The main problem discussed in this paper is: Why and how did animal cognition abilities arise? It is argued that investigations of the evolution of animal cognition abilities are very important from an epistemological point of view. A new direction for interdisciplinary researches – the creation and development of the theory of human logic origin – is proposed. The approaches to the origination of such a theory (mathematical models of ``intelligent invention'' of biological evolution, the cybernetic schemes of evolutionary progress (...) and purposeful adaptive behavior) as well as potential interdisciplinary links of the theory are described and analyzed. (shrink)

This is an introductory survey of the emerging theory of two new classes of (discrete, countable) groups, called hyperlinear and sofic groups. They can be characterized as subgroups of metric ultraproducts of families of, respectively, unitary groups U (n) and symmetric groups $S_{n},\ n\in {\Bbb N}$ . Hyperlinear groups come from theory of operator algebras (Connes' Embedding Problem), while sofic groups, introduced by Gromov, are motivated by a problem of symbolic dynamics (Gottschalk's Surjunctivity Conjecture). Open questions are numerous, in particular (...) it is still unknown if every group is hyperlinear and/or sofic. (shrink)

The notion of conditional entropy is extended to noncomposite systems. The \-deformed entropic inequalities, which usually are associated with correlations of the subsystem degrees of freedom in bipartite systems, are found for the noncomposite systems. New entropic inequalities for quantum tomograms of qudit states including the single qudit states are obtained. The Araki–Lieb inequality is found for systems without subsystems.

The problem of commensurability/incommensurability of different cultural codes is a key problem of modern civilizational development. This is the problem of the search for communicative unity in the world of cultural and biological diversity, which has to be protected, and the search for the cohesion of different Umwelten, of semiotically-defined artificial and natural environments, of ecological and cognitive niches, taking into account that each of them has their own identity and uniqueness. The purpose of the article is to draw attention (...) to the fact that the question of the so-called incommensurability of different conceptual schemes, paradigms, language consciousnesses is widely discussed not only in cross-cultural studies and philosophical problems of translation but also in connection with the problems of incommensurability between the language of classical physics and the language of relativistic quantum physics. Attention is drawn to the problem of the incommensurability and correlation of different languages that are used in debates about the foundations of quantum mechanics, its interpretation, comprehension and ontology. Two approaches stand out in this debate. The first approach is based on the language of the formed being, on the language of things localized in time and on the logic of Aristotle. The second approach is based on the language of the becoming, process and nonlocality, on the search for various processual-oriented temporal logics. In this regard, we discuss the processual approach to understanding quantum mechanics, proposed in the philosophical and physical works of D. Bohm. The authors argue that the experience of constructive understanding of the metaproblems of the interpretation of quantum mechanics, the critical reception of the legacy of such philosophers of the process as Peirce, Bergson and Whitehead, the deep reflection on the problems of commensurability/ incommensurability of linguistic consciousnesses of different cultures – will eventually create a common synergetic-interdisciplinary space of cooperation for the solutions of the above-mentioned issues. (shrink)

Tomographic approach to describing both the states in classical statistical mechanics and the states in quantum mechanics using the fair probability distributions is reviewed. The entropy associated with the probability distribution (tomographic entropy) for classical and quantum systems is studied. The experimental possibility to check the inequalities like the position–momentum uncertainty relations and entropic uncertainty relations are considered.

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

The dynamical equations of quantum mechanics are rewritten in the form of dynamical equations for the measurable, positive marginal distribution of the shifted, rotated, and squeezed quadrature introduced in the so-called “symplectic tomography”. Then the possibility of a purely classical description of a quantum system as well as a reinterpretation of the quantum measurement theory is discussed and a comparison with the well-known quasi-probabilities approach is given. Furthermore, an analysis of the properties of this marginal distribution, which contains all the (...) quantum information, is performed in the framework of classical probability theory. Finally, examples of the harmonic oscillator's states dynamics are treated. (shrink)

The small, the tiny, and the infinitesimal have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that written by Errett Bishop about Keisler’s book Elementary Calculus: an Infinitesimal Approach. In this skit we investigate both the argument itself, and some of its roots in Bishop George Berkeley’s criticism of Leibnizian and Newtonian Calculus. We also explore some of the consequences to students for whom the infinitesimal approach is congenial. The (...) casual mathematical reader may be satisfied to read the text of the five act play, whereas the others may wish to delve into the 130 footnotes, some of which contain elucidation of the mathematics or comments on the history. (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for (...) solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)

: El estudio de la percepción y significación de la variable tiempo por parte del profesorado ha sido un aporte en la comprensión de diversos fenómenos educativos. El objetivo de esta investigación cualitativa fue describir e interpretar las teorías subjetivas de docentes sobre el tiempo de enseñanza priorizada en el contexto de pandemia. Se realizaron entrevistas episódicas grupales a 11 profesores de establecimientos escolares de la Región de Coquimbo (Chile). Los datos fueron analizados utilizando procedimientos de la teoría fundamentada, lo (...) que permitió elaborar dos modelos axiales sobre el tiempo de enseñanza priorizada. Los resultados reconstruyeron diversas implicancias de la experiencia profesional en la percepción del tiempo psicológico del profesorado cuando éste gestiona la enseñanza priorizada. Asimismo, las teorías subjetivas de profesores(as) pertenecientes a los diversos grupos socioeconómicos estudiados, variaron en sus explicaciones sobre sus elecciones pedagógicas previo y durante el tiempo de clases remotas virtuales. Se concluyó que una mayor experiencia profesional y la orientación. (shrink)

We have synthesized a 582,970-base pair Mycoplasma genitalium genome. This synthetic genome, named M. genitalium JCVI-1.0, contains all the genes of wild-type M. genitalium G37 except MG408, which was disrupted by an antibiotic marker to block pathogenicity and to allow for selection. To identify the genome as synthetic, we inserted "watermarks" at intergenic sites known to tolerate transposon insertions. Overlapping "cassettes" of 5 to 7 kilobases (kb), assembled from chemically synthesized oligonucleotides, were joined by in vitro recombination to produce intermediate (...) assemblies of approximately 24 kb, 72 kb ("1/8 genome"), and 144 kb ("1/4 genome"), which were all cloned as bacterial artificial chromosomes in Escherichia coli. Most of these intermediate clones were sequenced, and clones of all four 1/4 genomes with the correct sequence were identified. The complete synthetic genome was assembled by transformation-associated recombination cloning in the yeast Saccharomyces cerevisiae, then isolated and sequenced. A clone with the correct sequence was identified. The methods described here will be generally useful for constructing large DNA molecules from chemically synthesized pieces and also from combinations of natural and synthetic DNA segments. 10.1126/science.1151721. (shrink)

Foundations of Science recently published a rebuttal to a portion of our essay it published 2 years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his 2005 book, Conflicts between generalization, rigor, and intuition. He further argues that our attempt to show that Cauchy is part of a long infinitesimalist tradition confuses text with context and thereby misunderstands the significance of Cauchy’s use of infinitesimals. Here we defend our original analysis of various misconceptions and misinterpretations concerning (...) the history of infinitesimals and, in particular, the role of infinitesimals in Cauchy’s mathematics. We show that Schubring misinterprets Proclus, Leibniz, and Klein on non-Archimedean issues, ignores the Jesuit context of Moigno’s flawed critique of infinitesimals, and misrepresents, to the point of caricature, the pioneering Cauchy scholarship of D. Laugwitz. (shrink)

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal (...) calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)

The article discusses the attitude of Georg Lukács and his adherents who formed a circle “Techeniye” toward the phenomenon of Stalinism. Despite the political nature of the topic, the authors are aspired to provide an unbiased research. G. Lukács’ views on the theory and practice of Stalinism evolved over time. In the 1920s Lukács welcomes the idea of creation of socialism in one country and abandons the former revolutionary ideas expressed in his bookHistory and Class Consciousness. This turn is grounded (...) by new interpretation of Hegel as “realistic” thinker whose “realism” was shown in the aspiration to find “reconciliation” with reality and in denial of any utopias. The philosophical evolution leading to “realism” assumes integration of revolutionaries into the hierarchy of existing society. The article “Hölderlin’s Hyperion” represents attempt to justify Stalinism as a necessary and “progressive” phase of revolutionary development of the proletariat. Nevertheless, events of the second half of the 1930s force Lukács to realize the catastrophic nature of political strategy of Stalinism. In his works, Lukács ceases to analyze political topics and concentrates on problems of aesthetics and literary criticism. However, his aesthetic position allows to reconstruct the changed political views and to understand why he had earned the reputation of the “internal opponent” to Stalinism. After 1956, Lukács turns to political criticism of Stalinism, which nevertheless remains unilateral. He sees in Stalinism a kind of the left sectarianism, the theory and practice of the implementation of civil war measures in the era of peaceful co-existence of two systems. (shrink)

It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation of quantum states. This can be regarded as a classical-like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator. The relevant concepts of quantum mechanics are then reconsidered and the epistemological implications of such approach discussed.

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to (...) teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms. (shrink)

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...) Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. (shrink)

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.

Deep brain stimulation of the subthalamic nucleus is a clinically effective tool for treating medically refractory Parkinson’s disease, but its neural mechanisms remain debated. Previous work has demonstrated that STN DBS results in evoked potentials in the primary motor cortex, suggesting that modulation of cortical physiology may be involved in its therapeutic effects. Due to technical challenges presented by high-amplitude DBS artifacts, these EPs are often measured in response to low-frequency stimulation, which is generally ineffective at PD symptom management. This (...) study aims to characterize STN-to-cortex EPs seen during clinically relevant high-frequency STN DBS for PD. Intraoperatively, we applied STN DBS to 6 PD patients while recording electrocorticography from an electrode strip over the ipsilateral central sulcus. Using recently published techniques, we removed large stimulation artifacts to enable quantification of STN-to-cortex EPs. Two cortical EPs were observed – one synchronized with DBS onset and persisting during ongoing stimulation, and one immediately following DBS offset, here termed the “start” and the “end” EPs respectively. The start EP is, to our knowledge, the first long-latency cortical EP reported during ongoing high-frequency DBS. The start and end EPs differ in magnitude and latency, and the end, but not the start, EP magnitude has a significant relationship to ongoing high gamma power during the EP. These contrasts may suggest mechanistic or circuit differences in EP production during the two time periods. This represents a potential framework for relating DBS clinical efficacy to the effects of a variety of stimulation parameters on EPs. (shrink)

Did Leibniz exploit infinitesimals and infinities à la rigueur or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Hidé Ishiguro defends the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz’s repeated assertions that infinitesimals violate the Archimedean property (i.e., Euclid’s Elements, V.4). We present textual evidence from Leibniz, as well as historical evidence from the early decades of the calculus, to undermine Ishiguro’s (...) interpretation. Leibniz frequently writes that his infinitesimals are useful fictions, and we agree, but we show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions. (shrink)

In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians’ approach to interpreting James Gregory’s expression ultimate terms in his paper attempting to prove the irrationality of \. Here (...) Gregory referred to the last or ultimate terms of a series. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? As well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an “unrigorous” practice, in fact finds close procedural proxies in modern infinitesimal theories. We analyze a mix of social and religious reasons that had led to the suppression of both the religious order of Gregory’s teacher degli Angeli, and Gregory’s books at Venice, in the late 1660s. (shrink)

We preface our publication of V.V. Rozanov's articles with a conversation between Nataliia Skorobogat'ko and V.G. Sukach, both of whom have spent much time studying the works of V.V. Rozanov.

There are methodological problems with the new techniques reviewed by Posner & Raichle. Some brain mechanisms are not detected by the temporal and spatial resolution. Questions are also raised by the stimulation paradigms.

This article considers the contribution of Ernst Mally to the theory of objects, namely he introduced a distinction of two types of properties (determinations) — formal and extraformal, that allowed Meinongian theory of objects to provide a satisfactory response to the criticism of Bertrand Russell, related to the question of the existence of the golden mountain. According to Meinong, object's so-being (Sosein) is logically independent of its ontic status (Sein). In this case Russell asks, how the existent golden mountain can (...) be existent, golden and a mountain, because golden mountains do not exit (as well as round squares). So it was obvious that the independence of Sosein from Sein thesis should be revised. That's why Mally made a remark that the properties that make object's Sosein are extraformal. Later these ideas have formed the basis of the contemporary study of nonexistent objects, the basis of the theories of Terens Parsons, Richard Routley etc. In particular Parsons's distinction of nuclear and extranuclear properties is a revised version of Mally's distinction.In addition, the article analyzes Ernst Mally's critics of epistemological idealism and its connection with the famous psychologism dispute in logic and semantics in the late 19th and early 20th century. This dispute is more associated with the works of G. Frege and E. Husserl, which accused of psychologism many representatives of Brentano's School. Particularly Alexius Meinong was also called a psychologist. That's it's important to find out, which side was Mally on in this dispute. (shrink)