Hilary Putnam once suggested that “the actual existence of sets as ‘intangible objects’ suffers… from a generalization of a problem first pointed out by Paul Benacerraf… are sets a kind of function or are functions a sort of set?” Sadly, he did not elaborate; my aim, here, is to do so on his behalf. There are well-known methods for treating sets as functions and functions as sets. But these do not raise any obvious philosophical or foundational puzzles. For that, we (...) first need to provide a full-fledged function theory. I supply such a theory: it axiomatizes the iterative notion of function in exactly the same sense that ZF axiomatizes the iterative notion of set. Indeed, this function theory is synonymous with ZF. It might seem that set theory and function theory present us with rival foundations for mathematics, since they postulate different ontologies. But appearances are deceptive. Set theory and function theory provide the very same judicial foundation for mathematics. They do not supply rival metaphysical foundations; indeed, if they supply metaphysical foundations at all, then they supply the very same metaphysical foundations. (shrink)

The use of tensed language and the metaphor of set ‘formation’ found in informal descriptions of the iterative conception of set are seldom taken at all seriously. Both are eliminated in the nonmodal stage theories that formalise this account. To avoid the paradoxes, such accounts deny the Maximality thesis, the compelling thesis that any sets can form a set. This paper seeks to save the Maximality thesis by taking the tense more seriously than has been customary (although not (...) literally). A modal stage theory, MST, is developed in a bimodal language, governed by a tenselike logic. Such a language permits a very natural axiomatisation of the iterative conception, which upholds the Maximality thesis. It is argued that the modal approach is consonant with mathematical practice and a plausible metaphysics of sets and shown that MST interprets a natural extension of Zermelo set theory less the axiom of Infinity and, when extended with a further axiom concerning the extent of the hierarchy, interprets Zermelo–Fraenkel set theory. (shrink)

이 글의 주된 목적은 반복적 집합 개념을 검토하여 그 개념이 지니는 내부적인 문제들을 드러내고 그 개념에 대한 구성주의적 비판을 제시하는 것이다. 내부적인 문제는 고전수학자들에게 반복적 집합 개념이 불분명한 데에서 기인한다. 이 점은 불로스의 단계이론을 반복적 집합 개념에 관한 다른 설명과 비교하고 치환공리와 선택공리에 관한 논란을 검토하여 드러낼 것이다. 이보다 훨씬 심각한 문제는 구성주의적 비판이다. 특히, 고전적인 반복적 집합 개념에 내재한 무한에 관한 실재론적 이해에 대한 비판과, 집합론에서 고전논리의 사용에 대한 비판 및 비서술성에 관한 비판이 제기될 것이며, 반복적 집합 개념이 구성적으로 (...) 이해될 때, 어떤 원칙들이 정당화될 수 있는지 검토될 것이다. (shrink)

The phrase ‘The iterative conception of sets’ conjures up a picture of a particular settheoretic universe – the cumulative hierarchy – and the constant conjunction of phrasewith-picture is so reliable that people tend to think that the cumulative hierarchy is all there is to the iterative conception of sets: if you conceive sets iteratively, then the result is the cumulative hierarchy. In this paper, I shall be arguing that this is a mistake: the iterative (...) class='Hi'>conception of set is a good one, for all the usual reasons. However, the cumulative hierarchy is merely one way among many of working out this conception, and arguments in favour of an iterative conception have been mistaken for arguments in favour of this one special instance of it. (This may be the point to get out of the way the observation that although philosophers of mathematics write of the iterative conception of set, what they really mean – in the terminology of modern computer science at least – is the recursive conception of sets. Nevertheless, having got that quibble off my chest, I shall continue to write of the iterative conception like everyone else.). (shrink)

Surveying and criticising attitudes towards the role and strength of the iterative conception of set—widely seen as the justificatory basis of Zermelo-Fraenkel set theory with Choice—this paper highlights a tension in both contemporary and historic accounts of the iterative conception’s justificatory role: on the one hand its advocates wish to claim that it justifies ZFC, but on the other hand they abstain from stating whether the preconditions for such justification exists. Expanding the number of axioms that (...) the conception is standardly charged with failing to justify, in the forms of the Emptyset and Powerset, this paper aims to extend the critique that the iterative conception does not justify ZFC to its subsystems. Exploring the historic and contemporary relationship between the iterative conception and set theory, the paper then attempts to defuse strategies that avoid the problems of intrinsic justification by weakening the iterative conception’s role to the ‘motivational’ or ‘heuristic’ by showing that what the iterative conception has been seen to motivate has changed over time. It is suggested that the conjunction of these arguments seriously weakens the programme of reasoning on an ‘atheoretic’ notion of set. (shrink)

According to the iterative conception of set, each set is a collection of sets formed prior to it. The notion of priority here plays an essential role in explanations of why contradiction-inducing sets, such as the Russell set, do not exist. Consequently, these explanations are successful only to the extent that a satisfactory priority relation is made out. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of (...) the empty set and raises serious epistemological concerns; but the leading realist interpretations---ontological and modal interpretations of priority---are deeply problematic as well. I conclude that the purported explanatory virtues of the iterative conception are, at present, unfounded. (shrink)

The iterative conception of set commonly is regarded as supporting the axioms of Zermelo-Fraenkel set theory (ZF). This paper presents a modified version of the iterative conception of set and explores the consequences of that modified version for set theory. The modified conception maintains most of the features of the iterative conception of set, but allows for some non-wellfounded sets. It is suggested that this modified iterative conception of set supports the (...) axioms of Quine's set theory NF. (shrink)

Arecent paper by George Boolos suggests that it is philosophically respectable to use monadic second order logic in one’s explication of the iterative concept of set. I shall here give a partial indication of the new range of theories of the iterative hierarchy which are thus madeavailable to philosophers of set theory.

Arecent paper by George Boolos suggests that it is philosophically respectable to use monadic second order logic in one’s explication of the iterative concept of set. I shall here give a partial indication of the new range of theories of the iterative hierarchy which are thus madeavailable to philosophers of set theory.

The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as (...) to allow for the existence of wide sets. Drawing upon Cantor’s notion of the absolute infinite, the paper argues that the modifications are warranted and preserve a robust iterative conception of set. The resulting theory is proved consistent relative to ZFC + “there exists an inaccessible cardinal number.”. (shrink)

We describe a first-order theory of generalized sets intended to allow a similar treatment of sets and proper classes. The theory is motivated by the iterative conception of set. It has a ternary membership symbol interpreted as membership relative to a set-building step. Set and proper class are defined notions. We prove that sets and proper classes with a defined membership form an inner model of Bernays-Morse class theory. We extend ordinal and cardinal notions to generalized sets and (...) prove ordinal and cardinal results in the theory. We prove that the theory is consistent relative to ZFC + (∃ x) [ x is a strongly inaccessible cardinal]. (shrink)

The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. One idea sometimes alluded to is that maximality considerations speak in favour of large cardinal axioms consistent with ZFC, since it appears to be `possible' to continue the hierarchy far enough to generate the relevant transfinite number. In this paper, we argue against this idea based on a priority of subset formation under the iterative conception. In particular, we argue (...) that there are several conceptions of maximality that justify the consistency but falsity of large cardinal axioms. We argue that the arguments we provide are illuminating for the debate concerning the justification of new axioms in iteratively-founded set theory. (shrink)

It is shown in this article in how far one has to have a clear picture of Gödel’s philosophy and scientific thinking at hand (and also the philosophical positions of other philosophers in the history of Western Philosophy) in order to interpret one single Philosophical Remark by Gödel. As a single remark by Gödel (very often) mirrors his whole philosophical thinking, Gödel’s Philosophical Remarks can be seen as a philosophical monadology. This is so for two reasons mainly: Firstly, because it (...) pictures a monadology already via its form. And secondly, because Gödel wanted to establish in his Philosophical Remarks a modern monadology in adapting the philosophical principles of Leibniz’s monadology to modern science. This article will only deal with the first aspect that has been mentioned namely that a single remark by Gödel (very often) mirrors his whole philosophy (at the time) for example: set theory, perception and intuition of mathematical objects and axioms, the nature of the human mind, the concept and existence of God, and so on. This renders it extremely difficult to interpret Gödel’s philosophical remarks but interpreting it provides also a deep insight in Gödel’s philosophical thoughts. (shrink)

Transfinite recursion is an essential component of set theory. In this paper, we seek intrinsically justified reasons for believing in recursion and the notions of higher computation that surround it. In doing this, we consider several kinds of recursion principles and prove results concerning their relation to one another. We then consider philosophical motivations for these formal principles coming from the idea that computational notions lie at the core of our conception of set. This is significant because, while the (...) iterative conception of set has been widely recognized as insufficient to establish replacement and recursion, its supplementation by considerations pertaining to algorithms suggests a new and philosophically well-motivated reason to believe in such principles. (shrink)

A neologicist set theory based on an abstraction principle (NewerV) codifying the iterative conception of set is investigated, and its strength is compared to Boolos's NewV. The new principle, unlike NewV, fails to imply the axiom of replacement, but does secure powerset. Like NewV, however, it also fails to entail the axiom of infinity. A set theory based on the conjunction of these two principles is then examined. It turns out that this set theory, supplemented by a principle (...) stating that there are infinitely many nonsets, captures all (or enough) of standard second-order ZFC. Issues pertaining to the axiom of foundation are also investigated, and I conclude by arguing that this treatment provides the neologicist with the most viable reconstruction of set theory he is likely to obtain. (shrink)

Discusses the metaphysics of the iterative conception of set.

As the story goes, the source of the paradoxes of naive set theory lies in a conflation of two distinct conceptions of set: the so-called iterative, or mathematical, conception, and the Fregean, or logical, conception. While the latter conception is provably inconsistent, the former, as Godel notes, "has never led to any antinomy whatsoever". More important, the iterative conception explains the paradoxes by showing precisely where the Fregean conception goes wrong by enabling us (...) to distinguish between sets and proper classes, collections that are "too big" to be sets. While I agree wholeheartedly with this distinction, in this paper I argue first that the iterative conception does not provide an explanation of all of the set theoretic paradoxes. I then argue that we need to reconsider the distinction between sets and proper classes rather more carefully. The result will be that ZFC does not capture the iterative conception in its full generality. I close by offering a more general theory that, arguably, does. (shrink)

The Question of Iterated Grounding (QIG) asks what grounds the grounding facts. Although the question received a lot of attention in the past few years, it is usually discussed independently of another important issue: the connection between metaphysical explanation and the relation or relations that supposedly “back” it. I will show that once we get clear on the distinction between metaphysical explanation and the relation(s) backing it, we can distinguish no fewer than four questions lumped under QIG. I will also (...) argue that given some plausible assumptions about what it would take for a relation to back metaphysical explanation, many salient views about grounding allow us to give “easy” answers to these questions—easy in the sense that we can straightforwardly derive them from the respective conception of grounding without getting into the sorts of complexities that typically inform answers to QIG. The paper's main upshot is that we cannot expect to make much progress on QIG without first addressing the difficult issue of how exactly grounding is related to metaphysical explanation. (shrink)

At first thought, iteration seems banal. It is about repeating the existing; nothing is changing. But this special issue shows that, in an era obsessed with the new, it is often the repetition of the old that creates social change. Iteration fosters persuasion. It affords opportunities for critical and creative engagement with meaning, values and knowledge. It invites collaboration, though its apparent simplicity often belies a tremendous amount of individual and collective labour involved in the practices of iteration. Through its (...) repetition of the existing, however, iteration also can be a mechanism for reproducing the status quo. Its pervasiveness and banality naturalises power, and its mimetic qualities shrink spaces for critical distance and care. The editors of this special issue have brought together a delightful and fascinating diversity of articles focussed on iteration in cultural production in the digital age. We hop across geographies to examine lockdown diaries, artists’ books, socialist memes, fake news, the design of social media platforms and artificial intelligence, activism, film, social media forum moderation, news website reader comments and more. Iterating through the collection as a whole, across its many disciplines, is a commitment to theorising through empirical evidence, to explaining with critique, and to providing pathways to praxis. These characteristics of this special issue, and the many concepts and arguments it puts forward, make this collection of work exceptionally rich material for seeing iteration and how it shapes the world we live in today, as well as the world we want it to be. In this preface, I take a media sociology approach to show how iteration can be usefully understood as collaborative communication for change. I see this understanding of iteration, whose ascendancy is related to the ascendancy of computer science, as baked into the form of communication technologies—and thus as shaping the kinds of iteration that are possible when we use these technologies. This understanding also prompts us to focus on the connection between iteration and social change. To explore how this works, I analytically slow down the practice of iteration to show that it is a communication practice of transmission. That transmission practice is itself constituted of cognate communication practices—the reception, evaluation and production of knowledge—in which visibility and persuasion are key. In the latter parts of the preface, I illustrate this through the example of witnessing as iteration, as the high-stakes nature of witnessing make it a canary in the coalmine, more generally, for mediated communication in the digital age. I show how breaking the witnessing practice down into its various parts allows us to see how power enters and inflects who and what are iterated, when—and who and what are not. Thinking critically with iteration and against unequal power relations, the praxis this preface suggests is one—much in line with the rest of this special issue—of explaining how iteration might move the grassroots towards their goals. (shrink)

In this paper, we investigate processes involving iterative information updating due to van Benthem (Int Game Theory Rev 9: 13—45, 2007), who characterized existent game-theoretic solution concepts by such processes in the framework of Plaza's public announcement logic. We refine this approach and clarify the relationship between stable strategies and information update processes. After extending Plaza's logic, we then derive the conditions under which a stable outcome is determined independently of the order of the iterative information updates. This (...) result gives an epistemic foundation for the order independence of iterated elimination of disadvantageous strategies. (shrink)

Belief revision theory concerns methods for reformulating an agent's epistemic state when the agent's beliefs are refuted by new information. The usual guiding principle in the design of such methods is to preserve as much of the agent's epistemic state as possible when the state is revised. Learning theoretic research focuses, instead, on a learning method's reliability or ability to converge to true, informative beliefs over a wide range of possible environments. This paper bridges the two perspectives by assessing the (...) reliability of several proposed belief revision operators. Stringent conceptions of minimal change are shown to occasion a limitation called inductive amnesia: they can predict the future only if they cannot remember the past. Avoidance of inductive amnesia can therefore function as a plausible and hitherto unrecognized constraint on the design of belief revision operators. (shrink)

In the advent of communication, Derrida finds that meaning through signification carries with it the possibility of mis-communication in which the intended meaning behind the text becomes undecidable and inevitably polysemic in its transference. In a short, yet fecund essay “Signature Event Context,” Derrida tackles the problem of communication and the supposed claim of the classical notion of writing’s conception of virtual permanence within the text. The classical notion of writing claims that writing as a medium or a species (...) of communication implies that the transference of meaning is conducted in a homogenous fashion. In the absence, of the author who has abandoned the fleeting singular instance of inscription, the text, according to the classical notion of writing, must be able to transfer meaning within the interplay of the text itself, for if it fails to do so, writing would cease to become useful in its intention to transfer meaning. (shrink)

This paper offers a comparative analysis of the uses and formulations of speech-act theory in Derrida's and Deleuze/Guattari's work. It begins by juxtaposing Derrida's concept/nonconcept of 'iterability' to Deleuze and Guattari's conception of the 'order-word' and then examines these theories of the speech act in terms of their implications and consequences for a politics of resistance. Whereas Deleuze and Guattari generate a detailed material stratum — an order-word plateau — for exploring the performative in socio-political contexts, Derrida attends to (...) the singular, wholly unique acts that occur when a repeatable formula such as a political declaration is uttered or otherwise takes place.The paper then concludes by drawing a tentative link between iterability and Deleuze and Guattari's notion of 'deterritorialisation' via the order-word plateau. (shrink)

This paper offers a comparative analysis of the uses and formulations of speech-act theory in Derrida's and Deleuze/guattari's work. It begins by juxtaposing Derrida's concept/nonconcept of ‘iterability’ to Deleuze and Guattari's conception of the ‘order-word’ and then examines these theories of the speech act in terms of their implications and consequences for a politics of resistance. Whereas Deleuze and Guattari generate a detailed material stratum — an order-word plateau — for exploring the performative in socio-political contexts, Derrida attends to (...) the singular, wholly unique acts that occur when a repeatable formula such as a political declaration is uttered or otherwise takes place. The paper then concludes by drawing a tentative link between iterability and Deleuze and Guattari's notion of ‘deterritorialisation’ via the order-word plateau. (shrink)

Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified (...) class='Hi'>conception and the graph conception. In addition, he presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics. (shrink)

This journey shall provide the reader with a simple, though complete, guide to Leibniz’s metaphysics, incidentally preventing him or her from common errors. I will start with unfolding Leibniz’s definition of a simple substance as a free acting individual substance, which, in doing so, constitutes its complete concept. This latter contains everything that happens to the individual substance, a process taking place in God’s mind by forming the possibilities as combinations of his attributes before his decision to create the best (...) world. The totality of possibilities is divided through the compatibility relation into possible worlds. A world is a collection of all compatible individual substances. God creates the best among all possible worlds. The journey will enlighten the reader on Leibniz’s technical distinction between “possible” and “contingent”. In short, “possible” is defined by Leibniz in logical terms, as what is contradiction-free. “Contingent” is something that is, but might have been not. Just the fact that the non-being of contingent things remains possible saves Leibniz from Descartes’ and Spinoza’s determinism. (shrink)

In my book, The Rights of Others, I developed a discourse-theoretic approach to questions of political membership in liberal democracies, which include practices of citizenship, as well as of immigration, refuge and asylum. This article revisits five issues in response to various criticisms. How can we justify democratic exclusions? Is there a `right to membership' and how can it be reconciled with the different practices of various constitutional democracies? Is there a distinction between normatively acceptable and normatively problematic restrictions on (...) political membership? Does the concept of `democratic iterations' describe normative or empirical processes? How plausible is the binarism of the national and the global? I argue that democratic exclusions can be justified by not discriminating against would-be citizens and immigrants on the basis of ascriptive criteria. Ascriptive characteristics, like one's sex and skin colour, are not the product of one's voluntary doings. Democratic iterations are empirical processes which can be judged in the light of normative criteria deriving from discourse theory. Furthermore, while the binarism of national and global is problematical, alternative configurations of political membership at the present are not more defensible. (shrink)

This paper argues for several related theses. First, the epistemological position that knowledge requires safe belief can be motivated by views in the philosophy of science, according to which good explanations show that their explananda are robust. This motivation goes via the idea—recently defended on both conceptual and empirical grounds—that knowledge attributions play a crucial role in explaining successful action. Second, motivating the safety requirement in this way creates a choice point—depending on how we understand robustness, we'll end up with (...) different conceptions of safety in epistemology. Lastly, and most controversially, there's an attractive choice at this point that will not vindicate some of the most influential applications of the safety-theoretic framework in epistemology, e.g., Williamson's arguments against the KK principle, and luminosity. (shrink)

In this paper we argue that ecological evolutionary developmental biology accounts of cognitive modernity are compatible with cultural evolution theories of language built upon iterated learning models. Cultural evolution models show that the emergence of near universal properties of language do not require the preexistence of strong specific constraints. Instead, the development of general abilities, unrelated to informational specificity, like the copying of complex signals and sharing of communicative intentions is required for cultural evolution to yield specific properties, such as (...) language structure. We argue that eco-evo-devo provides the appropriate conceptual background to ground an account for the many interconnected genetic, environmental and developmental factors that facilitated the emergence of an organic system able to develop language through the iterated transmission of information. We use the concept of niche construction to connect evolutionary developmental accounts for sensory guided motor capacities and cultural evolution guided by iterated learning models. This integrated theoretical model aims to build bridges between biological and cultural approaches. (shrink)

Two models of second-order ZFC need not be isomorphic to each other, but at least one is isomorphic to an initial segment of the other. The situation is subtler for impure set theory, but Vann McGee has recently proved a categoricity result for second-order ZFCU plus the axiom that the urelements form a set. Two models of this theory with the same universe of discourse need not be isomorphic to each other, but the pure sets of one are isomorphic to (...) the pure sets of the other. This paper argues that similar results obtain for considerably weaker second-order axiomatizations of impure set theory that are in line with two different conceptions of set, the iterative conception and the limitation of size doctrine. (shrink)

Maximizers in isolated Prisoner's Dilemmas are doomed to frustration. But in Braybrooke's view maximizers might do better in a series, securing Pareto-optimal arrangements if not from the very beginning, at least eventually. Given certain favourable special conditions, it can be shown according to Braybrooke and shown even without question-begging motivational or value assumptions, that in a series of Dilemmas maximizers could manage to communicate a readiness to reciprocate, generate thereby expectations of reciprocation, and so give rise to optimizing reciprocations which, (...) in due course, would reinforce expectations, the net result of all this being an increasingly stable practice to mutual benefit. In this way neighbours could learn to be good neighbours: they could learn to respect each other's property, to engage in reciprocal assistance, and perhaps even to make and keep promises covering a range of activities. So maximizers are, Braybrooke holds, capable of a society of sorts. Out of the ashes they might build something. But even under favourable conditions they could not build much and most conditions would defeat them almost entirely, for many-person Dilemmas whether isolated or in series would be quite beyond them, question-begging motivational assumptions aside. In settings at all like those in which we live, utility maximization assuming only ‘traditional’ motivation is self-defeating without remedy, and most certainly not an adequate basis for social life.' The probable inference from as a conception of individual rationality: for surely truly rational agents would be if not perfectly well designed at least better designed for communal living than utility maximizers would be assuming only ‘traditional’, non-question-begging values or motivation. Or, if this inference is denied and maximization is not challenged as a conception of rationality, we can conclude instead, Braybrooke has more recently suggested, that the advice, ‘Be rational,’ is incomplete, and when addressed to groups should be supplemented by, ‘And be somewhat ‘untraditional’ in your values; respect at least some rules that would make coordination and cooperation possible even in many-person interaction problems.’. (shrink)

This paper considers some aspects of the early conception and use of the infinite in ancient Greece, in the spirit of recent results in the history of ancient mathematics. It follows aspects of the practice of reasoning ad infinitum from the extant corpus of and about Zeno of Elea up to early Hellenistic examples in Aristotle and Euclid. Starting with the idea of ‘reasoning from indefinite iteration’, based on the metalogical recognition of the unachievability of an inference process, it (...) identifies several different classes of more or less sophisticated arguments that make use of this idea, and examines the logical devices and notions required for their acceptance in the philosophical practice. Those include ‘Infinite regress’ properly speaking, where Non-Contradiction is used in the formation of indirect infinitary arguments. (shrink)

In his influential paper ‘Two Types of Foundationalism,’ William Alston distinguishes two important conceptions of foundationalism: ‘simple foundationalism’ and ‘iterative foundationalism’. SF is the view that there are immediately justified beliefs of some kind or other. IF is the stronger view that certain epistemic propositions are immediately justified. Alston favors a reliability account of immediate justification of the kind defended by externalists such as Armstrong, Dretske, and Goldman. Alston rejects IF by appeal to what he calls the ‘second level (...) argument.’ He claims, further, that since IF is essential to the Cartesian project of reconstructing knowledge, the Cartesian project must be abandoned. (shrink)

In his influential paper ‘Two Types of Foundationalism,’ William Alston distinguishes two important conceptions of foundationalism: ‘simple foundationalism’ and ‘iterative foundationalism’. SF is the view that there are immediately justified beliefs of some kind or other. IF is the stronger view that certain epistemic propositions are immediately justified. Alston favors a reliability account of immediate justification of the kind defended by externalists such as Armstrong, Dretske, and Goldman. Alston rejects IF by appeal to what he calls the ‘second level (...) argument.’ He claims, further, that since IF is essential to the Cartesian project of reconstructing knowledge, the Cartesian project must be abandoned. (shrink)

The non-well-founded set theories described by Aczel (1988) have received attention from category theorists and computer scientists, but have been largely ignored by philosophers. At the root of this neglect might lie the impression that these theories do not embody a conception of set, but are rather of mere technical interest. This paper attempts to dispel this impression. I present a conception of set which may be taken as lying behind a non-well-founded set theory. I argue that the (...) axiom AFA is justified on the conception, which provides, contra Rieger (Mind 109:241–253, 2000), a rationale for restricting attention to the system based on this axiom. By making use of formal and informal considerations, I then make a case that most of the other axioms of this system are also justified on the conception. I conclude by commenting on the significance of the conception for the debate about the justification of the Axiom of Foundation. (shrink)

The present study develops an artificial agent that plays the iterative chicken game based on a computational model that describes human behavior in competitive social interactions in terms of fairness. The computational model we adopted in this study, named as the self-concept fairness model, decides the agent’s action according to the evaluation of fairness of both opponent and self. We implemented the artificial agent in a computer program with a set of parameters adjustable by researchers. These parameters allow researchers (...) to determine the extent to which the agent behaves aggressively or cooperatively. To demonstrate the use of the proposed method for the investigation of human behavior, we performed an experiment in which human participants played the iterative chicken game against the artificial agent. Participants were divided into two groups, each being informed to play with either a person or the computer. The behavioral analysis results showed that the proposed method can induce changes in the behavioral pattern of human players by changing the agent’s behavioral pattern. Also, we found that participants tended to be more sensitive to fairness when they played with a human opponent than with a computer opponent. These results support that the artificial agent developed in this study will be useful to investigate human behavior in competitive social interactions. (shrink)

Deep Neural Networks (DNNs) are becoming increasingly important as scientific tools, as they excel in various scientific applications beyond what was considered possible. Yet from a certain vantage point, they are nothing but parametrized functions fθ(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{f}_{\varvec{\theta }}(\varvec{x})$$\end{document} of some data vector x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{x}$$\end{document}, and their ‘learning’ is nothing but an iterative, algorithmic fitting of the parameters to data. Hence, what could be special (...) about them as a scientific tool or model? I will here suggest an integrated perspective that mediates between extremes, by arguing that what makes DNNs in science special is their ability to develop functional concept proxies (FCPs): Substructures that occasionally provide them with abilities that correspond to those facilitated by concepts in human reasoning. Furthermore, I will argue that this introduces a problem that has so far barely been recognized by practitioners and philosophers alike: That DNNs may succeed on some vast and unwieldy data sets because they develop FCPs for features that are not transparent to human researchers. The resulting breach between scientific success and human understanding I call the ‘Actually Smart Hans Problem’. (shrink)

Igal Kvart A Coding Conception in Action-Directed-Pragmatics -/- I present formal Pragmatics for a domain in Pragmatics that I call Action-Directed Pragmatics, which focuses on the Pragmatic riddle of how implicit contents are conveyed and understood, by adopting a coding model, in which the speaker and addressee simulate each other iteratively in a deliberative context (an ‘action-pregnant’ one). The implicit content, conveyed by a speaker and decoded by her addressee, in such cases, consists in the specified steered-to action, plus (...) modulations on the action-polarity (pro or con) and the degree of the so-called Steering Thrust that accompanies such assertions and is conveyed by verbal locutions, intonation, and/or bodily and facial gestures. There are two main tasks to model (in a given setup and conversational context): First, how is the speaker, with a steered-to action in mind (and Steering Thrust), to select an assertion so as to (say) optimize the successful transmission and decoding of its implicit pragmatic content by her and her addressee? Second, how does the addressee, given an assertion by the speaker, decipher the implicit pragmatic content conveyed via it? Both will invoke pertinent information they have about each other and the setup/context in order to best encode and best decode the implicit pragmatic content. (This coding mechanism proposed here is not offered for the ubiquitous ‘frozen’, ‘trivial’ or ‘routinized’ pragmatic messages.) A prelude to this formal Pragmatics is a general formal Pragmatic account of Assertibility in contexts that are multi-normative (which is the common case). I focus here only on Epistemic/Semantic and Instrumental Norms. A linear representation of degrees of Overall Assertibility (i.e., taking into account various operative norms) will be an n-dimensional qualitative vector space with a function that computes them given different degrees of sub-normative Assertibility on each axis. (shrink)

In this chapter, we philosophically reflect on the concept of innovation. To this end, we distinguish between the innovation process and outcome dimension, and between the ontic and ontological dimension of innovation. The ontic dimension of innovation concerns beings like new artefacts, and the ontological dimension concerns the being of these beings. These distinctions lead to four characteristics of our understanding of innovation with several implications for the object of innovation and its novelty, as well as for the temporality and (...) human involvement in innovation practices. It will turn out that innovation concerns the ontogenetic process in which primarily a world is constituted - i.e. the digital world - in which the invention of new artefacts - i.e. blockchain technology - is embedded. This ontogenetic process doesn’t follow a chronological temporal order, but is constituted by a temporal iterative process. Human creativity is involved in this ontogenetic process as co-creative capacity. In section three, we show the advantage of our conceptualization of innovation and its implications for contemporary and future alternative theories of innovation. (shrink)

According to the iterative conception of set, sets can be arranged in a cumulative hierarchy divided into levels. But why should we think this to be the case? The standard answer in the philosophical literature is that sets are somehow constituted by their members. In the first part of the paper, I present a number of problems for this answer, paying special attention to the view that sets are metaphysically dependent upon their members. In the second part of (...) the paper, I outline a different approach, which circumvents these problems by dispensing with the priority or dependence relation altogether. Along the way, I show how this approach enables the mathematical structuralist to defuse an objection recently raised against her view. (shrink)

The present article reports several applied experiments in the generation of aesthetic forms from algorithms and data. In these experiments algorithms and data are the driving morphogenetic force to such an extent that the role of the human creator must be reexamined case-by-case. Artists that program the graphics or sound generating algorithms may in turn be said to be programmed perceptually by the resulting artworks, in the sense that they must adapt their perception in a conscious or involuntary effort to (...) detect or project meaning in their form. While forms generated by algorithms or data are to a large extent unpredictable, an intimate relationship between programmer, performer/listener/viewer and the piece is created through instinctive, empirical familiarity with the tools and the creative process, which is afforded by the immediate and highly interactive nature of the frameworks employed. A simple iterative algorithm that produces a novel kind of chaotic geometric shapes is presented. Examples of an application of Paul Klee’s principles of visual form in programming interactive audio-visual dance performance are given. These principles are directly derived from the language-based metaphor of active, medial and passive voice. Finally, ongoing experiments in letting language structures emerge from data derived from a sound database are presented. (shrink)

This article attempts to broaden the phenomenologically motivated perspective of H. Weyl's Das Kontinuum in the hope of elucidating the differences between the intuitive and mathematical continuum and further providing a deeper phenomenological interpretation. It is known that Weyl sought to develop an arithmetically based theory of continuum with the reasoning that one should be based on the naturally accessible domain of natural numbers and on the classical first-order predicate calculus to found a theory of mathematical continuum free of impredicative (...) circularities only to stumble, to cite a key question, in the evident lack of intuitive support for the notion of points of the continuum. In this motivation, I set out to deal from a Husserlian viewpoint with the general notion of points as appearances reducible to individuals of pre-predicative experience in contrast with the notion of an interval of real numbers taken as an abstraction based on the intuition of time-flowing experience. I argue that the notions of points and of real intervals in the above sense are not by essence related to objective temporality and thus their incompatibility in mathematical terms is ultimately due to deeper constituting reasons independently of any causal and spatio-temporal constraints. (shrink)

George Boolos has argued that the iterative conception of set justifies most, but not all, the ZFC axioms, and that a second conception of set, the Frege-von Neumann conception (FN), justifies the remaining axioms. This article challenges Boolos's claim that FN does better than the iterative conception at justifying the axioms in question.