Logos and Mathema: Studies in the Philosophy of Mathematics and History of Logic

340 Pages·2011·2.113 MB·English

by  Roman Murawski

1
The volume contains twenty essays devoted to the philosophy of mathemat- Roman Murawski
ics and the history of logic. They have been divided into four parts: general
philosophical problems of mathematics, Hilbert’s program vs. the incomplete-
ness phenomenon, philosophy of mathematics in Poland, mathematical logic
Logos and Máthēma
in Poland. Among considered problems are: epistemology of mathematics,
the meaning of the axiomatic method, existence of mathematical objects,
distinction between proof and truth, undefinability of truth, Gödel’s theorems Studies in the Philosophy of Mathematics
and computer science, philosophy of mathematics in Polish mathematical and
and History of Logic
logical schools, beginnings of mathematical logic in Poland, contribution of
Polish logicians to recursion theory.
a
m
ē
h
t Polish Contemporary Philosophy and
á
M
d Philosophical Humanities
n
a
s
o
g Edited by Jan Hartman
o
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i ·
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Roman Murawski, born in 1949; studied mathematics and theology; MSc 1972, ur
M
PhD 1979, Habilitation 1992, Full professor 2001; Professor of logic and philoso-
n
phy of mathematics at Adam Mickiewicz University in Poznań (Poland); 2006–
a
m
2009 President of Polish Association for Logic and Philosophy of Science.
o
R
www.peterlang.de ISBN 978-3-631- 61804-2 PETER LANG
Internationaler Verlag der Wissenschaften
CPPPH 01-Murawski 261804-A5HCk-SH.indd 1 09.06.11 12:03:48 Uhr
1
The volume contains twenty essays devoted to the philosophy of mathemat- Roman Murawski
ics and the history of logic. They have been divided into four parts: general
philosophical problems of mathematics, Hilbert’s program vs. the incomplete-
ness phenomenon, philosophy of mathematics in Poland, mathematical logic
Logos and Máthēma
in Poland. Among considered problems are: epistemology of mathematics,
the meaning of the axiomatic method, existence of mathematical objects,
distinction between proof and truth, undefinability of truth, Gödel’s theorems Studies in the Philosophy of Mathematics
and computer science, philosophy of mathematics in Polish mathematical and
and History of Logic
logical schools, beginnings of mathematical logic in Poland, contribution of
Polish logicians to recursion theory.
a
m
ē
h
t Polish Contemporary Philosophy and
á
M
d Philosophical Humanities
n
a
s
o
g Edited by Jan Hartman
o
L
i ·
k Volume 1
s
w
a
Roman Murawski, born in 1949; studied mathematics and theology; MSc 1972, ur
M
PhD 1979, Habilitation 1992, Full professor 2001; Professor of logic and philoso-
n
phy of mathematics at Adam Mickiewicz University in Poznań (Poland); 2006–
a
m
2009 President of Polish Association for Logic and Philosophy of Science.
o
R
www.peterlang.de PETER LANG
Internationaler Verlag der Wissenschaften
CPPPH 01-Murawski 261804-A5HCk-SH.indd 1 09.06.11 12:03:48 Uhr
Logos and Máthe¯ma
Polish Contemporary Philosophy
and Philosophical Humanities
Edited by Jan Hartman
Volume 1
PETER LANG
Frankfurt am Main · Berlin · Bern · Bruxelles · New York · Oxford · Warszawa · Wien
Roman Murawski
Logos and Máthēma
Studies in the Philosophy of Mathematics
and History of Logic
PETER LANG
Internationaler Verlag der Wissenschaften
Bibliographic Information published by the Deutsche
Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the
Deutsche Nationalbibliografie; detailed bibliographic data is
available in the internet at http://dnb.d-nb.de.
Cover Design:
© Olaf Gloeckler, Atelier Platen, Friedberg
The publication was financially supported
by the Faculty of Mathematics and Computer Science
of Adam Mickiewicz University in Poznan´.

ISBN 978­3­653­01030­5 (eBook)
ISSN 2191-1878
ISBN 978-3-631-61804-2
© Peter Lang GmbH
Internationaler Verlag der Wissenschaften
Frankfurt am Main 2011
All rights reserved.
All parts of this publication are protected by copyright. Any
utilisation outside the strict limits of the copyright law, without
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www.peterlang.de
Dedicated to
my wife Hania and daughter Zosia

Foreword
Thevolumecontains20essaysdevotedtothephilosophyofmathematicsand
the history of logic. They have been divided into four parts. Part 1 contains
papersconsideringgeneralphilosophicalproblemsofmathematics.Inthees-
say“MathematicalKnowledge”basicepistemologicalproblemsofmathematics
areconsidered.Maindoctrinesintheepistemologyofmathematicshavebeen
presentedandanalyzed.Theinterrelationsbetweenlogicandphilosophyofma-
thematicsaswellassomecurrenttendenciesinthephilosophyofmathematics
havebeenstudied.Theessay“OnthePowerandWeaknessesoftheAxiomatic
Method”discussesthemeaningoftheaxiomaticmethodforthemethodologyof
mathematics.In“RemarksontheMathematicalUniverse”theproblemoftheexi-
stenceandthenatureofmathematicalentitiesisconsidered.Variousconceptions
thatappearedinthehistoryarepresentedandexamplesofmathematiciansand
logiciansdeclaringforthemareindicated.Consequencesofthoseconceptions
fordoingmathematicsareconsidered.Inthelastessayinthispart“Structuralism
and Category Theory in the Contemporary Philosophy of Mathematics” set-
theoretical(Bourbaki-style)andcategory-theoreticalapproachestostructuralism
inthephilosophyofmathematicsarecompared.Advantagesanddisadvantages
ofthemareindicated.
Part2isdevotedtoproblemsconcerningHilbert’sprogramandtheinfluence
onitofthediscoveryoftheincompletenessphenomenon.Intheessay“Hilbert’s
Program:IncompletenessTheoremsvs.PartialRealizations”thequestionwhe-
therGo¨dels’incompletenesstheoremsdidrejectHilbert’sprogramisstudied.
GeneralizationsandstrengtheningsofGo¨del’sresultsaswellasgeneralizedand
relativizedHilbert’sprogramsandtheirmeaningforthephilosophyofmathema-
ticsareconsidered.Theessay“OntheDistinctionProof–TruthinMathematics”
contains some historical, philosophical and logical considerations connected
withthedistinctionbetweenproofandtruthinmathematics.Thecrucialroˆle
of Go¨del’s incompleteness theorems as well as of the undefinability of truth
vs.definabilityofprovabilityandtheroˆleoffinitaryvs.infinitarymethodsare
stressed. The problem of the necessity of extending the available methods by
new rules of inference and new axioms is also considered. The discovery of
the incompleteness phenomenon destroyed the old conviction that axiomatic
methodistheidealmethodformathematics.Thereforeitwasnotimmediately
acceptedbylogicians.Reactionstothisdiscoveryareconsideredinthenextpaper.
Theessay“Go¨del’sIncompletenessTheoremsandComputerScience”indicates
someapplicationsofGo¨del’sresultstothediscussionofproblemsofcomputer
science.Inparticulartheproblemofrelationsbetweenthemindandmachine
(argumentsbyJ.J.C.SmartandJ.R.Lucas),Go¨del’sopiniononthisissueand
8 LogosandMa´the¯ma
someinterpretationsoftheincompletenesstheoremsfromthepointofviewof
theinformationtheoryarepresented.Thoughitseemsthathumanmindisnot
fullyequivalenttoamachine(computer),neverthelesssomeofitsfunctionscan
bemechanized.Nextessay“ThePresentStateofMechanizedDeduction,andthe
PresentKnowledgeofItsLimitations”tellsaboutattemptstodevelopprocedures
ofmechanizeddeduction.Itindicatesalsovariouslimitationsofthem.
OneoftheaimsofHilbert’sprogramwastoshowthattheclassicalmathe-
matics(referringtotheactualinfinity)issafeandfreeofanyinconsistencies.
Hencetheattemptstoprovetheconsistencyofbasicmathematicaltheoriesby
finitisticmethodsundertakeninHilbert’sschool(Ackermann,vonNeumann).
Go¨del’sresultsshowthatnewnon-finitisticmethodsmustbeappliedhere.Those
problemsareconsideredinthepaper“OnProofsoftheConsistencyofArithme-
tic”.Thepaper“Decidabilityvs.Undecidability.Logico-Philosophico-Historical
Remarks”presentsthedecidabilityproblemfromaphilosophicalandhistorical
perspective.Itindicatesalsobasicmathematicalandlogicalresultsconcerning
(un)decidabilityofparticulartheoriesandproblems.Inthepaper“Undefinability
of Truth. The Problem of Priority: Tarski vs. Go¨del” it is argued that Tarski
obtainedthetheoremontheundefinabilityindependentlyfromGo¨delthough
hemadeclearhisindebtednesstoGo¨del’smethods.OntheotherhandGo¨del
wasawareoftheformalundefinabilityoftruthin1931buthedidnotpublish
this result–reasons for that are considered. The problem of definability and
undefinability ofthe conceptsof satisfaction andtruthfrom amore technical
pointofviewisconsideredintheessay“TroublesWith(theConceptof)Truth
inMathematics”closingPart2.
Parts3and4aredevotedtotheworkofPolishlogiciansandmathematicians
inthephilosophyofmathematicsandinlogic.Part3beginsbyanessaypresenting
thephilosophicalsystemofdeepandinterestingbutunfortunatelyratherforgotten
andunderestimatedPolishphilosopherandmathematicianJózefMariaHoene-
Wroński.Intheessays“PhilosophicalReflectiononMathematicsinPolandinthe
InterwarPeriod”and“PhilosophyofMathematicsintheWarsawMathematical
School” the views and tendencies of the most outstanding representatives of
Lvov-WarsawPhilosophicalSchoolandofthefoundersofPolishMathematical
Schoolarepresentedandanalyzed.Theproblemwhetherthoseviewshadany
influenceonlogicalandmathematicalresearchisconsidered.Philosophicalviews
concerningmathematicsofAndrzejMostowski,anoutstandingrepresentativeof
thesecondgenerationoftheLvov-WarsawSchool,arepresentedanddiscussed
inthenextessay.
ThelastPart4containsthreeessaysdevotedtoPolishmathematicallogic.
Thefirstone“StanisławPiątkiewiczandtheBeginningsofMathematicalLogic
inPoland”presentsinformationonthelifeandworkofStanisławPiątkiewicz.
HisAlgebrawlogice(1888)containedanexpositionofthealgebraoflogicand
itsuseinrepresentingsyllogisms.ThiswasthefirstoriginalPolishpublication