Why prove it again? : alternative proofs in mathematical practice

209 Pages·2015·2.22 MB·English

by  Dawson

John W. Dawson, Jr.
Why Prove it
Again?
Alternative Proofs in Mathematical
Practice

John W. Dawson, Jr.
Why Prove it Again?
Alternative Proofs in Mathematical Practice
with the assistance of Bruce S. Babcock
and with a chapter by Steven H. Weintraub
JohnW.Dawson,Jr.
PennStateYork
York,PA,USA
ISBN978-3-319-17367-2 ISBN978-3-319-17368-9 (eBook)
DOI10.1007/978-3-319-17368-9
LibraryofCongressControlNumber:2015936605
MathematicsSubjectClassification(2010):00A35,00A30,01A05,03A05,03F99
SpringerChamHeidelbergNewYorkDordrechtLondon
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ToSolomonFeferman,
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whosuggestedIwritethisbook

Preface
ThisbookisanelaborationofthemesthatIpreviouslyexploredinmypaper“Why
do mathematicians re-prove theorems?” (Dawson 2006). It addresses two basic
questionsconcerningmathematicalpractice:
1. What rationales are there for presenting new proofs of previously established
mathematicalresults?
and
2. Howdomathematiciansjudgewhethertwoproofsofagivenresultareessentially
different?
Thediscoveryandpresentationofnewproofsofresultsalreadyprovenbyother
meanshasbeenasalientfeatureofmathematicalpracticesinceancienttimes.1 Yet
historiansandphilosophersofmathematicshavepaidsurprisinglylittleattentionto
that phenomenon, and mathematical logicians have so far made little progress in
developing formal criteria for distinguishing different proofs from one another, or
forrecognizingwhenproofsaresubstantiallythesame.
A number of books and papers have compared alternative proofs of particular
theorems(seethereferencesinsucceedingchapters),butnoextendedgeneralstudy
of the roles of alternative proofs in mathematical practice seems hitherto to have
beenundertaken.
Considerationofparticularcasestudiesis,ofcourse,anecessaryprerequisitefor
formulatingmoregeneralconclusions,andthatcoursewillbefollowedhereaswell.
Theaimisnot,however,toarriveatanyformalframeworkforanalyzingdifferences
amongproofs.Itisrather
a) tosuggestsomepragmaticcriteriafordistinguishingamongproofs,and
b) to enumerate reasons why new proofs of previously established results have so
longplayedaprominentandesteemedroleinmathematicalpractice.
1WilburKnorr,e.g.,notedthat“multipleproofswerefrequentlycharacteristicofpre-Euclidean
studies”(Knorr1975,p.9).
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viii Preface
Chapter1addressesthefirstofthoseaims,followingclarificationofsomeperti-
nentlogicalissues.Chapter2thenoutlinesvariouspurposesthatalternativeproofs
may serve. The remaining chapters provide detailed case studies of alternative
proofsofparticulartheorems.Thedifferentproofsconsideredthereinbothillustrate
themotivesforgivingalternativeproofsandserveasbenchmarksforevaluatingthe
worthofthepragmaticcriteriaintermsofwhichtheyareanalyzed.
York,PA,USA JohnW.Dawson,Jr.
References
Dawson, J.: Why do mathematicians re-prove theorems? Philosophia Mathematica (III) 14,
pp.269–286(2006)
Knorr,W.:TheEvolutionoftheEuclideanElements.AStudyoftheTheoryofIncommensurable
MagnitudesanditsSignificanceforEarlyGreekGeometry.Reidel,Dordrecht(1975)
Acknowledgments
I am indebted to Solomon Feferman, Akihiro Kanamori, and two anonymous
reviewersforsuggestingimprovementstoearlierversionsofchapters1–8.Ithank
the participants in the Philadelphia Area Seminar on the History of Mathematics,
as well as Andrew Arana and Jeremy Avigad, for their enthusiasm for and
encouragement of this endeavor. Above all, I am grateful to Bruce Babcock, for
preparingtheillustrationsthroughoutthisbookandforhiscarefulcopy-editing,and
toStevenH.Weintraub,forseveralenhancementstochapter6andforcontributing
chapter 11 on proofs of the irreducibility of the cyclotomic polynomials. Finally,
I thank Anthony Charles and the rest of the Birkhäuser production staff for their
cordialandefficienteffortstotransformmymanuscriptintoprint.
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