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Logic, Arithmetic, and Definitions.
Logic, Arithmetic, and Definitions.

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자료유형  
 학위논문
Control Number  
0017164334
International Standard Book Number  
9798346387954
Dewey Decimal Classification Number  
510
Main Entry-Personal Name  
Mackereth, Stephen Gary.
Publication, Distribution, etc. (Imprint  
[S.l.] : University of Pittsburgh., 2024
Publication, Distribution, etc. (Imprint  
Ann Arbor : ProQuest Dissertations & Theses, 2024
Physical Description  
216 p.
General Note  
Source: Dissertations Abstracts International, Volume: 86-05, Section: B.
General Note  
Advisor: Walsh, Sean;Avigad, Jeremy;Ricketts, Thomas;Shaw, James;Gupta, Anil.
Dissertation Note  
Thesis (Ph.D.)--University of Pittsburgh, 2024.
Summary, Etc.  
요약Arithmetic and logic seem to enjoy an especially close relationship. Frege once wrote that arithmetic is reason's nearest kin. To deny any of the basic laws of arithmetic seems tantamount to denying a basic law of logic. My dissertation is concerned with two great attempts to make something more of this informal idea. In one direction, Frege tried to reduce arithmetic to nothing but quantificational logic and definitions. Neo-logicists continue to follow in Frege's footsteps, pursuing a version of this program today. In the other direction, Godel tried to reduce certain applications of quantificational logic to nothing but arithmetic and definitions, by means of his Dialectica translation.In the first half of my dissertation, I prove new theorems (with Jeremy Avigad) that shed a surprising light on the prospects for neo-logicism. An important objection against neo-logicism is that it makes use of allegedly stipulative definitions that are not conservativeover pure logic, i.e., definitions that settle open questions that we could not have settled before. This violates a basic requirement on stipulative definitions. I argue that by passing to a richer logical and definitional framework, it is possible to overcome the conservativeness objection. However, there is a subtlety: the strategy succeeds only if conservativeness is understood semantically rather than deductively. This suggests that the viability of neo-logicism is highly sensitive to the way in which epistemic commitments are represented in formal theories.In the second half of my dissertation, I argue that Godel's Dialectica translation succeeds in assigning a constructive meaning to quantificational theories of arithmetic. Virtually all commentators have objected that Godel's translation makes use of definitions which presuppose the very quantificational logic that Godel was trying to eliminate. This, of course, would render the translation philosophically circular. Godel was adamant that there was no circularity here, but no one has been able to understand his defense of this claim. I vindicate Godel, showing that there is no circularity and answering a longstanding exegetical question in Godel scholarship.
Subject Added Entry-Topical Term  
Mathematics.
Subject Added Entry-Topical Term  
Set theory.
Subject Added Entry-Topical Term  
Mathematicians.
Subject Added Entry-Topical Term  
Theorems.
Subject Added Entry-Topical Term  
20th century.
Subject Added Entry-Topical Term  
Philosophy.
Subject Added Entry-Topical Term  
Logic.
Subject Added Entry-Topical Term  
Theoretical mathematics.
Added Entry-Corporate Name  
University of Pittsburgh.
Host Item Entry  
Dissertations Abstracts International. 86-05B.
Electronic Location and Access  
로그인을 한후 보실 수 있는 자료입니다.
Control Number  
joongbu:657254

MARC

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■1001  ▼aMackereth,  Stephen  Gary.
■24510▼aLogic,  Arithmetic,  and  Definitions.
■260    ▼a[S.l.]▼bUniversity  of  Pittsburgh.  ▼c2024
■260  1▼aAnn  Arbor▼bProQuest  Dissertations  &  Theses▼c2024
■300    ▼a216  p.
■500    ▼aSource:  Dissertations  Abstracts  International,  Volume:  86-05,  Section:  B.
■500    ▼aAdvisor:  Walsh,  Sean;Avigad,  Jeremy;Ricketts,  Thomas;Shaw,  James;Gupta,  Anil.
■5021  ▼aThesis  (Ph.D.)--University  of  Pittsburgh,  2024.
■520    ▼aArithmetic  and  logic  seem  to  enjoy  an  especially  close  relationship.  Frege  once  wrote  that  arithmetic  is  reason's  nearest  kin.  To  deny  any  of  the  basic  laws  of  arithmetic  seems  tantamount  to  denying  a  basic  law  of  logic.  My  dissertation  is  concerned  with  two  great  attempts  to  make  something  more  of  this  informal  idea.  In  one  direction,  Frege  tried  to  reduce  arithmetic  to  nothing  but  quantificational  logic  and  definitions.  Neo-logicists  continue  to  follow  in  Frege's  footsteps,  pursuing  a  version  of  this  program  today.  In  the  other  direction,  Godel  tried  to  reduce  certain  applications  of  quantificational  logic  to  nothing  but  arithmetic  and  definitions,  by  means  of  his  Dialectica  translation.In  the  first  half  of  my  dissertation,  I  prove  new  theorems  (with  Jeremy  Avigad)  that  shed  a  surprising  light  on  the  prospects  for  neo-logicism.  An  important  objection  against  neo-logicism  is  that  it  makes  use  of  allegedly  stipulative  definitions  that  are  not  conservativeover  pure  logic,  i.e.,  definitions  that  settle  open  questions  that  we  could  not  have  settled  before.  This  violates  a  basic  requirement  on  stipulative  definitions.  I  argue  that  by  passing  to  a  richer  logical  and  definitional  framework,  it  is  possible  to  overcome  the  conservativeness  objection.  However,  there  is  a  subtlety:  the  strategy  succeeds  only  if  conservativeness  is  understood  semantically  rather  than  deductively.  This  suggests  that  the  viability  of  neo-logicism  is  highly  sensitive  to  the  way  in  which  epistemic  commitments  are  represented  in  formal  theories.In  the  second  half  of  my  dissertation,  I  argue  that  Godel's  Dialectica  translation  succeeds  in  assigning  a  constructive  meaning  to  quantificational  theories  of  arithmetic.  Virtually  all  commentators  have  objected  that  Godel's  translation  makes  use  of  definitions  which  presuppose  the  very  quantificational  logic  that  Godel  was  trying  to  eliminate.  This,  of  course,  would  render  the  translation  philosophically  circular.  Godel  was  adamant  that  there  was  no  circularity  here,  but  no  one  has  been  able  to  understand  his  defense  of  this  claim.  I  vindicate  Godel,  showing  that  there  is  no  circularity  and  answering  a  longstanding  exegetical  question  in  Godel  scholarship.
■590    ▼aSchool  code:  0178.
■650  4▼aMathematics.
■650  4▼aSet  theory.
■650  4▼aMathematicians.
■650  4▼aTheorems.
■650  4▼a20th  century.
■650  4▼aPhilosophy.
■650  4▼aLogic.
■650  4▼aTheoretical  mathematics.
■690    ▼a0422
■690    ▼a0405
■690    ▼a0395
■690    ▼a0642
■71020▼aUniversity  of  Pittsburgh.
■7730  ▼tDissertations  Abstracts  International▼g86-05B.
■790    ▼a0178
■791    ▼aPh.D.
■792    ▼a2024
■793    ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17164334▼nKERIS▼z이  자료의  원문은  한국교육학술정보원에서  제공합니다.

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