The following three related contributions jointly serve to lift up elements of the thought of the anthropolo­gist Gregory Bateson that can be fruitfully compared with elements of Michael Polanyi’s thought. In a brief introduction, William Stillwell reviews Bateson’s life and developing interests. Stillwell also provides, in a creative dialog form akin to Bateson’s own dialogs, a short review article on Noel Charlton’s Understanding Gregory Bateson: Mind, Beauty and the Sacred Earth. The third piece is Jere Moorman’s short 1991 (...) essay (now out of print) discussing Polanyi’s ideas about tacit knowing and their connection with Bateson’s ideas about the double bind. (shrink)

This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse (...) Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics. (shrink)

I present a new interpretation of Wittgenstein's later philosophy of logic and mathematics. This interpretation, like others, emphasizes Wittgenstein's attempt to reconcile platonistic and constructivistic approaches. But, unlike other interpretations, mine explains that attempt in terms of Wittgenstein's position about the relations between our concepts of necessity and provability. If what I say here is correct, then we can rescue Wittgenstein from the charge of naive relativism. For his relativism extends only to provability, and not to necessity.

This paper explores different interpretations of the word ‘deep’ as it is used by mathematicians, with a large number of examples illustrating various criteria for depth. Most of the examples are theorems with ‘historical depth’, in the sense that many generations of mathematicians contributed to their proof. Some also have ‘foundational depth’, in the sense that they support large mathematical theories. Finally, concepts from mathematical logic suggest that it may be possible to order certain theorems or problems according to ‘logical (...) depth’. (shrink)

This Element aims to present an outline of mathematics and its history, with particular emphasis on events that shook up its philosophy. It ranges from the discovery of irrational numbers in ancient Greece to the nineteenth- and twentieth-century discoveries on the nature of infinity and proof. Recurring themes are intuition and logic, meaning and existence, and the discrete and the continuous. These themes have evolved under the influence of new mathematical discoveries and the story of their evolution is, to a (...) large extent, the story of philosophy of mathematics. (shrink)

Hilbert took to using ideal elements in the 1890's, in both algebraic number theory and geometry. His Zahlbericht of 1897 popularized the concept of the ideal introduced by Dedekind in 1871 (which in turn formalized the concept of "ideal number" introduced by Kummer in the 1840's). His geometric work likewise followed a long history of ideal elements, some that originated in geometry and others that originated elsewhere and were applied to geometry. Important examples were:Piero della Francesca's Ideal City.Points at infinity (...) in projective geometry, originating in the 1430s from the "vanishing points" in perspective drawing.Imaginary points in algebraic geometry, originating from the imaginary numbers used by .. (shrink)

How the concept of proof has enabled the creation of mathematical knowledge. The Story of Proof investigates the evolution of the concept of proof--one of the most significant and defining features of mathematical thought--through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his influence (...) on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as "infinitesimal algebra," and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved. Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field's power and progress. (shrink)

Ikujiro Nonaka, whose formative experience is Japanese, is an established scholar who has written about large business organizations. He sees knowledge at the heart of the organization and its products and aims to develop Michael Polanyi’s conception of tacit knowledge in a practical direction to enhance organizational “knowledge creation.” For Nonaka, what matters is the practice, the doing, the embodiment of knowledge. An organization can amplify and crystallize individuals’ tacit knowledge in a process that allows them to experience deeper understanding. (...) Nonaka holds that it is iimportant to explore the potential that knowledge holds. His spiral process describes disciplined practices that make tacit knowledge independent and available to restructure the organizational knowledge context. (shrink)

Prolific journalist, historian, political columnist, and practicing Catholic Wills (now 76) writes an intensely opinionated re-evaluation of leaders and celebrities he has encountered, among them Studs Terkel, Beverly Sills, William Buckley, Richard Nixon, and more.

The author attempts to encompass the self, or a self, that, while at some times appears to be his own, at other times not, thus encompassing and continually morphing. It is a mixture of poetry and prose.

Examines the history of speculative thought by focusing on such dominant personalities as Plato, Bacon, Spinoza, Kant, Schopenhauer, and Nietzsche.

For them, citizenship is by definition a matter of treating people as individuals with equal rights under the law. This is what distinguishes democratic citizenship from feudal and other pre-modern views that determined people's political status by ...

in a very different sense, to refer to the cultural community, or cultural structure, itself On this view, the cultural community continues to exist even when its members arc free to modify the character of the culture, should they find its traditional ...

This new edition of Will Kymlicka's best selling critical introduction to contemporary political theory has been fully revised to include many of the most significant developments in Anglo-American political philosophy in the last eleven years, particularly the new debates over issues of democratic citizenship and cultural pluralism. The book now includes two new chapters on citizenship theory and multiculturalism, in addition to updated chapters on utilitarianism, liberal egalitarianism, libertarianism, socialism, communitarianism, and feminism. The many thinkers discussed include G. A. (...) Cohen, Ronald Dworkin, William Galston, Carol Gilligan, R. M. Hare, Chandran Kukathas, Catherine Mackinnon, David Miller, Philippe Van Parijs, Susan Okin, Robert Nozick, John Rawls, John Roemer, Michael Sandel, Charles Taylor, Michael Walzer, and Iris Young. Extended guides to further reading have been added at the end of each chapter, listing the most important books and articles on each school of thought, as well as relevant journals and websites. Covering some of the most advanced contemporary thinking, Will Kymlicka writes in an engaging, accessible, and non-technical way to ensure that the book is suitable for students approaching these difficult concepts for the first time. This second edition promises to build on the original edition's success as a key text in the teaching of modern political theory. (shrink)

It is valuable for inquiry to have researchers who are committed advocates of their own theories. However, in light of pervasive disagreement, such a commitment is not well explained by the idea that researchers believe their theories. Instead, this commitment, the rational attitude to take toward one’s favored theory during the course of inquiry, is what I call endorsement. Endorsement is a doxastic attitude, but one which is governed by a different type of epistemic rationality. This inclusive epistemic rationality is (...) sensitive to reasons beyond those to think the particular proposition in question is true. Instead, it includes extrinsic epistemic reasons, which concern the health of inquiry more generally. Such extrinsic reasons include the distribution of cognitive labor that a researcher will contribute to by endorsing a particular theory. Recognizing endorsement and inclusive epistemic rationality thus allows us to smooth a tension between individual rationality and collective rationality. It does so by showing how it can be epistemically rational to endorse a theory on the basis of the way this endorsement will benefit collective inquiry. I provide a decision theoretic treatment for inclusive epistemic rationality and endorsement which illustrates how this can be accomplished. (shrink)

Discussion of epistemic responsibility typically focuses on belief formation and actions leading to it. Similarly, accounts of collective epistemic responsibility have addressed the issue of collective belief formation and associated actions. However, there has been little discussion of collective responsibility for preventing epistemic harms, particularly those preventable only by the collective action of an unorganized group. We propose an account of collective epistemic responsibility which fills this gap. Building on Hindriks' (2019) account of collective moral responsibility, we introduce the Epistemic (...) Duty to Join Forces. Our theory provides an account of the responsibilities of scientists to prevent epistemic harms during inquiry. (shrink)

The development of paper-based information technologies in the early modern period is a field of enquiry that has lately benefited from extensive studies by intellectual historians and historians o...

Scientists, philosophers, and other researchers commonly assert their theories. This is surprising, as there are good reasons for skepticism about theories in cutting-edge research. I propose a new account of assertion in research contexts that vindicates these assertions. This account appeals to a distinct propositional attitude called endorsement, which is the rational attitude of committed advocacy researchers have to their theories. The account also appeals to a theory of conversational pragmatics known as the Question Under Discussion model, or QUD. Hence, (...) I call the theory the EQUD model. Motivating this account is a recognition that the speech act of assertion has two roles to play in research contexts. The first is the advocacy role, in which researchers assert a theory in order to advocate for it. The second is the evidential role, which is used to add to the common stock of information available to a field of inquiry. The EQUD model provides an account of warranted assertion for both these roles in research contexts. This success provides support for the theory of endorsement. It also provides support for information updating accounts of assertion. (shrink)

I argue that recognizing a distinct doxastic attitude called endorsement, along with the epistemic norms governing it, solves the self-undermining problem for conciliationism about disagreement. I provide a novel account of how the self-undermining problem works by pointing out the auxiliary assumptions the objection relies on. These assumptions include commitment to certain epistemic principles linking belief in a theory to following prescriptions of that theory. I then argue that we have independent reason to recognize the attitude of endorsement. Endorsement is (...) the attitude of resilient and committed advocacy which is appropriate for researchers to have toward their own theory. Recognizing the importance of endorsement, and of its resiliency, gives us reason to deny the epistemic principles that serve as auxiliary assumptions in the self-undermining objection. This defuses the objection, and provides additional support for the theory of endorsement. (shrink)

This paper motivates and defends alethic nihilism, the theory that nothing is true. I first argue that alethic paradoxes like the Liar and Curry motivate nihilism; I then defend the view from objections. The critical discussion has two primary outcomes. First, a proof of concept. Alethic nihilism strikes many as silly or obviously false, even incoherent. I argue that it is in fact well-motivated and internally coherent. Second, I argue that deflationists about truth ought to be nihilists. Deflationists maintain that (...) the utility of the truth predicate is exhausted by its expressive role, and I argue that the truth predicate can still play this expressive role even if nothing is true. As such, deflationists do not stand to lose anything by accepting nihilism. Since they also stand to gain an elegant solution to the alethic paradoxes, on balance deflationists ought to be nihilists. (shrink)

Many AI systems that make important decisions are black boxes: how they function is opaque even to their developers. This is due to their high complexity and to the fact that they are trained rather than programmed. Efforts to alleviate the opacity of black box systems are typically discussed in terms of transparency, interpretability, and explainability. However, there is little agreement about what these key concepts mean, which makes it difficult to adjudicate the success or promise of opacity alleviation methods. (...) I argue for a unified account of these key concepts that treats the concept of understanding as fundamental. This allows resources from the philosophy of science and the epistemology of understanding to help guide opacity alleviation efforts. A first significant benefit of this understanding account is that it defuses one of the primary, in-principle objections to post hoc explainable AI (XAI) methods. This “rationalization objection” argues that XAI methods provide mere rationalizations rather than genuine explanations. This is because XAI methods involve using a separate “explanation” system to approximate the original black box system. These explanation systems function in a completely different way than the original system, yet XAI methods make inferences about the original system based on the behavior of the explanation system. I argue that, if we conceive of XAI methods as idealized scientific models, this rationalization worry is dissolved. Idealized scientific models misrepresent their target phenomena, yet are capable of providing significant and genuine understanding of their targets. (shrink)

Sometimes inquirers may rationally pursue a theory even when the available evidence does not favor that theory over others. Features of a theory that favor pursuing it are known as considerations of promise or pursuitworthiness. Examples of such reasons include that a theory is testable, that it has a useful associated analogy, and that it suggests new research and experiments. These reasons need not be evidence in favor of the theory. This raises the question: what kinds of reasons are provided (...) by pursuitworthiness considerations? Are they epistemic reasons or practical reasons? I argue that pursuitworthiness considerations are a kind of non-evidential epistemic reason, which I call an inquisitive reason. In support of this, I first point out two important similarities between the traditional pursuitworthiness considerations discussed in philosophy of science, which I call promise reasons, and certain social epistemic reasons that I call social inquisitive reasons. Specifically, both kinds of reason (1) favor pursuing a theory in a non-evidential way, and (2) concern promoting successful inquiry. I then propose recognition of a new category of normative reason: inquisitive reasons. This category contains both promise and social inquisitive reasons. Finally, I argue that inquisitive reasons share three essential features with previously recognized epistemic reasons: a connection to epistemic aims, explanatory independence, and the presence of a specific right-kind/wrong-kind reasons distinction. Each of these features have been used to argue that evidence should be treated as part of a distinct, independent domain of epistemic normativity. Since inquisitive reasons share these features, they too should be considered part of this independent epistemic domain. Thus, inquisitive reasons, including pursuitworthiness considerations, are epistemic reasons. (shrink)

Plato -- Aristotle and Greek science -- Francis Bacon -- Spinoza -- Voltaire and the French Enlightenment -- Immanuel Kant and German idealism -- Schopenhauer -- Herbert Spencer -- Friedrich Nietzsche -- Contemporary European philosophers : Henri Bergson ; Bennedetto Croce ; Bertrand Russell -- Contemporary American philosophers : George Santayana ; William James ; John Dewey.

Pulitzer Prize-winning author Will Durant chronicles the lives and ideas of key philosophers throughout history in this informative yet eminently readable text. Beginning with Socrates and Plato and concluding with Friedrich Nietzsche, Durant builds a history of philosophy by showing how each thinker's ideas informed and influenced the next generation.

Thought: A Journal of Philosophy, EarlyView.

This article surveys recent work on the idea of "citizenship", not as a legal category, but as a normative ideal of membership and participation. We focus on two emerging issues. First, whereas traditional notions of citizenship assume that membership and participation are promoted by the possession of rights, many theorists now emphasize civic responsibilities. Second, whereas traditional theories assume that citizenship provides a common status and identity, some theorists now argue that the distinctive needs and identities of certain groups -such (...) as women, ethnic minorities, the disabled - can only be accommodated through "group-differentiated citizenship". The writings of neo-conservatives, participatory democrats, civic republicans, feminists, civil society theorists, virtue theorists and cultural pluralists are surveyed. (shrink)

We combine two notions in AECs, tameness and good λ-frames, and show that they together give a very well-behaved nonforking notion in all cardinalities. This helps to fill a longstanding gap in classification theory of tame AECs and increases the applicability of frames. Along the way, we prove a complete stability transfer theorem and uniqueness of limit models in these AECs.

This study explores the primacy of interrelating and its ecopsychological significance. Grounded in evidence from everyday experience, and in dialogue with the phenomenology of Martin Buber, Emmanuel Levinas, and Maurice Merleau-Ponty, we discover that humans are inherently relational beings, not separate egoic subjects. When experienced intimately , this realization may transform our interrelationship with the beings and presences in the community of nature. Specifically, interrelating is primary in three ways: 1) interrelating is always already here, transpiring from the beginning of (...) the human species and human culture, from the beginning of every infant's life, and from the continuously arising beginning of every presencing moment; 2) the quality of our interrelating is truly what matters most, the most important expression of and facilitator of health, compassion, and justice for humankind and the rest of nature; and 3) interrelating is the ever-present path via which we discover/create and carry on our existence. Interrelating is an existential given—indeed we are our interrelating—yet we are summoned ethically to cultivate our way of interrelating so as to serve others and the non-human natural community. Thus interrelating is our essence, our calling, and our path. (shrink)

Intellectual courage requires acting to promote epistemic goods despite significant risk of harm. Courage is distinguished from recklessness and cowardice because the expected epistemic benefit of a courageous action outweighs (in some sense) the threatened harm. Sometimes, however, inquirers pursue theories that are not best supported by their current evidence. For these inquirers, the expected epistemic benefit of their actions cannot be explained by appeal to their evidence alone. The probability of pursuing the true theory cannot contribute enough to the (...) expected epistemic benefit for the action to count as courageous rather than reckless. Thus, there must be some other epistemic consideration which favors their action, besides evidence for their theory. I argue that the proper account of intellectual courage requires recognition of inquisitive reasons: a distinct category of epistemic reasons which concern successful inquiry. This category includes reasons to think a theory itself is promising, e.g., that the theory suggests potentially fruitful new research. It also includes social epistemic reasons, e.g., that pursuing a theory will improve the distribution of cognitive labor. Inquisitive reasons help explain why researchers who pursue improbable theories count as intellectually courageous, rather than reckless. The expected epistemic benefit of an action is partially determined by the inquisitive reasons in its favor. On my account, intellectually courageous inquiry requires acting in a way that is sensitive to inquisitive reasons. (shrink)