Bienvenidos !!!

Este blog no tiene ninguna otra finalidad que compartir y ayudar a reflexionar sobre lógica y filosofía de la lógica, filosofía de las matemáticas, de la ciencia etc.
El blog es absolutamente gratuito.Es importante difundir nuestras reflexiones, discusiones, investigaciones y logros en el campo de las disciplinas que nos apasionan .

Gracias por seguir el blog !!!

Claudio Conforti

jueves, 28 de junio de 2012

El día Tau 6.28 comparto The Tau Manifiesto


The Tau Manifesto
Michael Hartl
Tau Day, 2010
updated Tau Day, 2012

1 The circle constant
http://tauday.com/tau-manifesto#sec:getting_to_the_bottom_of_pi


The Tau Manifesto is dedicated to one of the most important numbers in
mathematics, perhaps the most important: the circle constant relating the
circumference of a circle to its linear dimension. For millennia, the circle
has been considered the most perfect of shapes, and the circle constant captures
the geometry of the circle in a single number. Of course, the traditional
choice for the circle constant is —but, as mathematician Bob Palais notes
in his delightful article “ Is Wrong!”,1 is wrong. It’s time to set things
right.
(Note: Mathematically sophisticated readers, including those already familiar
with The Tau Manifesto, can skip directly to Section 5, which is part
of a revision released on Tau Day, 2012. This new section builds an irrefutable
case against Pi.)

martes, 26 de junio de 2012

Russell and His Sources for Non-Classical Logics Irving H. Anellis


Abstract.

My purpose here is purely historical. It is not an attempt to
resolve the question as to whether Russell did or did not countenance
nonclassical logics, and if so, which nonclassical logics, and still less to
demonstrate whether he himself contributed, in any manner, to the development of nonclassical logic. Rather, I want merely to explore and insofar as possible document, whether, and to what extent, if any, Russell interacted with the various, either the various candidates or their, ideas that
Dejnoˇzka and others have proposed as potentially influential in Russell’s
intellectual reactions to nonclassical logic or to the philosophical concepts
that might contribute to his reactions to nonclassical logics.

Time in Philosophical Logic • Peter Øhrstrøm Aalborg University Aalborg Denmark • Per F. V. Hasle Aalborg University Aalborg Denmark


The aim of the study of time in philosophical logic is to provide a conceptual framework for an interdisciplinary study of the nature of time and to formalize and study various conceptions and systems of time. In addition, the introduction of time into logic has led to the development of formal systems, which are particularly well suited to represent and study temporal phenomena such as program execution, temporal databases, and argumentation in natural language.

Historical Background
The philosophy of time is based on a long tradition, going back to ancient thought. It is an accepted wisdom within the field that no attempt to clarify the concept of time can be more than an accentuation of some aspects of time at the expense of others. Plato's statement that time is the "moving image of eternity" and Aristotle's suggestion that "time is the number of motion with respect to earlier and later" are no exceptions (see [17]). According to St. Augustine (354-430) time cannot be satisfactorily described using just one single definition or explanation: "What, then, is time? If no one asks me, I know: if I wish to explain it to one that asketh, I know not." [5, p. 40] Time is not definable in terms of other concepts. On the other hand, according to the Augustinian insight, all human beings have a tacit knowledge of what time is. In a sense, the endeavor of the logic of time is to study important manifestations and structures of this tacit knowledge.
There were many interesting contributions to the study of time in Scholastic philosophy, e.g., the analysis of the notions of beginning and ending, the duration of the present, temporal ampliation, the logic of "while," future contingency, and the logic of tenses.Anselm of Canterbury (ca. 1033-1109), William of Sherwood (ca. 1200-1270), William of Ockham (ca. 1285-1349), John Buridan (ca. 1295-1358), and Paul of Venice (ca. 1369-1429) all contributed significantly to the development of the philosophical and logical analysis of time. With the Renaissance, however, the logical approach to the study of timefell into disrepute, although it never disappeared completely from philosophy.
However, the twentieth century has seen a very important revival of the philosophical study of time. The most important contribution to the modern philosophy of time was made in the 1950s and 1960s by A. N. Prior (1914-1969). In his endeavors, A. N. Prior took great inspiration from ancient and medieval thinkers and especially their work on time and logic.
The Aristotelian idea of time as the number of motion with respect to earlier and later actually unites two different pictures of time, the dynamic and the static view. On the one hand, time is linked to motion, i.e., changes in the world (the flow of time), and on the other hand time can be conceived as a stationary order of events represented by numbers. In his works, A. N. Prior logically analyzed the tension between the dynamic and the static approach to time, and developed four possible positions in regard to this tension. In particular, A. N. Prior used the idea of branching time to demonstrate that there is a model of time which is logically consistent with his ideas of free choice and indeterminism. (See [8, 189 ff.].)
After A. N. Prior's development of formalised temporal logic, a number of important concepts have been studied within this framework. In relation to temporal databases the studies of the topology of time and discussions regarding time in narratives are particularly interesting.

lunes, 25 de junio de 2012

Questions and Answers in an Orthoalgebraic Approach Reinhard Blutner


Abstract

Taking the lead from orthodox quantum theory, I will introduce a handy
generalization of the Boolean approach to propositions and questions: the orthoalgebraic
framework. I will demonstrate that this formalism relates to a formal theory of
questions (or ‘observables’ in the physicist’s jargon). This theory allows formulating
attitude questions, which normally are non-commuting, i.e., the ordering of the questions
affects the answer behavior of attitude questions. Further, it allows the expression
of conditional questions such as “If Mary reads the book, will she recommend it to
Peter?”, and thus gives the framework the semantic power of raising issues and being
informative at the same time. In the case of commuting observables, there are close
similarities between the orthoalgebraic approach to questions and the Jäger/Hulstijn
approach to question semantics. However, there are also differences between the two
approaches even in case of commuting observables. The main difference is that the
Jäger/Hulstijn approach relates to a partition theory of questions whereas the orthoalgebraic
approach relates to a ‘decorated’ partition theory (i.e. the elements of the partition
are decorated by certain semantic values). Surprisingly, the orthoalgebraic approach is
able to overcome most of the difficulties of the Jäger/Hulstijn approach. Furthermore,
the general approach is suitable to describe the different types of (non-commutative)
attitude questions as investigated in modern survey research. Concluding, I will suggest
that an active dialogue between the traditional model-theoretic approaches to
semantics and the orthoalgebraic paradigm is mandatory.

Bohrification of operator algebras and quantum logic Chris Heunen · Nicolaas P. Landsman · Bas Spitters


Abstract

Following Birkhoff and von Neumann, quantum logic has traditionally
been based on the lattice of closed linear subspaces of some Hilbert space, or, more
generally, on the lattice of projections in a von Neumann algebra A. Unfortunately,
the logical interpretation of these lattices is impaired by their nondistributivity and
by various other problems. We show that a possible resolution of these difficulties,
suggested by the ideas of Bohr, emerges if instead of single projections one considers
elementary propositions to be families of projections indexed by a partially ordered set
C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal
generality and ease of use within topos theory, we assume that A is a so-called Rickart
C*-algebra and that C(A) consists of all unital commutative Rickart C*-subalgebras
of A. Such families of projections form a Heyting algebra in a natural way, so that the
associated propositional logic is intuitionistic: distributivity is recovered at the expense
of the law of the excluded middle. Subsequently, generalizing an earlier computation
for n × n matrices, we prove that the Heyting algebra thus associated to A arises as
a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of
the “Bohrification” A of A, which is a commutative Rickart C*-algebra in the topos
of functors from C(A) to the category of sets. We explain the relationship of this

construction to partial Boolean algebras and Bruns–Lakser completions. Finally, we
establish a connection between probability measures on the lattice of projections on a
Hilbert space H and probability valuations on the internal Gelfand spectrum of A for
A = B(H).

The dynamic turn in quantum logic Alexandru Baltag · Sonja Smets

Qué es Lógica Dinámica Proposicional
Abstract:

In this paperweshowhowideas coming from two areas of research in logic
can reinforce each other. The first such line of inquiry concerns the “dynamic turn” in
logic and especially the formalisms inspired by Propositional Dynamic Logic (PDL);
while the second line concerns research into the logical foundations of Quantum Physics,
and in particular the area known as Operational Quantum Logic, as developed by
Jauch and Piron (Helve Phys Acta 42:842–848, 1969), Piron (Foundations of Quantum
Physics, 1976). By bringing these areas together we explain the basic ingredients
of Dynamic Quantum Logic, a new direction of research in the logical foundations of
physics.

Logic for physical space From antiquity to present days Marco Aiello · Guram Bezhanishvili · Isabelle Bloch · Valentin Goranko


Abstract

Since the early days of physics, space has called for means to represent,
experiment, and reason about it. Apart from physicists, the concept of space has
intrigued also philosophers, mathematicians and, more recently, computer scientists.
This longstanding interest has left us with a plethora of mathematical tools developed
to represent and work with space. Here we take a special look at this evolution by
considering the perspective of Logic. From the initial axiomatic efforts of Euclid,
we revisit the major milestones in the logical representation of space and investigate
current trends. In doing so, we do not only consider classical logic, but we indulge
ourselves with modal logics. These present themselves naturally by providing simple
axiomatizations of different geometries, topologies, space-time causality, and vector
spaces.

The logic of empirical theories revisited Johan van Benthem


Abstract

 Logic and philosophy of science share a long history, though contacts
have gone through ups and downs. This paper is a brief survey of some major themes
in logical studies of empirical theories, including links to computer science and current
studies of rational agency. The survey has no new results: we just try to make some
things into common knowledge.

A very brief history of logic and philosophy of science


Looking at famous 19th century authors, it is often hard to separate what we would
now call logicians from philosophers of science. Bolzano’sWissenschaftslehre (1937)
is mainly a classic in logical inference, while Mill’s famous work A System of Logic
(1843) is mainly a classic of scientific methodology. Likewise, Helmholtz’ theory
of transformations and invariants in the foundations of the empirical sciences (1868)
linked to the psychology of perception, reached mathematics, deeply influencing the
logical study of definability. But at the end of the 19th century, things changed. Modern
logic underwent an agenda contraction toward the foundations of mathematics: just
compare the small set of concerns in Frege’s Begriffsschrift (1879) as a model for the
field of logic with the Collected Papers of his contemporary Charles Sanders Peirce
(1933): a rich mixture of formal and informal themes, ranging from common sense
reasoning to science, that is still being mined today.The foundational turn made mathematics the paradigm for logical method (whichit still is) and also the major field of investigation for those methods.
 Even so, a new brand of philosophers of science soon picked up on the new developments, and in the
20th century, too, many major philosophers contributed to both areas, such as Carnap,
Beth, Lewis, Hintikka, or van Fraassen. The main insights and techniques from the
foundational phase concern mathematical proof and formal systems. But in the 1930s,
members of the Vienna Circle and other groups turned these modern tools to the
empirical sciences as well, with Reichenbach and Popper as famous examples. Interests
went both ways, and for instance, Carnap also played a role in logical discussions
at the time (van Benthem 1978a). Logical methods still dominated ‘neo-positivism’
in the 1950s.
This marriage came under attack from several sides around 1960. The external
criticism of Kuhn’s The Structure of Scientific Revolutions (1962) seemed to show
that logic paints a largely false picture of the reasoning underlying actual practice and
progress in science. Added to this, influential internal critics like Suppes observed that
the formal language methodology of logic is irrelevant to scientific practice, where
one goes for the relevant structures with any symbolism at hand, by-passing systemgenerated
issues like first- versus higher-order languages that logicians delight in.
Contacts did not break off, and Philosophical Logic kept many themes alive, such as
conditional reasoning and causality, that meander through logic and the philosophy of
science. But through the 1970s, logic became friends with disciplines where languages
do play a central role, in particular, computer science and linguistics. Simultaneously,
many philosophers of science defected to probabilistic methods. Contacts between the
fields atrophied—and sometimes, even a certain animosity could be observed.
Through the 1980s and 1990s, however, many themes have emerged that are again
common to the two fields, often with a new impetus from a shared interest in computation.
I will discuss a number of these and earlier themes in this paper, and show how
a new liaison may be in the air. My emphasis will be, not on shiny new logic tools that
philosophers of science should use, but more symmetrically, on shared interests.




New logical perspectives on physics Johan van Benthem · Sonja Smets


Synthese (2012) 186:615–617
DOI 10.1007/s11229-011-9911-y


This special issue is situated at the interface between Logic and the Foundations of
Physics. This interface, though not as active as the logical foundations of mathematics,
has long existed—with highlights such as “quantum logic”, or studies of the general
logical structure of physical theories. In recent years, more themes have come to the
fore, and we may be witnessing a revival. The papers presented here emanate from a
symposium held at the University of Utrecht in January 2008 with the aim of charting
established as well as new connections between the two fields. One of the main questions
discussedwas whether and howmodern techniques coming from logic, computer
science and information theory might be combined with state-of-the-art insights in the
philosophy of physics to gain a better understanding of the main foundational issues
and open problems in modern physics. The success of this symposium has shown that
there are several possible answers to this question. The invited papers in this issue
present the reader with an overview of the main topics at play right now. A common

feature is that all authors make essential use of logical and formal methods in physics
and point out new interesting connections between the two fields.
Research in logic has made essential progress in the last decades, along many
dimensions that seem relevant to the foundations of physics. One conspicuous strand
concerns mathematical depth. Much traditional research in the foundations of mathematics
has now begun to blend with powerful more mainstream mathematical developments
in algebra, category theory and other fields, making logical techniques more
widely available inmathematics and, in principle also, mathematical physics. Another
noticeable trend is an ever-growing amalgam of logic and formal theories of computation
and processes—perhaps the bulk of logic research as pursued today—ranging
from modal logics (spatial logic, dynamic logic and temporal logic of actions) to prooftheory-
inspired linear logic and other resource-sensitive logics, game logics, process
algebras, coalgebraic logics, etcetera. Finally, there has also been an extension of
descriptive coverage in another direction, with what has been called a “dynamic turn”
toward interaction and communication between intelligent agents, bringing logic in
touch with artificial intelligence, game theory, social choice theory, linguistics, cognitive
science, and other disciplines modeling human behaviour in an exact manner.
The papers in this volume testify to the vitality of logic in this modern sense. For
instance, modal and spatial logics can provide efficient formal tools to talk about the
qualitative temporal and spatial evolution of dynamical systems. These logics can
handle a large variety of interactive properties of processes and they can also be used
to formalize various conceptions of space. The paper by M. Aiello, G. Bezhanishvili,
I. Bloch and V. Goranko on “Logic for Physical Space” gives an overview of some
of these developments by highlighting new logical perspectives on spatial structures.
This reflects the larger emerging area documented in the “Handbook of Spatial Logics”,
edited by M. Aiello, J. van Benthem and I. Pratt-Hartmann (Springer, Dordrecht
2008).
Another current trend in logic of potential interest for physics and the philosophy
of physics, is reflected in the paper by H. Andréka, J. Madarasz, I. Németi and
G. Székely on “A Logic Road from Special Relativity to General Relativity”. Inspired
by the grand traditions of algebraic logic and classical model theory in the study of
geometry, the authors present a detailed first-order analysis of the structure of both
Special and general relativity theory, throwing surprising new light on their not always
evident logical connections. This paper is at the same time a characteristic sample of
the ‘Budapest School’ at the interface of logic, space-time geometry, and physics.
Another recent trend in Logic combines the use of proof theory and categorical
logic with insights from the foundations of physics, as pursued in the Oxford projects
of S. Abramsky and B. Coecke. The paper of B. Coecke and R.W. Spekkens on “Picturing
Classical and Quantum Bayesian Inference” takes this categorical line of work
one step further into the direction of a graphical representation of Bayesian inference
and quantum causal relations. Another topic in this line of research, is the paper by
S. Abramsky on “Big ToyModels” in which the author shows how Chu spaces can be
used to represent physical systems including both classical and quantum systems.
A next trend of interest relates to intuitionistic logic and Heyting algebra, i.e., the
constructive foundations of mathematics, now brought to bear on the foundations
of quantum physics. The paper by C. Heunen, N.P. Landsman and B. Spitters on

“Bohrification of Operator Algebras and Quantum Logic” shows how an intuitionistic
approach can shed new light on the difficulties and problems of traditional quantum
logic. While traditional quantum logic has its merits, it also confronts us with deep
questions that touch upon the roots of logic itself. In particular the original work
of Birkhoff and von Neumann has left both philosophers and logicians wondering
whether empirical theories like quantum physics can really provide principled weakening
of our classical logical principles.
While the preceding paper opts for the intuitionistic approach of weakening classical
logic, the paper “The Dynamic Turn in Quantum Logic” by A. Baltag and
S. Smets connects traditional quantum logicwith the above-mentioned dynamic turn in
logic. It shows how the non-classical character of quantum logic can also be diagnosed
quite differently, as the result of bringing dynamic actions of measurement on suitable
quantum information systems explicitly into the logic. The result is a classical propositional
logic with explicit operators for quantum measurement actions by observing
agents, creating one more interface between logic, physics and computation.
Finally, the latter direction of work ties in nicely with the survey paper of J. van
Benthem on “The Logic of Empirical Theories Revisited”. The author first looks at
the interface of logic and philosophy of science as it has functioned over the last
century, and recalls some of its main trends. He then argues that bringing in agency
and informational action explicitly into logical systems leads to an overhaul of formal
models in the philosophy of science, bringing it much closer to the actual dynamics
of observation, communication, and other activities that make up scientific inquiry.
All logical methods in this volume reach beyond traditional styles of formalizing
physical theories. It is our hope that by promoting work in this direction, physicists,
philosophers of physics and logicians can come closer together, and find that they
have much more common ground than is often supposed. But we do not just see this
as a one-way street of importing ideas. While new logical methods may be of relevance
to physics, we are also well aware that physical models of information flow and
even social behaviour may come to form a natural companion to existing logical and
computational ones.
These are just our editorial views: now the authors will take the stage, and we are
grateful for what they have provided so generously in the following pages. We also
take this opportunity to thank all participants of the 2008 Symposium on Logic and
Physics held at Utrecht University. Special thanks go to our fellow members of the
organizing committee : Dennis Dieks, Anne Kox, and Albert Visser. Finally, we are
happy tomention the sponsors who made this Symposium possible: The EvertWillem
Beth Stichting, The Heyting Stichting, and the Disciplinegroep Theoretische Filosofie
at the University of Utrecht.



sábado, 23 de junio de 2012

CHALLENGING TURING 2012 : NEW PERSPECTIVES ON COMPUTATION A Stanford University Conference

http://challengingturing.org/

Mi homenaje a Alan Turing desde este blog para el lógico, matematico, filósofo, padre de la Inteligencia Artificial, citando y remitiéndolos a este link excelente de Standford.

viernes, 15 de junio de 2012

Toward a Dynamic Logic of Questions

Johan van Benthem · ¸ Stefan Minic¢a


Abstract
Questions are triggers for explicit events of ‘issue management’.We
give a complete logic in dynamic-epistemic style for events of raising, refining,
and resolving an issue, all in the presence of information flow through observation
or communication. We explore extensions of the framework to multiagent
scenarios and long-term temporal protocols. We sketch a comparison
with some alternative accounts.

1 Introduction and Motivation
Questions are different from statements, but they are just as important in
driving reasoning, communication, and general processes of investigation. The
first logical studies merging questions and propositions seem to have come
from the Polish tradition:.... 

Artificial language philosophy of science Sebastian Lutz


Abstract

 Artificial language philosophy (also called ‘ideal language philosophy’)
is the position that philosophical problems are best solved or dissolved
through a reform of language. Its underlying methodology—the development
of languages for specific purposes—leads to a conventionalist view of language
in general and of concepts in particular. I argue that many philosophical
practices can be reinterpreted as applications of artificial language philosophy.
In addition, many factually occurring interrelations between the sciences and
philosophy of science are justified and clarified by the assumption of an
artificial language methodology.

Stephan Hartmann · Jan Sprenger The future of philosophy of science: introduction


Philosophy, perhaps more than any other academic discipline, likes to reflect
upon itself. Thus, it is no surprise that philosophers regularly ask questions
such as:What is the scope of philosophy, what are its important questions, and
what are the proper methods to address them? Asking these questions also
means to take stock and to enquire where the discipline is going.
This is an especially worthwhile activity in contemporary philosophy of
science as this field has been changing rapidly since its institutional consolidation
in the 1950s. For present purposes we may very roughly, but still
usefully, describe this change as having three phases. In the first phase, which
lasted until the mid 1960s, philosophy of science was dominated by Logical
Empiricism and formal approaches to philosophically analyzing science. The
second phase began in the late 1960s and lasted until the second half of the
1980s. It brought the naturalistic turn, a critique of the Logical Empiricist’s
picture of science as too far away from the actual practice of science, a focus
on the history and the social structure of science, a shift from theories as the
primary target of philosophical analysis to models and experimental practices,
and many detailed case studies.
While the Logical Empiricists gave us a grand general picture of science,
the naturalists’ working assumption has been that aiming at such a picture
underappreciates the complexity and diversity of real science. Moreover,
the naturalists moved normative questions into the background. In the third
phase, which began in the late 1980s, the dichotomy between normative
and descriptive approaches in philosophy of science still persists, but the
picture has become even more complex. So we can only list a number of
novel trends: Metaphysical questions, which were famously dismissed by the
Logical Empiricists, are gaining a considerable interest. Philosophies of the
special sciences are booming, and more and more subdisciplines are emerging.
Formal epistemologists are applying a variety of mathematical methods to
address normative questions in general philosophy of science. Social aspects
of science are systematically studied and mathematically modeled. Another
interesting development is the rise of experimental approaches to problems
from philosophy of science (e.g., causation) and its combination with formal
approaches.
This (incomplete) list shows that contemporary philosophers of science
address a large variety of topics. They also use many different methods, quite
similar to scientists who often use a combination of methods, or import a
method from another field to solve their problems. This is, to our mind, a
fruitful way of conducting “scientific philosophy”: a proper combination of
conceptual analysis, historical or contemporary case studies, formal modeling,
and experimental work that will lead to many new and exciting insights.
To find out whether our views were shared by our colleagues, and to
explore their expectations about the development of our discipline, we organized
two events at the Tilburg Center for Logic and Philosophy of Science
(TiLPS) in April 2010. The first event was a one-day workshop on “Scientific
Philosophy—Past and Future”; the second event was the three-day Sydney-
Tilburg conference “The Future of Philosophy of Science” with invited lectures
by Michael Friedman, ChristopherHitchcock, Hannes Leitgeb and Samir
Okasha. It is fair to say that the conference exceeded our expectations, and
that it was a full success. Not only because of the surprisingly high quality of
the papers and presentations, but also because of the cheerful atmosphere that
the participants brought to Tilburg, and because of the memorable volcano
eruption in Iceland that tied some guests involuntarily to the Netherlands,
leading to several fruitful collaborations.
The present special issue gathers five papers that were presented at the
conference. All of them address the conference topic from a different perspective,
and all of them represent a specific way of doing philosophy of science.
James Justus engages in a defense of Carnapian explication as a normative
account of concept determination for complex concepts in empirical science.
In support of his argument, Justus presents a case study from theoretical
biology. Sebastian Lutz defends artificial language philosophy—the view that
philosophical problems are best solved by the conventional prescription of a
new language—and explores the methodological implications of this position
for philosophy of science. Chris Hitchcock critically evaluates the status and
promise of philosophical experiments in philosophy of science and argues that
the expertise objection—the appeal to the privileged intuitions of professional
philosophers—is unsatisfactory in a number of ways. Markus Eronen blends
perspectives from philosophy of science and philosophy of mind: he argues
that adopting explanatory pluralism as well as the interventionist account of
causation—two positions that have recently become very popular—leads to
a novel, pluralistic variant of physicalism. Finally, William Bechtel expects
that mechanistic accounts of explanation must be expanded to incorporate
computational modeling, yielding dynamical mechanistic explanations.
We conclude with some words of thanks. First of all, to Mark Colyvan and
Paul Griffiths for co-organizing the conference. Second, to the Netherlands
Organization for Scientific Research (NWO) and TiLPS for generous financial
support. Third, to the authors and referees of the papers in this special issue for
their intellectual devotion and their generous advice. Last but not least, thanks
to Carl Hoefer, the editor-in-chief of EJPS, for his invaluable assistance with
editorial decisions, and for his continuous encouragement and support.

John Mumma · Marco Panza Diagrams in mathematics: history and philosophy



Synthese (2012) 186:1–5
DOI 10.1007/s11229-011-9988-3



1 Introduction

Diagrams are ubiquitous in mathematics. From the most elementary class to the most
advanced seminar, in both introductory textbooks and professional journals, diagrams
are present, to introduce concepts, increase understanding, and prove results. They
thus fulfill a variety of important roles in mathematical practice. Long overlooked by
philosophers focused on foundational and ontological issues, these roles have come
to receive attention in the past two decades, a trend in line with the growing philosophical
interest in actual mathematical practice. Seminal contributions include the
historical/philosophical analysis of diagrams in Euclid’s geometry offered byManders
in his 1995 paper (2008),The date of publication of Manders’ paper is misleading.
Though it was not published until 2008, it was written in 1995 and widely circulated,
and became highly influential as a manuscript. the logical studies of diagrammatic
reasoning contained in Gerard Allwein and Barwise’s compilation (1996), and Netz’s
historical study on the place of diagrams in Greek geometry (1999).
These works exhibit a broad range of intellectual perspectives. To bring people
from these separate perspectives together, two workshops devoted to the history and
philosophy of diagrams inmathematics were jointly organized by Departments of Philosophy
and Classics at the University of Stanford, and REHSEIS (a research center

of the CNRS and the University of Paris 7). The first workshop was held at Stanford
in the fall of 2007, the second in Paris in the fall of 2008. The present special issue is
the direct result of these workshops.
Because of the central position of Euclid’s geometry, in both the history and philosophy
of mathematics, the diagrams of Euclid’s geometry have been a major topic in
the recent research on diagrams inmathematics. This is reflected in themake-up of the
issue. Part I, which contains over half of the contributions to the issue, concerns Euclidean
diagrams, addressing either their place and nature in the manuscript tradition, their
role in Euclid’s geometry, or their relevance to early modern philosophy and mathematics.
The contributions of part II examine mathematical diagrams in more modern
and/or advanced settings. In the remainder of this introduction, we briefly describe the
content of these papers.
The two articles included in section I.a are devoted to geometric diagrams in the
manuscript tradition of ancient Greek texts. As Saito points out in the beginning of
his article, the diagrams we find in modern versions of ancient Greek mathematical
texts are far from representative of those appearing in the manuscripts from preceding
centuries. These manuscripts result from various scribal traditions that exhibit various
practices in the production of diagrams. The general aim of both Saito’s article and
Gregg De Young’s is to illuminate significant features of such practices.
Saito’s concern is with how the diagrams of the manuscript tradition relate to the
geometrical content of the arguments they accompany. They often satisfy conditions
not stipulated to hold (e.g. by instantiating a parallelogram as a square) or alternatively
misrepresent conditions stipulated or proven to hold (e.g. by depicting angles
stipulated as equal as clearly unequal). There are, further, notable features with the
diagrams representing reductio ad absurdam arguments, and the diagrams representing
arguments that range over many cases. To shed light on the different manuscript
traditions in these respects, Saito presents a case study of the manuscript diagrams
that accompany two propositions in book III of Euclid’s Elements.
De Young focuses on a specific manuscript tradition. He explores the basic “architecture”
(i.e. the relation of diagrams with white space and text) of medieval manuscripts
and early printed editions of Euclidean geometry in the Arabic transmission.
Many features of Euclidean diagrams remain constant through a long transmission history,
although differences in scribal abilities exist. Yet there also seems to be a degree
of freedom to adapt diagrams to the architectural context. Adaptation to print brings
subtle changes, such as title pages, page numbers, and fully pointed text. The demands
of a long-held calligraphic aesthetic ideal and the necessity to compete against traditional
manuscripts within the educational marketplace combined to favor the use of
lithography over typography in Arabic printed geometry.
The articles included in section I.b all share the general goal of deepening our philosophical
understanding of the diagrams of Euclid’s geometry. In his paper, Marco
Panza advances an account where geometric diagrams are fundamental to Euclid’s
plane geometry in fixing what the objects of the theory are. Specifically for Panza
geometric diagrams perform two indispensable theoretical roles. First, they provide
the identity conditions for the objects referred to in Euclid’s propositions; second,
they provide the basis via their concrete properties and relations for the attribution
of certain properties and relations to these objects. After explicating this conception

in general terms, Panza demonstrates what it amounts to vis-à-vis the Elements with
a detailed and thorough analysis of the definitions and the first 12 propositions of
book I.
Aconsequence of Panza’s account is that Euclid’s geometry is in a sense a constructive
theory. It does not concern a pre-existing domain of objects, but instead objects
brought into existence by the constructions of geometers. The issue of John Mumma’s
paper is the extent to which such an interpretation of Euclid can be given in formal
terms. At the center of his discussion is a proof system he developed where Euclid’s
diagrams are formalized as part of the system’s syntax. Mumma argues that his formalization
is superior to others in accounting for the spatial character of the theory’s
constructions. It is not immediate, however, that the resulting picture of the theory
qualifies as a constructive one. Mumma closes the article by pointing to the philosophical
work that is needed for it to be understood as such.
A prima facie feature of the diagram based reasoning of Euclid’s geometry is that
perception is involved in a distinctive and essential way. But how is it so involved? The
purpose of Annalisa Coliva’s paper is to develop the philosophical concepts necessary
to address the question. The key concept for Coliva, as the title of her paper indicates,
is that of seeing-as. A segment joining opposite vertices of a square in a diagram can,
for instance, be seen as a diagonal of the square, or the side of a triangle composing
the square. Coliva articulates seeing-as as philosophically sharp notion, and argues
that the ability is required to engage in diagram based geometric reasoning. She also
offers some initial thoughts on how her analysis bears on the status of geometrical
knowledge with respect to the a priori versus a posteriori and analytic versus synthetic
distinctions.
In her paper, Sun-Joo Shin considers Euclid’s diagrammatic arguments in order to
examine commonly held views on the strengths and weaknesses of diagrams in comparison
with symbolic/linguistic representations—namely the views that the latter are
superior for giving proofs and the former are superior for brainstorming and discovery.
Shin relates the first view to Locke and Berkeley’s opposing accounts of the generality
of Euclid’s proofs, and argues that Berkeley’s account offers a way for understanding
diagrams as a legitimate means for general proofs. She proposes that a major reason
behind the second view, aptly illustrated by Euclid’s proofs, is that diagrammatic representations
of individual mathematical objects are effective in generating the right
associations and connections with what is already known.
Section I.c contains papers that explore how the diagrammatic method of Euclid’s
geometry influenced and/or was interpreted by thinkers in the early modern period.
Though mathematics was pushing far beyond classical Greek geometry in this period,
Euclid’s Elements nevertheless maintained its position as a foundational text and thus
formed part of the intellectual context inwhich philosophical or methodological reflections
on mathematics were carried out.
Such reflections in Hume’s Treatise form the subject of Graciela De Pierris’s paper.
In part II of Book I of the Treatise Hume asserts that perceivable yet indivisible minima
compose geometric continua, but do so in a confused manner. Consequently, for
Hume, geometry does not possess the exactness of arithmetic, which concerns clearly
distinct discrete quantities. De Pierris argues that these doctrines follow in a principled
way from an epistemological model in which phenomenologically given sensory

images are fundamental. She also explains how Euclid’s geometry for Hume, despite
its relative inexactitude, qualifies as a demonstrative science in being based on simple,
easily surveyable diagrams.
In her paper, Katherine Dunlop explores the role of diagrams in Newton’s Principia
by considering Newton’s claim that the reasoning of the book is distinctively
geometrical. She notes that Newton conceives geometry as the science of measurement,
and argues that his characterization applies to the propositions of Section II,
Book I. As other scholars have noted, these constitute a foundation in the sense that
they show certain spatial quantities to be related as measures to nonspatial quantities,
in particular, force and time. Dunlop takes Newton’s point to be that the measures
he supplies can be manipulated in the same way as elements of diagrams in classical
geometry. In the case she examines in detail, Newton’s proof of Kepler’s area law, the
principal measure of time is just the diagrammed trace of an inertial motion. Dunlop
contends that Newton aims to share in geometry’s certainty by resting determinations
of equality and relative size (between nonspatial magnitudes) on relationships that are
open to sensory inspection.
In the Critique of Pure Reason Kant often illustrates his claims with geometric
proofs from Euclid’s Elements. One thus might look to recent work on the diagrammatic
character of these proofs to gain a better understanding of Kant’s philosophy of
geometry. In his article, Michael Friedman argues that such a project would be misdirected.
He argues specifically that Kant’s notion of a schema for geometrical concepts
and his distinction between pure and empirical intuition precludes a diagrammatic
interpretation of his theory of geometrical intuition. Such an interpretation would fail,
for Friedman, to explainwhy Kant takes the constructions of Euclid’s geometry to provide
an a priori framework for empirical space. Friedman relates these points to Kant’s
theory of space, and the role of geometry and spatial intuition in the transcendental
deduction of the categories.
Dominique Tournès’s paper, the first of section II.a, serves nicely as bridge between
the papers of part I and those of part II. In the article Tournès provides a survey of
the changing role of diagrams in the history of differential equations from the seventeenth
century—when there were still vestiges of the diagrammatic approach of
Euclid’s geometry—up until the time of Poincaré—when the influence of Euclid on
mathematics was entirely absent. The article is a useful resource for those interested in
understanding diagrams inmathematical subjects that are not ostensibly or exclusively
geometrical in content.
The other contributions of section II.a examine diagrams in the thought and work
of two giant figures in the history and philosophy of modern mathematics.
Danniele Macbeth’s proposes a new way of reading Frege’s logical notation, as
a sort of diagrammatic language within which to exhibit the contents of concepts.
Taking Frege’ s proof of theorem 133 in Part III of his 1879 treatise Begriffsschrift as
an illustrative example, Macbeth aims to show just how a proof in Frege’ s language
(so read) can extend our knowledge despite being strictly deductive.
In his paper, Ivahn Smadja seeks to clarify how Hilbert understood the relative
positions of diagrams and axioms in mathematics. An easy, and overly simplistic,
characterization of Hilbert on the issue would be just to attribute to him the view the
view Shin examines in her contribution—i.e. diagrams belong to the realm of heuris-

tics and discovery while axiomatics belong to the separate realm of justification. That
it would be overly simplistic is revealed by Smadja’s study of the connections between
the diagrams found in Minkowski’s work in number theory and Hilbert’s work in the
foundations of geometry. Smadja specifically describes the links the former have to
Hilbert’s axiomatic investigations of the notion of a straight line, and Hilbert’s concern
with the conceptual compatibility of geometry and arithmetic.
The papers included in section II.c, the last group of papers, concern contemporary
mathematics and linguistics. At the heart of Solomon Feferman’s discussion are
infinite diagrams for the proofs of central theorems in set theory, model theory and
homological algebra. The diagrams are infinite in the sense that they depict infinitely
iterated constructions. Feferman contends that they are more or less essential for
understanding and accepting the proofs, and considers whether this serves as evidence
against the so-called formalizability thesis: the idea that allmathematical proofs admit
of formalization. He argues for the conclusion that it does not in the light of recent
discussions of the formalizability thesis.
Sketch theory, a mathematical theory that treats the diagrams of category theory
as mathematical objects, is the topic of Brice Halimi’s paper. Halimi presents a technically
sophisticated discussion of the theory in order to bring out its relevance to a
variety of philosophical and foundational issues. These are: the capacity of a mathematical
diagram (in category theory and elsewhere) to represent not just a static object
but also the dynamics of a proof; the role and nature of diagrams within category
theory; and the alternative ways category theory and set theory provide a semantics
for mathematics.
A general point of many of the papers discussed so far is that diagrams can have
important theoretical functions within mathematics. Arancha San Gines’s paper provides
evidence that the same is true in linguistics, in the most direct way possible.
In it she presents a diagrammatic account of the role played by anaphoric pronouns
in natural discourse. The account, specifically, is a diagrammatic system of representation
for sentences containing such pronouns. After motivating and describing the
system, she shows how it accounts for a variety of linguistic phenomena that have so
far resisted uniform treatment.
John Mumma & Marco Panza
Stanford, Paris, May 2011
References
Allwein, G., & Barwise, J. (Eds.). (1996). Logical reasoning with diagrams. New York: Oxford University
Press.
Manders, K. (2008). The Euclidean diagram (1995). In P. Mancosu (Ed.), The philosophy of mathematical
practice (pp. 80–133). New York: Oxford University Press.
Netz, R. (1999). The shaping of deduction in Greek mathematics. A study in cognitive history. Cambridge,
USA: Cambridge U. P.




Handbook of Philosophical Logic Vol 11 2nd ed.

Gracias a los aportes de docentes y lógicos amigos he conseguido el Handbook of Philosophical Logic vol.11 2nd ed., tan difícil de encontrar. Lo tengo completo. Por supuesto, que como siempre a quien me solicite un artículo se lo envío.
Era el único handbook que faltaba en mi colección y notamos su ausencia cuando me solicitaron The Logic of Fiction de John Woods.