Formalism and Beyond: On the Nature of Mathematical Discourse

431 Pages·2014·2.202 MB·English

by  Godehard Link

Formalism and Beyond
Logos
Studien zur Logik, Sprachphilosophie und Metaphysik
Herausgegeben von/Edited by
Volker Halbach, Alexander Hieke, Hannes Leitgeb,
Holger Sturm
Volume / Band 23
Formalism and Beyond
On the Nature of Mathematical Discourse
Edited by
Godehard Link
ISBN 978-1-61451-829-7
e-ISBN (PDF) 978-1-61451-847-1
e-ISBN (EPUB) 978-1-61451-996-6
ISSN 2198-2201
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Contents
Preface ix
Duality, Epistemic Efficiency & Consistency
Michael Detlefsen 1
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Abstract Duality or Dualization? . . . . . . . . . . . . . 5
3 The Contentual Addition Model of Dualization. . . . . . 7
4 Proofs & Proof Developments . . . . . . . . . . . . . . . 8
5 TheContentualAdditionModel&TheTraditionalCon-
tentualist View of Proof. . . . . . . . . . . . . . . . . . . 10
6 Contentual Addition in an Abstract Setting . . . . . . . 12
7 Non-Trivial Axiom Systems . . . . . . . . . . . . . . . . 17
8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Frege on Quantities and Real Numbers in Consideration
of the Theories of Cantor, Russell and Others
Matthias Schirn 25
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 TheconceptofquantityinFrege’swritingsbetween1874
and 1884 . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Cantor’s theory of irrationalnumbers and Frege’s critique 49
4 Russell on quantities and real numbers in Principles of
Mathematics and Principia Mathematica . . . . . . . . . 56
5 Quantities and real numbers in Grundgesetze . . . . . . . 59
6 Frege’s plan carried out: von Kutschera’s account . . . . 89
Frege on Formality and the 1906 Independence-Test
Patricia A. Blanchette 97
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 97
2 The Proposal. . . . . . . . . . . . . . . . . . . . . . . . . 98
3 The Import of the 1910 Notes . . . . . . . . . . . . . . . 104
4 The Anti-Metatheory Explanation . . . . . . . . . . . . . 107
5 The Similarity with Hilbert. . . . . . . . . . . . . . . . . 110
6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 115
vi Contents
Formal Discourse in Russell: From Metaphysics to Philo-
sophical Logic
Godehard Link 119
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 119
2 Setting the Stage: Russell’s Early Ontology. . . . . . . . 124
3 On the Nature of Functions . . . . . . . . . . . . . . . . 139
4 The Substitutional Theory . . . . . . . . . . . . . . . . . 152
5 Principia Mathematica . . . . . . . . . . . . . . . . . . . 155
6 Ramification: Gödel’s Gestalt Switch . . . . . . . . . . . 163
7 Lessons for Ontology . . . . . . . . . . . . . . . . . . . . 166
8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 175
On Live and Dead Signs in Mathematics
Felix Mühlhölzer 183
1 A Mess Concerning the Reference, Interpretation and
Application. . . . . . . . . . . . . . . . . . . . . . . . . . 184
2 How can Intended Models be Singled Out? . . . . . . . . 195
3 Strings of Strokes in Hilbertian Finitism . . . . . . . . . 201
Generalization and the Impossible
Paul Ziche 209
1 “Contradictionsareemotions”: The exampleofthe com-
plex numbers . . . . . . . . . . . . . . . . . . . . . . . . . 209
2 Russell on symbolism: Making the simple complicated . 214
3 Ways into logic: Generalization and abstraction . . . . . 215
4 Pure logic and meta-scientific induction . . . . . . . . . . 219
5 Scientistic liberalism and interesting generalizations . . . 221
Assumptions of Infinity
Karl-Georg Niebergall 229
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 229
2 “T makes an assumption of infinity”, “T assumes merely
the finite” . . . . . . . . . . . . . . . . . . . . . . . . . . 234
3 Expressing infinity: a preliminary suggestion . . . . . . . 237
4 Axioms for and definitions of “finite” . . . . . . . . . . . 242
5 Elaboration of (DIiii) . . . . . . . . . . . . . . . . . . . . 250
6 The potentially infinite . . . . . . . . . . . . . . . . . . . 256
7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 261
8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
The Interpretation of Classes in Axiomatic Set Theory
Daniel Roth, Gregor Schneider 275
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 275
2 Set Theories . . . . . . . . . . . . . . . . . . . . . . . . . 275
Contents vii
3 Interpreting Classes . . . . . . . . . . . . . . . . . . . . . 296
4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . 308
Purity in Arithmetic: some Formal and Informal Issues
Andrew Arana 315
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 315
2 Topical purity . . . . . . . . . . . . . . . . . . . . . . . . 316
3 The infinitude of primes . . . . . . . . . . . . . . . . . . 318
4 Incompleteness and the possibility of purity . . . . . . . 331
5 Closing thoughts . . . . . . . . . . . . . . . . . . . . . . . 333
Domain Extensions and Higher-Order Syntactical Inter-
pretations
Marek Polański 337
1 Introductory remarks . . . . . . . . . . . . . . . . . . . . 337
2 Domain extensions: some paradigmatic examples . . . . 338
3 -operations and -constructions . . . . . . . . . . . . . 340
L L
4 Higher-order syntactical interpretations and their con-
structions. . . . . . . . . . . . . . . . . . . . . . . . . . . 344
5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . 349
Finite Methods in Mathematical Practice
Laura Crosilla, Peter Schuster 351
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 351
2 Hilbert’s programme now and then . . . . . . . . . . . . 352
3 Finite methods for constructive algebra . . . . . . . . . . 365
4 Geometric formulas and dynamical proofs. . . . . . . . . 369
5 Realising Hilbert’s programme in commutative algebra . 372
6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . 411
Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

Preface
The present volume grew out of a cooperation between German, Italian,
andUS-Americanresearchersfrom2005to2011,whichwasfundedinequal
parts by the Alexander-von-Humboldt Foundation and a matching fund
from the University of Notre Dame within the format of one of the Foun-
dation’s TransCoop projects. The general theme of the cooperation was
called: “Imaginary and Ideal Elements and Limit Concepts in Mathemat-
ics: Their Theory, History, and Philosophical Understanding”. The title
wasmotivatedby anongoingphilosophicalinterestinthe roleofformalist
aspectsinmathematicaltheorizingandpractice. Accordingly,theresearch
activities ranged from studies approaching historical topics with modern
logico-philosophicaltools to systematic conceptual and logical analyses.
The contributions presented here cover this ground and to some de-
gree extend it in various directions. Papers focussing on central historical
figures in the field, like Frege, Russell, Hilbert, and Wittgenstein, are ac-
companied by those dealing with issues like infinity, finiteness, and proof
procedures and ones putting formalist mathematics into historical per-
spective. More general information is given in the abstracts preceding
each paper.
Anoteonthenameindex: Namesofauthorsmightnotexplicitlyappear
on a page mentioned in the index if their work is merely referred to via
numbers in square brackets. In such a case the names can be retrieved
from the bibliography of the essay concerned.
In the name of all contributors, I wish to thank the Humboldt Founda-
tion and the University of Notre Dame for funding the project, and the
Munich Center for Mathematical Philosophy (MCMP) for generous addi-
tionalfinancialsupport. Ialsothankseveralanonymousreviewersfortheir
expert opinion; Jesse Tomalty for looking over the English of non-native
speakers;JohannesSternundRolandPoellingerforadministrationalwork
at various phases of the project. In particular, Mic Detlefsen is to be
thankedfororiginallyinitiating the transatlanticcooperation,andGregor
Schneider for serving as the general LATEX editor of the volume. Finally,
I thank the editors of the series and the publisher for agreeingto produce
the book with them.
Munich, June 2014, G. L.