Lajevardi and Salehi, in “There may be many arithmetical Gödel sentences”, argue against the use of the definite article in the expression “the Gödel sentence”, by claiming that any unsound theory has Gödelian sentences with different truth values. We show that their Theorems 1 and 2 are special cases (modulo Löb's theorem and the first incompleteness theorem) of general observations pertaining to fixed points of any formula, and argue that the false sentences of Lajevardi and Salehi are in fact not (...) Gödel sentences. (shrink)

Lajevardi and Salehi, in “There may be many arithmetical Gödel sentences”, argue against the use of the definite article in the expression “the Gödel sentence”, by claiming that any unsound theory has Gödelian sentences with different truth values. We show that their Theorems 1 and 2 are special cases (modulo Löb's theorem and the first incompleteness theorem) of general observations pertaining to fixed points of any formula, and argue that the false sentences of Lajevardi and Salehi are in fact not (...) Gödel sentences. (shrink)

Lajevardi and Salehi, in “There may be many arithmetical Gödel sentences”, argue against the use of the definite article in the expression “the Gödel sentence”, by claiming that any unsound theory has Gödelian sentences with different truth values. We show that their Theorems 1 and 2 are special cases (modulo Löb's theorem and the first incompleteness theorem) of general observations pertaining to fixed points of any formula, and argue that the false sentences of Lajevardi and Salehi are in fact not (...) Gödel sentences. (shrink)

An intensional logic is presented and suggested as a framework for a formal investigation of learning. The framework allows for discussing and comparing concepts and representations, and makes it possible to view learning processes as iterations of a certain type of functions. It is shown how this framework may be used to shed light on Meno’s paradox, but also on concepts such as Vygotsky’s ZPD and learning trajectories. In the case of mathematics, where there are recent attempts to merge ideas (...) from the philosophy of mathematics with ideas from the philosophy of education, a formal framework such as the one presented here, may constitute a common arena of discussion. (shrink)

Recently Carrie S. Jenkins formulated an epistemology of mathematics, or rather arithmetic, respecting apriorism, empiricism, and realism. Central is an idea of concept grounding. The adequacy of this idea has been questioned e.g. concerning the grounding of the mathematically central concept of set (or class), and of composite concepts. In this paper we present a view of concept formation in mathematics, based on ideas from Carnap, leading to modifications of Jenkins’s epistemology that may solve some problematic issues with her ideas. (...) But we also present some further problems with her view, concerning the role of proof for mathematical knowledge. (shrink)

Attempts have been made to analyse features in mathematics within a social constructivist context. In this paper we critically examine some of those attempts recently made with focus on problems of the objectivity, ontology, necessity, and atemporality of mathematics. Our conclusion is that these attempts fare no better than traditional alternatives, and that they, furthermore, create new problems of their own.

In this paper, I argue that no theory of consciousness can simultaneously respect four initially plausible metaphysical claims – namely, ‘first-person realism’, ‘non-solipsism’, ‘non-fragmentation’, and ‘one world’ – but that any three of the four claims are mutually consistent. So, theories of consciousness face a ‘quadrilemma’. Since it will be hard to achieve a consensus on which of the four claims to retain and which to give up, we arrive at a landscape of competing theories, all of which have pros (...) and cons. I will briefly indicate which kinds of theories correspond to the four horns of the quadrilemma. (shrink)

Spacetime as we know and love it is lost in most approaches to quantum gravity. For many of these approaches, as inchoate and incomplete as they may be, one of the main challenges is to relate what they take to be the fundamental non-spatiotemporal structure of the world back to the classical spacetime of GR. The present essay investigates how spacetime is lost and how it may be regained in one major approach to quantum gravity, loop quantum gravity.

_Making Sense of Tantric Buddhism_ fundamentally rethinks the nature of the transgressive theories and practices of the Buddhist Tantric traditions, challenging the notion that the Tantras were "marginal" or primitive and situating them instead--both ideologically and institutionally--within larger trends in mainstream Buddhist and Indian culture. Critically surveying prior scholarship, Wedemeyer exposes the fallacies of attributing Tantric transgression to either the passions of lusty monks, primitive tribal rites, or slavish imitation of Saiva traditions. Through comparative analysis of modern historical narratives--that depict (...) Tantrism as a degenerate form of Buddhism, a primal religious undercurrent, or medieval ritualism--he likewise demonstrates these to be stock patterns in the European historical imagination. Through close analysis of primary sources, Wedemeyer reveals the lived world of Tantric Buddhism as largely continuous with the Indian religious mainstream and deploys contemporary methods of semiotic and structural analysis to make sense of its seemingly repellent and immoral injunctions. Innovative, semiological readings of the influential _Guhyasamaja Tantra_ underscore the text's overriding concern with purity, pollution, and transcendent insight--issues shared by all Indic religions--and a large-scale, quantitative study of Tantric literature shows its radical antinomianism to be a highly managed ritual observance restricted to a sacerdotal elite. These insights into Tantric scripture and ritual clarify the continuities between South Asian Tantrism and broader currents in Indian religion, illustrating how thoroughly these "radical" communities were integrated into the intellectual, institutional, and social structures of South Asian Buddhism. (shrink)

_Making Sense of Tantric Buddhism_ fundamentally rethinks the nature of the transgressive theories and practices of the Buddhist Tantric traditions, challenging the notion that the Tantras were "marginal" or primitive and situating them instead -- both ideologically and institutionally -- within larger trends in mainstream Buddhist and Indian culture. Critically surveying prior scholarship, Wedemeyer exposes the fallacies of attributing Tantric transgression to either the passions of lusty monks, primitive tribal rites, or slavish imitation of Saiva traditions. Through comparative analysis of (...) modern historical narratives -- that depict Tantrism as a degenerate form of Buddhism, a primal religious undercurrent, or medieval ritualism -- he likewise demonstrates these to be stock patterns in the European historical imagination. Through close analysis of primary sources, Wedemeyer reveals the lived world of Tantric Buddhism as largely continuous with the Indian religious mainstream and deploys contemporary methods of semiotic and structural analysis to make sense of its seemingly repellent and immoral injunctions. Innovative, semiological readings of the influential _Guhyasamaja Tantra_ underscore the text's overriding concern with purity, pollution, and transcendent insight -- issues shared by all Indic religions -- and a large-scale, quantitative study of Tantric literature shows its radical antinomianism to be a highly managed ritual observance restricted to a sacerdotal elite. These insights into Tantric scripture and ritual clarify the continuities between South Asian Tantrism and broader currents in Indian religion, illustrating how thoroughly these "radical" communities were integrated into the intellectual, institutional, and social structures of South Asian Buddhism. (shrink)

One of the hallmark themes of Karl Popper’s approach to the social sciences was the insistence that when social scientists are members of the society they study, then they are liable to affect that society. In particular, they are liable to affect it in such a way that the claims they make lose their validity. “The interaction between the scientist’s pronouncements and social life almost invariably creates situations in which we have not only to consider the truth of such pronouncements, (...) but also their actual influence on future developments. The social scientist may be striving to find the truth; but, at the same time, he must always be exerting a definite influence upon society. The very fact that his pronouncements do exert an influence destroys their objectivity.” (Popper 1963. (shrink)

The problem of free will as it is know in Western philosophical traditions is hardly known in China. Considering how central the problem is in the West, this is a remarkable fact. We try to explain this, and we offer insights into discussions within Chinese traditions that we think are related, not historically but regarding the issues discussed. Thus we introduce four central Chinese concepts, namely: (1) xīn 心 (heart, heart-mind), (2) xìng 性 (human nature, characteristic tendencies, inborn capacity), (3) (...) mìng 命 (lifespan, fate, command, allotment, endowment), and (4) zìrán 自然 (self-so, so of itself, nature, spontaneity), and we indicate how these concepts and their discussions can be related to discussions on free will. (shrink)