Kochen-Specker Vectors

We give a constructive and exhaustive definition of Kochen-Specker (KS) vectors in a Hilbert space of any dimension as well as of all the remaining vectors of the space. KS vectors are elements of any set of orthonormal states, i.e., vectors in n-dim Hilbert space, H^n, n>3 to which it is impossible to assign 1s and 0s in such a way that no two mutually orthogonal vectors from the set are both assigned 1 and that not all mutually orthogonal vectors are assigned 0. Our constructive definition of such KS vectors is based on algorithms that generate MMP diagrams corresponding to blocks of orthogonal vectors in R^n, on algorithms that single out those diagrams on which algebraic 0-1 states cannot be defined, and on algorithms that solve nonlinear equations describing the orthogonalities of the vectors by means of statistically polynomially complex interval analysis and self-teaching programs. The algorithms are limited neither by the number of dimensions nor by the number of vectors. To demonstrate the power of the algorithms, all 4-dim KS vector systems containing up to 24 vectors were generated and described, all 3-dim vector systems containing up to 30 vectors were scanned, and several general properties of KS vectors were found.Comment: 19 pages, 6 figures, title changed, introduction thoroughly rewritten, n-dim rotation of KS vectors defined, original Kochen-Specker 192 (117) vector system translated into MMP diagram notation with a new graphical representation, results on Tkadlec's dual diagrams added, several other new results added, journal version: to be published in J. Phys. A, 38 (2005). Web page: http://m3k.grad.hr/pavici

Adding an Abstraction Barrier to ZF Set Theory

Much mathematical writing exists that is, explicitly or implicitly, based on set theory, often Zermelo-Fraenkel set theory (ZF) or one of its variants. In ZF, the domain of discourse contains only sets, and hence every mathematical object must be a set. Consequently, in ZF, with the usual encoding of an ordered pair ${\langle a, b\rangle}$, formulas like ${\{a\} \in \langle a, b \rangle}$ have truth values, and operations like ${\mathcal P (\langle a, b\rangle)}$ have results that are sets. Such 'accidental theorems' do not match how people think about the mathematics and also cause practical difficulties when using set theory in machine-assisted theorem proving. In contrast, in a number of proof assistants, mathematical objects and concepts can be built of type-theoretic stuff so that many mathematical objects can be, in essence, terms of an extended typed ${\lambda}$-calculus. However, dilemmas and frustration arise when formalizing mathematics in type theory. Motivated by problems of formalizing mathematics with (1) purely set-theoretic and (2) type-theoretic approaches, we explore an option with much of the flexibility of set theory and some of the useful features of type theory. We present ZFP: a modification of ZF that has ordered pairs as primitive, non-set objects. ZFP has a more natural and abstract axiomatic definition of ordered pairs free of any notion of representation. This paper presents axioms for ZFP, and a proof in ZF (machine-checked in Isabelle/ZF) of the existence of a model for ZFP, which implies that ZFP is consistent if ZF is. We discuss the approach used to add this abstraction barrier to ZF

Parity proofs of the Bell-Kochen-Specker theorem based on the 600-cell

The set of 60 real rays in four dimensions derived from the vertices of a 600-cell is shown to possess numerous subsets of rays and bases that provide basis-critical parity proofs of the Bell-Kochen-Specker (BKS) theorem (a basis-critical proof is one that fails if even a single basis is deleted from it). The proofs vary considerably in size, with the smallest having 26 rays and 13 bases and the largest 60 rays and 41 bases. There are at least 90 basic types of proofs, with each coming in a number of geometrically distinct varieties. The replicas of all the proofs under the symmetries of the 600-cell yield a total of almost a hundred million parity proofs of the BKS theorem. The proofs are all very transparent and take no more than simple counting to verify. A few of the proofs are exhibited, both in tabular form as well as in the form of MMP hypergraphs that assist in their visualization. A survey of the proofs is given, simple procedures for generating some of them are described and their applications are discussed. It is shown that all four-dimensional parity proofs of the BKS theorem can be turned into experimental disproofs of noncontextuality.Comment: 19 pages, 11 tables, 3 figures. Email address of first author has been corrected. Ref.[5] has been corrected, as has an error in Fig.3. Formatting error in Sec.4 has been corrected and the placement of tables and figures has been improved. A new paragraph has been added to Sec.4 and another new paragraph to the end of the Appendi

Parity proofs of the Kochen-Specker theorem based on 60 complex rays in four dimensions

It is pointed out that the 60 complex rays in four dimensions associated with a system of two qubits yield over 10^9 critical parity proofs of the Kochen-Specker theorem. The geometrical properties of the rays are described, an overview of the parity proofs contained in them is given, and examples of some of the proofs are exhibited.Comment: 17 pages, 13 tables, 3 figures. Several new references have been adde

Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem

Aravind and Lee-Elkin (1997) gave a proof of the Bell-Kochen-Specker theorem by showing that it is impossible to color the 60 directions from the center of a 600-cell to its vertices in a certain way. This paper refines that result by showing that the 60 directions contain many subsets of 36 and 30 directions that cannot be similarly colored, and so provide more economical demonstrations of the theorem. Further, these subsets are shown to be critical in the sense that deleting even a single direction from any of them causes the proof to fail. The critical sets of size 36 and 30 are shown to belong to orbits of 200 and 240 members, respectively, under the symmetries of the polytope. A comparison is made between these critical sets and other such sets in four dimensions, and the significance of these results is discussed.Comment: 2 new references added, caption to Table 9 correcte

Remembering the Sea: Personal and Communal Recollections of Maritime Life in Jizan and the Farasan Islands, Saudi Arabia

This is the final version of the article. Available from the publisher via the DOI in this record.People create narratives of their maritime past through the remembering and forgetting of seafaring experiences, and through the retention and disposal of maritime artefacts that function mnemonically to evoke or suppress those experiences. The sustenance and reproduction of the resulting narratives depends further on effective media of intergenerational transmission; otherwise, they are lost. Rapid socio-economic transformation across Saudi Arabia in the age of oil has disrupted longstanding seafaring economies in the Red Sea archipelago of the Farasan Islands, and the nearby mainland port of Jizan. Vestiges of wooden boatbuilding activity are few; long-distance dhow trade with South Asia, the Arabian-Persian Gulf and East Africa has ceased; and a once substantial pearling and nacre (mother of pearl) collection industry has dwindled to a tiny group of hobbyists: no youth dive today. This widespread withdrawal from seafaring activity among many people in these formerly maritime-oriented communities has diminished the salience of such activity in cultural memory, and has set in motion narrative creation processes, through which memories are filtered and selected, and objects preserved, discarded, or lost. This paper is a product of the encounter of the authors with keepers of maritime memories and objects in the Farasan Islands and Jizan. An older generation of men recall memories of their experiences as boat builders, captains, seafarers, pearl divers and fishermen. Their recounted memories are inscribed, and Arabic seafaring terms recorded. The extent of the retention of maritime material cultural items as memorials is also assessed, and the rôle of individual, communal and state actors in that retention is considered. Through this reflection, it becomes clear that the extra-biological memory and archive of the region’s maritime past is sparse; that intergenerational transmission is failing; that the participation of state agencies in maritime heritage creation is highly limited; and that, as a result, memories current among the older generation have limited prospect of survival. These memories, recorded and interpreted here, identify the Farasan Islands as a former centre of the pearling industry in the Red Sea, and identify them and Jizan as open to far-reaching maritime-mediated cultural influences in an era before the imposition of the attributes of the modern nation-state.This study was funded by the Golden Web Foundation (UK registered charity number 1100608), with additional support from the Seven Pillars of Wisdom Trust (UK registered charity number 208669)

Research strategies for organizational history:a dialogue between historical theory and organization theory

If history matters for organization theory, then we need greater reflexivity regarding the epistemological problem of representing the past; otherwise, history might be seen as merely a repository of ready-made data. To facilitate this reflexivity, we set out three epistemological dualisms derived from historical theory to explain the relationship between history and organization theory: (1) in the dualism of explanation, historians are preoccupied with narrative construction, whereas organization theorists subordinate narrative to analysis; (2) in the dualism of evidence, historians use verifiable documentary sources, whereas organization theorists prefer constructed data; and (3) in the dualism of temporality, historians construct their own periodization, whereas organization theorists treat time as constant for chronology. These three dualisms underpin our explication of four alternative research strategies for organizational history: corporate history, consisting of a holistic, objectivist narrative of a corporate entity; analytically structured history, narrating theoretically conceptualized structures and events; serial history, using replicable techniques to analyze repeatable facts; and ethnographic history, reading documentary sources "against the grain." Ultimately, we argue that our epistemological dualisms will enable organization theorists to justify their theoretical stance in relation to a range of strategies in organizational history, including narratives constructed from documentary sources found in organizational archives. Copyright of the Academy of Management, all rights reserved