Introduction to the philosophy of mathematics:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Blackwell
1979
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| Series: | America Philosophical quarterly / Library of philosophy.
2. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 177 S. |
| ISBN: | 0631115803 |
Staff View
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| 245 | 1 | 0 | |a Introduction to the philosophy of mathematics |c Hugh Lehmann |
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Record in the Search Index
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| adam_text | TABLE OF CONTENTS PREFACE XI INTRODUCTLON: 1. THE AIM OF THIS WORK IS TO
DISEUSS ONTOLOGIEAL AND EPISTEMOLOGIEAL ISSUES. 2. IT WILL AVOID
ASSUMING EXTENSIVE KNOWLEDGE OF MATHEMATIES OR LOGIE. 3. CONEERNING THE
ONTOLOGIEAL AND EPISTEMOLOGICAL 2 ISSUES AND OF THE PRESENT APPROAEH IN
SUPPORTING THESE VIEWS. 4. STATEMENTS OF MATHEMATIEAL THEORIES EARRY
ONTOLOGI- 3 EAL IMPLIEATIONS, FOR EXAMPLE, THE AXIOMS OF REAL NUMBERS.
5. NUMBERS ARE UNOBSERVABLE, NEITHER PHYSIEAL NOR 5 MENTAL AND
UNIVERSALS, I.E., NUMBERS ARE QUEER ENTITIES. 6. CONSIDERATION OF THE
SUBSTITUTIONAL INTERPRETATION AS 6 A WAY OF AEEEPTING MATHEMATIEAL
TRUTHS WITHOUT QUEER ENTITIES. PART I: EXISTENCE IN MATHEMATICS CHAPTER
ONE: IF-THENISM 7. MATHEMATIEAL STATEMENTS ARE ALLEGED TO BE EON- 11
DITIONAL. THERE ARE TWO VERSIONS OF THIS VIEW. 8. BUT MATHEMATIEAL
STATEMENTS ARE NOT MATERIAL IMPLI- 11 EATIONS. 9. A SEEOND VERSION OF
IF-THENISM IS THE VIEW THAT 12 MATHEMATIEAL STATEMENTS ARE LOGICAL
IMPLIEATIONS. THIS VIEW IS OPEN TO TWO SORTS OF ERITIEISMS. WE MAY ASK
WHETHER AEEEPTANEE OF LOGICAL TRUTHS INVOLVES MAKING ONTOLOGIEAL
EOMMITMENTS. V VI PHILOSOPHY OF MATHEMATICS 10. WE MAYAISO ASK WHETHER
MATHEMATIEAL STATEMENTS 14 ARE LOGICAL IMPLIEATIONS. DEFINITIONS OF
REAL NUMBER MUST BE EONSIDERED. 11. SOMEONE MAY OBJEET TO MY CLAIM
BEEAUSE HE SUB- 16 SERIBES TO A MORE INCLUSIVE NOT ION OF LOGIEAL TRUTH.
BUT, I ASK, HOW DOES HE DISTINGUISH LOGIEAL FROM NON-LOGIEAL TRUTHS. IN
MY VIEW A LOGICAL TRUTH IS ONE WHIEH IS AN INSTANEE OF A FORMULA WHIEH
IS TRUE IN ALL POSSIBLE WORLDS . 12. ONE FURTHER WAY OF DEFINING REAL
NUMBER IS 17 EONSIDERED. 13. CONCLUSION OF EHAPTER ONE. 17 CHAPTER TWO:
POSTULATIONISM 14. THE DEFINITION OF REAL NUMBER EONSIDERED IN SEETION
19 12 SUGGESTS A VIEW OF MATHEMATIES HELD (AT ONE TIME) BY BERTRAND
RUSSELL AND ALSO BY HENRI POINEARE, NAMELY THE VIEW THAT MATHEMATIEAL
STATEMENTS ASSERT ONLY THAT THEOREMS ARE EONSEQUENEES OF EERTAIN
ASSUMPTIONS. 15. BUT, EVEN IF POSTULATIONISM IS TRUE WITH RESPEET TO 20
PURE MATHEMATIES, IN APPLIEATIONS OF MATHEMATIES EATEGORIEAL
MATHEMATIEAL ASSERTIONS ARE MADE. THUS, POSTULATIONISM DOES NOT ENABLE
US TO USE MATHEMATI- EAL KNOWLEDGE WHILE AVOIDING THE IMPLIEATION THAT
QUEER ENTITIES EXIST. 16. BUT POSTULATIONISM DOES NOT GIVE A EORREET
DESERIPTION 21 OF THE AETIVITY OF THE PURE MATHEMATIEIAN. EX- AMPLE
REGARDING EONVEXITY. 17. NOR DOES IT EORREETLY DESERIBE THE CLASS OF 25
MATHEMATIEAL TRUTHS, AS WAS POINTED OUT BY QUINE. 18. CONCLUSION OF
EHAPTER TWO. 26 CHAPTER THREE: MATHEMATICAL PRINCIPLES AS ANALYTIC 19.
SOME PHILOSOPHERS HAVE CLAIMED THAT MATHEMATIEAL 27 PROPOSITIONS ARE
ANALYTIE. IN PARTIEULAR THEY HAVE ASSERTED THAT MATHEMATIEAL TRUTHS ARE
TRUE BY DEFINITION AND LAEKING IN FAETUAL IMPORT. A PRELIMINARY
STATEMENT OF THESES VII 20. IS 3 + 2 = 5 TRUE BY VIRTUE OF DEFINITIONS
OF 3 , 2 , 28 + , 5 AND = ? WE ARE NOT COMPELLED TO SAY THAT IT
ISO BUT EVEN IF IT IS TRUE BY DEFINITION, THE THEORY PRESUPPOSED BY
THESE DEFINITIONS HAS ONTOLOGICAL IMPLICATIONS. 21. DEFENDERS OF THE
VIEW THAT MATHEMATICAL PROPOSITIONS 30 ARE ANALYTIC HAVE CLAIMED THAT
CONSIDERATIONS OF THE WAYS IN WHICH SUCH PROPOSITIONS ARE LEARNED ARE
IRRELEVANT TO EPISTEMOLOGY. BUT THIS POSITION IS WRONG. IT LEADS TO AN
ABSURD CONSEQUENCE AS CONSIDERATION OF THE NOTION LEARNING SHOWS. 22.DEFENDERS OF THE VIEW THAT MATHEMATICAL PROPOSITIONS 33 ARE ANALYTIC
HAVE CLAIMED THAT ARITHMETIC TRUTHS CANNOT BE REFUTED BY OBSERVED
COUNTER-INSTANCES. BUT THIS CLAIM IS APPARENTLY UNTENABLE. 23. SOME
PHILOSOPHERS, NOTABLY C.!. LEWIS AND E. NAGEL 35 HAVE MAINTAINED THAT
MATHEMATICAL AND LOGICAL PROPOSITIONS ARE PRESCRIPTIONS AND THEREFORE
HAVE NO ONTOLOGICAL IMPORT. WE ARGUE THAT WHILE SUCH PRO- POSITIONS HAVE
A PRESCRIPTIVE ROLE IT DOES NOT FOLLOW THAT THEY HAVE NO ONTOLOGICAL
IMPORT. 24. CONCLUSION OF CHAPTER THREE. 39 CHAPTER FOUR: CARNAP S
THEORIES 25. IN THIS SECTION WE TRY TO EXPLAIN SOME OF THE BASIC 40
IDEAS OF CARNAP S THEORIES. IN PARTICULAR WE EXPLAIN THE NOTIONS OF
SYNTACTICAL AND SEMANTICAL SYSTEMS. 26. AN INCOMPLETE EXAMPLE OF A
SYNTACTICAL AND SEMANTI- 42 CAL SYSTEM IS GIVEN. A COMPLETE SYSTEM OF
THE SORT GIVEN CONSTITUTES, ACCORDING TO CARNAP, THE RULES OF ORDINARY
LOGICAL REASONING. 27. HERE WE CONSIDER SOME TRUTHS OF MATHEMATICS, 45
NAMELY PEANO S POSTULATES. CARNAP PRESENTED A SYNTACTIC SYSTEM FOR THESE
POSTULATES AND ALSO GAVE SEMANTICAL RULES. HE ALSO GAVE DEFINITIONS SO
THAT STATEMENTS OF PEANO S POSTULATES COULD BE OBTAINED IN THE
SEMANTICAL SYSTEM OF ORDINARY LOGIC. VIII PHI LOS 0 P H Y 0 F M A T H E
M A T I C S 28. WHILE THE TRANSLATIONS OF PEANO S POSTULATES INTO THE 48
SEMANTICAL SYSTEM DEVELOPED BY CARNAP ARE L-TRUE, WE MAY ASK WHETHER
CARNAP HAS SHOWN THAT MATHEMATICAL PRINCIPLES MAKE NO ONTOLOGICAL COM-
MITMENTS. FOR ONE THING WE MAY ASK WHETHER CARNAP S TRANSLATIONS REALLY
EXPRESS PEANO S POS- TULATES. ALSO, THE AXIOMS OF THE SEMANTICAL SYSTEM
STATED BY CARNAP APPEAR TO HAVE ONTOLOGICAL IMPLI- CATIONS. FURTHER, THE
AXIOMS NEEDED TO COMPLETE THE SEMANTICAL SYSTEM, NAMELY THE AXIOMS
NEEDED FOR DERIVATION OF PRINCIPLES OF MATHEMATICS CONTAIN EXISTENTIAL
IMPLICATIONS AND ARE NOT L-TRUE. THIS IS SHOWN THROUGH CONSIDERATION OF
THE AXIOMS OF CHOICE AND INFINITY. THUS, EVEN THOUGH MATHEMATICAL TRUTHS
MAY BE TRANSLATABLE INTO STATEMENTS OF THIS SEMANTI- CAL SYSTEM, THAT
DOES NOT SHOW THAT THE MATHEMATICAL TRUTHS CONTAIN NO ONTOLOGICAL
IMPLICATIONS. 29. CONSIDERATION OF PREDICATIVE DEFINITIONS, PREDICATIVE
53 FUNCTIONS, THE VICIOUS CIRCLE PRINCIPLE AND THE AXIOM OF
REDUCIBILITY. 30. CARNAP HAS ARGUED THAT THERE IS A DISTINCTION 59
BETWEEN INTERNAL AND EXTERNAL QUESTIONS AND THAT EXTERNAL QUESTIONS
REGARDING EXISTENCE ARE MEAN- INGLESS. WE CRITICIZE THE DISTINCTION AND
ARGUE THAT HIS CONCLUSION THAT EXTERNAL QUESTIONS REGARDING EXISTENCE
ARE MEANINGLESS IS FALSE. PART 11: MATHEMATICAL KNOWLEDGE CHAPTER FIVE:
FICTIONALISM, PROOF, GOEOEL S VIEW OF MATHEMATICAL KNOWLEOGE 31.
CONSIDERATION OF THE VIEW OF HANS VAIHINGER THAT 66 MATHEMATICAL
CONCEPTS ARE FICTIONS. HIS THEORY THAT MATHEMATICAL STATEMENTS ARE ALL
FICTIONS OR SEMI- FICTIONS SEEMS MISTAKEN SINCE IT IS INCOMPATIBLE WITH
THE FACT THAT SOME PEOPLE HAVE MATHEMATICAL KNOW- LEDGE. ON OUR VIEW A
PRAGMATIC THEORY OF MATHEMATI- CAL KNOWLEDGE IS CORRECL 32.
CONSIDERATION OF THE NATURE OF MATHEMATICAL PROOF. 70 A PR E LI M I N A
R Y S TAT E M E N T 0 F T H ES ES IX 33. SOME MATHEMATIEAL PRINEIPLES
MUST BE KNOWN 77 WITHOUT PROOF, SINEE THERE IS MATHEMATIEAL KNOWLEDGE
THROUGH PROOFS AND THE NUMBER OF PREMISSES OF SUEH KNOWLEDGE IS FINITE.
34. CONSIDERATION OF THE VIEW OF KURT GOEDEL THAT 78 MATHEMATIEAL
KNOWLEDGE RESTS ON PRINEIPLES KNOWN VIA INTUITION OF MATHEMATIEAL
OBJEETS. OBJEETIONS TO THIS VIEW. 35. MATHEMATIEAL INTUITION AND THEEAUSAL THEORY OF 85 PEREEPTION. VIEWS OF MARK STEINER. 36. CONCLUSION OF
EHAPTER FIVE. 89 CHAPTER SIX: INTUITIONISM 37. EXPLANATION OF
INTUITIONIST VIEWS ON MATHEMATIEAL 91 EXISTENEE AND KNOWLEDGE. 38.
CONSIDERATION OF INTUITIONIST AND WITTGENSTEINIAN 97 OBJEETIONS TO THE
LAW OF EXELUDED MIDDLE. THEIR OBJEETIONS ARE NOT SOUND. 39.
CONSIDERATION OF INTUITION IST REAL NUMBER THEORY (OF 105 INTUITIONIST
THEORY OF THE EONTINUUM). THE INTUITION- ISTS EANNOT AEEOUNT FOR ALL OF
OUR MATHEMATIEAL KNOWLEDGE. 40. CRITIEISM OF THE INTUITIONIST VIEW THAT
MATHEMATIEAL 110 KNOWLEDGE RESTS ON SELF-EVIDENT PRINEIPLES AND OF THE
INTUITIONIST THEORY OF THE REFERENEE OF MATHEMATIEAL TERMS. 41.
INTUITION AND LEARNING OF MATHEMATIES. 114 42. CONCLUSION. 118 CHAPTER
SEVEN: MATHEMATICAL KNOWLEDGE AS EMPIRICAL: J.S. MILL AND D. HILBERT 43.
1.S. MILL S VIEW OF MATHEMATIEAL KNOWLEDGE. 120 44. CRITIEISMS OF MILL S
VIEW BY POSITIVISTS. 121 45. GOTTLOB FREGE S ERITICISMS OF MILL. 123 46.
HILBERT S THEORY OF MATHEMATIEAL KNOWLEDGE. 127 47. CRITIEISMS OF
HILBERT S THEORY. 130 48. DISEUSSION AND ERITIEISM OF HASKELL CURRY S
VERSION 132 OF FORMALISM. X PHILOSOPHY OF MATHEMATICS 49. CRITICAL
DISCUSSION OF THE FORMALIST THEORY OF 137 ABRAHAM ROBINSON. ROBINSON
AVOIDS THE OBJECTIONS DIRECTED AGAINST HILBERT S EPISTEMOLOGY. BUT HIS
VIEW IS ESSENTIALLY INCOMPLETE. 50. CONCLUSION. 141 CHAPTER EIGHT: AN
EMPIRICIST THEORY OF KNOWLEDGE 51. SKEPTICISM REVISITED. DISCUSSION OF
THE VIEW OF 144 STEPHAN KOERNER. 52. DISCUSSION OF THE SIGNIFICANCE OF
NON-EUCLIDEAN 147 GEOMETRIES AND OF ALTERNATIVE SET THEORIES WITH
RESPECT TO THE EXISTENCE OF MATHEMATICAL KNOWLEDGE. 53. BRIEF
EXPLANATION OF OUR THEORY OF MATHEMATICAL 149 KNOWLEDGE. MATHEMATICAL
PRINCIPLES ARE CONFIRMED VIA HYPOTHETICO-DEDUCTIVE INFERENCES. 54.
CONSIDERATION OF AN OBJECTION: IT IS ALLEGED 152 SOMETIMES THAT
MATHEMATICAL KNOWLEDGE IS CERTAIN AND SO CANNOT BE EMPIRICALLY
CONFIRMED. 55. COULD SCIENCE DISPENCE WITH REAL NUMBER THEORY AND 155
MAKE DO WITH RATIONAL NUMBER THEORY INSTEAD? 56. DOES CONFIRMATION OF
MATHEMATICAL PRINCIPLES SHOW 156 THAT WHILE WE ARE WARRANTED IN USING
SUCH PRINCIPLES IN NATURAL SCIENCE WE ARE NOT WARRANTED IN BELIEVING
THAT THEY ARE TRUE? 57. CONSIDERATION OF THE VIEW THAT KNOWLEDGE OF 159
MATHEMATICAL PRINCIPLES PRESUPPOSES THAT (I) NO IMPREDICATIVE
DEFINITIONS OCCUR IN SUCH PRINCIPLES, (2) THERE IS NO REFERENCE TO
INFINITE TOTALITIES IN SUCH PRINCIPLES, (3) THERE IS NO REFERENCE TO THE
EXISTENCE OF SETS IN SUCH PRINCIPLES OR (4) THAT SUCH PRINCIPLES BE
CONSTRUCTIVE . REFERENCES NAME INDEX SUBJECT INDEX 165 171 173
|
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| id | DE-604.BV002256341 |
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| indexdate | 2024-07-09T15:42:53Z |
| institution | BVB |
| isbn | 0631115803 |
| language | English |
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| physical | XI, 177 S. |
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| spelling | Lehman, Hugh 1936- Verfasser (DE-588)130156426 aut Introduction to the philosophy of mathematics Hugh Lehmann Oxford Blackwell 1979 XI, 177 S. txt rdacontent n rdamedia nc rdacarrier America Philosophical quarterly / Library of philosophy. 2. Filosofie van de wiskunde gtt Mathematik Philosophie Mathematics Philosophy Mathematik (DE-588)4037944-9 gnd rswk-swf Philosophie (DE-588)4045791-6 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Mathematik (DE-588)4037944-9 s Philosophie (DE-588)4045791-6 s DE-604 Library of philosophy. America Philosophical quarterly 2. (DE-604)BV001899526 2. V:DE-604 application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001482710&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lehman, Hugh 1936- Introduction to the philosophy of mathematics Filosofie van de wiskunde gtt Mathematik Philosophie Mathematics Philosophy Mathematik (DE-588)4037944-9 gnd Philosophie (DE-588)4045791-6 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4045791-6 (DE-588)4151278-9 |
| title | Introduction to the philosophy of mathematics |
| title_auth | Introduction to the philosophy of mathematics |
| title_exact_search | Introduction to the philosophy of mathematics |
| title_full | Introduction to the philosophy of mathematics Hugh Lehmann |
| title_fullStr | Introduction to the philosophy of mathematics Hugh Lehmann |
| title_full_unstemmed | Introduction to the philosophy of mathematics Hugh Lehmann |
| title_short | Introduction to the philosophy of mathematics |
| title_sort | introduction to the philosophy of mathematics |
| topic | Filosofie van de wiskunde gtt Mathematik Philosophie Mathematics Philosophy Mathematik (DE-588)4037944-9 gnd Philosophie (DE-588)4045791-6 gnd |
| topic_facet | Filosofie van de wiskunde Mathematik Philosophie Mathematics Philosophy Einführung |
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