For each positive n , two alternative axiomatizations of the theory of strings over n alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the n characters and concatenation as primitives. The other class involves using n character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each n, the two theories are definitionally equivalent [or synonymous in the sense of deBouvere]. It (...) is further shown that each member of one class is synonymous with each member of the other class; thus that all of the theories are definitionally equivalent with each other and with Peano arithmetic. Categoricity of Peano arithmetic then implies categoricity of each of the above theories. (shrink)

Scale invariance provides a principled reason for the physical importance of Hilbert space, the Virasoro algebra, the string mode expansion, canonical commutators and Schrödinger evolution of states, independent of the assumptions of string theory and quantum theory. The usual properties of dimensionful fields imply an infinite, projective tower of conformal weights associated with the tangent space to scale-invariant spacetimes. Convergence and measurability on this tangent tower are guaranteed using a scale-invariant norm, restricted to conformally self-dual vectors. Maps on (...) the resulting Hilbert space are correspondingly restricted to semi-definite conformal weight. We find the maximally- and minimally-commuting, complete Lie algebras of definite-weight operators. The projective symmetry of the tower gives these algebras central charges, giving the canonical commutator and quantum Virasoro algebras, respectively. Using a continuous, m-parameter representation for rank-m tower tensors, we show that the parallel transport equation for the momentum-vector of a particle is the Schrödinger equation, while the associated definite-weight operators obey canonical commutation relations. Generalizing to the set of integral curves of general timelike, self-dual vector-valued weight maps gives a lifting such that the action of the curves parallel transports arbitrary tower vectors. We prove that the full set of Schrödinger-lifted integral curves of a general self-dual map gives an immersion of its 2-dim parameter space into spacetime, inducing a Lorentzian metric on the parameter space. This immersion is shown to satisfy the full variational equations of open string. (shrink)

Does the materiality of a three-dimensional model have an effect on how this model operates in an exploratory way, how it prompts discovery of new mathematical results? Material mathematical models were produced and used during the second half of the nineteenth century, visualizing mathematical objects, such as curves and surfaces—and these were produced from a variety of materials: paper, cardboard, plaster, strings, wood. However, the question, whether their materiality influenced the status of these models—considered as exploratory, technical, or representational—was hardly (...) touched upon. This article aims to approach this question by investigating two case studies: Beltrami’s paper models vs. Dyck’s plaster ones of the hyperbolic plane; and Chisini’s string models of braids vs. Artin’s and Moishezon’s algebraization of these braids. These two case studies indicate that materiality might have a decisive role in how the model was taken into account mathematically: either as an exploratory or rather as a technical or pedagogical object. (shrink)

We study a theory PTO of polynomial time computability on the type of binary strings, as embedded in full lambda calculus with total application and extensionality. We prove that the closed terms of type W → W are exactly the polynomial time operations. This answers a conjecture of Strahm [13].

A structure has a (finite-string) automatic presentation if the elements of its domain can be named by finite strings in such a way that the coded domain and the coded atomic operations are recognised by synchronous multitape automata. Consequently, every structure with an automatic presentation has a decidable first-order theory. The problems surveyed here include the classification of classes of structures with automatic presentations, the complexity of the isomorphism problem, and the relationship between definability and recognisability.

The powerset operator, [MATHEMATICAL SCRIPT CAPITAL P], is compared with other operators of similar type and logical complexity. Namely we examine positive operators whose defining formula has a canonical form containing at most a string of universal quantifiers. We call them ∀-operators. The question we address in this paper is: How is the class of ∀-operators generated ? It is shown that every positive ∀-operator Γ such that Γ ≠ ∅, is finitely generated from [MATHEMATICAL SCRIPT CAPITAL P], the (...) identity operator Id, constant operators and certain trivial ones by composition, ∪ and ∩. This extends results of [3] concerning bounded positive operators. (shrink)

Coordinate formalism on Hilbert manifolds developed in \cite{Kryukov}, \cite{Kryukov1} is further analyzed. The main subject here is a comparison of the ordinary and the string bases of eigenvectors of a linear operator as introduced in \cite{Kryukov}. It is shown that the string basis of eigenvectors is a natural generalization of its classical counterpart. It is also shown that the developed formalism forces us to consider any Hermitian operator with continuous spectrum as a restriction to a space of square (...) integrable functions of a self-adjoint operator defined on a space of generalized functions. In the formalism functional coordinate transformations preserving the norm of strings are now linear isometries rather than the unitary transformations. (shrink)

As is well known, it was only in 1926 that a comprehensive mathematical theory of braids was published—that of Emil Artin. That said, braids had been researched mathematically before Artin’s treatment: Alexandre Theophile Vandermonde, Carl Friedrich Gauß and Peter Guthrie Tait had all attempted to introduce notations for braids. Nevertheless, it was only Artin’s approach that proved to be successful. Though the historical reasons for the success of Artin’s approach are known, a question arises as to whether other approaches to (...) deal with braids existed, approaches that were developed after Artin’s article and were essentially different from his approach. The answer, as will be shown, is positive: Modesto Dedò developed in 1950 another notation for braids, though one, which was afterward forgotten or ignored. This raises a more general question: what was the role of Artin’s notation, or, respectively, Dedò’s, that enabled either the acceptance or the neglect of their theories? More philosophically, can notation be an epistemic technique, prompting new discoveries, or rather, can it also operate an as obstacle? The paper will analyze the method introduced by Dedò to notate braids, and also its history and implications. It aims to show that Dedò, in contrast to Artin, focused on factorizations of braids and the algebraic relations between the operations done on these factorizations. Dedò’s research was done against the background of Oscar Chisini’s research of algebraic curves on the one hand and of Artin’s successful notation of braids on the other hand. Taking this into account, the paper will in addition look into the epistemic role of notation, comparing Dedò’s work with Artin’s, as both presented different notations of braids and their deformations. (shrink)

Winner, 2020 Body and Embodiment Best Publication Award, given by the American Sociological Association Honorable Mention, 2020 Sociology of Sex and Gender Distinguished Book Award, given by the American Sociological Association How the female body has been racialized for over two hundred years There is an obesity epidemic in this country and poor Black women are particularly stigmatized as “diseased” and a burden on the public health care system. This is only the most recent incarnation of the fear of fat (...) Black women, which Sabrina Strings shows took root more than two hundred years ago. Strings weaves together an eye-opening historical narrative ranging from the Renaissance to the current moment, analyzing important works of art, newspaper and magazine articles, and scientific literature and medical journals—where fat bodies were once praised—showing that fat phobia, as it relates to Black women, did not originate with medical findings, but with the Enlightenment era belief that fatness was evidence of “savagery” and racial inferiority. The author argues that the contemporary ideal of slenderness is, at its very core, racialized and racist. Indeed, it was not until the early twentieth century, when racialized attitudes against fatness were already entrenched in the culture, that the medical establishment began its crusade against obesity. An important and original work, Fearing the Black Body argues convincingly that fat phobia isn’t about health at all, but rather a means of using the body to validate race, class, and gender prejudice. (shrink)

This paper proposes an extension of Chaitin's halting probability Ω to a measurement operator in an infinite dimensional quantum system. Chaitin's Ω is defined as the probability that the universal self-delimiting Turing machine U halts, and plays a central role in the development of algorithmic information theory. In the theory, there are two equivalent ways to define the program-size complexity H of a given finite binary string s. In the standard way, H is defined as the length of the (...) shortest input string for U to output s. In the other way, the so-called universal probability m is introduced first, and then H is defined as –log2m without reference to the concept of program-size.Mathematically, the statistics of outcomes in a quantum measurement are described by a positive operator-valued measure in the most general setting. Based on the theory of computability structures on a Banach space developed by Pour-El and Richards, we extend the universal probability to an analogue of POVM in an infinite dimensional quantum system, called a universal semi-POVM. We also give another characterization of Chaitin's Ω numbers by universal probabilities. Then, based on this characterization, we propose to define an extension of Ω as a sum of the POVM elements of a universal semi-POVM. The validity of this definition is discussed.In what follows, we introduce an operator version equation image of H in a Hilbert space of infinite dimension using a universal semi-POVM, and study its properties. (shrink)

I discuss a model inspired from the string/brane framework, in which our Universe is represented (after perhaps appropriate compactification) as a three brane, propagating in a bulk space time punctured by D0-brane (D-particle) defects. As the D3-brane world moves in the bulk, the D-particles cross it, and from an effective observer on D3 the situation looks like a “space-time foam” with the defects “flashing” on and off (“D-particle foam”). The open strings, with their ends attached on the brane, which (...) represent matter in this scenario, can interact with the D-particles on the D3-brane universe in a topologically non-trivial manner, involving splitting and capture of the strings by the D0-brane defects. Such processes are consistently described by logarithmic conformal field theories on the world-sheet of the strings. Physically, they result in effective decoherence of the string matter on the D3 brane, and as a result, of CPT Violation, but of a type that implies an ill-defined nature of the effective CPT operator. Due to electric charge conservation, only electrically neutral (string) matter can exhibit such interactions with the D-particle foam. This may have unique, experimentally detectable (in principle), consequences for electrically-neutral entangled quantum matter states on the brane world, in particular the modification of the pertinent Einstein-Podolsky-Rosen (EPR) Correlation in neutral mesons in an appropriate meson factory. For the simplest scenarios, the order of magnitude of such effects might lie within the sensitivity of upgraded φ-meson factories. (shrink)

This paper deals with Tarski's first axiomatic presentations of the syntax of deductive system. Andrzej Grzegorczyk's significant results which laid the foundations for the formalization of metalogic, are touched upon briefly. The results relate to Tarski's theory of concatenation, also called the theory of strings, and to Tarski's ideas on the formalization of metamathematics. There is a short mention of author's research in the field. The main part of the paper surveys research on the theory of deductive systems initiated by (...) Tarski, in particular research on the axiomatization of the general notion of consequence operation, axiom systems for the theories of classic consequence and for some equivalent theories, and axiom systems for the theories of nonclassic consequence. In this paper the results of Jerzy Supecki's research are taken into account, and also the author's and other people belonging to his circle of scientific research. Particular study is made of his dual characterization of deductive systems, both as systems in regard to acceptance and systems in regard to rejection . Comparison is made, therefore, with axiomatizations of the theories of rejection and dual consequence, and the theory of the usual consequence operation. (shrink)

The crossing property is perhaps the most subtle aspect of the particle-field relation. Although it is not difficult to state its content in terms of certain analytic properties relating different matrixelements of the S-matrix or formfactors, its relation to the localization- and positive energy spectral principles requires a level of insight into the inner workings of QFT which goes beyond anything which can be found in typical textbooks on QFT. This paper presents a recent account based on new ideas derived (...) from “modular localization” including a mathematic appendix on this subject. Its main achievement is the proof of the crossing property from a two-algebra generalization of the KMS condition.The main content is an in-depth criticism of the dual model and its string theoretic extension. The conceptual flaws of these models are closely related to misunderstandings of crossing. The correct interpretation of string theory is that of a dynamic infinite component one particle space where “dynamic” means that, unlike a mere collection of independent free fields, the formalism contains also operators which communicate between the different irreducible Poincaré representations and set the mass/spin spectrum. Whereas in pre-string times there were unsuccessful attempts to achieve this in analogy to the O(4,2) hydrogen spectrum by the use of higher noncompact groups, the superstring in d=9+1, which uses instead (bosonic/fermionic) oscillators obtained from multicomponent chiral currents, is the only known solution of the dynamical infinite component pointlike field (or pointlike generated wave function) project. (shrink)

Under the presupposition that human time perception is distorted in intertemporal choice, this study constructs a time scale in the framework of axiomatic measurement. First, the conditions (homogeneity of degree one or two) to identify the form of a time scale are proposed so that one can determine whether the hyperbolic or exponential is a more suitable function for modeling people’s discounting. Homogeneity of degree one implies that subjective time delay is measured by a power scale and its discount function (...) becomes a power-exponential type, whereas homogeneity of degree two implies that subjective time delay is measured by a log scale and its discount function becomes a hyperbolic type. Second, a method is shown for constructing a model that can reflect subadditive and superadditive discounting. A non-commutative and non-associative concatenation operation is provided to generate a time string consisting of subdivided durations that incorporates the effect of repeated delay with a subdivided duration. The discounting model is yielded by substituting these strings in an exponential model equipped with a generalized time scale, and the determination of subadditivity or superadditivity depends on whether the discount function value of a product by the non-commutative and non-associative operation is, respectively, smaller or greater than that of a product by an operation of the usual extensive structure. (shrink)

Given two strings, text t of length n, and pattern p = p1…pk of length k, and given a natural number w, the subsequence matching problem consists in finding the number of size w windows of text t which contain pattern p as a subsequence, i.e. the letters p1,…,pk occur in the window, in the same order as in p, but not necessarily consecutively . Subsequence matching is used for finding frequent patterns and association rules in databases. We generalize the (...) Knuth–Morris–Pratt pattern matching algorithm; we define a non-conventional kind of RAM, the MP-RAMs which model more closely the microprocessor operations; we design an O on-line algorithm for solving the subsequence matching problem on MP-RAMs. (shrink)

This paper introduces a framework for direct surface composition by online update. The surface string is interpreted as is, with each morpheme in turn updating the input state of information and attention. A formal representation language, Logic of Centering, is defined and some crosslinguistic constraints on lexical meanings and compositional operations are formulated.

Church's Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turing-computable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church's (...) Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing's Thesis, characterizing the effective string functions, and--in particular--the effectively-computable functions on string representations of numbers. (shrink)

Two ways of describing a group are considered. 1. A group is finite-automaton presentable if its elements can be represented by strings over a finite alphabet, in such a way that the set of representing strings and the group operation can be recognized by finite automata. 2. An infinite f.g. group is quasi-finitely axiomatizable if there is a description consisting of a single first-order sentence, together with the information that the group is finitely generated. In the first part of (...) the paper we survey examples of FA-presentable groups, but also discuss theorems restricting this class. In the second part, we give examples of quasi-finitely axiomatizable groups, consider the algebraic content of the notion, and compare it to the notion of a group which is a prime model. We also show that if a structure is bi-interpretable in parameters with the ring of integers, then it is prime and quasi-finitely axiomatizable. (shrink)

We characterize the polynomial time operations as those which are provably total in a first order system, which comprises (untyped) combinatory logic with extensionality, together with positive “safe induction” on the set of binary strings. The formalization of safe induction is inspired by Leivants idea of ramification. We also show how to replace ramification by means of modal logic.

Despite its long history and stunning experimental successes, the mathematical foundation of perturbative quantum field theory is still a subject of ongoing research. This book aims at presenting some of the most recent advances in the field, and at reflecting the diversity of approaches and tools invented and currently employed. Both leading experts and comparative newcomers to the field present their latest findings, helping readers to gain a better understanding of not only quantum but also classical field theories. Though the (...) book offers a valuable resource for mathematicians and physicists alike, the focus is more on mathematical developments. This volume consists of four parts: The first Part covers local aspects of perturbative quantum field theory, with an emphasis on the axiomatization of the algebra behind the operator product expansion. The second Part highlights Chern-Simons gauge theories, while the third examines (semi-)classical field theories. In closing, Part 4 addresses factorization homology and factorization algebras. (shrink)

This article addresses the question of the mechanisms of the emergence of structure and meaning in the biological and physical sciences. It proceeds from an examination of the concept of intentionality and proposes a model of intentional behavior on the basis of results of computer simulations of structural and functional self-organization. Current attempts to endow intuitive aspects of meaningful complexity with operational content are analyzed and the metaphor of DNA as a computer program (the `genetic program') is critically examined in (...) relation to an alternative metaphor of DNA as data. It is argued that relatively simple networks of boolean automata can classify and recognize patterns of binary strings on the basis of non-programmed, self-generated criteria, but lack a capacity for self-observation and interpretation. To overcome this problem it is necessary to clarify the relationships between the goals and underlying mechanisms of a process and between a system and its environment. It will be shown that memory devices that record the histories of interactions are essential for models of conscious and unconscious intentional behavior and that the possibility of infinitely sophisticated - and therefore unprogrammable - machines cannot be avoided. It will be argued that the notion of infinite sophistication allows the ideas of self-organization and physical determinism to be reconciled. These models will be used to suggest how the voluntary aspect of decision-making in general can emerge out of functional self-organizing processes. The conclusion will introduce the notion of `underdetermination' of theories, which imposes an intrinsic limitation on models of complex natural systems - a limitation that, at the same time, may be precisely what makes possible mutual understanding and intersubjectivity. (shrink)

Let us start by some general definitions of the concept of complexity. We take a complex system to be one composed by a large number of parts, and whose properties are not fully explained by an understanding of its components parts. Studies of complex systems recognized the importance of “wholeness”, defined as problems of organization (and of regulation), phenomena non resolvable into local events, dynamics interactions in the difference of behaviour of parts when isolated or in higher configuration, etc., in (...) short, systems of various orders (or levels) not understandable by investigation of their respective parts in isolation. In a complex system it is essential to distinguish between ‘global’ and ‘local’ properties. Theoretical physicists in the last two decades have discovered that the collective behaviour of a macro-system, i.e. a system composed of many objects, does not change qualitatively when the behaviour of single components are modified slightly. Conversely, it has been also found that the behaviour of single components does change when the overall behaviour of the system is modified. There are many universal classes which describe the collective behaviour of the system, and each class has its own characteristics; the universal classes do not change when we perturb the system. The most interesting and rewarding work consists in finding these universal classes and in spelling out their properties. This conception has been followed in studies done in the last twenty years on second order phase transitions. The objective, which has been mostly achieved, was to classify all possible types of phase transitions in different universality classes and to compute the parameters that control the behaviour of the system near the transition (or critical or bifurcation) point as a function of the universality class. This point of view is not very different from the one expressed by Thom in the introduction of Structural Stability and Morphogenesis (1975). It differs from Thom’s program because there is no a priori idea of the mathematical framework which should be used. Indeed Thom considers only a restricted class of models (ordinary differential equations in low dimensional spaces) while we do not have any prejudice regarding which models should be accepted. One of the most interesting and surprising results obtained by studying complex systems is the possibility of classifying the configurations of the system taxonomically. It is well-known that a well founded taxonomy is possible only if the objects we want to classify have some unique properties, i.e. species may be introduced in an objective way only if it is impossible to go continuously from one specie to another; in a more mathematical language, we say that objects must have the property of ultrametricity. More precisely, it was discovered that there are conditions under which a class of complex systems may only exist in configurations that have the ultrametricity property and consequently they can be classified in a hierarchical way. Indeed, it has been found that only this ultrametricity property is shared by the near-optimal solutions of many optimization problems of complex functions, i.e. corrugated landscapes in Kauffman’s language. These results are derived from the study of spin glass model, but they have wider implications. It is possible that the kind of structures that arise in these cases is present in many other apparently unrelated problems. Before to go on with our considerations, we have to pick in mind two main complementary ideas about complexity. (i) According to the prevalent and usual point of view, the essence of complex systems lies in the emergence of complex structures from the non-linear interaction of many simple elements that obey simple rules. Typically, these rules consist of 0–1 alternatives selected in response to the input received, as in many prototypes like cellular automata, Boolean networks, spin systems, etc. Quite intricate patterns and structures can occur in such systems. However, what can be also said is that these are toy systems, and the systems occurring in reality rather consist of elements that individually are quite complex themselves. (ii) So, this bring a new aspect that seems essential and indispensable to the emergence and functioning of complex systems, namely the coordination of individual agents or elements that themselves are complex at their own scale of operation. This coordination dramatically reduces the degree of freedom of those participating agents. Even the constituents of molecules, i.e. the atoms, are rather complicated conglomerations of subatomic particles, perhaps ultimately excitations of patterns of superstrings. Genes, the elementary biochemical coding units, are very complex macromolecular strings, as are the metabolic units, the proteins. Neurons, the basic elements of cognitive networks, themselves are cells. In those mentioned and in other complex systems, it is an important feature that the potential complexity of the behaviour of the individual agents gets dramatically simplified through the global interactions within the system. The individual degrees of freedom are drastically reduced, or, in a more formal terminology, the factual space of the system is much smaller than the product of the state space of the individual elements. That is one key aspect. The other one is that on this basis, that is utilizing the coordination between the activities of its members, the system then becomes able to develop and express a coherent structure at a higher level, that is, an emergent behaviour (and emergent properties) that transcends what each element is individually capable of. (shrink)

The article presents the first results we have obtained studying natural reasoning from a proof-theoretic perspective. In particular we focus our attention on monotonic reasoning. Our system consists of two parts: (i) A Formal Grammar – a multimodal version of classical Categorial Grammar – which while syntactically analysing linguistic expressions given as input, computes semantic information (In particular information about the monotonicity properties of the components of the input string are displayed.); (ii) A simple Natural Logic which derives (monotonicity) (...) inferences using as vehicle the parsed output. The monotonicity markers assigned in the lexicon are propagated through the proofs via a combination of the structural and the logical rules for the unary operators of Multimodal Categorial Grammar (MMCG) [Moo97]. We have chosen to work with an expressive ‘grammar logic’, in order to avoid the use of extra-logical marking devices and extra-logical structural reasoning. Having MMCG as parser, our system is able to make the derivations simply within the logic. This new approach makes the implementation of the theory an easier task. We have implemented the theoretical results, so far obtained, using Grail, a theorem prover for Categorial Grammar Logics [Moo98]. (shrink)

One of Noam Chomsky's earliest contributions is the idea that a theory of the unbounded construction of hierarchical structures should incorporate a computational system that generates the structures. This chapter focuses on the structure building system, what is sometimes called the computational system, as a source of explanation. In some sense it is the fundamental source of explanation in generative grammar, as it accounts for the central question of the unbounded hierarchical nature of the syntax of human language. The architecture (...) of the computational system of early generative grammar involves a device that carries out elementary operations following a procedure that maps strings of symbols to strings of symbols in two passes (a phrase structure sub‐procedure and a transformational sub‐procedure). (shrink)

This paper discusses DNA-based stochastic optimizations under the constraint that the search starts from a given point in a search space. Generally speaking, a stochastic optimization method explores a search space and finds out the optimum or a sub-optimum after many cycles of trials and errors. This search process could be implemented efficiently by ``molecular computing'', which processes DNA molecules by the techniques of molecular biology to generate and evaluate a vast number of solution candidates at a time. We assume (...) the exploration starting from a single point, and propose a method to embody DNA-based optimization under this constraint, because this method has a promising application in the research field of protein engineering. In this application, a string of nucleotide bases encodes a protein possessing a specific activity, which could be given as a value of an objective function. Thus, a problem of obtaining a protein with the optimum or a sub-optimum about the desired activity corresponds to a combinatorial problem of obtaining a base sequence giving the optimum or a sub-optimum in the sequence space. Biologists usually modify a base sequence corresponding to a naturally occurring protein into another sequence giving a desired activity. In other words, they explore the space in the proximity of a natural protein as a start point. We first examined if the optimization methods that involve a single start point, such as simulated annealing, Gibbs sampler, and MH algorithms, can be implemented by DNA-based operations. Then, we proposed an application of genetic algorithm, and examined the performance of this application on a model fitness landscape by computer experiments. These experiments gave helpful guidelines in the embodiments of DNA-based stochastic optimization, including a better design of crossover operator. (shrink)

In our previous work we have introduced loop-type sequent calculi for propositional linear discrete tense logic and proved that these calculi are sound and complete. Decision procedures using the calculi have been constructed for the considered logic. In the present paper we restrict ourselves to the logic with the unary temporal operators “next” and “henceforth always”. Proof-theory of the sequent calculus of this logic is considered, focusing on loop specification in backward proof-search. We describe cyclic sequents and prove that any (...) loop consists of only cyclic sequents. A class of sequents for which backward proof-search do not require loop-check is presented. It is shown how sequents can be coded by binary strings that are used in backward proof-search for the sake of more efficient loop-check. (shrink)

This article focuses on the relevance of computational complexity for cognition. The syntactic items may be expressions that are surface strings. But in general, strings are syntactically ambiguous in that they can be generated in more than one way from atomic expressions and operations. The semantic function must take disambiguated items as arguments. When expressions are ambiguous, expressions cannot be the arguments. Instead, it is common to take the arguments to be terms, whose surface syntax reflects the derivation of the (...) string. The semantic function differs in one other important respect from an arithmetic function, since it maps entities between domains, from a syntactic to an ontic or conceptual domain of meanings. Compositionality helps to explain the rate of success in linguistic communication when the sentence used or the content communicated is new. (shrink)

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of _which number_ is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of _notation_. The purpose of this article is to explore the relationship between (...) the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects. (shrink)

Sociological, economic and evolutionary paradigms of human agency have often seen social agents either as the rational controllers of their fate or as marionettes on the strings of historical, functional or adaptive necessity. They found it therefore difficult to account for the variability, intentionality and creativity of human behaviour and for its frequently redundant or harmful results. This paper argues that human agency is a product of evolution, but that genetic variation and inheritance can only provide a limited explanation of (...) its complex nature. The primary evolutionary problem which human agents face while they are alive is not to adapt to stable environments, but to respond flexibly and creatively to a contingent, uncertain world. Variation and selection therefore take two connected but distinct forms, one external, genetic, and inherited across generations, the other internal and cognitive, and operating during the lifetime of individuals. An examination of this lived part of evolution provides a better understanding of key properties of agency. (shrink)

In this AV column we will have a look at the DUMAND project, a new $10 million detector funded by the US Department of Energy for the detection of ultra-high energy neutrinos. DUMAND stands for Deep Underwater Muon And Neutrino Detector. It is now under construction in Hawaii and will come into operation in 1993-94. It is to be placed almost 3 miles deep on a level stretch of Pacific Ocean bottom about 18 miles west of Keahole Point on (...) the Island of Hawaii. Floats anchored to the ocean bottom about 40 meters apart and arranged in an octagon around a central junction point will support nine long vertical strings of sensitive light detectors. DUMAND will be connected to its land-based laboratory by bundles of fiber optics cables. AT&T will lay the cables to shore, and the US Navy's manned deep submersible DSV Sea Cliff will be used to connect, service, and repair the parts of the detector. (shrink)

A structure has a automatic presentationif the elements of its domain can be named by finite strings in such a way that the coded domain and the coded atomic operations are recognised by synchronous multitape automata. Consequently, every structure with an automatic presentation has a decidable first-order theory. The problems surveyed here include the classification of classes of structures with automatic presentations, the complexity of the isomorphism problem, and the relationship between definability and recognisability.

We reformulate minimalist grammars as partial functions on term algebras for strings and trees. Using filler/role bindings and tensor product representations, we construct homomorphisms for these data structures into geometric vector spaces. We prove that the structure-building functions as well as simple processors for minimalist languages can be realized by piecewise linear operators in representation space. We also propose harmony, i.e. the distance of an intermediate processing step from the final well-formed state in representation space, as a measure of processing (...) complexity. Finally, we illustrate our findings by means of two particular arithmetic and fractal representations. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Baudelaire’s Satanic VersesJonathan Culler (bio)Paul Verlaine was perhaps the first to declare the centrality of Baudelaire to what we may now call modern French studies: Baudelaire’s profound originality is to “représenter puissament et essentiellement l’homme moderne” [599–600]. Whether Baudelaire embodies or portrays modern man, Les Fleurs du mal is seen as exemplary of modern experience, of the possibility of experiencing or dealing with what, taking Paris as the exemplary (...) modern city, we have come to call the modern world. T. S. Eliot wrote, “Baudelaire is indeed the greatest exemplar in modern poetry in any language, for his verse and language is the nearest thing to a complete renovation that we have experienced. But his renovation of an attitude towards life is no less radical and no less important” [426]. And outside the field of literature we find such affirmations as Harold Rosenberg’s dating of “the tradition of the New” to Baudelaire, “who exactly one hundred years ago invited fugitives from the too-narrow world of memory to come aboard with him in search of the new” [11]. Baudelaire, writes another critic, “did more than anyone else in the nineteenth century to make the men and women of his century aware of themselves as moderns.... If we had to nominate a first modernist, Baudelaire would surely be the man” [Berman 132–33].There seems to be considerable agreement on this point, but, surprisingly, there is great difference of opinion about what it is that makes Baudelaire modern and worthy of special attention. Is it, as Albert Thibaudet and Walter Benjamin argue, that he was the first true poet of the city, the first to take the alienated experience of life in the modern city as the norm? Or is it, as Leo Bersani claims, that Baudelaire discovered and displayed the mobility of fantasy and of the desiring imagination? Or is it, as Paul de Man maintains, that Baudelaire invents modern self-consciousness about poetry itself, producing poems that allegorically expose the operations of the lyric?There are many competing accounts of what is most particularly modern and important about Baudelaire, but the one thing on which contentious critics seem to agree is that there is a side of Baudelaire that is of no interest today, that belongs to a bas romantisme and is the very antithesis of Baudelaire’s modernity, of Baudelaire the founder of modern poetry: this is the Baudelaire who invokes demons and the Devil. Most critics today pass over this in silence, but even those who explicitly address this Baudelaire seem to find him an embarrassment. Even the author of a book entitled The Demonic Imagination: Style and Theme in French Romantic Poetry begins his chapter on Baudelaire: “Baudelaire has, by now, ceased to interest us for the reasons which once appeared important: his diabolical Catholicism is a familiar, historical mode of sensibility which neither shocks nor has morbid appeal...” [Houston 85]. And Fredric Jameson distinguishes the modernist and the postmodernist Baudelaires—both worthy of our attention—from what he calls the “second-rate post-Romantic Baudelaire, the Baudelaire [End Page 86] of diabolism and of cheap frisson, the poet of blasphemy and of a creaking and musty religious machinery that was no more interesting in the mid-nineteenth century than it is today” [427].But Baudelaire called his collection “Les Fleurs du mal” and opens it with a poem that declares, “C’est le Diable qui tient le fils qui nous remuent [it’s the Devil who holds the strings that move us]” [OC 1: 5; FE 5]. Can this be dismissed as an irrelevancy—something mistakenly appended to this quintessentially modern poetry? That critics of such different orientations should agree in shunting aside the Satanic Baudelaire suggests that there is something worth investigating here, something disquieting and embarrassing, which may not in fact be merely trivial—which may complicate the story of modernity that has come to depend on Baudelaire as its originator. Perhaps the Satanic Baudelaire would tell us things about modernity we don’t want to know.Certainly the idea of the Devil seems fundamentally at odds with accounts of modernity. Even Christianity itself seems to... (shrink)

I examine notions of equivalence between logics (understood as languages interpreted model-theoretically) and develop two new ones that invoke not only the algebraic but also the string-theoretic structure of the underlying language. As an application, I show how to construe modal operator languages as what might be called typographical notational variants of _bona fide_ first-order languages.

This paper gives a simple method for providing categorial brands of feature-based unification grammars with a model-theoretic semantics. The key idea is to apply the paradigm of fibred semantics (or layered logics, see Gabbay (1990)) in order to combine the two components of a feature-based grammar logic. We demonstrate the method for the augmentation of Lambek categorial grammar with Kasper/Rounds-style feature logic. These are combined by replacing (or annotating) atomic formulas of the first logic, i.e. the basic syntactic types, by (...) formulas of the second. Modelling such a combined logic is less trivial than one might expect. The direct application of the fibred semantics method where a combined atomic formula like np (num: sg & pers: 3rd) denotes those strings which have the indicated property and the categorial operators denote the usual left- and right-residuals of these string sets, does not match the intuitive, unification-based proof theory. Unification implements a global bookkeeping with respect to a proof whereas the direct fibring method restricts its view to the atoms of the grammar logic. The solution is to interpret the (embedded) feature terms as global feature constraints while maintaining the same kind of fibred structures. For this adjusted semantics, the anticipated proof system is sound and complete. (shrink)

This paper provides a foundation for the polarity marking technique introduced by David Dowty [3] in connection with monotonicity reasoning in natural language and in linguistic analyses of negative polarity items based on categorial grammar. Dowty's work is an alternative to the better-known algorithmic approach first proposed by Johan van Benthem [11], and elaborated by Víctor Sánchez Valencia [10]. Dowty's system internalized the monotonicity/polarity markings by generating strings using a categorial grammar whose categories already contain the markings that the earlier (...) system would obtain by separate steps working on an already-derived string. Despite the linguistic advantages of the internalized system, no soundness proof has yet been given for it. This paper offers an account. The leading mathematical idea is to interpret categorial types as preorders (in order to talk about monotonicity in the first place), and then to add a primitive operation to the type hierarchy of taking the opposite of a preorder (in order to capture monotone decreasing functions). At the same time, the use of internalized categories also raises issues. Although these will not be addressed in full, the paper points out possible approaches to them. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 62.2 (2001) 193-210 [Access article in PDF] Zeno Against Mathematical Physics Trish Glazebrook Galileo wrote in The Assayer that the universe "is written in the language of mathematics," and therein both established and articulated a foundational belief for the modern physicist. 1 That physical reality can be interpreted mathematically is an assumption so fundamental to modern physics that chaos and super-strings are examples (...) of physical theories developed on the basis of success in mathematics. The mathematics came first, then the physical theory. Quantum theory is an awkward and similar case in point. Despite the fact that there is much agreement among physicists about the mathematical theory, its physical interpretation remains a matter of controversy. There are several interpretations, all of which challenge our everyday assumptions about reality. This led Bohr in 1935 to call for "a radical revision of our attitude towards the problem of physical reality." 2 Arthur Fine, originally a staunch defender of realist interpretations of quantum theory, saw the success of Bohr's Copenhagen interpretation as the end of Einsteinian realism, and subsequently Fine declared that "realism is well and truly dead." 3 Mathematical operators simply do not correspond with real entities in any conceivable way.Ilkka Niiniluoto has argued convincingly, however, that realism is alive and well in quantum theory. 4 The problem is that Bohr is right: quantum theory can be considered to concern the real only upon radical revision of ordinary conceptions of reality. Quantum theory does not describe the real in any meaningful sense of "the real." Rather, it is mathematical projection: ideal rather than real. [End Page 193]Aristotle did not hold that the physicist deciphers nature mathematically. He distinguished the physicist from the mathematician at Physics 2.2 by analogy to the definitions of "snub nose" and "curved." 5 The definition of "snub nose" requires the mention of a nose, that is, of matter which is inseparable from motion, whereas "curved" and other mathematical concepts like " 'odd' and 'even', 'straight'... 'number,' 'line,' and 'figure,' " which are separable from matter and from motion, are investigated by the mathematician. 6 Aristotle claims that physics is different from mathematics because the mathematician investigates physical lines, but not qua physical, and the physicist investigates mathematical lines "but qua physical, not qua mathematical." 7A peculiar consequence of my argument is that it establishes a common principle between two thinkers whose beliefs are otherwise so fundamentally in opposition. Zeno's paradoxes are commonly read as a denial of motion. For Aristotle, that move is axiomatic to the physicist. At 185a15 he considers arguments against this assumption outside the realm of objections to which the physicist must reply. If I am right that Zeno's paradoxes of motion challenge the very notion of mathematical physics, then he and Aristotle agree, contra Galileo, that the universe is not written in the language of mathematics. They simply come to this point from opposite directions, Zeno from an Eleatic thesis that physical reality is illusion, Aristotle from the natural scientist's pragmatic commitment to a truth about nature.I intend to use Zeno's paradoxes of motion to argue that the application of mathematical concepts to the physical world results in paradox. Zeno's arguments are limited within mathematics to geometry. This paper is accordingly a beginning from which I hope more work can be done later. Furthermore, Zeno's paradoxes are standardly read as directed against anyone who disagrees with Parmenides' thesis that there is but one thing which is motionless and indivisible. The paradoxes achieve this end by means of a reductio against the possibility of dividing space and time. That there are four paradoxes is accordingly explicable by the fact that there are four possibilities for the divisibility of space and time. The Achilles argues against the possibility that space and time are both infinitely divisible; the Dichotomy, against the infinite divisibility of space and finite divisibility of time; the Arrow, against the infinite divisibility of time and the finite divisibility of space; and finally, the Stadium shows that time and space cannot both... (shrink)

The ‘syntax’ and ‘combinatorics’ of my title are what Curry (1961) referred to as phenogrammatics and tectogrammatics respectively. Tectogrammatics is concerned with the abstract combinatorial structure of the grammar and directly informs semantics, while phenogrammatics deals with concrete operations on syntactic data structures such as trees or strings. In a series of previous papers (Muskens, 2001a; Muskens, 2001b; Muskens, 2003) I have argued for an architecture of the grammar in which finite sequences of lambda terms are the basic data structures, (...) pairs of terms syntax, semantics for example. These sequences then combine with the help of simple generalizations of the usual abstraction and application operations. This theory, which I call Lambda Grammars and which is closely related to the independently formulated theory of Abstract Categorial Grammars (de Groote, 2001; de Groote, 2002), in fact is an implementation of Curry’s ideas: the level of tectogrammar is encoded by the sequences of lambda-terms and their ways of combination, while the syntactic terms in those sequences constitute the phenogrammatical level. In de Groote’s formulation of the theory, tectogrammar is the level of abstract terms, while phenogrammar is the level of object terms. (shrink)

Extensive training with a musical instrument results in the automatisation of the bodily operations needed to manipulate the instrument: the performer no longer has to consciously think about the instrument while playing. The ability of the performer to automate operations on the instrument is due to sensorimotor mechanisms that can predict changes in the state of the body and the instrument in response to motor commands. But how strong are these mechanisms? To what extent can we alter the structure of (...) the instrument before they disappear? We performed an exploratory study to understand whether and how sensorimotor predictions survive instrument modification. We asked seven professional violinists to perform repertoire pieces and sight-reading exercises on four different violins: their own, a cheap violin, a small violin, and a violin whose strings had been put on in reverse order. We performed a series of quantitative investigations on performance intonation and duration, and on bowing gestures and errors. The analysis revealed that participants struggled adapting to the altered instruments, suggesting that prediction mechanisms are a function of instrument configuration. In particular, the analysis of bowing errors, intonation, and of performance duration suggested that the performance with the reverse violin was much less fluent and precise than the performer's own instrument; the performance with the small violin was also sub-standard though to a lesser extent. We also observed that violinists were differently affected by instrument modifications, suggesting that the capability to adapt to a new instrument is highly personal. (shrink)

We show how to encode context-free string grammars, linear context-free tree grammars, and linear context-free rewriting systems as Abstract Categorial Grammars. These three encodings share the same constructs, the only difference being the interpretation of the composition of the production rules. It is interpreted as a first-order operation in the case of context-free string grammars, as a second-order operation in the case of linear context-free tree grammars, and as a third-order operation in the case of (...) linear context-free rewriting systems. This suggest the possibility of defining an Abstract Categorial Hierarchy. (shrink)

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between (...) the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects. (shrink)

Scatological shock-merchants, untrained social workers, conflict-zone tourists: from a certain standpoint the relationship between contemporary art and ethics involves a string of negative conjunctions. At their center stands the artist, whose personality and intentions often serve as an ethical measure of the work. This book operates on the basis of a different premise: that artworks themselves have ethical effects, and looking at these effects can tell us about wider processes of social change. As the first full-length study of its (...) kind to outline a positive vision of the ethics of contemporary art, this book distances itself from previous accounts that focus on transgression. The critique of transgressive art is not made on the basis that it is wrong, but that it no longer succeeds on its own terms in societies where language, prohibition and morality are more plastic than they once were. By drawing on the work of Félix Guattari and Jacques Lacan, the book develops a novel theoretical framework that emphasizes the effect of art on subjectivity. Along the way artworks by Kader Attia, Artur Zmijewski, Dora Garcia and others serve as springboards launching discussions of the varied pathways along which a renewed ethics of contemporary art might develop. (shrink)

This work-in-progress paper consists of four points which relate to the foundations and physical realization of quantum computing. The first point is that the qubit cannot be taken as the basic unit for quantum computing, because not every superposition of bit-strings of length n can be factored into a string of n-qubits. The second point is that the “No-cloning” theorem does not apply to the copying of one quantum register into another register, because the mathematical representation of this copying (...) is the identity operator, which is manifestly linear. The third point is that quantum parallelism is not destroyed only by environmental decoherence. There are two other forms of decoherence, which we call measurement decoherence and internal decoherence, that can also destroy quantum parallelism. The fourth point is that processing the contents of a quantum register “one qubit at a time” destroys entanglement. (shrink)

We show that in Kabbalah, the esoteric teaching of Judaism, there were developed ideas of unconventional automata in which operations over characters of the Hebrew alphabet can simulate all real processes producing appropriate strings in accordance with some algorithms. These ideas may be used now in a syllogistic extension of Lindenmayer systems, where we deal also with strings in the Kabbalistic-Leibnizean meaning. This extension is illustrated by the behavior of Physarum polycephalum plasmodia which can implement, first, the Aristotelian syllogistic and, (...) second, a Talmudic syllogistic by qal wa-homer. (shrink)

We consider the question of randomness of the probability ΩU[X] that an optimal Turing machine U halts and outputs a string in a fixed set X. The main results are as follows: ΩU[X] is random whenever X is $\Sigma _{n}^{0}$-complete or $\Pi _{n}^{0}$-complete for some n ≥ 2. However, for n ≥ 2, ΩU[X] is not n-random when X is $\Sigma _{n}^{0}$ or $\Pi _{n}^{0}$ Nevertheless, there exists $\Delta _{n+1}^{0}$ sets such that ΩU[X] is n-random. There are $\Delta _{2}^{0}$ (...) sets X such that ΩU[X] is rational. Also, for every n ≥ 1, there exists a set X which is $\Delta _{n+1}^{0}$ and $\Sigma _{n}^{0}$-hard such that ΩU[X] is not random. We also look at the range of ΩU as an operator. We prove that the set {ΩU[X]: X ⊆ 2<ω} is a finite union of closed intervals. It follows that for any optimal machine U and any sufficiently small real r, there is a set X ⊆ 2<ω recursive in ∅′ ⊕ r, such that ΩU[X] = r. The same questions are also considered in the context of infinite computations, and lead to similar results. (shrink)

We demonstrate ways to incorporate nondeterminism in a system designed to formalize the reasoning of agents concerning their abilities and the results of the actions that they may perform. We distinguish between two kinds of nondeterministic choice operators: one that expresses an internal choice, in which the agent decides what action to take, and one that expresses an external choice, which cannot be influenced by the agent. The presence of abilities in our system is the reason why the usual approaches (...) towards nondeterminism cannot be used here. The semantics that we define for nondeterministic actions is based on the idea that composite actions are unravelled in the strings of atomic actions and tests that constitute them. The main notions used in defining this semantics are finite computation sequences and finite computation runs of actions. The results that we obtain meet our intuitions regarding events and abilities in the presence of nondeterminism. (shrink)