Realizability and recursive mathematics

Section 1: Philosophy, logic and constructivityPhilosophy, formal logic and the theory of computation all bear on problems in the foundations of constructive mathematics. There are few places where these, often competing, disciplines converge more neatly than in the theory of realizability structures. Uealizability applies recursion-theoretic concepts to give interpretations of constructivism along lines suggested originally by Heyting and Kleene. The research reported in the dissertation revives the original insights of Kleene—by which realizability structures are viewed as models rather than proof-theoretic interpretations—to solve a major problem of classification and to draw mathematical consequences from its solution.Section 2: Intuitionism and recursion: the problem of classificationThe internal structure of constructivism presents an interesting problem. Mathematically, it is a problem of classification; for philosophy, it is one of conceptual organization. Within the past seventy years, constructive mathematics has grown into a jungle of fullydeveloped "constructivities," approaches to the mathematics of the calculable which range from strict finitism through hyperarithmetic model theory. The problem we address is taxonomic: to sort through the jungle, set standards for classification and determine those features which run through everything that is properly "constructive."There are two notable approaches to constructivity; these must appear prominently in any proposed classification. The most famous is Brouwer's intuitioniam. Intuitionism relies on a complete constructivization of the basic mathematical objects and logical operations. The other is classical recursive mathematics, as represented by the work of Dekker, Myhill, and Nerode. Classical constructivists use standard logic in a mathematical universe restricted to coded objects and recursive operations.The theorems of the dissertation give a precise answer to the classification problem for intuitionism and classical constructivism. Between these realms arc connected semantically through a model of intuitionistic set theory. The intuitionistic set theory IZF encompasses all of the intuitionistic mathematics that does not involve choice sequences. (This includes all the work of the Bishop school.) IZF has as a model a recursion-theoretic structure, V(A7), based on Kleene realizability. Since realizability takes set variables to range over "effective" objects, large parts of classical constructivism appear over the model as inter¬ preted subsystems of intuitionistic set theory. For example, the entire first-order classical theory of recursive cardinals and ordinals comes out as an intuitionistic theory of cardinals and ordinals under realizability. In brief, we prove that a satisfactory partial solution to the classification problem exists; theories in classical recursive constructivism are identical, under a natural interpretation, to intuitionistic theories. The interpretation is especially satisfactory because it is not a Godel-style translation; the interpretation can be developed so that it leaves the classical logical forms unchanged.Section 3: Mathematical applications of the translation:The solution to the classification problem is a bridge capable of carrying two-way mathematical traffic. In one direction, an identification of classical constructivism with intuitionism yields a certain elimination of recursion theory from the standard mathematical theory of effective structures, leaving pure set theory and a bit of model theory. Not only are the theorems of classical effective mathematics faithfully represented in intuitionistic set theory, but also the arguments that provide proofs of those theorems. Via realizability, one can find set-theoretic proofs of many effective results, and the set-theoretic proofs are often more straightforward than their recursion-theoretic counterparts. The new proofs are also more transparent, because they involve, rather than recursion theory plus set theory, at most the set-theoretic "axioms" of effective mathematics.Working the other way, many of the negative ("cannot be obtained recursively") results of classical constructivism carry over immediately into strong independence results from intuitionism. The theorems of Kalantari and Retzlaff on effective topology, for instance, turn into independence proofs concerning the structure of the usual topology on the intuitionistic reals.The realizability methods that shed so much light over recursive set theory can be applied to "recursive theories" generally. We devote a chapter to verifying that the realizability techniques can be used to good effect in the semantical foundations of computer science. The classical theory of effectively given computational domains a la Scott can be subsumed into the Kleene realizability universe as a species of countable noneffective domains. In this way, the theory of effective domains becomes a chapter (under interpre¬ tation) in an intuitionistic study of denotational semantics. We then show how the "extra information" captured in the logical signs under realizability can be used to give proofs of classical theorems about effective domains.Section 4: Solutions to metamathematical problems:The realizability model for set theory is very tractible; in many ways, it resembles a Boolean-valued universe. The tractibility is apparent in the solutions it offers to a number of open problems in the metamathematics of constructivity. First, there is the perennial problem of finding and delimiting in the wide constructive universe those features that correspond to structures familiar from classical mathematics. In the realizability model, it is easy to locate the collection of classical ordinals and to show that they form, intuitionistically, a set rather than a proper class. Also, one interprets an argument of Dekker and Myhill to prove that the classical powerset of the natural numbers contains at least continuum-many distinct cardinals.Second, a major tenet of Bishop's program for constructivity has been that constructive mathematics is "numerical:" all the properties of constructive objects, including the real numbers, can be represented as properties of the natural numbers. The realizability model shows that Bishop's numericalization of mathematics can, in principle, be accomplished. Every set over the model with decidable equality and every metric space is enumerated by a collection of natural numbers

\u27Struggling with Language\u27 : Indigenous movements for Linguistic Security and the Politics of Local Community

In this article, I explore the relationship between linguistic diversity and political power. Specifically, I outline some of the ways that linguistic diversity has served as a barrier to the centralization of power, thus constraining, for example, the political practice of empire-formation. A brief historical example of this dynamic is presented in the case of Spanish colonialism of the 16th-century. The article proceeds then to demonstrate how linguistic diversity remains tied to struggles against forms of domination. I argue that in contemporary indigenous movements for linguistic security, the languages themselves are not merely conceived of as the object of the political struggle, but also as the means to preserve a space for local action and deliberation – a ‘politics of local community’. I show that linguistic diversity and the devolution of political power to the local level are in a mutually reinforcing relationship. Finally, I consider the implications of this thesis for liberal theorizing on language rights, arguing that such theory cannot fully come to terms with this political-strategic dimension of language struggles

Subsequent Surgery After Revision Anterior Cruciate Ligament Reconstruction: Rates and Risk Factors From a Multicenter Cohort

BACKGROUND: While revision anterior cruciate ligament reconstruction (ACLR) can be performed to restore knee stability and improve patient activity levels, outcomes after this surgery are reported to be inferior to those after primary ACLR. Further reoperations after revision ACLR can have an even more profound effect on patient satisfaction and outcomes. However, there is a current lack of information regarding the rate and risk factors for subsequent surgery after revision ACLR. PURPOSE: To report the rate of reoperations, procedures performed, and risk factors for a reoperation 2 years after revision ACLR. STUDY DESIGN: Case-control study; Level of evidence, 3. METHODS: A total of 1205 patients who underwent revision ACLR were enrolled in the Multicenter ACL Revision Study (MARS) between 2006 and 2011, composing the prospective cohort. Two-year questionnaire follow-up was obtained for 989 patients (82%), while telephone follow-up was obtained for 1112 patients (92%). If a patient reported having undergone subsequent surgery, operative reports detailing the subsequent procedure(s) were obtained and categorized. Multivariate regression analysis was performed to determine independent risk factors for a reoperation. RESULTS: Of the 1112 patients included in the analysis, 122 patients (11%) underwent a total of 172 subsequent procedures on the ipsilateral knee at 2-year follow-up. Of the reoperations, 27% were meniscal procedures (69% meniscectomy, 26% repair), 19% were subsequent revision ACLR, 17% were cartilage procedures (61% chondroplasty, 17% microfracture, 13% mosaicplasty), 11% were hardware removal, and 9% were procedures for arthrofibrosis. Multivariate analysis revealed that patients aged <20 years had twice the odds of patients aged 20 to 29 years to undergo a reoperation. The use of an allograft at the time of revision ACLR (odds ratio [OR], 1.79; P = .007) was a significant predictor for reoperations at 2 years, while staged revision (bone grafting of tunnels before revision ACLR) (OR, 1.93; P = .052) did not reach significance. Patients with grade 4 cartilage damage seen during revision ACLR were 78% less likely to undergo subsequent operations within 2 years. Sex, body mass index, smoking history, Marx activity score, technique for femoral tunnel placement, and meniscal tearing or meniscal treatment at the time of revision ACLR showed no significant effect on the reoperation rate. CONCLUSION: There was a significant reoperation rate after revision ACLR at 2 years (11%), with meniscal procedures most commonly involved. Independent risk factors for subsequent surgery on the ipsilateral knee included age <20 years and the use of allograft tissue at the time of revision ACLR

Astrocytes: biology and pathology

Astrocytes are specialized glial cells that outnumber neurons by over fivefold. They contiguously tile the entire central nervous system (CNS) and exert many essential complex functions in the healthy CNS. Astrocytes respond to all forms of CNS insults through a process referred to as reactive astrogliosis, which has become a pathological hallmark of CNS structural lesions. Substantial progress has been made recently in determining functions and mechanisms of reactive astrogliosis and in identifying roles of astrocytes in CNS disorders and pathologies. A vast molecular arsenal at the disposal of reactive astrocytes is being defined. Transgenic mouse models are dissecting specific aspects of reactive astrocytosis and glial scar formation in vivo. Astrocyte involvement in specific clinicopathological entities is being defined. It is now clear that reactive astrogliosis is not a simple all-or-none phenomenon but is a finely gradated continuum of changes that occur in context-dependent manners regulated by specific signaling events. These changes range from reversible alterations in gene expression and cell hypertrophy with preservation of cellular domains and tissue structure, to long-lasting scar formation with rearrangement of tissue structure. Increasing evidence points towards the potential of reactive astrogliosis to play either primary or contributing roles in CNS disorders via loss of normal astrocyte functions or gain of abnormal effects. This article reviews (1) astrocyte functions in healthy CNS, (2) mechanisms and functions of reactive astrogliosis and glial scar formation, and (3) ways in which reactive astrocytes may cause or contribute to specific CNS disorders and lesions

All the mathematics in the world: logical validity and classical set theory

A recognizable topological model construction shows that any consistent principles of classical set theory, including the validity of the law of the excluded third, together with a standard class theory, do not suffice to demonstrate the general validity of the law of the excluded third. This result calls into question the classical mathematician's ability to offer solid justifications for the logical principles he or she favors

What are the limits of mathematical explanation? Interview with Charles McCarty by Piotr Urbańczyk

An interview with Charles McCarty by Piotr Urbańczyk concerning mathematical explanation

Initial Testing of the Uncrewed Aerial System Pilot Kit (UASP-Kit) in Operational Settings

Pilots for small uncrewed aerial systems (sUAS) are at a disadvantage for building situation awareness of the remote airspace in which they are flying, simply because they are distant from their vehicles. A tool to provide increased air traffic situation awareness for an sUAS pilot is being developed. The UAS pilot kit, “UASP-kit,” is small and self-contained, with its chief capability being to collect and display Automatic Dependent Surveillance-Broadcast reports from local aircraft. UASP-kits were taken into the field, introduced to users during a training course, and then left with them for use throughout the summer fire season. sUAS pilots used the prototypes when it was appropriate during the summer. The UASP-kits were operational for a total of 79 flight-days. Users reported that the UASP-kits supported their situation awareness but also identified several usability issues. The findings contribute to the validation of the UASP-kit, and support continuing the work to improve the tool and develop additional functionality