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 These previously unpublished intimate and informal conversations, however, bring to light and amplify Gödel's other major contributions to logic and philosophy.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)

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 ...

This paper examines how professionals (N = 32) working on artificial intelligence (AI) view structural AI ethics challenges like injustices and inequalities beyond individual agents' direct intention and control. This paper answers the research question: What are professionals’ perceptions of the structural challenges of AI (in the agri-food sector)? This empirical paper shows that it is essential to broaden the scope of ethics of AI beyond micro- and meso-levels. While ethics guidelines and AI ethics often focus on the responsibility of (...) designers and the competencies and skills of designers to take this responsibility, our results show that many structural challenges are beyond their reach. This result means that while ethics guidelines and AI ethics frameworks are helpful, there is a risk that they overlook more complicated, nuanced, and intersected structural challenges. In addition, it highlights the need to include diverse stakeholders, such as quadruple helix (QH) participants, in discussions around AI ethics rather than solely focusing on the obligations of AI developers and companies. Overall, this paper demonstrates that addressing structural challenges in AI is challenging and requires an approach that considers four requirements: (1) multi-level, (2) multi-faceted, (3) interdisciplinary, and (4) polycentric governance. (shrink)

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)

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)

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)

The development of artificial perception technologies has surpassed the limitations of human natural senses, expanding humanity's perspectives on the world and self-awareness. However, it has simultaneously introduced new ethical challenges. These emerging challenges necessitate a philosophical reexamination of concepts such as 'nature,' 'artificial,' and ' invasiveness ' to better address associated issues. The advancement of artificial perception technologies not only involves scientific and practical applications but also profoundly influences human values, social ethics, and the future trajectory of human civilization. Amidst (...) the current transformation and transition in modes of cognition, reevaluating the relationship between technology and humanity becomes a crucial task. (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)

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 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)

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)

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)

This essay discusses a number of questions which arise from attempts to reduce the mental to the physical or the mental and the physical to the computational. It makes, in an organized way, several basic distinctions between different kinds of accounts of the mind. It reconstructs and elaborates many discussions between Gödel and the author on the nature of the human mind, with special emphasis on its mathematical capabilities.

With the rapid expansion of parameters in large language models (LLMs) and the application of Reinforcement Learning with Human Feedback (RLHF), there has been a noticeable growth in the moral competence of LLMs. However, several questions warrant further exploration: Is it really possible for LLMs to fully align with human values through RLHF? How can the current moral growth be philosophically contextualized? We identify similarities between LLMs’ moral growth and Deweyan ethics in terms of the discourse of human moral development. (...) We then attempt to use Dewey’s theory on an experimental basis to examine and further explain the extent to which the current alignment pathway enables the development of LLMs. A beating experiment serves as the foundational case for analyzing LLMs’ moral competence across various parameters and stages, including basic moral cognition, moral dilemma judgment, and moral behavior. The results demonstrate that the moral competence of the GPT series has seen a significant improvement, and Dewey’s Impulse-Habit-Character theory of moral development can be used to explain this: the moral competence of LLMs has been enhanced through experience-based learning, supported by human feedback. Nevertheless, LLMs’ moral development through RLHF remains constrained and does not reach the character stage described by Dewey, possibly due to their lack of self-consciousness. This fundamental difference between humans and LLMs underscores both the limitations of LLMs’ moral growth and the challenges of applying RLHF for AI alignment. It also emphasizes the need for external societal governance and legal regulation. (shrink)

The main purpose of this paper is to present a formal systemPin which we enjoy a smooth-running technique and which countenances a universe of classes which is symmetrical as between large and small. More exactly,Pis a system which differs from the inconsistent system of [1] only in the introduction of a rather natural new restrictive condition on the defining formulas of the elements. It will be proved that if the weaker system of [2] is consistent, thenPis also consistent.After the discovery (...) of paradoxes, it may be recalled, Russell and Zermelo in the same year proposed two different ways of safeguarding logic against contradictions. Since then various simplifications and refinements of these systems have been made. However, in the resulting systems of Zermelo set theory, generation of classes still tends to be laborious and uncertain; and in the systems of Russell's theory of types, complications in the matter of reduplication of classes and meaningfulness of formulas remain. In [2], Quine introduced a system which seems to be free from all these complications. But later it was found out that in it there appears to be an unavoidable difficulty connected with mathematical induction. Indeed, we encounter the curious situation that although we can prove in it the existence of a class V of all classes, and we can also prove particular existence theorems for each of infinitely many classes, nobody has so far contrived to prove in it that V is an infinite class or that there exists an infinite class at all. (shrink)

Logic as an activity deals with the interplay or the dialectic, as one thinks, between the known and the unknown, form and content, or the formal and the intuitive. For this purpose it is useful to select from what is taken to be known a universal part which remains fixed throughout all particular instances of the interplay. The propositions in such a universal part make up the logical truths. There are alternative answers to the question: What is to be required (...) of the concepts and the propositions of this universal part? Different choices can be and have been made with regard to the kind and the degree of their universality, and the degree of precision and systematic character of their codification. These different choices lead to different conceptions of logic, such as those of William of Occam, Kant, Hegel, Frege, Brouwer, the early Wittgenstein, the later Wittgenstein, and Gödel. (shrink)

I construe mathematical philosophy not in the narrow sense of philosophy of mathematics but in a broad indefinite sense of different manners of giving mathematics a privileged place in the study of philosophy. For example, in one way or another, mathematics plays an important part in the philosophy of Plato, Descartes, Spinoza, Leibniz, and Kant. In contrast, history plays a central role in the philosophy of Vico, Hegel, and Marx. In more recent times, Frege, Husserl, Russell, Ramsey, and Gödel all (...) began as mathematicians. One way of viewing Kant’s system of philosophy might be to stress that he was struck by the synthetic a priori character of mathematical propositions. He went on to offer a remarkable account of this fact and also look for and propose synthetic a priori foundations of physics, morality, and esthetics. (shrink)