Mathematics, Computer Science and Logic - A Never Ending Story: The Bruno Buchberger Festschrift

120 Pages·2013·4.586 MB·English

by  Henk Barendregt (auth.), Peter Paule (eds.)

Peter Paule Editor
Mathematics,
Computer Science
and Logic - A Never
Ending Story
The Bruno Buchberger Festschrift
Mathematics, Computer Science and Logic -
A Never Ending Story

Peter Paule
Editor
Mathematics, Computer
Science and Logic -
A Never Ending Story
The Bruno Buchberger Festschrift
123
Editor
PeterPaule
ResearchInstituteforSymbolic
Computation
JohannesKeplerUniversity
Linz,Austria
ISBN978-3-319-00965-0 ISBN978-3-319-00966-7(eBook)
DOI10.1007/978-3-319-00966-7
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013949176
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Preface
BrunoBuchbergerpassedthemilestoneofhis60thbirthdayonOctober22,2002.
All the contributors to this book helped to celebrate this event, by presenting
invited talks at the birthday conference “Logic, Mathematics and Computer Sci-
ence-LMCS2002”presentedinProfessorBuchberger’srenovatedmedievalcastle
at RISC in Hagenberg, Austria. Because of the superb spirit and the success of
this symposium, the idea was launched to make these talks available to a larger
audience.Aftermorethana decade,the planhasfinallycometrue,in the formof
thiscollectionofmathematicalessays.Twoofthemarealmostunchangedversions
of the LMCS2002 talks: Stephen Wolfram’s “New Directions in the Foundations
of Mathematics” and Doron Zeilberger’s “Towards a Symbolic Computational
Philosophy (and Methodology!) for Mathematics”. The essay “On the Role of
LogicandAlgebrain SoftwareEngineering”byManfredBroyisa slightlyedited
version of his LMCS2002 talk. Henk Barendregt significantly expanded his talk
on “Foundations of Mathematics from the Perspective of Computer Verifcation”.
In their mathematicalessence all these contributionsare still fully up-to-date,and
theyrekindletheinspiringatmosphereoftheBuchbergerSymposium.
I wantto take the opportunityto thankRalf Hemmeckefor editorialassistance
and,lastbutnotleast,MartinPetersandRuthAlleweltfromSpringerfortheirhelp
andalmostinfinitepatience.
Hagenberg,Austria PeterPaule
May2013
v

Contents
FoundationsofMathematicsfromthePerspectiveofComputer
Verification........................................................................ 1
HenkBarendregt
OntheRoleofLogicandAlgebrainSoftwareEngineering ................ 51
ManfredBroy
NewDirectionsintheFoundationsofMathematics(2002).................. 69
StephenWolfram
Towards a Symbolic Computational Philosophy
(andMethodology!)forMathematics.......................................... 101
DoronZeilberger
vii
Foundations of Mathematics
from the Perspective of Computer Verification
HenkBarendregt
ToBrunoBuchbergerindependentlyofanybirthday
Abstract In the philosophy of mathematics one speaks about Formalism,
Logicism, Platonism and Intuitionism. Actually one should add also Calculism.
Thesefoundationalviewscanbegivenacleartechnologicalmeaninginthecontext
of Computer Mathematics, that has as aim to represent and manipulate arbitrary
mathematical notions on a computer. We argue that most philosophical views
over-emphasizeaparticularaspectofthemathematicalendeavor.
1 Mathematics
The ongoing creation of mathematics, that started 5 or 6 millennia ago and is
still continuing at present, may be described as follows. By looking around and
abstractingfromthenatureofobjectsandthesizeofshapeshomosapienscreated
thesubjectsofarithmeticandgeometry.Highermathematicslateraroseasatower
of theories above these two, in order to solve questions at the basis. It turned out
thatthese moreadvancedtheoriesoften areable to modelpartof realityand have
applications. By virtue of the quantitative, and even more qualitative, expressive
forceofmathematics,everyscienceneedsthisdiscipline.Thisisthecaseinorder
toformulatestatements,butalsotocorrectconclusions(Fig.1).
H.Barendregt((cid:2))
NijmegenUniversity,Nijmegen,TheNetherlands
e-mail:[email protected]
P.Paule(ed.),Mathematics,ComputerScienceandLogic-ANeverEndingStory, 1
DOI10.1007/978-3-319-00966-7 1,©SpringerInternationalPublishingSwitzerland2013
2 H.Barendregt
Fig.1 Thetriangleof (cid:129)
mathematicalactivities
Calculating Reasoning
Computability Logic
Math
(cid:129) (cid:129)
Defining
Ontology
The mathematical endeavor consists in a stylized way of three activities1:
defining, calculating and proving.2 The three started in this order, but over the
centuries they became more and more intertwined. Indeed, before one can do
arithmetic,onehastohavenumbersandananalogousstatementholdsforgeometry.
Havingnumbers onewantstoaddandmultiplythese;havingpolygons onewants
to calculate their area. At some point the calculations became complex and one
discoveredshortcuts.Oneroleofproofsisthattheyareanessentialtooltoestablish
thecorrectnessofcalculationsandconstructions.
1.1 Egyptian-Chinese-Babylonianvs Greek Mathematics
Differentappreciationsof thethree sidesofthe triangleofmathematicalactivities
gaverisetovariousexplicitfoundationalviews.Beforeenteringintothesewewill
argue that different implicit emphases on the sides of the triangle also did lead
to different forms of mathematics. In the Egyptian-Chinese-Babylonian tradition
emphasis was put on calculation. One could solve e.g. linear and quadratic
equations. This was done in a correct way, but a developed notion of proof was
lacking. In the Greek tradition the emphasis was on proofs. Using these one can
p
show that there are infinitely many primes, or that 2 is irrational, something
impossibletodobymerecomputationalone.Buttherigorcomingfromgeometric
proofsalsohaditslimitations.Euclid3[51]givesageometricproofthat.xCy/2 D
x2C2xyCy2,butnosimilarresultsfor.xCy/3(althoughsucharesultcouldhave
beenprovedgeometrically)or.xCy/4,letalone.xCy/n.
1IlearnedthisfromGillesBarthe(1996,personalcommunication).
2Theactivityofsolvingcanbeseenasaparticularinstanceofcomputing(orofproving,namely
thatofanexistentialstatement9x:P.x/inaconstructivesetting).
3App.325–265BC.