL]HV WKH GHVFULSWLRQ RI WKH FRQFUHWH .DQW VWDQGV RQ WKH WRS RI WKLV WKRXJKW ZKHQKHVD\VWKDWPDWKHPDWLFDOVHQVHLVSRVLWLYHEHLQJDVHQVHRIWKHa prioriVKDSHVRIFRQVFLRXVQHVVVSDFHDQGWLPH7KLVVHQVHLVDSSOLFDEOH since the a prioriVKDSHVRIFRQVFLRXVQHVVÀUVWHQDEOHWKHPHHWLQJZLWK UHDOLW\i.e.WKHVHDUHFRQVWLWXHQWVRIUHDOLW\7KHRSSRVLWLRQDEVWUDFW FRQFUHWHLVUHSODFHGE\WKHRSSRVLWLRQWUDQVFHQGHQWDOLPPDQHQWDQG DSSOLFDELOLW\VKRXOGQRWEHXQGHUVWRRGDVDSSOLFDELOLW\RIWKHDEVWUDFW in the comprehension of the transcendental since the transcendental is QRWFRQFUHWH%RWKWKHFRQFUHWHDQGWKHDEVWUDFWDUHLPPDQHQWDQGWKDW LVKRZWKH\DUHUHODWHGWRHDFKRWKHU 7KH ROG SKLORVRSK\ RI PDWKHPDWLFV
Load more

Recommended publications
  • Logic in Action: Wittgenstein's Logical Pragmatism and the Impotence of Scepticism
    This is the final, pre-publication draft. Please cite only from published paper in Philosophical Investigations 26:2 (April 2003), 125-48. LOGIC IN ACTION: WITTGENSTEIN'S LOGICAL PRAGMATISM AND THE IMPOTENCE OF SCEPTICISM DANIÈLE MOYAL-SHARROCK UNIVERSITY OF GENEVA 1. The Many Faces of Certainty: Wittgenstein's Logical Pragmatism So I am trying to say something that sounds like pragmatism. (OC 422) In his struggle to uncover the nature of our basic beliefs, Wittgenstein depicts them variously in On Certainty: he thinks of them in propositional terms, in pictorial terms and in terms of acting. As propositions, they would be of a peculiar sort – a hybrid between a logical and an empirical proposition (OC 136, 309). These are the so-called 'hinge propositions' of On Certainty (OC 341). Wittgenstein also thinks of these beliefs as forming a picture, a World-picture – or Weltbild (OC 167). This is a step in the right (nonpropositional) direction, but not the ultimate step. Wittgenstein's ultimate and crucial depiction of our basic beliefs is in terms of a know-how, an attitude, a way of acting (OC 204). Here, he treads on pragmatist ground. But can Wittgenstein be labelled a pragmatist, having himself rejected the affiliation because of its utility implication? But you aren't a pragmatist? No. For I am not saying that a proposition is true if it is useful. (RPP I, 266) Wittgenstein resists affiliation with pragmatism because he does not want his use of use to be confused with the utility use of use. For him, it is not that a proposition is true if it is useful, but that use gives the proposition its sense.
    [Show full text]
  • Weyl's Predicative Classical Mathematics As a Logic-Enriched
    Weyl’s Predicative Classical Mathematics as a Logic-Enriched Type Theory? Robin Adams and Zhaohui Luo Dept of Computer Science, Royal Holloway, Univ of London {robin,zhaohui}@cs.rhul.ac.uk Abstract. In Das Kontinuum, Weyl showed how a large body of clas- sical mathematics could be developed on a purely predicative founda- tion. We present a logic-enriched type theory that corresponds to Weyl’s foundational system. A large part of the mathematics in Weyl’s book — including Weyl’s definition of the cardinality of a set and several re- sults from real analysis — has been formalised, using the proof assistant Plastic that implements a logical framework. This case study shows how type theory can be used to represent a non-constructive foundation for mathematics. Key words: logic-enriched type theory, predicativism, formalisation 1 Introduction Type theories have proven themselves remarkably successful in the formalisation of mathematical proofs. There are several features of type theory that are of particular benefit in such formalisations, including the fact that each object carries a type which gives information about that object, and the fact that the type theory itself has an inbuilt notion of computation. These applications of type theory have proven particularly successful for the formalisation of intuitionistic, or constructive, proofs. The correspondence between terms of a type theory and intuitionistic proofs has been well studied. The degree to which type theory can be used for the formalisation of other notions of proof has been investigated to a much lesser degree. There have been several formalisations of classical proofs by adapting a proof checker intended for intuitionistic mathematics, say by adding the principle of excluded middle as an axiom (such as [Gon05]).
    [Show full text]
  • "There Is No Right to Be Stupid": a Voegelinian Analysis of Islamist
    "There is No Right to Be Stupid": A Voegelinian Analysis of Islamist Terrorism with Reference to the "Elements of Reality" Copyright 2005 Celestino Perez, Jr. In the following essay I make three claims. First, in agreement with Eric Voegelin and Barry Cooper, I illustrate why a comprehensive study of politics requires that attention be paid to the symbols "God and man," "the Good", and "the soul" (Section I). I then briefly wonder if Voegelin and Cooper, despite offering cogent analyses of pernicious ideologies, ask too much of their readers? If so, what are the prospects for a competent political science and a healthy democratic populace (Section II)? In Section III, I suggest a different way of applying Voegelin's theory to an analysis of Islamist ideology, wherein I lay out ten "elements of reality" and five distilled "fundamental truths" derived from these elements. To the degree that someone distorts or neglects these elements of reality, he is a pneumopath. He may be very smart. But, alas, he is a "philosophical and religious ignoramus,"1 [1] and hence stupid. And, as Voegelin reminds us in Hitler and the Germans, "There is no right to be stupid." To be sure, if we allow ourselves to submit to stupidity, the gangster animals will be sure to mockingly let us know: 1 [1] Barry Cooper, New Political Religions, or An Analysis of Modern Terrorism ( Columbia : University of Missouri Press , 2004), xii. Cooper describes Sayyid Qutb's and other terrorists' "dogmatic certainty" as resulting from their being "philosophical and religious ignoramuses." "Yoohoo, silly ass!"2 [2] In the Concluding Notes, I lay out what I consider to be the gist of Cooper's argument, and I briefly comment on his Appendix in New Political Religions.
    [Show full text]
  • Logic Model Workbook
    Logic Model Workbook INNOVATION NETWORK, INC. www.innonet.org • [email protected] Logic Model Workbook Table of Contents Page Introduction - How to Use this Workbook .....................................................................2 Before You Begin .................................................................................................................3 Developing a Logic Model .................................................................................................4 Purposes of a Logic Model ............................................................................................... 5 The Logic Model’s Role in Evaluation ............................................................................ 6 Logic Model Components – Step by Step ....................................................................... 6 Problem Statement: What problem does your program address? ......................... 6 Goal: What is the overall purpose of your program? .............................................. 7 Rationale and Assumptions: What are some implicit underlying dynamics? ....8 Resources: What do you have to work with? ......................................................... 9 Activities: What will you do with your resources? ................................................ 11 Outputs: What are the tangible products of your activities? ................................. 13 Outcomes: What changes do you expect to occur as a result of your work?.......... 14 Outcomes Chain ......................................................................................
    [Show full text]
  • Introduction to Philosophy. Social Studies--Language Arts: 6414.16. INSTITUTION Dade County Public Schools, Miami, Fla
    DOCUMENT RESUME ED 086 604 SO 006 822 AUTHOR Norris, Jack A., Jr. TITLE Introduction to Philosophy. Social Studies--Language Arts: 6414.16. INSTITUTION Dade County Public Schools, Miami, Fla. PUB DATE 72 NOTE 20p.; Authorized Course of Instruction for the Quinmester Program EDRS PRICE MF-$0.65 HC-$3.29 DESCRIPTORS Course Objectives; Curriculum Guides; Grade 10; Grade 11; Grade 12; *Language Arts; Learnin4 Activities; *Logic; Non Western Civilization; *Philosophy; Resource Guides; Secondary Grades; *Social Studies; *Social Studies Units; Western Civilization IDENTIFIERS *Quinmester Program ABSTRACT Western and non - western philosophers and their ideas are introduced to 10th through 12th grade students in this general social studies Quinmester course designed to be used as a preparation for in-depth study of the various schools of philosophical thought. By acquainting students with the questions and categories of philosophy, a point of departure for further study is developed. Through suggested learning activities the meaning of philosopky is defined. The Socratic, deductive, inductive, intuitive and eclectic approaches to philosophical thought are examined, as are three general areas of philosophy, metaphysics, epistemology,and axiology. Logical reasoning is applied to major philosophical questions. This course is arranged, as are other quinmester courses, with sections on broad goals, course content, activities, and materials. A related document is ED 071 937.(KSM) FILMED FROM BEST AVAILABLE COPY U S DEPARTMENT EDUCATION OF HEALTH. NAT10N41
    [Show full text]
  • 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.
    [Show full text]
  • Number Theory
    “mcs-ftl” — 2010/9/8 — 0:40 — page 81 — #87 4 Number Theory Number theory is the study of the integers. Why anyone would want to study the integers is not immediately obvious. First of all, what’s to know? There’s 0, there’s 1, 2, 3, and so on, and, oh yeah, -1, -2, . Which one don’t you understand? Sec- ond, what practical value is there in it? The mathematician G. H. Hardy expressed pleasure in its impracticality when he wrote: [Number theorists] may be justified in rejoicing that there is one sci- ence, at any rate, and that their own, whose very remoteness from or- dinary human activities should keep it gentle and clean. Hardy was specially concerned that number theory not be used in warfare; he was a pacifist. You may applaud his sentiments, but he got it wrong: Number Theory underlies modern cryptography, which is what makes secure online communication possible. Secure communication is of course crucial in war—which may leave poor Hardy spinning in his grave. It’s also central to online commerce. Every time you buy a book from Amazon, check your grades on WebSIS, or use a PayPal account, you are relying on number theoretic algorithms. Number theory also provides an excellent environment for us to practice and apply the proof techniques that we developed in Chapters 2 and 3. Since we’ll be focusing on properties of the integers, we’ll adopt the default convention in this chapter that variables range over the set of integers, Z. 4.1 Divisibility The nature of number theory emerges as soon as we consider the divides relation a divides b iff ak b for some k: D The notation, a b, is an abbreviation for “a divides b.” If a b, then we also j j say that b is a multiple of a.
    [Show full text]
  • Chapter 1 Logic and Set Theory
    Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic.
    [Show full text]
  • Pure Mathematics Professors Teaching and Leading Research CO-OP OR REGULAR 28
    PURE MATHEMATICS Pure Mathematics professors teaching and leading research CO-OP OR REGULAR 28 Mathematician ranked among top 10 TOP 10 jobs from 2011-2017 – Comcast.com of grads are employed Search for a deeper 96.6% within 2 years understanding of mathematics Pure mathematics is at the foundation of all mathematical reasoning. If first-year calculus ALEX teaches you how to drive the car, Pure Mathematics teaches you how to build one. 3B, PURE MATHEMATICS AND Mathematicians know that there could be no general relativity without differential COMBINATORICS AND geometry, and no computer security without advanced number theory. OPTIMIZATION Pure Mathematics at Waterloo is a small, cohesive, and challenging program that will open countless doors for you. Our graduates have used the program as a springboard into careers WHAT DO YOU LOVE ABOUT in information technology, finance, business, science, education, and insurance, often by way PURE MATHEMATICS? The satisfaction from understanding of some of the most prestigious graduate programs in the world. an idea at a deeper level and tying together unrelated branches of ALEX’S FAVOURITE COURSES mathematics or physics for the › PMATH 320 Euclidean Geometry: This course is everything you love about Geometry: first time is the most rewarding Euclid’s axioms, isometries of the Euclidean plane and of Euclidean space, polygons, part of learning and understanding polyhedral, polytopes, and the kissing problem. mathematics. What I really enjoy is › PMATH 351 Real Analysis: It’s a very intuitive and natural approach to real analysis, and the developing a deep understanding of complexity of the course builds very naturally to the end of the semester.
    [Show full text]
  • Video Lessons for Illustrative Mathematics: Grades 6-8 and Algebra I
    Video Lessons for Illustrative Mathematics: Grades 6-8 and Algebra I The Department, in partnership with Louisiana Public Broadcasting (LPB), Illustrative Math (IM), and SchoolKit selected 20 of the critical lessons in each grade level to broadcast on LPB in July of 2020. These lessons along with a few others are still available on demand on the SchoolKit website. To ensure accessibility for students with disabilities, all recorded on-demand videos are available with closed captioning and audio description. Updated on August 6, 2020 Video Lessons for Illustrative Math Grades 6-8 and Algebra I Background Information: Lessons were identified using the following criteria: 1. the most critical content of the grade level 2. content that many students missed due to school closures in the 2019-20 school year These lessons constitute only a portion of the critical lessons in each grade level. These lessons are available at the links below: • Grade 6: http://schoolkitgroup.com/video-grade-6/ • Grade 7: http://schoolkitgroup.com/video-grade-7/ • Grade 8: http://schoolkitgroup.com/video-grade-8/ • Algebra I: http://schoolkitgroup.com/video-algebra/ The tables below contain information on each of the lessons available for on-demand viewing with links to individual lessons embedded for each grade level. • Grade 6 Video Lesson List • Grade 7 Video Lesson List • Grade 8 Video Lesson List • Algebra I Video Lesson List Video Lessons for Illustrative Math Grades 6-8 and Algebra I Grade 6 Video Lesson List 6th Grade Illustrative Mathematics Units 2, 3,
    [Show full text]
  • John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected]
    John P. Burgess Department of Philosophy Princeton University Princeton, NJ 08544-1006, USA [email protected] LOGIC & PHILOSOPHICAL METHODOLOGY Introduction For present purposes “logic” will be understood to mean the subject whose development is described in Kneale & Kneale [1961] and of which a concise history is given in Scholz [1961]. As the terminological discussion at the beginning of the latter reference makes clear, this subject has at different times been known by different names, “analytics” and “organon” and “dialectic”, while inversely the name “logic” has at different times been applied much more broadly and loosely than it will be here. At certain times and in certain places — perhaps especially in Germany from the days of Kant through the days of Hegel — the label has come to be used so very broadly and loosely as to threaten to take in nearly the whole of metaphysics and epistemology. Logic in our sense has often been distinguished from “logic” in other, sometimes unmanageably broad and loose, senses by adding the adjectives “formal” or “deductive”. The scope of the art and science of logic, once one gets beyond elementary logic of the kind covered in introductory textbooks, is indicated by two other standard references, the Handbooks of mathematical and philosophical logic, Barwise [1977] and Gabbay & Guenthner [1983-89], though the latter includes also parts that are identified as applications of logic rather than logic proper. The term “philosophical logic” as currently used, for instance, in the Journal of Philosophical Logic, is a near-synonym for “nonclassical logic”. There is an older use of the term as a near-synonym for “philosophy of language”.
    [Show full text]
  • 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.
    [Show full text]