Structures and Algorithms: Mathematics and the Nature of Knowledge

145 Pages·2018·1.7 MB·English

by  Jens Erik Fenstad

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
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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
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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
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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