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.)

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.

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.