Table Of ContentFields Institute Communications 82
The Fields Institute for Research in Mathematical Sciences
Nicolas Fillion
Robert M. Corless
Ilias S. Kotsireas
Editors
Algorithms and
Complexity in
Mathematics,
Epistemology,
and Science
Proceedings of 2015 and 2016
ACMES Conferences
Fields Institute Communications
Volume 82
FieldsInstituteEditorialBoard:
IanHambleton,DirectoroftheInstitute
HuaxiongHuang,DeputyDirectoroftheInstitute
JamesG.Arthur,UniversityofToronto
KennethR.Davidson,UniversityofWaterloo
LisaJeffrey,UniversityofToronto
BarbaraLeeKeyfitz,OhioStateUniversity
ThomasS.Salisbury,YorkUniversity
JurisSteprans,YorkUniversity
NorikoYui,Queen’sUniversity
The Communications series features conference proceedings, surveys, and lecture
notes generated from the activities at the Fields Institute for Research in the
MathematicalSciences.Thepublicationsevolvefromeachyear’smainprogramand
conferences.Manyvolumesareinterdisciplinaryinnature,coveringapplicationsof
mathematicsinscience,engineering,medicine,industry,andfinance.
Moreinformationaboutthisseriesathttp://www.springer.com/series/10503
Nicolas Fillion • Robert M. Corless
Ilias S. Kotsireas
Editors
Algorithms and Complexity
in Mathematics,
Epistemology, and Science
Proceedings of 2015 and 2016 ACMES
Conferences
123
Editors
NicolasFillion RobertM.Corless
DepartmentofPhilosophy TheRotmanInstituteofPhilosophy,
SimonFraserUniversityPhilosophy TheOntarioResearchCenterforComputer
Burnaby,BC,Canada AlgebraandTheSchoolofMathematical
andStatisticalSciences
TheUniversityofWesternOntario
London,ON,Canada
IliasS.Kotsireas
DepartmentofPhysics&ComputerScience
WilfridLaurierUniversity
Waterloo,ON,Canada
ISSN1069-5265 ISSN2194-1564 (electronic)
FieldsInstituteCommunications
ISBN978-1-4939-9050-4 ISBN978-1-4939-9051-1 (eBook)
https://doi.org/10.1007/978-1-4939-9051-1
LibraryofCongressControlNumber:2019930430
MathematicsSubjectClassification(2010):00
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partofSpringerNature.
Theregisteredcompanyaddressis:233SpringStreet,NewYork,NY10013,U.S.A.
ThisproceedingsisdedicatedtothememoriesofJonBorweinandAnnJohnson,distinguished
participantsoftheACMESmeetings.Theircontributionsshallnotbeforgotten.
Preface
Mathematicians, scientists, and philosophers have historically been involved in a
multidisciplinaryendeavorofseekingtounderstandandrepresentaspectsofreality
withtheoriesandmodels.Thispracticeisnowadaysmorefruitfulthanatanyother
point before, thanks to an expanding understanding of the methods that allow us
to extract information from the theories and models we advance. Indeed, as Alan
Turing remarked, “the assumption that as soon as a fact is presented to a mind
all consequences of that fact spring into the mind simultaneously with it [...] is
a very useful assumption under many circumstances, but one too easily forgets
thatitisfalse.”Inotherwords,understandingtheimplicationsofscientificmodels
and theories typically requires ingenious and skillful computation. Indeed, from a
computational point of view, to have theoretical knowledge worthy of the name,
we typically need to have an efficient computational algorithm to answer a range
of questions, within specified constraints on time and computational resources. In
some cases, the required computations are straightforward, and the theories are,
in a sense, inferentially transparent, but this is the exception rather than the rule.
Moreover, in order to assess the validity of results obtained thusly, one needs to
interpret the computational task as a component of a broader modeling practice,
whichincludesotherimportantthemesinappliedmathematics,suchasrobustness
underperturbation,inferencefromdata,anduncertaintyquantification.
Thisvolumebringstogetherpapersthateachcontributetothisbroadtaskinone
way or another fromdisciplinary perspectives that include mathematics, computer
science,philosophyofscience,andhistoryofscience.Thevolumeisbasedonthe
contributionsmadeatthetwoACMESconferencesorganizedatWesternUniversity
in 2015 and 2016, in which about 150 academics from multiple countries partici-
pated. ACMES (Algorithms and Complexity in Mathematics, Epistemology, and
Science) is a multidisciplinary conference that covers a combination of numerical
analysisanditsunderlyingphilosophy,computeralgebra,reliabilityanduncertainty
quantification, computation and complexity theory, error analysis, perturbation
theory, experimental mathematics, scientific epistemology, machine learning, and
foundations of mathematics, with the aim of furthering our understanding of the
multifacetedroleofmathematicsinmodernscience.Bybringingcontributionsfrom
vii
viii Preface
researcherswhoapproachthemathematicalsciencesfromdifferentperspectives,the
volumeaimstodosoinawaythatisinformedbythestateoftheartinmathematics,
scientificcomputing,andcurrentmodelingtechnique,withthehopethatitleadsto
a self-reflective outlook on modern applied mathematics that draws from theory
and practice and situates it in its proper philosophical, sociological, and historical
context. In line with its multidisciplinary commitment, the conferences have been
co-fundedprimarilybytheFieldsInstituteforResearchinMathematicalSciences
andbytheRotmanInstituteofPhilosophy.
Thecollectionfeaturestheworkofdistinguishedscientistsandphilosophers.Its
mostimportantfeaturesarethedepthofindividualworksandbreadthoftopicsin
computationalmathematicsanditsunderlyingphilosophy.Thedepthoftheresearch
contributions included in this paper goes hand in hand with their accessibility, as
authors were asked to write for a multidisciplinary audience. Several papers are,
in part, presented as tutorials for readers in other areas. Even though research
productivity requires academics to focus on more or less narrow research areas,
setting the core methods and results of a subdiscipline in a broader context also
benefitsresearchersacrossdisciplines.
Burnaby,BC,Canada NicolasFillion
London,ON,Canada RobertM.Corless
Waterloo,ON,Canada IliasS.Kotsireas
Contents
EthicsandtheContinuumHypothesis......................................... 1
JamesRobertBrown
HowtoGenerateAllPossibleRationalWilf-ZeilbergerPairs?............. 17
ShaoshiChen
BackwardErrorAnalysisforPerturbationMethods........................ 35
RobertM.CorlessandNicolasFillion
ProofVerificationTechnologyandElementaryPhysics ..................... 81
ErnestDavis
AnApplied/ComputationalMathematician’sViewofUncertainty
QuantificationforComplexSystems........................................... 133
MaxGunzburger
DynamicalSymmetriesandModelValidation................................ 153
BenjaminC.Jantzen
ModelingtheBiasesinLastDigitDistributionsofConsecutivePrimes ... 177
DanielLichtblau
ComputationalAspectsofHamburger’sTheorem........................... 195
YuriMatiyasevich
EffectiveValidity:AGeneralizedLogicforStableApproximate
Inference .......................................................................... 225
RobertH.C.Moir
CounterfactualsintheRealWorld............................................. 269
JamesWoodwardandMarkWilson
ix
Contributors
JamesRobertBrown DepartmentofPhilosophy,UniversityofToronto,Toronto,
ON,Canada
Shaoshi Chen KLMM, Academy of Mathematics and Systems Science, Chinese
AcademyofSciences,Beijing,China
School of Mathematical Sciences, University of Chinese Academy of Sciences,
Beijing,China
Robert M. Corless The Rotman Institute of Philosophy, The Ontario Research
Center for Computer Algebra and The School of Mathematical and Statistical
Sciences,TheUniversityofWesternOntario,London,ON,Canada
ErnestDavis DepartmentofComputerScience,NewYorkUniversity,NewYork,
NY,USA
Nicolas Fillion Department of Philosophy, Simon Fraser University Philosophy,
Burnaby,BC,Canada
Max Gunzburger Department of Scientific Computing, Florida State University,
Tallahassee,FL,USA
BenjaminC.Jantzen DepartmentofPhilosophy,VirginiaTech,Blacksburg,VA,
USA
DanielLichtblau WolframResearch,Champaign,IL,USA
YuriMatiyasevich St.PetersburgDepartmentofV.A.SteklovMathematicalInsti-
tute,SaintPetersburg,Russia
Robert H.C. Moir Department of Computer Science, The University of Western
Ontario,London,ON,Canada
MarkWilson UniversityofPittsburgh,Pittsburgh,PA,USA
JamesWoodward UniversityofPittsburgh,Pittsburgh,PA,USA
xi
