Classical Notion Research Articles

Does AI Think?

This article raises a very relevant question today: “Does AI think?” To come up with an answer, it explores several conceptions of knowledge, from classical Aristotelian notions to modern philosophers’ takes on it. For Aristotle, knowledge starts with the senses, and, through epagoge and nous, it grows into an understanding of the nature of the thing known, which is something that does not happen with AI. Modern schools of thoughts are either rationalist or empiricist. For rationalists, knowledge is innate, while empiricists believe all knowledge comes from the senses. As Kant views it, knowledge is “built” on sensorial data with the use of theoretical reason categories. Based on these approaches, it is harder to claim that AI does not think by itself. The article also reviews current philosophers’ views on AI and AI thinking. Summing up, the answer to the initial question depends on the concept of knowledge adopted. As this article shares the classical notion of thinking, it concludes that AI does not think.

Fixed Point Results for Fuzzy Enriched Contraction in Fuzzy Banach Spaces with Applications to Fractals and Dynamic Market Equillibrium

We introduce fuzzy enriched contraction, which extends the classical notion of fuzzy Banach contraction and encompasses specific fuzzy non-expansive mappings. Our investigation establishes both the presence and uniqueness of fixed points considering this broad category of operators using a Krasnoselskij iterative scheme for their approximation. We also show the graphical representation of fuzzy enriched contraction and analyze its graph for different values of beta. The implications of these findings extend to significant results within fuzzy fixed-point theory, enriching the understanding of iterative processes in fuzzy metric spaces. To demonstrate the versatility of our innovative concepts and the associated fixed-point theorems, we provide illustrative examples that showcase their applicability across diverse domains, including the generation of fractals. This demonstrates the relevance of fuzzy enriched contraction to iterated function systems, enabling the study of fractal structures under various contractive conditions. Additionally, we explore practical applications of fuzzy enriched contraction in dynamic market equilibrium, offering new insights into stability and convergence in economic models. Through this unified framework, we open new avenues for both theoretical advancements and real world applications in fuzzy systems.

Relative injective modules, superstability and noetherian categories

We study classes of modules closed under direct sums, [Formula: see text]-submodules and [Formula: see text]-epimorphic images where [Formula: see text] is either the class of embeddings, RD-embeddings or pure embeddings. We show that the [Formula: see text]-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, RD-injective modules, pure injective modules, flat cotorsion modules and [Formula: see text]-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah’s non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability.

Lightly chaotic dynamical systems

In this paper we introduce some weak dynamical properties by using subbases for the phase space. Among them, the notion of light chaos is the most significant. Several examples, which clarify the relationships between this kind of chaos and some classical notions, are given. Particular attention is also devoted to the connections between the dynamical properties of a system and the dynamical properties of the associated functional envelope. We show, among other things, that a continuous map f : X → X , where X is a metric space, is chaotic (in the sense of Devaney) if and only if the associated functional dynamical system is lightly chaotic.

On tame ramification and centers of F$F$‐purity

AbstractWe introduce a notion of tame ramification for general finite covers. When specialized to the separable case, it extends to higher dimensions the classical notion of tame ramification for Dedekind domains and curves and sits nicely in between other notions of tame ramification in arithmetic geometry. However, when applied to the Frobenius map, it naturally yields the notion of center of ‐purity (aka compatibly ‐split subvariety). As an application, we describe the behavior of centers of ‐purity under finite covers — it all comes down to a transitivity property for tame ramification in towers.

Local complementation in Banach spaces and its preservation under free constructions

In this work we consider natural generalizations of local complementation in Banach spaces, which include Lipschitz-local complementation. We show that all these notions are indeed equivalent to the classical notion of local complementation of Banach spaces. As an application, we show that local complementation is naturally preserved under certain free constructions in Functional Analysis, including Lipschitz-free spaces and free Banach lattices.

Regularity of the Siciak-Zaharjuta extremal function on compact Kähler manifolds

We prove that the regularity of the extremal function of a compact subset of a compact Kähler manifold is a local property, and that the continuity and Hölder continuity are equivalent to classical notions of the local L L -regularity and the locally Hölder continuous property in pluripotential theory. As a consequence we give an effective characterization of the ( C α , C α ′ ) (\mathscr {C}^\alpha , \mathscr {C}^{\alpha ’}) -regularity of compact sets, the notion introduced by Dinh, Ma and Nguyen [Ann. Sci. Éc. Norm. Supér. (4) 50 (2017), pp. 545–578]. Using this criterion all compact fat subanalytic sets in R n \mathbb {R}^n are shown to be regular in this sense.

Weaker forms of increasingness of binary operations and their role in the characterization of meet and join operations

Due to the lack of transitivity of the pseudo-order relation of a proper trellis, its meet and join operations are not increasing. In this paper, we identify weaker forms of increasingness of binary operations that are satisfied by the meet and/or join operation of a trellis or lattice. Some of these forms coincide with the classical notion of increasingness when the trellis is a lattice. Furthermore, we demonstrate the role of these weaker forms in the characterization of the meet and join operations of a trellis, lattice or chain as specific idempotent operations.

Unveiling the Journey from the Gut to the Brain: Decoding Neurodegeneration-Gut Connection in Parkinson's Disease.

Parkinson's disease, a classical motor disorder affecting the dopaminergic system of the brain, has been as a disease of the brain, but this classical notion has now been viewed differently as the pathology begins in the gut and then gradually moves up to the brain regions. The microorganisms in the gut play a critical role in maintaining the physiology of the gut from maintaining barrier integrity to secretion of microbial products that maintain a healthy gut state. The pathology subsequently alters the normal composition of gut microbes and causes deleterious effects that ultimately trigger strong neuroinflammation and nonmotor symptoms along with characteristic synucleopathy, a pathological hallmark of the disease. Understanding the complex pathomechanisms in distinct and established preclinical models is the primary goal of researchers to decipher how exactly gut pathology has a central effect; the quest has led to many answered and some open-ended questions for researchers. We summarize the popular opinions and some contrasting views, concise footsteps in the treatment strategies targeting the gastrointestinal system.

Chunk boundaries disrupt dependency processing in an AG: Reconciling incremental processing and discrete sampling.

Language is rooted in our ability to compose: We link words together, fusing their meanings. Links are not limited to neighboring words but often span intervening words. The ability to process these non-adjacent dependencies (NADs) conflicts with the brain's sampling of speech: We consume speech in chunks that are limited in time, containing only a limited number of words. It is unknown how we link words together that belong to separate chunks. Here, we report that we cannot-at least not so well. In our electroencephalography (EEG) study, 37 human listeners learned chunks and dependencies from an artificial grammar (AG) composed of syllables. Multi-syllable chunks to be learned were equal-sized, allowing us to employ a frequency-tagging approach. On top of chunks, syllable streams contained NADs that were either confined to a single chunk or crossed a chunk boundary. Frequency analyses of the EEG revealed a spectral peak at the chunk rate, showing that participants learned the chunks. NADs that cross boundaries were associated with smaller electrophysiological responses than within-chunk NADs. This shows that NADs are processed readily when they are confined to the same chunk, but not as well when crossing a chunk boundary. Our findings help to reconcile the classical notion that language is processed incrementally with recent evidence for discrete perceptual sampling of speech. This has implications for language acquisition and processing as well as for the general view of syntax in human language.

Vacuum Genesis and Spontaneous Emergence of the Universe from Nothing in Reference to the Classical Notion of Causality and Creation ex nihilo*

ABSTRACT The article addresses philosophical and theological reflections inspired by the cosmological models of the origin of the universe from the quantum vacuum (through quantum tunneling) and through a spontaneous emergence from an a-temporal initial state of non-being. A thesis about the possibility of cosmogenesis ex nihilo without the need of God the creator—inspired by these models—is put into question. Its philosophical and theological presuppositions are verified in reference to the classical thought of Aristotle and Aquinas and the categories of potency and act, causality and chance, and metaphysically defined nothingness.

Microglial Dyshomeostasis: A Common Substrate in Neurodevelopmental and Neurodegenerative Diseases

Neurodevelopmental disorders such as autism spectrum disorder (ASD) and attention-deficit hyperactivity disorder (ADHD) are clinically distinct, yet share synaptic dysfunction as a common brain pathophysiology. Neurodegenerative diseases such as Huntington’s disease (HD) entail a neuroinflammatory cascade of molecular and cellular events which can contribute to the death of neurons. Emerging roles for supportive glial cells such as microglia and astrocytes in the ongoing regulation of neural synapses and brain excitability raise the possibility that some of the synaptic pathology and/or inflammatory events could be a direct consequence of malfunctioning glial cells. Focusing on microglia, we cross-examined 12 recently published studies in which microglial dysfunction was induced/identified in a cell-autonomous manner and its functional consequence on neural development, brain volume, functional connectivity, inflammatory response and synaptic regulation were evaluated; in many cases, the onset of symptoms relevant to all three neurodevelopmental disorders were assessed behaviorally. Challenging the classic notion of microglial activation as an inflammatory response to neuropathology, our compilation clarifies that microglial dyshomeostasis itself can consequently disrupt neural homeostasis, leading to neuropathology and symptom onset. This further warranted defining the molecular signatures of context-specific microglial pathology relevant to human diseases.

Collective behavior from surprise minimization

Collective motion is ubiquitous in nature; groups of animals, such as fish, birds, and ungulates appear to move as a whole, exhibiting a rich behavioral repertoire that ranges from directed movement to milling to disordered swarming. Typically, such macroscopic patterns arise from decentralized, local interactions among constituent components (e.g., individual fish in a school). Preeminent models of this process describe individuals as self-propelled particles, subject to self-generated motion and "social forces" such as short-range repulsion and long-range attraction or alignment. However, organisms are not particles; they are probabilistic decision-makers. Here, we introduce an approach to modeling collective behavior based on active inference. This cognitive framework casts behavior as the consequence of a single imperative: to minimize surprise. We demonstrate that many empirically observed collective phenomena, including cohesion, milling, and directed motion, emerge naturally when considering behavior as driven by active Bayesian inference-without explicitly building behavioral rules or goals into individual agents. Furthermore, we show that active inference can recover and generalize the classical notion of social forces as agents attempt to suppress prediction errors that conflict with their expectations. By exploring the parameter space of the belief-based model, we reveal nontrivial relationships between the individual beliefs and group properties like polarization and the tendency to visit different collective states. We also explore how individual beliefs about uncertainty determine collective decision-making accuracy. Finally, we show how agents can update their generative model over time, resulting in groups that are collectively more sensitive to external fluctuations and encode information more robustly.

On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms

Abstract We show that, generically, the unique invariant measure of a sufficiently regular piecewise smooth circle homeomorphism with irrational rotation number and zero mean nonlinearity (e.g. piecewise linear) has zero Hausdorff dimension. To encode this generic condition, we consider piecewise smooth homeomorphisms as generalized interval exchange transformations (GIETs) of the interval and rely on the notion of combinatorial rotation number for GIETs, which can be seen as an extension of the classical notion of rotation number for circle homeomorphisms to the GIET setting.

Nonlocality activation in a photonic quantum network

Bell nonlocality refers to correlations between two distant, entangled particles that challenge classical notions of local causality. Beyond its foundational significance, nonlocality is crucial for device-independent technologies like quantum key distribution and randomness generation. Nonlocality quickly deteriorates in the presence of noise, and restoring nonlocal correlations requires additional resources. These often come in the form of many instances of the input state and joint measurements, incurring a significant resource overhead. Here, we experimentally demonstrate that single copies of Bell-local states, incapable of violating any standard Bell inequality, can give rise to nonlocality after being embedded into a quantum network of multiple parties. We subject the initial entangled state to a quantum channel that broadcasts part of the state to two independent receivers and certify the nonlocality in the resulting network by violating a tailored Bell-like inequality. We obtain these results without making any assumptions about the prepared states, the quantum channel, or the validity of quantum theory. Our findings have fundamental implications for nonlocality and enable the practical use of nonlocal correlations in real-world applications, even in scenarios dominated by noise.

The formal theory of relative monads

We develop the theory of relative monads and relative adjunctions in a virtual equipment, extending the theory of monads and adjunctions in a 2-category. The theory of relative comonads and relative coadjunctions follows by duality. While some aspects of the theory behave analogously to the non-relative setting, others require new insights. In particular, the universal properties that define the algebra object and the opalgebra object for a monad in a virtual equipment are stronger than the classical notions of algebra object and opalgebra object for a monad in a 2-category. Inter alia, we prove a number of representation theorems for relative monads, establishing the unity of several concepts in the literature, including the devices of Walters, the j-monads of Diers, and the relative monads of Altenkirch, Chapman, and Uustalu. A motivating setting is the virtual equipment ▪ of categories enriched in a monoidal category ▪, though many of our results are new even for ▪.

VIPurPCA: Visualizing and Propagating Uncertainty in Principal Component Analysis.

Variables obtained by experimental measurements or statistical inference typically carry uncertainties. When an algorithm uses such quantities as input variables, this uncertainty should propagate to the algorithm's output. Concretely, we consider the classic notion of principal component analysis (PCA): If it is applied to a finite data matrix containing imperfect (i.e., uncertain) multidimensional measurements, its output-a lower-dimensional representation-is itself subject to uncertainty. We demonstrate that this uncertainty can be approximated by appropriate linearization of the algorithm's nonlinear functionality, using automatic differentiation. By itself, however, this structured, uncertain output is difficult to interpret for users. We provide an animation method that effectively visualizes the uncertainty of the lower dimensional map. Implemented as an open-source software package, it allows researchers to assess the reliability of PCA embeddings.

Introduction: Emerging rights and commons in the Global South

The commons are a new concept and a new institution that goes beyond old traditions such as communism, communitarianism, and communalist anarchism. Moreover, the theories of the commons imply a novelty compared to the classical notion of Common Good or to the historical commons. In the first place, they do not understand the commons as a sort of regulative horizon towards which society must tend ––as is distilled from the concept of Common Good; secondly, they start from a materialist approach according to which the community is not something pre-existing for cultural or historical reasons ––as communitarianism would mind–– but is the fruit of a shared activity. Given the number of studies that have been produced on the commons in recent years ––reproductive commons, natural commons, urban commons, global commons, digital commons–– this issue proposes to analyze them from two approaches that have been less emphasized than others: on the one hand, it highlights the role of the commons in the Global South; on the other hand, it seeks to analyze the link between the commons and emerging rights.

Radical Conventionalism and Hinge Epistemology

In the paper I explore some hints one can find in an updated version of Ajdukiewicz’s radical conventionalism that may help to resolve some controversies within hinge epistemology, i.e. a family of positions that invoke Wittgenstein’s idea of groundless grounds of knowledge. In particular I put into doubt whether there is a real difference between pragmatic and transcendental justification of hinges, I reject epistemological disjunctivism, and I argue for anti-realistic reading of truth in a context determined by particular hinges while retaining the classical notion of truth as a regulative idea of knowledge that governs the possible revisions of hinges.

High-Dimensional Analysis for Generalized Nonlinear Regression: From Asymptotics to Algorithm

Overparameterization often leads to benign overfitting, where deep neural networks can be trained to overfit the training data but still generalize well on unseen data. However, it lacks a generalized asymptotic framework for nonlinear regressions and connections to conventional complexity notions. In this paper, we propose a generalized high-dimensional analysis for nonlinear regression models, including various nonlinear feature mapping methods and subsampling. Specifically, we first provide an implicit regularization parameter and asymptotic equivalents related to a classical complexity notion, i.e., effective dimension. We then present a high-dimensional analysis for nonlinear ridge regression and extend it to ridgeless regression in the under-parameterized and over-parameterized regimes, respectively. We find that the limiting risks decrease with the effective dimension. Motivated by these theoretical findings, we propose an algorithm, namely RFRed, to improve generalization ability. Finally, we validate our theoretical findings and the proposed algorithm through several experiments.