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)

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)

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)

The orthodox monadic determination thesis holds that we represent colour relations by virtue of representing colours. Against this orthodoxy, I argue that it is possible to represent colour relations without representing any colours. I present a model of iconic perceptual content that allows for such primitive relational colour representation, and provide four empirical arguments in its support. I close by surveying alternative views of the relationship between monadic and relational colour representation.

This paper resolves a paradox concerning colour constancy. On the one hand, our intuitive, pre-theoretical concept holds that colour constancy involves invariance in the perceived colours of surfaces under changes in illumination. On the other, there is a robust scientific consensus that colour constancy can persist in cerebral achromatopsia, a profound impairment in the ability to perceive colours. The first stage of the solution advocates pluralism about our colour constancy capacities. The second details the close relationship between colour constancy and (...) contrast. The third argues that achromatopsics retain a basic type of colour constancy associated with invariants in contrast processing. The fourth suggests that one person-level, conscious upshot of such processing is the visual awareness of chromatic contrasts ‘at’ the edges of surfaces, implicating the ‘colour for form’ perceptual function. This primitive type of constancy sheds new light on our most basic perceptual capacities, which mark the lower borders of representational mind. (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.

Levinas, Storytelling and Anti-Storytelling explores the troubling nature of storytelling through a reading of the work of Emmanuel Levinas. Levinas is a thinker who has a complex relationship with literature and with storytelling.

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.

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 ...

Psychiatry widely assumes an internalist biomedical model of mental illness. I argue that many of psychiatry’s diagnostic categories involve an implicit commitment to constitutive externalism about mental illness. Some of these categories are socially externalist in nature.

Colour constancy is a foundational and yet puzzling phenomenon. Standard appearance invariantism is threatened by the psychophysical matching argument, which is taken to favour variantism. This argument, however, is inconclusive. The data at best support a pluralist view: colour constancy is sometimes variantist, sometimes invariantist. I add another potential explanation of these data, complex invariantism, which adopts an atypical six-dimensional model of colour appearance. Finally I prospect for a unifying conception of constancy among two neglected notions: discriminatory colour constancy and (...) relational colour constancy. The former arguably marks a common core capacity that is present across widely differing viewing contexts. (shrink)

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 paper presents a new response to the colour similarity argument, an argument that many people take to pose the greatest threat to colour physicalism. The colour similarity argument assumes that if colour physicalism is true, then colour similarities should be scrutable under standard physical descriptions of surface reflectance properties such as their spectral reflectance curves. Given this assumption, our evident failure to find such similarities at the reducing level seemingly proves fatal to colour physicalism. I argue that we should (...) dispense with this assumption, and thus endorse the inscrutability of colour similarity. This strategy is inspired by parallels between the colour similarity argument and the explanatory gap between mind and body made vivid by Jackson’s (1986) knowledge argument, and in particular by type-B physicalist responses to that argument. This inscrutability response is further motivated by cases in chemistry and biochemistry in which analogous scrutability theses fail to hold. Along the way, I present a challenge to standard formulations of the colour similarity argument based on the extreme context sensitivity of the similarity relation. Most presentations of the argument fail to control for such contextual variation, which raises the distinct possibility that the argument equivocates on the similarity relation across its premises. Although ultimately inconclusive, this context challenge forces a significant reformulation of the colour similarity argument, and highlights the need for much greater care in handling claims about colour similarity. (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.

Colour vision plays a foundational explanatory role in the philosophy of colour, and serves as perennial quarry in the wider philosophy of perception. I present two contributions to our understanding of this notion. The first is to develop a constitutive approach to characterizing colour vision. This approach seeks to comprehend the nature of colour vision qua psychological kind, as contrasted with traditional experiential approaches, which prioritize descriptions of our ordinary visual experience of colour. The second contribution is to argue that (...) colour vision does not constitutively involve the ability to see colours, in a natural and categorically committed sense. I argue that two subjects exactly alike in respect of their constitutive colour vision abilities could differ in respect of whether or not they have categorical perception of colour. The argument is supported by thought experiment and dissociations observed in cognitive neuropsychology. The argument also bears connections to recent neo-Whorfian accounts of colour categorization. _1._ Introduction _2._ Colour Vision, Experience, and Natures _3._ The Central Argument _3.1._ Colour constancy and discrimination _3.2._ Dissociating colour vision and categorical perception for colour _4._ Whorf and the View from Cognitive Neuropsychology _5._ Conclusion. (shrink)

Generics about social kinds (or GSKs) frequently propagate descriptions that carry normative force (e.g., 'women are emotional'). Some philosophers of language attribute this to GSKs’ tendency to transmit essentialist beliefs about social kinds. According to these accounts, utterances of GSKs implicate that there is something in the nature of social kinds that causes them to possess the properties described, and that individual members of these social kinds therefore ought to exhibit (or be expected to exhibit) these properties. Here, I draw (...) on empirical evidence to suggest an alternative account. According to my framework, an utterance of a GSK implicates a distinction between the social kind described and its salient conceptual opposite, producing what I call a dichotomizing perspective. For example, ‘women are emotional’ suggests that men are not. Importantly, such distinctions frequently persist in the societal common ground as a function of social power, in part due to their alignment with hierarchical social structures between dichotomized social kinds. This enables such GSKs to perpetuate biased patterns of attention, expectation, and behavior even in the absence of essentialist belief. (shrink)

We show that Shelah’s Eventual Categoricity Conjecture for successors follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC withLS below a strongly compact cardinalκis <κ-tame and applying the categoricity transfer of Grossberg and VanDieren [11]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property (...) to tameness, calledtype shortness, and show that it follows similarly from large cardinals. (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)