We study the bi-embeddability and elementary bi-embeddability relation on graphs under Borel reducibility and investigate the degree spectra realized by these relations. We first give a Borel reduction from embeddability on graphs to elementary embeddability on graphs. As a consequence we obtain that elementary bi-embeddability on graphs is a $\boldsymbol {\Sigma }^1_1$ complete equivalence relation. We then investigate the algorithmic properties of this reduction. We obtain that elementary bi-embeddability on the class of computable graphs is $\Sigma ^1_1$ complete with respect (...) to computable reducibility and show that the elementary bi-embeddability and bi-embeddability spectra realized by graphs are related. (shrink)

We calculate the complexity of Scott sentences of scattered linear orders. Given a countable scattered linear order L of Hausdorff rank $\alpha $ we show that it has a ${d\text {-}\Sigma _{2\alpha +1}}$ Scott sentence. It follows from results of Ash [2] that for every countable $\alpha $ there is a linear order whose optimal Scott sentence has this complexity. Therefore, our bounds are tight. We furthermore show that every Hausdorff rank 1 linear order has an optimal ${\Pi ^{\mathrm {c}}_{3}}$ (...) or ${d\text {-}\Sigma ^{\mathrm {c}}_{3}}$ Scott sentence and give a characterization of those linear orders of rank $1$ with ${\Pi ^{\mathrm {c}}_{3}}$ optimal Scott sentences. At last we show that for all countable $\alpha $ the class of Hausdorff rank $\alpha $ linear orders is $\boldsymbol {\Sigma }_{2\alpha +2}$ complete and obtain analogous results for index sets of computable linear orders. (shrink)

We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, \ bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of \ bi-embeddable categoricity and relative \ bi-embeddable categoricity coincide for equivalence structures for \. We also prove that computable equivalence structures have degree of bi-embeddable categoricity \, or \. We furthermore obtain results (...) on the index set complexity of computable equivalence structure with respect to bi-embeddability. (shrink)

In this paper, we give a characterization of the strong degrees of categoricity of computable structures greater or equal to [Formula: see text]. They are precisely the treeable degrees — the least degrees of paths through computable trees — that compute [Formula: see text]. As a corollary, we obtain several new examples of degrees of categoricity. Among them we show that every degree [Formula: see text] with [Formula: see text] for [Formula: see text] a computable ordinal greater than [Formula: see (...) text] is the strong degree of categoricity of a rigid structure. Using quite different techniques we show that every degree [Formula: see text] with [Formula: see text] is the strong degree of categoricity of a structure. Together with the above example this answers a question of Csima and Ng. To complete the picture we show that there is a degree [Formula: see text] with [Formula: see text] that is not the degree of categoricity of a rigid structure. (shrink)

We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $\omega $ and that non-standard models of true arithmetic must have Scott rank greater than $\omega $. Other than that there are no restrictions. By giving a reduction via $\Delta ^{\mathrm {in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $\omega $ -jump of models of an arbitrary completion T of $\mathrm {PA}$ we (...) show that every countable ordinal $\alpha>\omega $ is realized as the Scott rank of a model of T. (shrink)

We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the bi-embeddable categoricity spectrum of a structure A as the family of Turing degrees that compute embeddings between any computable bi-embeddable copies of A; the degree of bi-embeddable categoricity of A is the least degree in this spectrum (if it exists). We extend many known results about categoricity spectra to the case of bi-embeddability. In particular, we exhibit structures without degree of bi-embeddable categoricity, and (...) we show that every degree d.c.e above 0(α) for α a computable successor ordinal and 0(λ) for λ a computable limit ordinal is a degree of bi-embeddable categoricity. We also give examples of families of degrees that are not bi-embeddable categoricity spectra. (shrink)

We study the algorithmic complexity of isomorphic embeddings between computable structures.

We study degree spectra of structures with respect to the bi-embeddability relation. The bi-embeddability spectrum of a structure is the family of Turing degrees of its bi-embeddable copies. To facilitate our study we introduce the notions of bi-embeddable triviality and basis of a spectrum. Using bi-embeddable triviality we show that several known families of degrees are bi-embeddability spectra of structures. We then characterize the bi-embeddability spectra of linear orderings and study bases of bi-embeddability spectra of strongly locally finite graphs.

Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and bi-reducibility. These spectra provide a natural way of measuring the complexity of reductions between equivalence relations. We prove that any upward closed collection of Turing degrees with a countable basis can be realised as a reducibility spectrum or as a bi-reducibility spectrum. We show also that there is a reducibility (...) spectrum of computably enumerable equivalence relations with no countable basis and a reducibility spectrum of computably enumerable equivalence relations which is downward dense, thus has no basis. (shrink)

Critical Theory: The Key Concepts introduces over 200 widely-used terms, categories and ideas drawing from new historicism, postmodernism, psychoanalysis, Marxism, narratology and other approaches. Entries range from concise definitions to longer explanatory essays and include terms such as :EgoDesireConsumptionHypertextKitschMisogynyQueer StudiesSymbolSuperstructureRaceFeaturing cross-referencing throughout, a substantial bibliography and index, this accessible and easy-to-use guide is an invaluable introduction for anyone studying critical theory.

What contribution can visual art practice bring to interdisciplinary research? And how to give an account of practice-led research that acknowledges the need for interdisciplinary intelligibility? I consider these two questions by reflecting on the methodology--which I call "critical practical analogy" (CPA)--that I have developed while investigating the metaethical implications of French philosopher Simone Weil's notion of attention, during my practice-led PhD. In order to address the first question, I consider as a case study a research art project that employs (...) CPA, and I explain how CPA proved instrumental in overcoming the impasse that I reached by purely theoretical investigation of Weil's discourse on attention and how it opened a distinctly artistic way forward in my research. In order to address the second question, I consider a problem posed by the interdisciplinary nature of my research (covering art and philosophy). I show how, through the application of CPA to the case study, I articulated an exegesis of my research that was intelligible across these two heterogeneous fields of investigation. In conclusion, I give some reasons for my hope that CPA may possess some heuristic and exegetical applicability in practice-led interdisciplinary research beyond my own research. (shrink)

This book is intended as a historical and critical study on the origin of the equations of motion as established in Newton's Principia. The central question that it aims to answer is whether it is indeed correct to ascribe to Galileo the inertia principle and the law of falling bodies. In order to accomplish this task, the study begins by considering theories on the motion of bodies from classical antiquity, and especially those of Aristotle. The theories developed during the Middle (...) Ages and the Renaissance are then reviewed, with careful analysis of the contributions of, for example, the Merton and Parisian Schools and Galileo's immediate predecessors, Tartaglia and Benedetti. Finally, Galileo's work is examined in detail, starting from the early writings. Excerpts from individual works are presented, to allow the texts to speak for themselves, and then commented upon. The book provides historical evidence both for Galileo's dependence on his forerunners and for the major breakthroughs that he achieved. It will satisfy the curiosity of all who wish to know when and why certain laws have been credited to Galileo. (shrink)

In Boccacio's Decameron, Cervigni sees a parodic echo of the circles of Dante's Divine Comedy, and asks whether Bocaccio envisions the voyage of the brigata as similar to Dante the Pilgrim's journey toward the center, first the abysmal center of Lucifer, then towards the highest center, God.

This book describes in detail the various theories on the shape of the Earth from classical antiquity to the present day and examines how measurements of its form and dimensions have evolved throughout this period. The origins of the notion of the sphericity of the Earth are explained, dating back to Eratosthenes and beyond, and detailed attention is paid to the struggle to establish key discoveries as part of the cultural heritage of humanity. In this context, the roles played by (...) the Catholic Church and the philosophers of the Middle Ages are scrutinized. Later contributions by such luminaries as Richer, Newton, Clairaut, Maupertuis, and Delambre are thoroughly reviewed, with exploration of the importance of mathematics in their geodetic enterprises. The culmination of progress in scientific research is the recognition that the reference figure is not a sphere but rather a geoid and that the earth’s shape is oblate. Today, satellite geodesy permits the solution of geodetic problems by means of precise measurements. Narrating this fascinating story from the very beginning not only casts light on our emerging understanding of the figure of the Earth but also offers profound insights into the broader evolution of human thought. (shrink)

This article examines Paul Guyer’s claim that we should attempt to ground the principle of religious freedom on the basis of Kant’s arguments for religious liberty. I problematise Guyer’s suggestion by investigating a hypothetical ‘dogmatic conflict’ between a scientifically and a religiously grounded belief. I further suggest that considering Christian Wolff’s philosophy might provide us with an approach which shares the benefits that Guyer identifies in Kant, while at the same time avoiding the issues Kant might run into that result (...) from the occurrence of the dogmatic conflict. I start by providing a background to Wolff’s philosophy and explaining the notion of the dogmatic conflict. Then I present a potential contemporary case of the dogmatic conflict and try to see how it would be dealt with based on Guyer’s proposal. Finally, I consider what a Wolffian solution would look like, arguing that Guyer’s project might benefit from considering Wolff. (shrink)

This paper presents Christian Wolff’s claim that philosophy, undertaken on the basis of a proper method, cannot contradict revealed religion. The paper first provides a context of Wolff’s banishment from Halle for holding views in conflict with religious doctrines. Next, it proceeds, on the basis of Wolff’s Discursus præliminaris de philosophia in genere prefixed to his 1728 Latin Logic, to explain the principles of Wolff’s method, and to show how his conception of method enables him to disallow the possibility of (...) a genuine conflict between philosophical and religious dogmas. For Wolff, doctrinal conflicts between philosophy and revealed religion can only occur as a result of terminological disagreements, disagreements between dogmas and hypotheses, or disagreements between dogmas and theological misinterpretations. The actual conflict of dogmas, understood as religious or philosophical truths, Wolff holds to be impossible. (shrink)

This thesis investigates the development of ontology as a philosophical discipline in the German philosophical tradition. It starts from what can be considered the invention of ontology and proceeds to the way it was received in the philosophy of Hegel. It is separated into two parts. The first part argues that what can be called the ‘traditional’ form of ontology is developed by Christian Wolff in his 1730 monograph Philosophia prima sive Ontologia, and it traces both the history of the (...) name ‘ontology’, as well as the history of the conception which led to Wolff’s formulation of it. The history of the name tracks the use of the concept ‘ontology’ from its first occurrence in 1606 up to Wolff. The history of the conception tracks the conceptions of various philosophical disciplines, found in thinkers such as Aristotle, Aquinas, and Spinoza, that ultimately give rise to Wolff’s conception of ontology as a science of an entity qua entity. The second part traces the development of this Wolffian conception through the philosophical systems of Kant and Hegel. The aim of this thesis is to argue that Wolff’s philosophy should be seen as the original formulation of the philosophical discipline of ontology and that the Wolffian conception of ontology is the one shared by subsequent German thinkers up to, and including, Hegel. I refer to this shared understanding of what ontology is as ‘the German ontological tradition’. The title of the thesis, The Possibility of Ontology, refers to the way in which this traditional understanding of what ontology is, is treated throughout the German ontological tradition. Specifically, Kant argues that the traditional conception is effectively impossible, while in Hegel one can find arguments that are intended to show that some aspects of this traditional discipline are in fact possible. Besides focusing on a fairly under-researched topic of the early history of ontology as a philosophical discipline, this thesis attempts to utilise its historical findings in order to provide novel ways in which the systems of the thinkers such as Kant and Hegel can be understood. There is a serious disregard for, or underplaying of, the Wolffian influence on the philosophical thought of Kant and Hegel, and it is my aim to contribute to the rectification of this situation by demonstrating the frequently overlooked dialogue these thinkers had with Wolff’s conception of ontology. (shrink)

The origin of the genetic code has been attributed in part to an accidental assignment of codons to amino acids. Although several lines of evidence indicate the subsequent expansion and improvement of the genetic code, the hypothesis of Francis Crick concerning a frozen accident occurring at the early stage of genetic code evolution is still widely accepted. Considering Crick’s hypothesis, mathematical descriptions of hypothetical scenarios involving a huge number of possible coexisting random genetic codes could be very important to explain (...) the origin and evolution of a selected genetic code. This work aims to contribute in this regard, that is, it provides a theoretical framework in which statistical parameters of error functions are calculated. Given a genetic code and an amino acid property, the functional code robustness is estimated by means of a known error function. In this work, using analytical calculations, general expressions for the average and standard deviation of the error function distributions of completely random codes with standard stop codons were obtained. As a possible biological application of these results, any set of amino acids and any pure or mixed amino acid properties can be used in the calculations, such that, in case of having to select a set of amino acids to create a genetic code, possible advantages of natural selection of the genetic codes could be discussed. (shrink)

The new wave of ‘foundation models’—general-purpose generative AI models, for production of text (e.g., ChatGPT) or images (e.g., MidJourney)—represent a dramatic advance in the state of the art for AI. But their use also introduces a range of new risks, which has prompted an ongoing conversation about possible regulatory mechanisms. Here we propose a specific principle that should be incorporated into legislation: that any organization developing a foundation model intended for public use must demonstrate a reliable detection mechanism for the (...) content it generates, as a condition of its public release. The detection mechanism should be made publicly available in a tool that allows users to query, for an arbitrary item of content, whether the item was generated (wholly or partly) by the model. In this paper, we argue that this requirement is technically feasible and would play an important role in reducing certain risks from new AI models in many domains. We also outline a number of options for the tool’s design, and summarize a number of points where further input from policymakers and researchers would be required. (shrink)

Il termineha cominciato a essere impiegato nel nostro paese con riferimento all'integrazione sociale di gruppi minoritari solo a partire dagli anni sessanta, per sostituire il concetto ormai logoro di. Prendendo in esame le piů recenti normative italiane ed europee l'articolo mostra come le nuove politiche disegnino un ritorno, al di sotto di una terminologia rimasta immutata, alla concettualitÀ assimilazionista. Questo ritorno, in controtendenza rispetto alla progressiva giuridificazione e proceduralizzazione dell'appartenenza messa in moto dall'emergere nel secondo dopoguerra della politica dei diritti (...) umani, sembra spingere le nostre societÀ politiche indietro verso una pericolosa ri-sostanzializzazione del concetto di cittadinanza in un processo che, lungi dall'aprire spazi di convivenza culturalmente plurali, sembra condurre concretamente verso una sempre piů spregiudicata e discriminante selezione migratoria funzionale a perpetuare quella sistematica inferiorizzazione economica e simbolica degli immigrati in cui consiste e di cui si nutre il razzismo istituzionale europeo. (shrink)

Since its inception in the late 2000s, supporters of Quentin Meillassoux tended to oppose the movement he founded, speculative materialism, to Derrida and the Derridean community. The arc of Meillassoux's later publications, however, did not support that opposition, especially when it became clear that he neither dismissed correlationism nor set aside literary concerns. These approaches might not be as incompatible as has been supposed. Still, if any affinity is to be established, a first need is to reassess the core of (...) Meillassoux's argument in After Finitude, which has continued to influence his publications. In this article, I attempt not merely to comment upon Meillassoux's argument, but to justify it from within, and so to demonstrate its logic. Then, using a method of re-reading that will be familiar to many, I will attempt to show how that logic reveals speculative materialism and Derridean concerns to be better treated as logical affiliates. Doing so will not presuppose agreement, but will set the stage for more thorough comparisons in future. (shrink)

Derrida became increasingly overt in later years in suggesting that his work displays a rigour, and even a “logic.” Further, it is becoming accepted that deconstruction arose in dialogue with Husserl. In support of these views, this article points out that in 1990 Derrida told us that his first work of 1954 revealsa “law” which guides his career, and that some responses had already arisen there. The work of 1954 is examined, and an interrelated “system” developed by which the responses (...) relate to the law, to help find a common, early and systematic base to apply to Derrida’s oeuvre as it develops. Brief examples will be pointed to in closing to show that this basis subsists, at least in part, in later work. (shrink)