We study model theoretic tree properties and their associated cardinal invariants. In particular, we obtain a quantitative refinement of Shelah’s theorem for countable theories, show that [Formula: see text] is always witnessed by a formula in a single variable and that weak [Formula: see text] is equivalent to [Formula: see text]. Besides, we give a characterization of [Formula: see text] via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are (...) indeed [Formula: see text]. (shrink)

In this article, we develop and clarify some of the basic combinatorial properties of the new notion of n-dependence recently introduced by Shelah. In the same way as dependence of a theory means its inability to encode a bipartite random graph with a definable edge relation, n-dependence corresponds to the inability to encode a random -partite -hypergraph with a definable edge relation. We characterize n-dependence by counting φ-types over finite sets, and in terms of the collapse of random ordered -hypergraph (...) indiscernibles down to order-indiscernibles. (shrink)

We initiate a systematic study of the class of theories without the tree property of the second kind — NTP2. Most importantly, we show: the burden is “sub-multiplicative” in arbitrary theories ; NTP2 is equivalent to the generalized Kimʼs lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters — so the dp-rank of a 1-type in any theory is (...) always witnessed by sequences of singletons; in NTP2 theories, simple types are co-simple, characterized by the co-independence theorem, and forking between the realizations of a simple type and arbitrary elements satisfies full symmetry; a Henselian valued field of characteristic is NTP2 if and only if the residue field is NTP2 , so in particular any ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2 theory preserves NTP2. (shrink)

We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP 2.

Discoverers of “market failures” as well as advocates of the general efficiency of a “true, unhampered market” sometimes seem to disregard the fundamental fact that there is no such thing as a “market as such.” What we call a market is always a system of social interaction characterized by a specific institutional framework, that is, by a set of rules defining certain restrictions on the behavior of the market participants, whether these rules are informal, enforced by private sanctions, or formal, (...) enforced by a particular agency, the “protective state,” in J. M. Buchanan's terminology. (shrink)

Sociological theories of space have so far not provided an in-depth analysis of online spaces. The paper addresses this issue by means of Löw’s relational theory of space. As this theory mainly focuses on material spaces, it is necessary to embrace the phenomenological perspective in order to apply it to the virtual realm. More recent phenomenological research has highlighted the ongoing mediatization or virtualization of the life-world. These theories, and presence research more generally, are useful for examining the layers of (...) virtual presence. This paper focuses on two emblematic types of spaces: multiplayer online role-playing games and Skype video chats. The first represents an online version of a Schutzian finite province of meaning, while Skyping is an example of how the paramount reality of everyday life expands into the virtual realm. Albeit differently, actors in both cases constitute hybrid, virtual-material spaces with various forms and degrees of virtual presence. User experiences in these spaces are in line with contemporary sociological diagnoses indicating the vanishing experience of living in space and the general tendency of late modernity to question previous social forms of modernity. (shrink)

We prove that every ultraproduct of p-adics is inp-minimal. More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic 0 in the RV language.

We develop the theory of Kim-independence in the context of NSOP $_{1}$ theories satisfying the existence axiom. We show that, in such theories, Kim-independence is transitive and that -Morley sequences witness Kim-dividing. As applications, we show that, under the assumption of existence, in a low NSOP $_{1}$ theory, Shelah strong types and Lascar strong types coincide and, additionally, we introduce a notion of rank for NSOP $_{1}$ theories.

We paraphrase Descartes’ famous dictum in the area of AI ethics where the “I doubt and therefore I am” is suggested as a necessary aspect of morality. Therefore AI, which cannot doubt itself, cannot possess moral agency. Of course, this is not the end of the story. We explore various aspects of the human mind that substantially differ from AI, which includes the sensory grounding of our knowing, the act of understanding, and the significance of being able to doubt ourselves. (...) The foundation of our argument is the discipline of ethics, one of the oldest and largest knowledge projects of human history, yet, we seem only to be beginning to get a grasp of it. After a couple of thousand years of studying the ethics of humans, we (humans) arrived at a point where moral psychology suggests that our moral decisions are intuitive, and all the models from ethics become relevant only when we explain ourselves. This recognition has a major impact on what and how we can do regarding AI ethics. We do not offer a solution, we explore some ideas and leave the problem open, but we hope somewhat better understood than before our study. (shrink)

We introduce and study (weakly) semi-equational theories, generalizing equationality in stable theories (in the sense of Srour) to the NIP context. In particular, we establish a connection to distality via one-sided strong honest definitions; demonstrate that certain trees are semi-equational, while algebraically closed valued fields are not weakly semi-equational; and obtain a general criterion for weak semi-equationality of an expansion of a distal structure by a new predicate.

Dr. Viktor Dörfler combines his background in developing and implementing AI with scholarly research on knowledge and cultivating talent to address misconceptions about AI. The Element explains what AI can and cannot do, carefully delineating facts from beliefs or wishful thinking. Filled with examples, this practical Element is thought-provoking. The purpose is to help CEOs figure out how to make the best use of AI, suggesting how to extract AI’s greatest value through appropriate task allocation between human experts and (...) AI. The author challenges the attribution of characteristics like understanding, thinking, and creativity to AI, supporting his argument with the ideas of the finest AI philosophers. He also discusses in depth one of the most sensitive AI-related topics: ethics. The readers are encouraged to make up their own minds about AI and draw their own conclusions rather than accepting opinions from people with vested interests or an agenda. (shrink)

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such (...) as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised. (shrink)

Journal of Philosophy of Education, EarlyView.

We give examples of (i) a simple theory with a formula (with parameters) which does not fork over [Formula: see text] but has [Formula: see text]-measure 0 for every automorphism invariant Keisler measure [Formula: see text] and (ii) a definable group [Formula: see text] in a simple theory such that [Formula: see text] is not definably amenable, i.e. there is no translation invariant Keisler measure on [Formula: see text]. We also discuss paradoxical decompositions both in the setting of discrete groups (...) and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups. (shrink)

Drawing from the literatures on the ethics of nudging and moral bioenhancement, I elaborate several pairs of cases in which one intervention is classified as an environmental behavioral intervention (EBI) and the other as a neurochemical behavioral intervention (NBI) in order to morally compare them. The intuition held by most is that NBIs are by far the more morally troubling kind of influence. However, if this intuition cannot be vindicated, we should at least entertain the _Similarity Thesis_, according to which (...) EBIs and NBIs share relevant moral features to the extent that moral conclusions about one are implied about the other in the described pairs of cases. I test this thesis by putting forward a number of possible moral grounds for setting EBIs and NBIs apart, including three of the most promising ones – physical invasiveness, disclosure and avoidance, and inevitability. I conclude that although these promising grounds might not bear the full burden of vindicating the intuition against _Similarity_ by themselves, clustering them together can establish discernible moral separation. (shrink)

The geometrical algebra hypothesis was once the received interpretation of Greek mathematics. In recent decades, however, it has become anathema to many. I give a critical review of all arguments against it and offer a consistent rebuttal case against the modern consensus. Consequently, I find that the geometrical algebra interpretation should be reinstated as a viable historical hypothesis.

The exacerbation of ethnic [natsional'nye] conflicts in the USSR is ruinous not only for our country. Polemics on the ethnic question are being waged at all levels and with all means, but what they lack most is a sober analysis of explosive problems. Even persons who in every other respect are distinguished by tolerance and common sense make an unworthy show of themselves on this sensitive question.

Mathematical texts raise particular dilemmas for the translator. With its arm’s-length relation to verbal expression and long-standing “mathematics is written for mathematicians” ethos, mathematics lends itself awkwardly to textually centered analysis. Otherwise sound standards of historical scholarship can backfire when rigidly upheld in a mathematical context. Mathematically inclined historians have had more faith in a purported empathic sixth sense—and there is a case to be made that this is how mathematical authors have generally expected their works to be read—but it (...) is difficult to pin down exact evidentiary standards for this supposed instinct. This essay urges that both of these points of view, for all the tension between them, be kept in the historian’s toolbox. It illustrates these considerations with a case study from the Ptolemaic astronomical tradition on computing lunar model parameters from eclipse data. (shrink)

We present results from a single case study based on semi-structured interviews with a student (a boy in school year 3) diagnosed with autism spectrum disorder and his school staff after participating in a short and small-scale intervention carried out in a socio-economically disadvantaged Swedish elementary school in 2019. The student participated in a seven week long intervention with a total of 12 philosophical dialogues (ranging from 45 to 60 minutes). Two facilitators, both with years of facilitation experience and teacher (...) degree and at least BA in philosophy, facilitated the majority of the dialogues and mainly followed a ”routine” procedure. The student was interviewed in direct connection to the end of the intervention about his experiences from the dialogues and his perceptions about whether and how the dialogues had influenced him. The student’s two teachers, who had participated in the dialogues as participants, were interviewed as a pair, also in direct connection to the end of the intervention, while the school principal was interviewed two years after the study. These staff interviews concerned the staff’s experiences of the influence of the dialogues on the students within the intervention as well as transfer effects to other contexts in school. The data from the study include detailed elaborations from a student perspective of different effects on the student’s communicative and cognitive development, which are in several respects supported also by staff reports. The results show that the student was able, interested, and willing to participate in philosophical dialogues, and our data point to several positive outcomes for the student in the communicative and cognitive domains. (shrink)

We prove a generalization of the Elekes–Szabó theorem [G. Elekes and E. Szabó, How to find groups?, Combinatorica 32 537–571 ] for relations defina...

The article has two main goals. 1. To reveal the psychological mechanism of the assimilation of faith and the concept of its relationship with reason in the philosophy of B. Pascal, as well as the role of faith both in knowledge in general and in knowledge of God. 2. To determine the degree of influence of Augustine's theological and philosophical concepts on the formation of Pascal's views. By comparing the concepts of will, understanding, faith and reason, the author comes to (...) the conclusion that Augustine, like Pascal, does not oppose faith to reason, but includes reason in the "mechanism" of acquiring faith. In this way, the reason in the act of faith becomes a direct partner of consciousness and reason. The logic of faith, according to Pascal and Augustine, differs from the logic of understanding in that understanding is based on the material reality of earthly existence, while faith is based on the supernatural reality of divine inspiration. (shrink)

Nudges have been criticized for working ‘in the dark’, influencing people without their full awareness. To assess whether this property renders nudging an illegitimate policy tool in liberal democracies, we argue that in scrutinizing nudge transparency, we should adequately divide our focus between nudging techniques, the nudgers employing them, and the nudgees subjected to them. We develop an account of what it means for nudgees to be ‘watchful’, a disposition that enables them to resist and circumvent nudges. We argue that (...) such ‘watchfulness’ should be cultivated if we want to implement nudges in legitimate, accountable, and democratic ways. (shrink)

Analyses in Greek geometry are traditionally seen as heuristic devices. However, many occurrences of analysis in formal treatises are difficult to justify in such terms. I show that Greek analysies of geometrics can also serve formal mathematical purposes, which are arguably incomplete without which their associated syntheses are arguably incomplete. Firstly, when the solution of a problem is preceded by an analysis, the analysis latter proves rigorously that there are no other solutions to the problem than those offered in the (...) synthesis. Secondly, whenever some construction assumption beyond ruler and compass is made, the problem is not only solvable by that assumption but is in fact equivalent to that assumption in a rigorous sense. (shrink)