Noted logician and philosopher addresses various forms of mathematical logic, discussing both theoretical underpinnings and practical applications. After historical survey, lucid treatment of set theory, model theory, recursion theory and constructivism and proof theory. Place of problems in development of theories of logic, logic’s relationship to computer science, more. Suitable for readers at many levels of mathematical sophistication. 3 appendixes. Bibliography. 1981 edition.

In the name of safeguarding public interests and ethical principles, China’s National Health Commission bans unmarried women from using assisted reproductive technology (ART), including egg freezing. Supported by local governments, the ban has restricted single women’s reproductive rights nationwide. Although some courts bypassed the ban to allow widowed single women to use ART, they have not adopted a position in favour of single women’s reproductive autonomy, but quite the contrary. Faced with calls to relax the ban and allow single women (...) to freeze eggs electively, the National Health Commission refused to amend their policy, partly to protect women’s well-being paternalistically and partly to implement the central government’s policies to boost the birthrate and maintain traditional family structures. While the government’s concerns about elective egg freezing are not entirely unfounded, they have failed to demonstrate that banning single women’s egg freezing is a suitable, necessary and proportionate means to safeguard societal interests and ethical principles. The authority’s assumptions that women cannot make rational decisions for their health even with adequate informed consent procedures, that banning egg freezing by single women promotes a culture of having children ‘at a proper age’, and that egg freezing by single women offends China’s public moralities have not been substantiated. (shrink)

In his doctor's thesis [1], Henkin has shown that if a formal logic is consistent, and sufficiently complex, then it must admit a non-standard model. In particular, he showed that there must be a model in which that portion of the model which is supposed to represent the positive integers of the formal logic is not in fact isomorphic to the positive integers; indeed it is not even well ordered by what is supposed to be the relation of ≦.For the (...) purposes of the present paper, we do not need a precise definition of what is meant by a standard model of a formal logic. The non-standard models which we shall discuss will be flagrantly non-standard, as for instance a model of the sort whose existence is proved by Henkin. It will suffice if we and our readers are in agreement that a model of a formal logic is not a standard model if either: The relation in the model which represents the equality relation in the formal logic is not the equality relation for objects of the model. That portion of the model which is supposed to represent the positive integers of the formal logic is not well ordered by the relation ≦. That portion of the model which is supposed to represent the ordinal numbers of the formal logic is not well ordered by the relation ≦. (shrink)

Hao Wang was one of the few confidants of the great mathematician and logician Kurt Gödel. _A Logical Journey_ is a continuation of Wang's _Reflections on Gödel_ and also elaborates on discussions contained in _From Mathematics to Philosophy_. A decade in preparation, it contains important and unfamiliar insights into Gödel's views on a wide range of issues, from Platonism and the nature of logic, to minds and machines, the existence of God, and positivism and phenomenology. The impact of (...) Gödel's theorem on twentieth-century thought is on par with that of Einstein's theory of relativity, Heisenberg's uncertainty principle, or Keynesian economics. These previously unpublished intimate and informal conversations, however, bring to light and amplify Gödel's other major contributions to logic and philosophy. They reveal that there is much more in Gödel's philosophy of mathematics than is commonly believed, and more in his philosophy than his philosophy of mathematics. Wang writes that "it is even possible that his quite informal and loosely structured conversations with me, which I am freely using in this book, will turn out to be the fullest existing expression of the diverse components of his inadequately articulated general philosophy." The first two chapters are devoted to Gödel's life and mental development. In the chapters that follow, Wang illustrates the quest for overarching solutions and grand unifications of knowledge and action in Gödel's written speculations on God and an afterlife. He gives the background and a chronological summary of the conversations, considers Gödel's comments on philosophies and philosophers, and his attempt to demonstrate the superiority of the mind's power over brains and machines. Three chapters are tied together by what Wang perceives to be Gödel's governing ideal of philosophy: an exact theory in which mathematics and Newtonian physics serve as a model for philosophy or metaphysics. Finally, in an epilog Wang sketches his own approach to philosophy in contrast to his interpretation of Gödel's outlook. (shrink)

First published in 1974. Despite the tendency of contemporary analytic philosophy to put logic and mathematics at a central position, the author argues it failed to appreciate or account for their rich content. Through discussions of such mathematical concepts as number, the continuum, set, proof and mechanical procedure, the author provides an introduction to the philosophy of mathematics and an internal criticism of the then current academic philosophy. The material presented is also an illustration of a new, more general method (...) of approach called substantial factualism which the author asserts allows for the development of a more comprehensive philosophical position by not trivialising or distorting substantial facts of human knowledge. (shrink)

First published in 1974. Despite the tendency of contemporary analytic philosophy to put logic and mathematics at a central position, the author argues it failed to appreciate or account for their rich content. Through discussions of such mathematical concepts as number, the continuum, set, proof and mechanical procedure, the author provides an introduction to the philosophy of mathematics and an internal criticism of the then current academic philosophy. The material presented is also an illustration of a new, more general method (...) of approach called substantial factualism which the author asserts allows for the development of a more comprehensive philosophical position by not trivialising or distorting substantial facts of human knowledge. (shrink)

In this first extended treatment of his life and work, Hao Wang, who was in close contact with Godel in his last years, brings out the full subtlety of Godel's ideas and their connection with grand themes in the history of mathematics and ...

Automated algorithms are silently making crucial decisions about our lives, but most of the time we have little understanding of how they work. To counter this hidden influence, there have been increasing calls for algorithmic transparency. Much ink has been spilled over the informational account of algorithmic transparency—about how much information should be revealed about the inner workings of an algorithm. But few studies question the power structure beneath the informational disclosure of the algorithm. As a result, the information disclosure (...) itself can be a means of manipulation used by a group of people to advance their own interests. Instead of concentrating on information disclosure, this paper examines algorithmic transparency from the perspective of power, explaining how algorithmic transparency under a disciplinary power structure can be a technique of normalizing people’s behavior. The informational disclosure of an algorithm can not only set up some de facto norms, but also build a scientific narrative of its algorithm to justify those norms. In doing so, people would be internally motivated to follow those norms with less critical analysis. This article suggests that we should not simply open the black box of an algorithm without challenging the existing power relations. (shrink)

Love is often seen as the most intimate aspect of our lives, but it is increasingly engineered by a few programmers with Artificial Intelligence (AI). Nowadays, numerous dating platforms are deploying so-called smart algorithms to identify a greater number of potential matches for a user. These AI-enabled matchmaking systems, driven by a rich trove of data, can not only predict what a user might prefer but also deeply shape how people choose their partners. This paper draws on Jürgen Habermas’s “colonization (...) of the lifeworld” thesis to critically explore the insidious influence of delegating romantic decision-making to an algorithm. The love lifeworld is colonized inasmuch as online dating algorithms encroach into our romantic relations to the extent that communicative action within romance is replaced by the technocratic rules of an algorithm. (shrink)

This cogent and knowledgeable critique of the tradition of modern analytic philosophy focuses on the work of its central figures -- Russell, Carnap, and Quine -- and finds it wanting. In its place, Hao Wang unfolds his own original view of what philosophy could and should be. The base of any serious philosophy, he contends, should take as its point of departure the actual state of human knowledge. He explains the relation of this new tradition to mathematical logic and (...) reveals the crucial transitions and mistakes in mainstream Anglo-American philosophy that make a new approach so compelling.Equally at home in philosophy and mathematics, Wang is uniquely qualified to take on the task of critically examining modern philosophy. He carefully traces the path of ideas from Russell and Wittgenstein through the Vienna Circle to modern British and American philosophy, and makes use of his familiarity with the profound thought of Kurt Gödel with whom he has had numerous discussions. He also presents the broader significance of Russell's philosophy, provides a comprehensive and unified treatment of Quine's work in logic and in philosophy, and delineates what is common between Carnap and Quine. (shrink)

Franke Philosophy & Technology, 35(4), 1-7, (2022) offers an interesting claim that algorithmic transparency as manipulation does not necessarily follow that it is good or bad. Different people can have good reasons to adopt different evaluative attitudes towards this manipulation. Despite agreeing with some of his observations, this short reply will examine three crucial misconceptions in his arguments. In doing so, it defends why we are morally obliged to care about the manipulative potential of algorithmic transparency. It suggests that we (...) as society have a moral duty to incorporate the value of transparency into algorithmic systems while keeping algorithmic transparency itself sensitive to power relations. (shrink)

A survey is made of work since 1879 on foundational problems viewed as an analysis, by reduction and formalization, of the concepts proof, feasible, number, set, and constructivity. It is suggested that there are five domains of concepts and methods, viz., anthropologism, finitism, intuitionism, predicativism, and platonism. It is also suggested that the central problem is to characterize these domains by formalization and to determine their interrelations by different forms of reduction. Finally, the range of logic in the narrower sense (...) is discussed, and applications of mathematical logic are briefly outlined.ZusammenfassungDie Entwicklung der Grundlagenforschung seit dem Jahre 1879 wird hier im Überblick dargestellt und zwar vom Standpunkt einer Analyse durch Reduktion and Formalisierung der Begriffe Beweis, Zahl, Menge und Konstruktivität aus gesehen. Fünf Bereiche von Begriffsbildungen und Methoden werden zu Grunde gelegt, nämlich: Anthropologismus, Finitismus, Intuitionismus, Prädikativismus und Platonismus. Als zentrale Aufgabenstellung wird hier die Charakterisierung dieser Bereiche durch Formalisierung betrachtet und weiterhin die Bestimmung ihrer gegenseitigen Zusammenhänge durch verschiedene Arten von Reduktionen. Schliesslich wird das Gebiet der Logik im engeren Sinne diskutiert und Anwendungen der mathematischen Logik skizziert.RésuméL'article réexamine les travaux faits depuis 1879 sur les problèmes du fondement des mathématiques, que P'on considère comme des analyses par réduction et formalisation des concepts de démonstration, de nombre, d'ensemble, et de constructivité. On suggère qu'il existe cinq domaines de concepts et de méthodes, à savoir: anthropologisme, finitisme, intuitionisme, « predicativism » et platonisme. On suggère aussi que le problème essentiel consiste à caractériser ces domaines par formalisation et à déterminer leurs relations réciproques par différentes formes de réduction. Finalement, on étudie la portée de la logique prise dans son sens le plus étroit et on esquisse rapidement les applications de la logique mathématique. (shrink)

I propose to sketch my views on several aspects of the philosophy of mathematics that I take to be especially relevant to philosophy as a whole. The relevance of my discussion would, I think, become more evident, if the reader keeps in mind the function of (the philosophy of) mathematics in philosophy in providing us with more transparent aspects of general issues. I shall consider: (1) three familiar examples; (2) logic and our conceptual frame; (3) communal agreement and objective certainty; (...) (4) the transcommunal universality of mathematics; (5) the big jump to the potential infinite; (6) the reconciliation of local creation with the hypothesis of discovery; (7) Platonism as realism plus conceptualism; (8) foundational studies and mathematical practice; and (9) the decomposition of philosophical disagreements. The views of Gödel and Wittgenstein are emphasized in order to add specificity to the discussions. (shrink)

The essay centers on Gödel's views on the place of our intuitive concept of time in philosophy and in physics. It presents my interpretation of his work on the theory of relativity, his observations on the relationship between Einstein's theory and Kantian philosophy, as well as some of the scattered remarks in his conversations with me in the seventies — namely, those on the philosophies of Leibniz, Hegel and Husserl — as a successor of Kant — in relation to their (...) conceptions of time. (shrink)

First published in 1974. Despite the tendency of contemporary analytic philosophy to put logic and mathematics at a central position, the author argues it failed to appreciate or account for their rich content. Through discussions of such mathematical concepts as number, the continuum, set, proof and mechanical procedure, the author provides an introduction to the philosophy of mathematics and an internal criticism of the then current academic philosophy. The material presented is also an illustration of a new, more general method (...) of approach called substantial factualism which the author asserts allows for the development of a more comprehensive philosophical position by not trivialising or distorting substantial facts of human knowledge. (shrink)

Security issue against different attacks is the core topic of cyberphysical systems. In this paper, optimal control theory, reinforcement learning, and neural networks are integrated to provide a brief overview of optimal robust control strategies for a benchmark power system. First, the benchmark power system models with actuator and sensor attacks are considered. Second, we investigate the optimal control issue for the nominal system and review the state-of-the-art RL methods along with the NN implementation. Third, we propose several robust control (...) strategies for different types of cyberphysical attacks via the optimal control design, and stability proofs are derived through Lyapunov theory. Furthermore, the stability analysis with the NN approximation error, which is rarely discussed in the previous works, is studied in this paper. Finally, two different simulation examples demonstrate the effectiveness of our proposed methods. (shrink)

Labels provide a quick and effective solution to obtain people interesting content from large-scale social network information. The current interest label extraction method based on the subgraph stream proves the feasibility of the subgraph stream for user label extraction. However, it is extremely time-consuming for constructing subgraphs. As an effective mathematical method to deal with fuzzy and uncertain information, rough set-based representations for subgraph stream construction are capable of capturing the uncertainties of the social network. Therefore, we propose an effective (...) approach called RS_UNITE_SS, which is suitable for large-scale social network user interest label extraction. Specifically, we first propose the subgraph division algorithm to construct a subgraph stream by incorporating a rough set. Then, the algorithm for user real-time interest label extraction based on upper approximation is proposed by using sequentially characteristics of the subgraph. Empirically, we evaluate RS_UNITE_SS over real-world datasets, and experimental results demonstrate that our proposed approach is more computationally efficient than existing methods while achieving higher precision value and MRR value. (shrink)