Logic, Argumentation & Reasoning 15 Jens Erik Fenstad Structures and Algorithms Mathematics and the Nature of Knowledge Logic, Argumentation & Reasoning Interdisciplinary Perspectives from the Humanities and Social Sciences Volume 15 Series editor Shahid Rahman Logic,Argumentation&ReasoningexploresthelinksbetweenHumanitiesandthe Social Sciences, with theories including, decision and action theory as well as cognitivesciences,economy,sociology,law,logic,andphilosophyofsciences.It’s twomainambitionsaretodevelopatheoreticalframeworkthatwillencourageand enable interaction between disciplines as well as to federate the Humanities and Social Sciences around their main contributions to public life: using informed debate, lucid decision-making and action based on reflection. The series welcomes research from the analytic and continental traditions, putting emphasis on four main focus areas: (cid:129) Argumentation models and studies (cid:129) Communication, language and techniques of argumentation (cid:129) Reception of arguments, persuasion and the impact of power (cid:129) Diachronic transformations of argumentative practices TheSeriesisdevelopedinpartnershipwiththeMaisonEuropéennedesSciences del’HommeetdelaSociété(MESHS)atNord-PasdeCalaisandtheUMR-STL: 8163 (CNRS). Proposals should include: (cid:129) A short synopsis of the work or the introduction chapter (cid:129) The proposed Table of Contents (cid:129) The CV of the lead author(s) (cid:129) If available: one sample chapter We aim to make a first decision within 1 month of submission. In case of a positive first decision the work will be provisionally contracted: the final decision aboutpublicationwilldependupontheresultoftheanonymouspeerreviewofthe complete manuscript. 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More information about this series at http://www.springer.com/series/11547 Jens Erik Fenstad Structures and Algorithms Mathematics and the Nature of Knowledge 123 Jens ErikFenstad Institute of Mathematics University of Oslo Oslo Norway ISSN 2214-9120 ISSN 2214-9139 (electronic) Logic, Argumentation & Reasoning ISBN978-3-319-72973-2 ISBN978-3-319-72974-9 (eBook) https://doi.org/10.1007/978-3-319-72974-9 LibraryofCongressControlNumber:2018932528 ©SpringerInternationalPublishingAG,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface This book is a collection of essays on what there is and how we know. We are interestedinunderstandingthegeneralnatureofhumanknowledge.Structuresand algorithms will be essential tools for this understanding: what we see and how we can make use of what we see. Mathematics will be the link between the two. But mathematics is not exactly what common man believes; this is the theme ofthefirstessay.Whenreadintheproperway,thisessayisanattempttoextendthe current art and science of mathematical modeling to all fields of knowledge. The introductory text will be followed by several essays on the “mathematical way”, withexamplesfromart,history,languageaswellasthenaturalandsocialsciences. Inthelastessay,wereturntothequestionofwhatthereis.Weshallarguefora culturalfoundationofknowledge.Cultureisgroundedinmanandnature,butexists outside of space and time. This means that numbers, language, and mind are abstractobjectsinthe“collectivemindofthespecies,”tousethelanguageofsome anthropologists. Structures are what we see. And being abstract cultural objects, theycanalmostalwaysbeseeninmanydifferentways.Butcertainstructures,such asthenaturalnumbersandthebasicnotionsofgrammar,seemtohaveanabsolute character. Any theory of knowledge grounded in human culture must explain how thisispossible.Inthelastessay,wepresentouranalysisofthisculturalinvariance, combing insights from evolutionary theory and current neuroscience. Thebookisnotaresearchmonograph,noratextbook.Itisacollectionofessays offered to any reader who seeks a deeper understanding of knowledge in present-day society. The mathematical and computational sciences have gained in importance, but the understanding of “the mathematical way” is not always as it shouldbeinoursociety.Thisis,aboveall,achallengetooureducationalsystem— from elementary school to research training. Note on the essays. After an introductory essay on mathematics and the nature of knowledge, I have in Part I included five essays on structure and knowledge fromtheEuropeanReview,thejournalofAcademiaEuropaea.Theyarewrittenfor a general audience and illustrate through a number of examples how “the mathe- maticalway”givesinsightandunderstanding.ThisisathemethatIhavedeveloped over many years. As a young postdocin theearly 1960s, Itook anactivepart ina v vi Preface seriesoflecturesarrangedbytheNorwegianMathematicalSocietyonthegrowing importance of mathematical modeling and computations in science and industry; thereportIwroteofthiseventcouldhavebeenpartofthiscollectionofessays.As ChairoftheNaturalScienceResearchCouncilofNorway,Icouldinthemid-1980s contribute to the financing of the first Cray supercomputer to Norwegian research andthetextofmylectureattheopeningceremonycouldequallyhavebeenpartof this collection.The same istrue oftheinput onmathematics and computingtothe 5th Framework Program of the EU that I developed as Chair of the Standing Committee for Physics and the Engineering Sciences of the European Science Foundation in the mid-1990s. I stressed, in particular, the applicability of “the mathematicalway”tolanguageandthehumansciences,aswellastothesocialand medical sciences. The essays of Part I are followed in Part II by three essays on language, mind, and number. The first two of these essays are included in order to give a simple exampleof“themathematicalway”asappliedtolanguageandmeaning.Grammar, geometry, and mind have been major research interests, starting with a first paper on “Models for Natural Languages” from the 1970s. There is, however, no math- ematical way, connecting structures and algorithms, without numbers, either directlyasobjectsormoregenerallyastools.Understandingknowledgethusmeans to know what numbers are. This is the topic of the last essay of this collection. Ihavetriedinthisbooktonavigatearoundtheactualformalismofmathematics, but in the two essays on language and meaning (Chaps. 7 and 8), the reader will find a few formulas and some diagrams to illustrate the general analysis. No other item in this collection, including the last essay on the nature of numbers, presup- poses knowledge of the mathematical details of these essays. Theessayshavebeenwrittenoveranumberofyears.Thismeansthatwhatwas “recent”atthetime,nowsometimesneedstobeupdated.The“mathematicalway” and the basic analysis have remained unchanged over time, but some examples related to computing and “big data” need to be revised. The reader will find the necessary updates in the introductory essay. Theessayscanbereadinalmostanyorder;thismeansthatthereissomeoverlap between the chapters, in particular, in the introductory parts to Chaps. 5 and 7. Permission to reprint. Chapters 2–9 have previously been published and are here reprinted, sometimes with a few corrections and updates, as follows: (i)Chapters2,3,5,and6arereprintedwithpermissionfromCambridgeUniversity Press,thepublisheroftheEuropeanReview.(ii)Chapter4wasoriginallypublished inFormativeYearsofScholars,London:PortlandPress,andisherereprintedwith permission from Cambridge University Press, the publisher of the European Review. (iii) Chapter 7 is reprinted with permission from Elsevier, the publisher oftheAnnalsofPureandAppliedLogic.(iv)Chapter8isreprintedwithpermission fromSpringer-Verlag,thecurrentcopyrightholderofX.Arrazolaetal,Discourse, Interaction and Communication. (v) Chapter 9 is reprinted with permission from Taylor & Francis, the publisher of Inquiry. Preface vii AcknowledgementsManypeoplehavegivengoodadviceandusefulcomments in the written of this book. I want above all to thank Dagfinn Føllesdal, Øystein Linnebo, Richard Tieszen, and the Springer referee for many helpful and critical comments on several drafts of the text. It is also a pleasure to acknowledge the expert and helpful assistance of my Springer editors in production the book. Oslo, Norway Jens Erik Fenstad This page intentionally left blank Contents 1 Mathematics and the Nature of Knowledge—An Introductory Essay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 What Common Man Believes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Mathematical Way. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Mathematics and Knowledge. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Part I Structures and Algorithms 2 The Miraculous Left Hand—On Leonardo Da Vinci and the Search for a Common Understanding of Man and Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 Relationships Between the Social and the Natural Sciences . . . . . . . 27 3.1 On the Use of Models in Anthropology. . . . . . . . . . . . . . . . . . . . 28 3.2 A Digression on General Methodology . . . . . . . . . . . . . . . . . . . . 30 3.3 A Taxonomy of Mathematical Models. . . . . . . . . . . . . . . . . . . . . 33 3.3.1 Mechanical Man and Nature. . . . . . . . . . . . . . . . . . . . . . . 33 3.3.2 Adding Chance and Uncertainties. . . . . . . . . . . . . . . . . . . 34 3.3.3 Chaos and Catastrophes . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 The Case of Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 One or Several Sciences? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Advice on Literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4 Changes of the Knowledge System and Their Implication for the Formative Stage of Scholars: Experiences in the Natural Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1 Science Is Shaped by Ignorance . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Beyond Reductionism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ix
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