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Tool and Objecthttp://www.springerlink.com/content/v42840/?sortorder=asc&v=expanded
A History and Philosophy of Category Theory
Why I am a Platonist
David Mumford Like the previous authors of this ongoing debate,1 I have to begin by clarifying what “Platonism” means to me. Here’s my phrase: "The belief that there is a body of mathematical objects, relations and facts about them that is independent of and unaffected by human endeavors to discover them." This is essentially Davies’ fi rst fl avor of Platonism, but in his article he isn’t content with my phrase “there is” a body of objects etc., but feels he must characterize this belief as existence in a realm outside or beyond spacetime. I think using these prepositions already implies certain philosophical, specifi cally ontological, assumptions. Hersh is more tolerant, merely adding the qualification that this body of objects, etc. is objective, which still puts a special ontological spin on the belief. Mazur seems closest to my simple statement above when he appropriates Huck Finn’s words saying that this body of objects etc. just happened (instead of being invented by people). “Just happened” implies that, one way or another, they are there, without further characterization of how they exist or especially ‘where’ they exist.
JSTOR: Proceedings of the Aristotelian Society, Supplementary Volumes, Vol. 70 (1996), pp. 243-265
www.jstor.org/pss/4107008 (article consists of 23 pages) StrictRevMath110709.08.pdf (application/pdf Object)
www.math.ohio-state.edu/%7Efriedman/pdf/StrictRevM... STRICT REVERSE MATHEMATICS
by Harvey M. Friedman Ohio State University November 7, 2009 Reverse Mathematics Workshop University of Chicago November 6-8, 2009 M athematics, the
U ltimate C hallenge to E mbodiment: T ruth and the G rounding of A xiomatic S ystems c00017 The human body is an animal body. A body that has evolved over millions of years coping with real-world properties such as temperature, gravity, humidity, color, space, texture and so on. With this same body humans have been able to create concepts—and think with them—in a way that transcends immediate bodily experience. Today, millions of modern humans effortlessly operate in everyday life with abstract notions like “ democracy, ” “ black humor, ” “ infl ation, ” and the “ fl ow of time. ” In technical domains, like mathematics, humans have created abstract concepts, such as “ square root of minus one ” and “ transfi nite numbers ” — rich and precise entities that lack any concrete instantiation in the real world. These entities are the product of the human imagination, and exist in the realm of mental abstractions and social practices. How do humans achieve this with the body of a primate? In what sense are the abstract ideas humans create embodied ? And then there is a question of what is mathematics in the fi rst place? What is the nature of this body of knowledge that appears to be timeless, eternal, absolute, and effective to the point that many scholars fi rmly believe it is part of the very fabric of the universe, transcending human existence? The Mathematical Universe
Max Tegmark Dept. of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 (Dated: Submitted to Found. Phys. April 7 2007, revised September 6, accepted September 30) I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, parallel universes and G¨odel incompleteness. I hypothesize that only computable and decidable (in G¨odel’s sense) structures exist, which alleviates the cosmological measure problem and may help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems. New Waves in Philosophy of Mathematics
Edited by Otavio Bueno and Oystein Linnebo
A History of Abstract Algebrahttp://www.springerlink.com/content/p3020j/
Creativity, Freedom, and Authority: A New Perspective On the Metaphysics of Mathematics -
www.informaworld.com/smpp/content~content=a9068571... Creativity, Freedom, and Authority: A New Perspective On the Metaphysics of Mathematics
Author:
Julian C. Cole a
DOI:
10.1080/00048400802598629
Publication Frequency:
4 issues per year
Published in:
First Published:
December
2009
Australasian Journal of Philosophy,
Volume
87,
Issue
4
December
2009
, pages 589
- 608
Subject:
Philosophy;
Formats available:
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(English)
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(English)
Previously published as:
Australasian Journal of Psychology and Philosophy
(1832-8660)
until 1947
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Mathematical knowledgeBy Mary Leng, Alexander Paseau, Michael D. PotterOn 'Average' -- Kennedy and Stanley 118 (471): 583 -- Mind
mind.oxfordjournals.org/cgi/content/short/118/471/... On ‘Average’Christopher KennedyDepartment of Linguistics University of Chicago 1010 E. 59th St. Chicago, IL 60637 USA ck@uchicago.edu Jason Stanley Department of Philosophy Rutgers University 1 Seminary Place New Brunswick, NJ 08901-1107 USA jasoncs@ruccs.rutgers.edu
Whitehead and Russell on Points -- Bostock, 10.1093/philmat/nkp017 -- Philosophia Mathematica
philmat.oxfordjournals.org/cgi/content/short/nkp01... Whitehead and Russell on Points
David Bostock*
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PDF (183.9 KB)
PDF (186.4 KB)
| Book Series | The Western Ontario Series in Philosophy of Science |
| ISSN | 1566-659X |
| Subject | Philosophy, Philosophy of Science, Logic and Mathematical Logic and Foundations |
| Volume | Volume 71 |
| Subject | Philosophy, Philosophy of Science, Logic and Mathematical Logic and Foundations |
| Publisher | Springer Netherlands |
| DOI | 10.1007/978-1-4020-4265-2 |
| Copyright | 2008 |
| ISBN | 978-1-4020-4264-5 (Print) 978-1-4020-4265-2 (Online) |
| Subject Collection | Humanities, Social Sciences and Law |
| Subject | Philosophy, Philosophy of Science, Logic and Mathematical Logic and Foundations |
| SpringerLink Date | Tuesday, November 27, 2007 |
Intuition and the Axiomatic Method
| Book Series | The Western Ontario Series in Philosophy of Science |
| ISSN | 1566-659X |
| Subject | Philosophy, Philosophy, History of Philosophy and Philosophy of Science |
| Volume | Volume 70 |
| Subject | Philosophy, Philosophy, History of Philosophy and Philosophy of Science |
| Publisher | Springer Netherlands |
| DOI | 10.1007/1-4020-4040-7 |
| Copyright | 2006 |
| ISBN | 978-1-4020-4039-9 (Print) 978-1-4020-4040-5 (Online) |
| Subject Collection | Humanities, Social Sciences and Law |
| Subject | Philosophy, Philosophy, History of Philosophy and Philosophy of Science |
| SpringerLink Date | Sunday, July 02, 2006 |
(article consists of 23 pages)
JSTOR: Proceedings and Addresses of the American Philosophical Association, Vol. 73, No. 2 (Nov.,
www.jstor.org/pss/3131087
www.jstor.org/pss/3131087
(article consists of 30 pages)
Confirming Mathematical Theories: an Ontologically Agnostic Stance
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Anthony Peressini Abstract The Quine/Putnam indispensability approach to the confirmation of mathematical theories in recent times has been the subject
of significant criticism. In this paper I explore an alternative to the Quine/Putnam indispensability approach. I begin with
a van Fraassen-like distinction between accepting the adequacy of a mathematical theory and believing in the truth of a mathematical
theory. Finally, I consider the problem of moving from the adequacy of a mathematical theory to its truth. I argue that the
prospects for justifying this move are qualitatively worse in mathematics than they are in science.
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(article consists of 25 pages)
JSTOR: The Journal of Philosophy, Vol. 89, No. 6 (Jun., 1992), pp. 275-289
www.jstor.org/pss/2026712
www.jstor.org/pss/2026712
(article consists of 15 pages)
What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
Existence, Freedom, Identity, and the Logic
http://mind.oxfordjournals.org/cgi/reprint/116/461/23
of Abstractionist Realism
Peter Milne
From the point of view of proof-theoretic semantics, we examine the logical background
invoked by Neil Tennant’s abstractionist realist account of mathematical existence.
To prepare the way, we must first look closely at the rule of existential
elimination familiar from classical and intuitionist logics and at rules governing
identity. We then examine how well free logics meet the harmony and uniqueness
constraints familiar from the proof-theoretic semantics project. Tennant assigns a
special role to atomic formulas containing singular terms. This, we find, secures harmony
and uniqueness but militates against the putative realism.
Neil Tennant’s constructive logicism (Tennant 1999) has mutated into
‘abstractionist realism’ (Tennant 2004). Against the background of a
free logic, abstraction terms, of which the paradigms in the wider literature
are definite descriptions and set abstracts, are introduced in identity
statements. That is, variable-binding, term-forming abstraction
operators are used to form abstraction terms, singular terms; identity
statements in which on one side of the identity sign stands an abstraction
term are subject to introduction and elimination rules.
http://mind.oxfordjournals.org/cgi/reprint/116/461/23
of Abstractionist Realism
Peter Milne
From the point of view of proof-theoretic semantics, we examine the logical background
invoked by Neil Tennant’s abstractionist realist account of mathematical existence.
To prepare the way, we must first look closely at the rule of existential
elimination familiar from classical and intuitionist logics and at rules governing
identity. We then examine how well free logics meet the harmony and uniqueness
constraints familiar from the proof-theoretic semantics project. Tennant assigns a
special role to atomic formulas containing singular terms. This, we find, secures harmony
and uniqueness but militates against the putative realism.
Neil Tennant’s constructive logicism (Tennant 1999) has mutated into
‘abstractionist realism’ (Tennant 2004). Against the background of a
free logic, abstraction terms, of which the paradigms in the wider literature
are definite descriptions and set abstracts, are introduced in identity
statements. That is, variable-binding, term-forming abstraction
operators are used to form abstraction terms, singular terms; identity
statements in which on one side of the identity sign stands an abstraction
term are subject to introduction and elimination rules.
'Tarski, Truth and Model Theory', Proceedings of the Aristotelian Society XCIX (1998-9), 141-167.
Implicit Epistemic Aspects of Constructive Logic
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Göran Sundholm1
Abstract In the present paper I wish to regard constructivelogic as a self-contained system for the treatment ofepistemological issues; the explanations of theconstructivist logical notions are cast in anepistemological mold already from the outset. Thediscussion offered here intends to make explicit thisimplicit epistemic character of constructivism.Particular attention will be given to the intendedinterpretation laid down by Heyting. This interpretation, especially as refined in the type-theoretical work of Per Martin-Löf, puts thesystem on par with the early efforts of Frege andWhitehead-Russell. This quite recent work, however,has proved valuable not only in the philosophy andfoundations of mathematics, but has also foundpractical application in computer science, where thelanguage of constructivism serves as an implementableprogramming language, and within the philosophy oflanguage.\footnote{Nordstr\"{o}m et al. (1990) give an overview of the work in computerscience, whereas Ranta (1995) provides an impressiveconstructivist alternative to Montague Grammar usingthe richer type structure of Martin-L\"{o}f in placeof the simple classical type theory of Church.} Mypresentation will be carried out through a contrastwith standard metamathematical work.\footnote{Troelstra and van Dalen (1988) give an encyclopedictreatment of the metamathematics of constructivism.}In the course of the development I have occasion tooffer some novel considerations (in Sections~6 and 8) on thenature of proof and inference(-acts). |
Quantization and Cohomology
In these lectures I spoke about:- the Lagrangian approach to classical mechanics,
- the path-integral approach to quantum mechanics,
- symplectic geometry,
- geometric quantization,
- how going from point particles to strings makes us "categorify" all the above.
Model theory of the regularity and reflection schemes
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Ali Enayat1
Received: 3 May 2007 Revised: 19 March 2008 Published online: 15 July 2008 Abstract This paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme and its strengthening REF
 (the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here  is a language with a distinguished linear order <, and REF
 consists of formulas of the form |
One hundred years of Russell's ... - Google Books
books.google.com/books?id=Xg6QpedPpcsC&dq=reflecti...
books.google.com/books?id=Xg6QpedPpcsC&dq=reflecti...
| One hundred years of Russell's paradox: mathematics, logic, philosophy Volume 6 of De Gruyter series in logic and its applications | |
| Author | Godehard Link |
| Editor | Godehard Link |
| Publisher | Walter de Gruyter, 2004 |
| ISBN | 3110174383, 9783110174380 |
| Length | 662 pages |
| Subjects | Liar paradox Mathematics Mathematics / General Mathematics / History & Philosophy Mathematics / Logic Mathematics / Research Paradox Philosophy / General Philosophy / History & Surveys / Modern Philosophy / Logic |
Extending the Language of Set Theory
Authors:
Dmytro Taranovsky
(Submitted on 19 Apr 2005)
Abstract: We discuss the problems of incompleteness and inexpressibility. We introduce almost self-referential formulas, use them to extend set theory, and relate their expressive power to that of infinitary logic. We discuss the nature of proper classes. Finally, we introduce and axiomatize a powerful extension to set theory.
| Comments: | 11 pages, HTML, UTF-8 encoding |
| Subjects: | Logic (math.LO) |
| MSC classes: | 03E99 |
| Cite as: | arXiv:math/0504375v1 [math.LO] |
Cardinality without Enumeration
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Athanassios Tzouvaras1
Received: 23 November 2003 Abstract We show that the notion of cardinality of a set is independent from that of wellordering, and that reasonable total notions of cardinality exist in every model of ZF where the axiom of choice fails. Such notions are either definable in a simple and natural way, or non-definable, produced by forcing. Analogous cardinality notions exist in nonstandard models of arithmetic admitting nontrivial automorphisms. Certain motivating phenomena from quantum mechanics are also discussed in the Appendix. |
Working foundations
| Journal | Synthese |
| Publisher | Springer Netherlands |
| ISSN | 0039-7857 (Print) 1573-0964 (Online) |
| Issue | Volume 62, Number 2 / February, 1985 |
| DOI | 10.1007/BF00486048 |
| Pages | 229-254 |
| Subject Collection | Humanities, Social Sciences and Law |
| SpringerLink Date | Tuesday, December 07, 2004 |
Description:
First published in 1980 (with an Italian translation in 1983) Peano: Life and Works of Giuseppe Peano has remained the standard biography of this great mathematician of the late 19th century. In addition to his mathematics, the book also discusses Peano’s role in the international auxiliary language movement. Long out of print, this corrected Definitive Edition (2006) makes it available once more in an attractive, low-priced edition that also includes numerous illustrations.
THE EMPTY SET, THE SINGLETON, AND THE ORDERED PAIR AKIHIRO KANAMORI Dedicated to the memory of Burton S. Dreben For the modern set theorist the empty set 0, the singleton {a}, and the ordered pair (x, y) are at the beginning of the systematic, axiomatic de- velopment of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest build- ing blocks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice are broached in the formal elaboration of the 'set of' {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano.
Douglas Michael Jesseph - Leibniz on the Foundations of the Calculus: The Question of the Reality of
muse.jhu.edu/journals/perspectives_on_science/v006...
muse.jhu.edu/journals/perspectives_on_science/v006...
Leibniz on the Foundations of the Calculus:
The Question of the Reality of Infinitesimal Magnitudes
Douglas M. Jesseph
Figures
Among his achievements in all areas of learning, Leibniz's contributions to the development of European mathematics stand out as especially influential. His idiosyncratic metaphysics may have won few adherents, but his place in the history of mathematics is sufficiently secure that historians of mathematics speak of the "Leibnizian school" of analysis and delineate a "Leibnizian tradition" in mathematics that extends well past the death of its founder. This great reputation rests almost entirely on Leibniz's contributions to the calculus. Whether he is granted the status of inventor or co-inventor, there is no question that Leibniz was instrumental in instituting a new method, and his contributions opened up a vast new field of mathematical research.
Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis
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Geoffrey Hellman1
Received: 1 September 2005 Accepted: 4 February 2006 Published online: 12 July 2006 Abstract A remarkable development in twentieth-century mathematics is smooth infinitesimal analysis (‘SIA’), introducing nilsquare and nilpotent infinitesimals, recovering the bulk of scientifically applicable classical analysis (‘CA’) without resort to the method of limits. Formally, however, unlike Robinsonian ‘nonstandard analysis’, SIA conflicts with CA, deriving, e.g., ‘not every quantity is either = 0 or not = 0.’ Internally, consistency is maintained by using intuitionistic logic (without the law of excluded middle). This paper examines problems of interpretation resulting from this ‘change of logic’, arguing that standard arguments based on ‘smoothness’ requirements are question-begging. Instead, it is suggested that recent philosophical work on the logic of vagueness is relevant, especially in the context of a Hilbertian structuralist view of mathematical axioms (as implicitly defining structures of interest). The relevance of both topos models for SIA and modal-structuralism as appled to this theory is clarified, sustaining this remarkable instance of mathematical pluralism. |
Category Theory as a Framework for an in re Interpretation of Mathematical Structuralism
Structures And Structuralism In Contemporary Philosophy Of Mathematics
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Erich H. Reck1
Abstract In recent philosophy of mathematics avariety of writers have presented ``structuralist''views and arguments. There are, however, a number ofsubstantive differences in what their proponents take``structuralism'' to be. In this paper we make explicitthese differences, as well as some underlyingsimilarities and common roots. We thus identifysystematically and in detail, several main variants ofstructuralism, including some not often recognized assuch. As a result the relations between thesevariants, and between the respective problems theyface, become manifest. Throughout our focus is onsemantic and metaphysical issues, including what is orcould be meant by ``structure'' in this connection. |
The Identity Problem for Realist Structuralism -- KERÄNEN 9 (3): 308 -- Philosophia Mathematica
philmat.oxfordjournals.org.proxy.lib.uiowa.edu/cgi...
philmat.oxfordjournals.org.proxy.lib.uiowa.edu/cgi...
The Identity Problem for Realist Structuralism
JUKKA KERÄNEN*
* Department of Philosophy, 1001 Cathedral of Learning, University of Pittsburgh Pittsburgh, Pennsylvania 15260 U. S. A. keranen+@pitt.edu
According to realist structuralism, mathematical objects are places in abstract structures. We argue that in spite of its many attractions, realist structuralism must be rejected. For, first, mathematical structures typically contain intra-structurally indiscernible places. Second, any account of place-identity available to the realist structuralist entails that intra-structurally indiscernible places are identical. Since for her mathematical singular terms denote places in structures, she would have to say, for example, that 1 = – 1 in the group (Z, +). We call this the identity problem and conclude that nominalism is presently the safest route for the structuralist.
HOW APPLIED MATHEMATICS BECAME PURE
http://www.lps.uci.edu/maddy/how%20applied%20math%20rsl.pdf
PENELOPE MADDY
University of California, Irvine
Abstract. This paper traces the evolution of thinking on how mathematics relates to the world—
from the ancients, through the beginnings of mathematized science in Galileo and Newton, to the
rise of pure mathematics in the nineteenth century. The goal is to better understand the role of
mathematics in contemporary science.
http://www.lps.uci.edu/maddy/how%20applied%20math%20rsl.pdf
PENELOPE MADDY
University of California, Irvine
Abstract. This paper traces the evolution of thinking on how mathematics relates to the world—
from the ancients, through the beginnings of mathematized science in Galileo and Newton, to the
rise of pure mathematics in the nineteenth century. The goal is to better understand the role of
mathematics in contemporary science.
Ontology and the Word ‘Exist’: Uneasy Relations
http://philmat.oxfordjournals.org/cgi/content/short/nkp011v1?rss=1
Jody Azzouni*
* Department of Philosophy, Tufts University, Medford, Mass. 02155-7068, U.S.A. jodyazzouni@mindspring.com
An extensive exploration of the special properties of ‘exist’ is here undertaken. Two of several results are: Denying that `exist’ has associated with it a set of necessary and sufficient conditions has seemed to a number of philosophers to imply metaphysical nihilism. This is because it has seemed that without such conditions the target domain of `existence’ is arbitrarily open. I show this is wrong. Second, my analysis sheds light on the puzzling question of what we are asking when we ask of something, `Does it exist?’ and mean that question in an ontically relevant way.
http://philmat.oxfordjournals.org/cgi/content/short/nkp011v1?rss=1
Jody Azzouni*
* Department of Philosophy, Tufts University, Medford, Mass. 02155-7068, U.S.A. jodyazzouni@mindspring.com
An extensive exploration of the special properties of ‘exist’ is here undertaken. Two of several results are: Denying that `exist’ has associated with it a set of necessary and sufficient conditions has seemed to a number of philosophers to imply metaphysical nihilism. This is because it has seemed that without such conditions the target domain of `existence’ is arbitrarily open. I show this is wrong. Second, my analysis sheds light on the puzzling question of what we are asking when we ask of something, `Does it exist?’ and mean that question in an ontically relevant way.
Subsystems of Second Order Arithmetic - Cambridge University Press
www.cambridge.org/uk/catalogue/catalogue.asp?isbn=...
www.cambridge.org/uk/catalogue/catalogue.asp?isbn=...
Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are examined to prove particular theorems in core mathematical areas such as algebra, analysis, and topology, focusing on the language of second-order arithmetic, the weakest language rich enough to express and develop the bulk of mathematics. In many cases, if a mathematical theorem is proved from appropriately weak set existence axioms, then the axioms will be logically equivalent to the theorem. Furthermore, only a few specific set existence axioms arise repeatedly in this context, which in turn correspond to classical foundational programs. This is the theme of reverse mathematics, which dominates the first half of the book. The second part focuses on models of these and other subsystems of second-order arithmetic.
Conceptual Mathematics - Cambridge University Press
www.cambridge.org/catalogue/catalogue.asp?isbn=978...
www.cambridge.org/catalogue/catalogue.asp?isbn=978...
In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics introduces this tool for the learning, development, and use of mathematics, to beginning students and also to practising mathematical scientists. This book provides a skeleton key that makes explicit some concepts and procedures that are common to all branches of pure and applied mathematics. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories. This second edition provides links with more advanced topics of possible study. In the new appendices and annotated bibliography the reader will find concise introductions to adjoint functors and geometrical structures, as well as sketches of relevant historical developments.
• Authors are world class authorities on the subject • Only text at this elementary level - requires only high-school algebra • Applications in pure and applied mathematics, computer science, physics, linguistics, logic and philosophy
http://jmilne.org/math/xnotes/MOT.pdf
James S. Milne
June 7, 2009; v2.03
Abstract
Grothendieck introduced the notion of a “motif” in a letter to Serre in 1964. Later
he wrote that, among the objects he had been privileged to discover, they were the most
charged with mystery and formed perhaps the most powerful instrument of discovery.1
In this talk, I shall explain what motives are, and why Grothendieck valued them so
highly.
These are my notes for a “popular” talk in the ‘What is . . . ?’ seminar at the University
of Michigan, Feb 3, 2009. 2
James S. Milne
June 7, 2009; v2.03
Abstract
Grothendieck introduced the notion of a “motif” in a letter to Serre in 1964. Later
he wrote that, among the objects he had been privileged to discover, they were the most
charged with mystery and formed perhaps the most powerful instrument of discovery.1
In this talk, I shall explain what motives are, and why Grothendieck valued them so
highly.
These are my notes for a “popular” talk in the ‘What is . . . ?’ seminar at the University
of Michigan, Feb 3, 2009. 2
This paper is based on lectures given to meetings of the Society for Exact Philosophy in Toronto, May 17, 1985, and of the Association for Symbolic Logic in Washington, DC, December 29, 1985. Some of the material had been presented in lectures at Dartmouth College in 1981 and at the University of Padua in 1983. I am indebted to comments from all four audiences, and also to correspondence with Michael Resnik and especially W. W. Tait. This paper was written while I was at Columbia University, to which I and it owe much. Support of the John Simon Guggenheim Memorial Foundation is gratefully acknowledged.
Abstract Recent semantic approaches to scientific structuralism, aiming to make
precise the concept of shared structure between models, formally frame a model as a
type of set-structure. This framework is then used to provide a semantic account of
(a) the structure of a scientific theory, (b) the applicability of a mathematical theory
to a physical theory, and (c) the structural realist’s appeal to the structural continuity
between successive physical theories. In this paper, I challenge the idea that, to
be so used, the concept of a model and so the concept of shared structure between
models must be formally framed within a single unified framework, set-theoretic or
other. I first investigate the Bourbaki-inspired assumption that structures are types of
set-structured systems and next consider the extent to which this problematic assumption
underpins both Suppes’ and recent semantic views of the structure of a scientific
theory. I then use this investigation to show that, when it comes to using the concept
of shared structure, there is no need to agree with French that “without a formal
framework for explicating this concept of ‘structure-similarity’ it remains vague, just
as Giere’s concept of similarity between models does…” (French, 2000, Synthese, 125,
pp. 103–120, p. 114).Neither concept is vague; either can bemade precise by appealing
to the concept of a morphism, but it is the context (and not any set-theoretic type) that
determines the appropriate kind of morphism. I make use of French’s (1999, From
physics to philosophy (pp. 187–207). Cambridge: Cambridge University Press) own
example from the development of quantum theory to show that, for both Weyl and
Wigner’s programmes, it was the context of considering the ‘relevant symmetries’ that
determined that the appropriate kind of morphism was the one that preserved the
shared Lie-group structure of both the theoretical and phenomenological models.
precise the concept of shared structure between models, formally frame a model as a
type of set-structure. This framework is then used to provide a semantic account of
(a) the structure of a scientific theory, (b) the applicability of a mathematical theory
to a physical theory, and (c) the structural realist’s appeal to the structural continuity
between successive physical theories. In this paper, I challenge the idea that, to
be so used, the concept of a model and so the concept of shared structure between
models must be formally framed within a single unified framework, set-theoretic or
other. I first investigate the Bourbaki-inspired assumption that structures are types of
set-structured systems and next consider the extent to which this problematic assumption
underpins both Suppes’ and recent semantic views of the structure of a scientific
theory. I then use this investigation to show that, when it comes to using the concept
of shared structure, there is no need to agree with French that “without a formal
framework for explicating this concept of ‘structure-similarity’ it remains vague, just
as Giere’s concept of similarity between models does…” (French, 2000, Synthese, 125,
pp. 103–120, p. 114).Neither concept is vague; either can bemade precise by appealing
to the concept of a morphism, but it is the context (and not any set-theoretic type) that
determines the appropriate kind of morphism. I make use of French’s (1999, From
physics to philosophy (pp. 187–207). Cambridge: Cambridge University Press) own
example from the development of quantum theory to show that, for both Weyl and
Wigner’s programmes, it was the context of considering the ‘relevant symmetries’ that
determined that the appropriate kind of morphism was the one that preserved the
shared Lie-group structure of both the theoretical and phenomenological models.
Continuity and Infinitesimals (Stanford Encyclopedia of Philosophy)
plato.stanford.edu/entries/continuity/
plato.stanford.edu/entries/continuity/
The usual meaning of the word continuous is
“unbroken” or “uninterrupted”: thus a
continuous entity—a continuum—has no
“gaps.” We commonly suppose that space and time are
continuous, and certain philosophers have maintained that all natural
processes occur continuously: witness, for example, Leibniz's famous
apothegm natura non facit saltus—“nature makes no
jump.” In mathematics the word is used in the same general sense,
but has had to be furnished with increasingly precise definitions. So,
for instance, in the later 18th century continuity of a function was
taken to mean that infinitesimal changes in the value of the argument
induced infinitesimal changes in the value of the function. With the
abandonment of infinitesimals in the 19th century this definition came
to be replaced by one employing the more precise concept of
limit.
Logicism, Intuitionism, and Formalism
http://www.springerlink.com/content/978-1-4020-8925-1
What has Become of Them?
| Book Series | Synthese Library |
| Subject | Mathematics, Mathematical Logic and Foundations, Logic, Philosophy of Language, Epistemology, Ontology and History of Mathematics |
| Volume | Volume 341 |
| Subject | Mathematics, Mathematical Logic and Foundations, Logic, Philosophy of Language, Epistemology, Ontology and History of Mathematics |
| Publisher | Springer Netherlands |
| DOI | 10.1007/978-1-4020-8926-8 |
| Copyright | 2009 |
| ISBN | 978-1-4020-8925-1 (Print) 978-1-4020-8926-8 (Online) |
| Subject Collection | Mathematics and Statistics |
| Subject | Mathematics, Mathematical Logic and Foundations, Logic, Philosophy of Language, Epistemology, Ontology and History of Mathematics |
| SpringerLink Date | Tuesday, November 25, 2008 |
What is mathematics about? In what does the reality of mathematics consist? How can we know this reality? These questions are natural but also tendentious. They presuppose the widely but not universally accepted view that mathematics has a reality, and that we can know (something of) that reality. I disagree with but shall not argue in this paper against those who do not accept that the subject matter of mathematics constitutes a reality, for example Wittgenstein (who held that the theorems of mathematics do not establish truth), and Quine (who accepted that a mathematical theory establishes truth but only in virtue of its participation in
some part of science as a whole that has been con¯rmed as true by observation of the physical world, i.e. there is no reality particular to mathematics1). I also
disagree with constructivist accounts of the reality of mathematics, of which the most in°uential, radical, and highly developed is intuitionism as developed from the
philosophical ideas and mathematical results of L.E.J. Brouwer. Constructivism as a philosophy of mathematics is prescriptive. I reject this prescriptivity but will not
give arguments against it in this paper. Regardless, constructivism is a way of doing mathematics that has undoubted use in mathematics and undoubted correctness, and the realist understanding of mathematics being developed in this paper must be able to account for this correctness. I sketch elements of such an account in section 2.
some part of science as a whole that has been con¯rmed as true by observation of the physical world, i.e. there is no reality particular to mathematics1). I also
disagree with constructivist accounts of the reality of mathematics, of which the most in°uential, radical, and highly developed is intuitionism as developed from the
philosophical ideas and mathematical results of L.E.J. Brouwer. Constructivism as a philosophy of mathematics is prescriptive. I reject this prescriptivity but will not
give arguments against it in this paper. Regardless, constructivism is a way of doing mathematics that has undoubted use in mathematics and undoubted correctness, and the realist understanding of mathematics being developed in this paper must be able to account for this correctness. I sketch elements of such an account in section 2.
Computational complexity is the study of what resources, such as time and memory, are needed to carry out given computational tasks, with a particular focus on lower bounds for the amount needed of these resources. Proving any result of this kind is notoriously difficult, and includes the famous problem of whether P = N P . This course will be focused on two major results in the area. The first is a lower bound, due to Razborov, for the number of steps needed to determine whether a graph contains a large clique, if only “monotone” computations are allowed. This is perhaps the strongest result in the direction of showing that P and N P are distinct (though there is unfortunately a very precise sense in which the proof cannot be developed to a proof of the whole conjecture). The second is Peter Shor’s remarkable result that a quantum computer can factorize large integers in
polynomial time. In order to present these two results, it will be necessary to spend some time discussing some of the basic concepts of computational complexity, such as the relationship between Turing machines and the more obviously mathematical notion of circuit complexity, and an introduction to what a quantum computation actually is. For the latter, no knowledge of quantum mechanics will be expected, and scarcely any will be imparted during the course: it is possible to understand quantum computation in a very “pure mathematics” way. The reason this is a graduate course rather than a Part III course is that I intend to give several lectures in an informal style that would be hard to examine. It is not because the material will be more advanced: indeed, my aim will be to make allowances for the fact that people will not be working on it with an exam in mind, and to make the course as easy to follow as I can. Having said that, the main results will be proved in full: the informal discussion will be with a view to making these proofs more comprehensible.
The collection will have 12 graduate level lectures which are currently being given during the Easter term 2009. Many thanks to Adrian Callum-Hinshaw for his help with these video lectures.
polynomial time. In order to present these two results, it will be necessary to spend some time discussing some of the basic concepts of computational complexity, such as the relationship between Turing machines and the more obviously mathematical notion of circuit complexity, and an introduction to what a quantum computation actually is. For the latter, no knowledge of quantum mechanics will be expected, and scarcely any will be imparted during the course: it is possible to understand quantum computation in a very “pure mathematics” way. The reason this is a graduate course rather than a Part III course is that I intend to give several lectures in an informal style that would be hard to examine. It is not because the material will be more advanced: indeed, my aim will be to make allowances for the fact that people will not be working on it with an exam in mind, and to make the course as easy to follow as I can. Having said that, the main results will be proved in full: the informal discussion will be with a view to making these proofs more comprehensible.
The collection will have 12 graduate level lectures which are currently being given during the Easter term 2009. Many thanks to Adrian Callum-Hinshaw for his help with these video lectures.
Logical structuralism and Benacerraf’s problem--Audrey Yap
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Abstract There are two general questions which many views in the philosophy of mathematics can be seen as addressing: what aremathematical objects, and how do we have knowledge of them? Naturally, the answers given to these questions are linked, since whatever account we give of how we have knowledge of mathematical objects surely has to take into account what sorts of things we claim they are; conversely, whatever account we give of the nature of mathematical objects must be accompanied by a corresponding account of how it is that we acquire knowledge of those objects. The connection between these problems results in what is often called “Benacerraf’s Problem”, which is a dilemma that many philosophical views about mathematical objects face. It will be my goal here to present a view, attributed to Richard Dedekind, which approaches the initial questions in a differentway than many other philosophical views do, and in doing so, avoids the dilemma given by Benacerraf’s problem.
Predicativity and Structuralism in Dedekind’s Construction of the Reals--Audrey Yap
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Abstract It is a commonly held view that Dedekind’s construction of the real numbers is impredicative. This naturally raises the question
of whether this impredicativity is justified by some kind of Platonism about sets. But when we look more closely at Dedekind’s
philosophical views, his ontology does not look Platonist at all. So how is his construction justified? There are two aspects
of the solution: one is to look more closely at his methodological views, and in particular, the places in which predicativity
restrictions ought to be applied; another is to take seriously his remarks about the reals as things created by the cuts,
instead of considering them to be the cuts themselves. This can lead us to make finer-grained distinctions about the extent
to which impredicative definitions are problematic, since we find that Dedekind’s use of impredicative definitions in analysis
can be justified by his (non-Platonist) philosophical views.
Empty de re Attitudes About Numbers -- Azzouni 17 (2): 163 -- Philosophia Mathematica
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I dub a certain central tradition in philosophy of language (and mind) the de re tradition. Compelling thought experiments show that in certain common cases the truth conditions for thoughts and public-language expressions categorically turn on external objects referred to, rather than on linguistic meanings and/or belief assumptions. However, de re phenomena in language and thought occur even when the objects in question don't exist. Call these empty de re phenomena. Empty de re thought with respect to numeration is explored in this paper, and such thought with respect to hallucinations is touched on.
On Formally Measuring and Eliminating Extraneous Notions in Proofs -- Arana 17 (2): 189 --
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Many mathematicians and philosophers of mathematics believe some proofs contain elements extraneous to what is being proved. In this paper I discuss extraneousness generally, and then consider a specific proposal for measuring extraneousness syntactically. This specific proposal uses Gentzen's cut-elimination theorem. I argue that the proposal fails, and that we should be skeptical about the usefulness of syntactic extraneousness measures.
I argue that it may well be the case that space and time do not consist of points, indeed that they have no smallest parts. I examine two different approaches to such pointless spaces (to 'gunky' spaces): a topological approach and a measure theoretic approach. I argue in favor of the measure theoretic approach.
Probably there is no position in Goodman's corpus that has generated greater perplexity and criticism than Goodman's "nominalism". As is abundantly clear from Goodman's writings,' it is not "abstract entities" generally that he questions-indeed, he takes sensory qualia as "basic" in his Carnap-inspired constructional system in Structure-but rather just those abstract that are so crystal clear in their identity conditions, so fundamental to our thought, so prevalent and seemingly unavoidable in our discourse and theorizing that they have come to form the generally accepted framework for the most time- honored, exact, sophisticated, refined, central, and secure branch of human knowledge yet devised, mathematics itself! Of all the abstract to question, why sets? Of course, Goodman gave his "reasons", the unintelligibility of "generating" an infinitude of "constructed objects" automatically from any given object or objects. But critics have been quick to point out that set theory is intended not as a theory of what can be "generated" or "constructed" from given objects in any literal sense but rather as a theory of a certain realm of objects independently existing in their own right. "Construction" is a metaphor.
The aim of this paper is to defend mathematical trivialism|the view that the truths of
pure mathematics have trivial truth-conditions and the falsities of pure mathematics have
trivial falsity-conditions.
Which Abstraction Principles are Acceptable? Some Limitative Results -- Linnebo and Uzquiano,
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Neo-Fregean logicism attempts to base mathematics on abstraction principles. Since not all abstraction principles are acceptable, the neo-Fregeans need an account of which ones are. One of the most promising accounts is in terms of the notion of stability; roughly, that an abstraction principle is acceptable just in case it is satisfiable in all domains of sufficiently large cardinality. We present two counterexamples to stability as a sufficient condition for acceptability and argue that these counterexamples can be avoided only by major departures from the existing neo-Fregean programme.
Intuition Between the Analytic-Continental Divide: Hermann Weyl's Philosophy of the
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Though logical positivism is part of Kant's complex legacy, positivists rejected both Kant's theory of intuition and his classification of mathematical knowledge as synthetic a priori. This paper considers some lingering defenses of intuition in mathematics during the early part of the twentieth century, as logical positivism was born. In particular, it focuses on the difficult and changing views of Hermann Weyl about the proper role of intuition in mathematics. I argue that it was not intuition in general, but his commitment to twodifferent types of intuition, which explains his rather unusual and tormented philosophy of the mathematical continuum.
Multiple Reductions Revisited -- Clarke-Doane 16
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Paul Benacerraf’s argument from multiple reductions consists of a general argument against realism about the natural numbers (the view that numbers are objects), and a limited argument against reductionism about them (the view that numbers are identical with prima facie distinct entities). There is a widely recognized and severe difficulty with the former argument, but no comparably recognized such difficulty with the latter. Even so, reductionism in mathematics continues to thrive. In this paper I develop a difficulty for Benacerraf’s argument against reductionism that is of comparable severity to the now widely
recognized difficulty with his general argument against realism.
recognized difficulty with his general argument against realism.
Abstraction and Additional Nature -- Hale and Wright
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In ‘What is wrong with abstraction’, Michael Potter and Peter Sullivan explain a further objection to the abstractionist programme in the foundations of mathematics which they first presented in their ‘Hale on Caesar’ and which they believe our discussion in The Reason’s Proper Study misunderstood. The aims of the present note are: 1. To get the character of this objection into sharper focus; 2. To explore further certain of the assumptions—primarily, about reference-fixing in mathematics, about certain putative limitations of abstractionist set theory, and about the effects of impredicativity in abstraction principles—which underlie it; and
3. To advance the debate of the issues thereby raised.
3. To advance the debate of the issues thereby raised.
I
Introduction to synthetic differential geometry, and a synthetic theory of dislocations
| Book Series | Lecture Notes in Mathematics |
| Publisher | Springer Berlin / Heidelberg |
| ISSN | 0075-8434 (Print) 1617-9692 (Online) |
| Volume | Volume 1174/1986 |
| Book | Categories in Continuum Physics |
| DOI | 10.1007/BFb0076928 |
| Copyright | 1986 |
| ISBN | 978-3-540-16096-0 |
| DOI | 10.1007/BFb0076934 |
| Pages | 52-68 |
| Subject Collection | Mathematics and Statistics |
| SpringerLink Date | Tuesday, November 14, 2006 |
Neo-Fregeanism: An Embarrassment of Riches--Weir
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Abstract Neo-Fregeans argue that substantial mathematics can be derived from a priori abstraction principles, Hume’s Principle connecting numerical identities with one:one correspondences being a prominent example. The embarrassment of riches objection is that there is a plurality of consistent but pairwise inconsistent abstraction principles, thus not all consistent abstractions can be true. This paper considers and criticizes various further criteria on acceptable abstractions proposed by Wright settling on another one—stability—as the best bet for neo-Fregeans. However, an analogue of the embarrassment of riches objection resurfaces in the metatheory and I conclude by arguing that the neo-Fregean program, at least insofar as it includes a platonistic ontology, is fatally wounded by it.
Toward a Topic-Specific Logicism? Russell's Theory of Geometry in the Principles of
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Russell’s philosophy is rightly described as a programme of reduction of mathematics to logic. Nowthe theory of geometry developed in 1903 does not fit this picture well, since it is deeply rooted in the purely synthetic projective approach, which conflicts with all the endeavours to reduce geometry to analytical geometry. The first goal of this paper is to present an overview of this conception. The second aim ismore far-reaching. The fact that such a theory of geometry was sustained by Russell compels us to question the meaning of logicism: how is it possible to reconcile Russell’s global reductionist standpoint with his local defence of the specificities of geometry?
Toward a Topic-Specific Logicism? Russell's Theory of Geometry in the Principles of
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Russell’s philosophy is rightly described as a programme of reduction of mathematics to logic. Nowthe theory of geometry developed in 1903 does not fit this picture well, since it is deeply rooted in the purely synthetic projective approach, which conflicts with all the endeavours to reduce geometry to analytical geometry. The first goal of this paper is to present an overview of this conception. The second aim ismore far-reaching. The fact that such a theory of geometry was sustained by Russell compels us to question the meaning of logicism: how is it possible to reconcile Russell’s global reductionist standpoint with his local defence of the specificities of geometry?
As any mathematics undergraduate knows, in the hierarchy of
number systems that goes N, Z, Q, R, C, (that is, positive integers,
integers, rationals, reals, complex numbers) the biggest jump in
sophistication is that between Q and R. In order to understand
the real numbers properly, you are forced to think about limits,
whereas the other jumps (including the one from R to C) just
involve finite algebraic manipulations. Why do we bother to
make this effort? How "real" are the real numbers anyway? Most
of the dialogue to follow is between three imagined characters:
M, a mathematician who takes the real numbers for granted, S, a
sceptic who is not prepared to learn about anything without
being absolutely convinced that it is necessary, and U, an
undergraduate who has recently learnt basic analysis. Towards
the end, a logician, L, tries to sort out some of the mess.
(My apologies if, in my ignorance, I put words into the logician's
mouth that no self-respecting logician would utter.)
What anti-realism in philosophy of mathematics must offer
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Abstract This article attempts to motivate a new approach to anti-realism (or nominalism) in the philosophy of mathematics. I will
explore the strongest challenges to anti-realism, based on sympathetic interpretations of our intuitions that appear to support
realism. I will argue that the current anti-realistic philosophies have not yet met these challenges, and that is why they
cannot convince realists. Then, I will introduce a research project for a new, truly naturalistic, and completely scientific
approach to philosophy of mathematics. It belongs to anti-realism, but can meet those challenges and can perhaps convince
some realists, at least those who are also naturalists.
Abstract The article deals with Cantor’s argument for the non-denumerability of reals somewhat in the spirit of Lakatos’ logic of mathematical
discovery. At the outset Cantor’s proof is compared with some other famous proofs such as Dedekind’s recursion theorem, showing
that rather than usual proofs they are resolutions to do things differently. Based on this I argue that there are “ontologically”
safer ways of developing the diagonal argument into a full-fledged theory of continuum, concluding eventually that famous
semantic paradoxes based on diagonal construction are caused by superficial understanding of what a name is.
Three di erent styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they di ffer? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each system into the others and examining the preservation and loss of mathematical content thereby.
Platonism and Aristotelianism in Mathematics -- Pettigrew 16 (3): 310 -- Philosophia Mathematica
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Philosophers of mathematics agree that the only interpretation of arithmetic that takes that discourse at ‘face value’ is one on which the expressions ‘N’, ‘0’, ‘1’, ‘+’, and ‘×’ are treated as proper names. I argue that the interpretation on which these expressions are treated as akin to free variables has an equal claim to be the default interpretation of arithmetic. I show that no purely syntactic test can distinguish proper names from free variables, and I observe that any semantic test that can
must beg the question. I draw the same conclusion concerning areas of mathematics beyond arithmetic.
must beg the question. I draw the same conclusion concerning areas of mathematics beyond arithmetic.
"Putting Structuralism in Its Place"
Talk given to the Stanford logic colloquium in January, 2009, and the conference on philosophy of mathematics at NYU in April, 2009. May appear in an RSL volume of proceedings of the latter event.
To download as a Word document click here.
"Two Undecidable Questions About Group Actions"
A short note revisiting some basic questions of invariant descriptive set theory from my student years. Written for a special issue of Sankhya, the journal of the Indian Statistical Institute, dedicated to the memory of the institute's former director Ashok Maitra, who was one of my teachers.
To download as a Word document click here.
Abstract This paper offers a novel method for nominalizing metalogic without transcending first-order reasoning about physical tokens (inscriptions, etc.) of proofs. A kind of double-negation scheme is presented which helps construct, for any platonistic statement in metalogic, a nominalistic statement which has the same assertability condition as the former. For instance, to the platonistic statement 
there is a (platonistic) proof of A in deductive system D
corresponds the nominalistic statement there is no (metalogical) proof token in (possibly informal) set theory for the claim that there is no proof of A in D. And it is argued that the nominalist can use all the platonistic results by transforming them into such nominalistic correlates.
there is a (platonistic) proof of A in deductive system D corresponds the nominalistic statement there is no (metalogical) proof token in (possibly informal) set theory for the claim that there is no proof of A in D. And it is argued that the nominalist can use all the platonistic results by transforming them into such nominalistic correlates.Quine, Putnam, and the ‘Quine–Putnam’ Indispensability Argument
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Abstract Much recent discussion in the philosophy of mathematics has concerned the indispensability argument—an argument which aims
to establish the existence of abstract mathematical objects through appealing to the role that mathematics plays in empirical
science. The indispensability argument is standardly attributed to W. V. Quine and Hilary Putnam. In this paper, I show that
this attribution is mistaken. Quine’s argument for the existence of abstract mathematical objects differs from the argument
which many philosophers of mathematics ascribe to him. Contrary to appearances, Putnam did not argue for the existence of
abstract mathematical objects at all. I close by suggesting that attention to Quine and Putnam’s writings reveals some neglected
arguments for platonism which may be superior to the indispensability argument.
Abstract Some mathematicians and philosophers contend that set theory plays a foundational role in mathematics. However, the development
of category theory during the second half of the twentieth century has encouraged the view that this theory can provide a
structuralist alternative to set-theoretical foundations. Against this tendency, criticisms have been made that category theory
depends on set-theoretical notions and, because of this, category theory fails to show that set-theoretical foundations are
dispensable. The goal of this paper is to show that these criticisms are misguided by arguing that category theory is entirely
autonomous from set theory.
ABSTRACT. In this paper, I present a summary of the philosophical relationship between Wittgenstein and Brouwer, taking as my point of departure Brouwer's lecture on March 10, 1928 in Vienna. I argue that Wittgenstein having at that stage not done serious philosoph ical work for years, if one is to understand the impact of that lecture on him, it is better to compare its content with the remarks on logics and mathematics in the Tractactus. I thus show that Wittgenstein's position, in the Tractactus, was already quite close to Brouwer's and that the points of divergence are the basis to Wittgenstein's later criticisms of intu itionism. Among the topics of comparison are the role of intuition in mathematics, rule following, choice sequences, the Law of Excluded Middle, and the primacy of arithmetic over logic. During
Solving the Caesar Problem Without Categorical Sortals
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Abstract The neo-Fregean account of arithmetical knowledge is centered around the abstraction principle known as Hume’s Principle: for any concepts X and Y, the number of X’s is the same as the number of Y’s just in case there is a 1–1 correspondence between X and Y. The Caesar Problem, originally raised by Frege in §56 of Die Grundlagen der Arithmetik, emerges in the context of the neo-Fregean programme, because, though Hume’s Principle provides a criterion of identity for
objects falling under the concept of Number–namely, 1–1 correspondence—the principle fails to deliver a criterion of application.
That is, it fails to deliver a criterion that will tell us which objects fall under the concept Number, and so, leaves unanswered
the question whether Caesar could be a number. Hale and Wright have recently offered a neo-Fregean solution to this problem.
The solution appeals to the notion of a categorical sortal. This paper offers a reconstruction of their solution, which has
the advantage over Hale and Wright’s original proposal of making clear what the structure of the background ontology is. In
addition, it is shown that the Caesar Problem can be solved in a framework more minimal than that of Hale and Wright, viz. one that dispenses with categorical sortals. The paper ends by discussing an objection to the proposed neo-Fregean solutions,
based on the idea that Leibniz’s Law gives a universal criterion of identity. This is an idea that Hale and Wright reject.
However, it is shown that a solution very much in keeping with their own proposal is available, even if it is granted that
Leibniz’s Law provides a universal criterion of identity.
This book looks at the Quine-Putnam indispensability argument in the philosophy of mathematics.
This argument urges us to place mathematical entities on the same ontological footing as other theoretical
entities indispensable to our best scientific theories. The argument has come under serious scrutiny in recent
times with many influential philosophers unconvinced of its cogency. This book outlines the indispensability argument
in considerable detail before defending it against various challenges.
In his Grundgesetze, Frege hints that prior to his theory that cardinal numbers are objects (courses-of-values) he had an “almost completed” manuscript on cardinals. Taking this early theory to have been an account of cardinals as second-level functions, this paper works out the significance of the fact that Frege’s cardinal numbers (as objects) is a theory of concept-correlates. Frege held that, where n>2, there is a one–one correlation between each n-level function and an n−1 level function, and a one–one correlation between each first-level function and an object (a course-of-values of the function). Applied to cardinals, the correlation offers new answers to some perplexing features of Frege’s philosophy. It is shown that within Frege’s concept-script, a generalized form of Hume’s Principle is equivalent to Russell’s Principle of Abstraction – a principle Russell employed to demonstrate the inadequacy of definition by abstraction. Accordingly, Frege’s rejection of definition of cardinal number by Hume’s Principle parallels Russell’s objection to definition by abstraction. Frege’s correlation thesis reveals that he has a way of meeting the structuralist challenge (later revived by Benacerraf, 1965) that it is arithmetic, and not a privileged progression of objects, that matters to the finite cardinals.
From absolute to local mathematics
Wittgenstein on Circularity in the Frege-Russell Definition of Cardinal Number -- De Bruin 16 (3):
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Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of number turns out to be valid on its own terms, even though it depends on two epistemological principles the logicist may find too ‘constructivist’.
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Does Category Theory Provide a Framework for Mathematical Structuralism? -- HELLMAN 11 (2): 129 --
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Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out relative to such domains; puzzles about ‘large categories’ and ‘proper classes’ are handled in a uniform way, by relativization, sustaining insights of Zermelo.
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Structuralism Without Structures -- GEOFFREY 4 (2): 100 -- Philosophia Mathematica
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Recent technical developments in the logic of nominalism make it possible to improve and extend significantly the approach to mathematics developed in Mathematics without Numbers. After reviewing the intuitive ideas behind structuralism in general, the modal-structuralist approach as potentially class-free is contrasted broadly with other leading approaches. The machinery of nominalistic ordered pairing (Burgess-Hazen-Lewis) and plural quantification (Boolos) can then be utilized to extend the core systems of modal-structural arithmetic and analysis respectively to full, classical, polyadic third- and fourthorder number theory. The mathenatics of many structures of central importance in functional analysis, measure theory, and topology can be recovered within essentially these frameworks.
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Learning from Questions on Categorical Foundations -- McLarty 13 (1): 44 -- Philosophia Mathematica
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We can learn from questions as well as from their answers. This paper urges some things to learn from questions about categorical foundations for mathematics raised by Geoffrey Hellman and from ones he invokes from Solomon Feferman.
Categories in Context: Historical, Foundational, and Philosophical -- Landry and Marquis 13 (1): 1
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The aim of this paper is to put into context the historical, foundational and philosophical significance of category theory. We use our historical investigation to inform the various category-theoretic foundational debates and to point to some common elements found among those who advocate adopting a foundational stance. We then use these elements to argue for the philosophical position that category theory provides a framework for an algebraic in re interpretation of mathematical structuralism. In each context, what we aim to show is that, whatever the significance of category theory, it need not rely upon any set-theoretic underpinning.
Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-mathematics -- Shapiro
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There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the slogan that mathematics is the science of structure?
PA-weird: 2nd order PA plus a formal statement of inconsistency for 2nd order PA... is formally consistent! Shapiro employs this fact to make the point that we need an assertory meta-mathematics (e.g. set-theoretic satisfiability) to evaluate the coherence of a proposed structure.
Are the axioms of set theory assertoric or algebraic? Related questions. Is BLV subject to the Caesar objection? Are sets "quasi-concrete" in Parsons' sense? Does quasi-concreteness imply an adequate degree of completeness?
Shapiro: "So if we insist on the letter of our second theoretical option, and if statements of relative consistency are assertory, as seems obvious, then they are not mathematical. But it seems equally obvious that relative consistency is a mathematical matter." Perhaps statements of relative consistency express logical knowledge, BUT they are known using mathematical methods (Field's worry in "Is Mathematical Knowledge Just Logical Knowledge"). So, what about Wittgenstein's point here that mathematical methods and mathematical propositions need not be identified? That is, the use of mathematical methods to prove logical propositions about mathematical structures is RULE circular, not LOGICALLY circular, one might argue.
Labels:
philosophy of math, category theory, Shapiro, structuralism, Hilbert, Frege
Non-actual model systems discussed in scientific theories are compared to fictions in literature. This comparison may help
with the understanding of similarity relations between models and real-world target systems. The ontological problems surrounding
fictions in science may be particularly difficult, however. A comparison is also made to ontological problems that arise in
the philosophy of mathematics.
Horwich in "Stipulation, Meaning, and Apriority" factors the neoFregean use of Hume's Principle into two distinct steps. The first is: (HM) (|e|%)(|a|F)(|a|G)(%F=%G <--> F~G). This says that there is some function from concepts to objects that holds IFF the concepts may be placed in 1-1 correspondence. The second step is: (HS) (|e|%)(|a|F)(|a|G)(%F=%G <--> F~G) --> (|a|F)(|a|G)(the number of Fs = the number of Gs <--> F~G). Horwich notes that it is possible to be a skeptic about (HM), which is an instance of Frege's schema (BLV), while accepting (HS) as providing an implicit/contextual meaning for the locution "the number of _s".
Labels:
philosophy, philosophy of math, abstracta, Hume's Principle, Basic Law V
The Philosophical Sense of Theaetetus' Mathematics
Author(s): M. F. Burnyeat
Author(s): M. F. Burnyeat
Degrees of Belief
| Book Series | Synthese Library |
| Subject | Philosophy, Epistemology, Philosophy of Science, Probability Theory and Stochastic Processes, Artificial Intelligence (incl. Robotics), Logic and Probability and Statistics in Computer Science |
| Volume | Volume 342 |
| Subject | Philosophy, Epistemology, Philosophy of Science, Probability Theory and Stochastic Processes, Artificial Intelligence (incl. Robotics), Logic and Probability and Statistics in Computer Science |
| Publisher | Springer Netherlands |
| DOI | 10.1007/978-1-4020-9198-8 |
| Copyright | 2009 |
| ISBN | 978-1-4020-9197-1 (Print) 978-1-4020-9198-8 (Online) |
| Subject Collection | Humanities, Social Sciences and Law |
| Subject | Philosophy, Epistemology, Philosophy of Science, Probability Theory and Stochastic Processes, Artificial Intelligence (incl. Robotics), Logic and Probability and Statistics in Computer Science |
| SpringerLink Date | Sunday, December 21, 2008 |