Gödel’s philosophical conceptions bear striking similarities to Cantor’s. Although there is no conclusive evidence that Gödel deliberately used or adhered to Cantor’s views, one can successfully reconstruct and see his “Cantorianism” at work in many parts of his thought. In this paper, I aim to describe the most prominent conceptual intersections between Cantor’s and Gödel’s thought, particularly on such matters as the nature and existence of mathematical entities (sets), concepts, Platonism, the Absolute Infinite, the progress and inexhaustibility of mathematics.
A cognitive science perspective of yoga system of thought will be developed in conjugation with the Samkhya Darsana. This development will be further advanced using Advaita Vedanta and will be translated into modern scientific terms to arrive at an idea about cognition process. The stalling of the cognitive process and stilling the mind will be critically discussed in the light of this perspective. This critical analysis and translation into cognitive science and modern scientific terms will be presented together with (...) its implications and applications to the disciplines of mind-machine modeling, natural language comprehension branch of artificial intelligence and physiological psychology. (shrink)
Yoga is a unique form of expert movement that promotes an increasingly subtle interpenetration of thought and movement. The mindful nature of its practice, even at expert levels, challenges the idea that thought and mind are inevitably disruptive to absorbed coping. Building on parallel phenomenological and ethnographic studies of skilful performance and embodied apprenticeship, we argue for the importance in yoga of mental access to embodied movement during skill execution by way of a case study of instruction and (...) practice in two related traditions, Iyengar and Anusara. Sharing a pose repertoire, they are based on distinctive philosophical systems with different teaching styles and metaphoric structures. To address relations between pedagogy and practice in embodied expertise, and to investigate the reciprocal influences of embodiment and thought, we explore in detail the linguistically mediated learning context where practitioners work with yoga teachers. Here, the mind/body problem comes to practical life. We demonstrate the effects of words on bodies, as knowledge is literally incorporated. We show why interpersonal influence on our movement capacities is sometimes needed to enhance expertise. We theorize and identify ?signature patterns of tension? among practitioners. These patterns have four sources: ghost gestures, innate differences in bodily form, functional fusing, and signature patterns of affective experience, modulation and expression. These patterns of tension produce ?silent zones?, cognitively impenetrable actions, functional fusing of a skilful, compensatory form, and signature patterns of pain and damage. We show how instruction can disrupt these silent zones, enhancing mental and physical flexibility. (shrink)
An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match (...) with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question. (shrink)
The author of this paper discusses some major points vital for two classical Indian schools of philosophy: (1) a significant feature of linguistic analysis in the Yoga tradition; (2) the role of the religious practice (iśvara-pranidhana) in the search for true self-identity in Samkhya and Yoga darśanas with special reference to their gnoseological purposes; and (3) some possible readings of ‘ahamkara’ and ‘asmita’ displayed in the context of Samkhya-Yoga phenomenology and metaphysics. The collision of language and metaphysics (...) refers to the risk of paralogism caused by the common linguistic procedures making the subject define its identity within the semantic order (i.e. verbal conventions and grammatical rules) which does not reflect the actual metaphysical situation of the self, though it determines one’s self-understanding in the empirical sense. Whereas, Samkhya-Yoga aims at recognizing, reorganizing and, finally, going beyond these procedures regarded as the obstacles on the path towards self-knowledge and liberation form metaphysical ignorance. (shrink)
In §8 of Remarks on the Foundations of Mathematics (RFM), Appendix 3 Wittgenstein imagines what conclusions would have to be drawn if the Gödel formula P or ¬P would be derivable in PM. In this case, he says, one has to conclude that the interpretation of P as “P is unprovable” must be given up. This “notorious paragraph” has heated up a debate on whether the point Wittgenstein has to make is one of “great philosophical interest” revealing “remarkable insight” in (...) Gödel’s proof, as Floyd and Putnam suggest (Floyd (2000), Floyd (2001)), or whether this remark reveals Wittgenstein’s misunderstanding of Gödel’s proof as Rodych and Steiner argued for recently (Rodych (1999, 2002, 2003), Steiner (2001)). In the following the arguments of both interpretations will be sketched and some deficiencies will be identified. Afterwards a detailed reconstruction of Wittgenstein’s argument will be offered. It will be seen that Wittgenstein’s argumentation is meant to be a rejection of Gödel’s proof but that it cannot satisfy this pretension. (shrink)
This chapter describes Kurt Gödel's paper on the incompleteness theorems. Gödel's incompleteness results are two of the most fundamental and important contributions to logic and the foundations of mathematics. It had been assumed that first-order number theory is complete in the sense that any sentence in the language of number theory would be either provable from the axioms or refutable. Gödel's first incompleteness theorem showed that this assumption was false: it states that there are sentences of number theory that are (...) neither provable nor refutable. The first theorem is general in the sense that it applies to any axiomatic theory, which is ω-consistent, has an effective proof procedure, and is strong enough to represent basic arithmetic. Their importance lies in their generality: although proved specifically for extensions of system, the method Gödel used is applicable in a wide variety of circumstances. Gödel's results had a profound influence on the further development of the foundations of mathematics. It pointed the way to a reconceptualization of the view of axiomatic foundations. (shrink)
Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel’s 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church, Kleene, and Rosser. Church and his students studied Gödel 1931, and Gödel taught a seminar (...) at Princeton in 1934. Seen in the historical context, Gödel was an important catalyst for the emergence of computability theory in the mid 1930s. (shrink)
The paper starts with some textual distinctions concerning the concept of God in the metaphysical framework of two classical schools of Hindu philosophy, Sāṃkhya and Yoga. Then the author focuses on the functional and pedagogical aspects of prayer as well as practical justification of “religious meditation” in both philosophical schools. A special attention is put on the practice called īśvarapraṇidhāna, recommended in Yoga school, which is interpreted by the author as a form of non-theistic devotion. The meaning of (...) the central object of this concentration, that is puruṣa-viśeṣa, is reconsidered in detail. The subject matter is discussed in the wider context of yogic self-discipline that enables a practitioner to overcome ignorance ( avidyā) and the narrowness of egotic perspective (asmitā), recognized in the Hindu darśanas as the root-cause of all suffering or never-fulfilled-satisfaction ( duḥkha). The non-theistic devotion and spiritual pragmatism assumed by the adherents of Sāṃkhya-Yoga redefines the concept of “God” ( īśvara) as primarily an object of meditative practice and a special tool convenient for spiritual pedagogy. (shrink)
Since the millennium, first person accounts of experience have been accepted as philosophically valid, potentially useful sources of information about the nature of mind and self. Several Vedic sciences rely on such first person accounts to discuss experience and consciousness. This paper shows that their insights define the information structure of experience in agreement with a scientific theory of mind fulfilling all presently known philosophical and scientific conditions. Experience has two separate components, its information content, and a separate ‘witness aspect’, (...) which can reflect on all forms of experience, and with training be strengthened until its power of reflection identifies it as the innermost aspect of ‘self’. The Vedic sciences, Sankhya, Yoga and Vedanta develop these themes. Sankhya identifies the different aspects of experience, outer and inner; Yoga practices lead the mind to inner states without information content (samadhi) in which the experience of the witness (sakshi) is strengthened and deepened. Vedanta states the nature of the ‘self’ is to know itself directly without intermediary. All this requires the witness to have a singular loop structure. The information structure of experience therefore has two aspects, information content plus a singular loop endowing it with a subjective sense of ‘Self’. (shrink)
It is shown that Gqp↑, the quantified propositional Gödel logic based on the truth-value set V↑ = {1 - 1/n : n≥1}∪{1}, is decidable. This result is obtained by reduction to Büchi's theory S1S. An alternative proof based on elimination of quantifiers is also given, which yields both an axiomatization and a characterization of Gqp↑ as the intersection of all finite-valued quantified propositional Gödel logics.
In his "Ontological proof", Kurt Gödel introduces the notion of a second-order value property, the positive property P. The second axiom of the proof states that for any property φ: If φ is positive, its negation is not positive, and vice versa. I put forward that this concept of positiveness leads into a paradox when we apply it to the following self-reflexive sentences: (A) The truth value of A is not positive; (B) The truth value of B is positive. Given (...) axiom 2, sentences A and B paradoxically cannot be both true or both false, and it is also impossible that one of the sentences is true whereas the other is false. (shrink)
Entailment in propositional Gödel logics can be defined in a natural way. While all infinite sets of truth values yield the same sets of tautologies, the entailment relations differ. It is shown that there is a rich structure of infinite-valued Gödel logics, only one of which is compact. It is also shown that the compact infinite-valued Gödel logic is the only one which interpolates, and the only one with an r.e. entailment relation.
It is shown that the infinite-valued first-order Gödel logic G° based on the set of truth values {1/k: k ε w {0}} U {0} is not r.e. The logic G° is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kröger's temporal logic of programs (even of the fragment without the nexttime operator O) and of the authors' temporal (...) logic of linear discrete time with gaps follows. (shrink)
Self-knowledge, at first glance, seems to be naturally and easily accessible to each of us. We commonly believe that we need much less effort to understand ourselves than to understand the world. The authoress of the paper uncovers the fallacy of this popular view referring to the fundamental conceptions and philosophical ideas of the classical Yoga. She tries to demystify our deceptive self-understanding explaining the definitions of ignorance (avidya), I-am-ness (asmita), desire (raga), aversion (dvesha) and fear of death (abhinivesha) (...) given by the author of the oldest Yoga treatise. Besides, the paper discusses briefly how we can make use of our limited, incorrect self-knowledge as far as we are aware that it needs to be transcended. In the final part of the paper, the issue of self-discipline consisting basically in cultivation of detachment and the practice of meditation is addressed. (shrink)
All first-order Gödel logics G_V with globalization operator based on truth value sets V C [0,1] where 0 and 1 lie in the perfect kernel of V are axiomatized by Ciabattoni’s hypersequent calculus HGIF.
Comparative philosophy has been subjected to much criticism in the latter half of the last century, though some of these criticisms were appropriate and justified. However, in our present cultural milieu, where traditions and culture transcend their geographical boundaries, seeping through the global network of views and ideas, it seems to be a legitimate enterprise to understand one’s own traditions and culture through the critical lens of the ‘other culture’. It is such cross-cultural understanding that paved the way towards legitimizing (...) “human rights” as a universal discourse. So also, the discourse on “environmental ethics” has gained acceptance in a similar manner across cultures and traditions. The paper attempts at an understanding of “Indian Philosophy” as a theoretical practice by exemplifying the notion of “sādhanā” in the Yoga system of Indian Philosophy through a reading of the notion of ‘phenomenological reduction’ as espoused in the transcendental phenomenology of Husserl. Thus, we reject the claim that philosophy as a pure rational activity was unique to the West as proclaimed in some Eurocentric readings. It is pertinent to recall here that the Sāṅkhya philosophy, which the Yoga system accepts as an allied school of thought was atheistic in its origin and attempted to explain the universe in terms of an evolutionary theory—the “prakṛti pariṇāma vāda”. (shrink)
A Husserlian phenomenological approach to logic treats concepts in terms of their experiential meaning rather than in terms of reference, sets of individuals, and sentences. The present article applies such an approach in turn to the reasoning operative in various paradoxes: the simple Liar, the complex Liar paradoxes, the Grelling-type paradoxes, and Gödel’s Theorem. It finds that in each case a meaningless statement, one generated by circular definition, is treated as if were meaningful, and consequently as either true or false, (...) although in fact it is neither. The situation is further complicated by the fact that the sentence used to express the meaningless statement is ambiguous, and may also be used to express a meaningful statement. The paradoxes result from a failure to distinguish between the two meanings the sentence may have. (shrink)
It is commonly thought that such topics as Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason are disparate scientific physical or mathematical issues having little or nothing in common. I suggest that they are largely standard philosophical problems (i.e., language games) which were resolved by Wittgenstein over 80 years ago. -/- Wittgenstein also demonstrated the fatal error in regarding mathematics or language or our behavior in general as a unitary coherent logical ‘system,’ rather than as (...) a motley of pieces assembled by the random processes of natural selection. “Gödel shows us an unclarity in the concept of ‘mathematics’, which is indicated by the fact that mathematics is taken to be a system” and we can say (contra nearly everyone) that is all that Gödel and Chaitin show. Wittgenstein commented many times that ‘truth’ in math means axioms or the theorems derived from axioms, and ‘false’ means that one made a mistake in using the definitions, and this is utterly different from empirical matters where one applies a test. Wittgenstein often noted that to be acceptable as mathematics in the usual sense, it must be useable in other proofs and it must have real world applications, but neither is the case with Godel’s Incompleteness. Since it cannot be proved in a consistent system (here Peano Arithmetic but a much wider arena for Chaitin), it cannot be used in proofs and, unlike all the ‘rest’ of PA it cannot be used in the real world either. As Rodych notes “…Wittgenstein holds that a formal calculus is only a mathematical calculus (i.e., a mathematical language-game) if it has an extra- systemic application in a system of contingent propositions (e.g., in ordinary counting and measuring or in physics) …” Another way to say this is that one needs a warrant to apply our normal use of words like ‘proof’, ‘proposition’, ‘true’, ‘incomplete’, ‘number’, and ‘mathematics’ to a result in the tangle of games created with ‘numbers’ and ‘plus’ and ‘minus’ signs etc., and with -/- ‘Incompleteness’ this warrant is lacking. Rodych sums it up admirably. “On Wittgenstein’s account, there is no such thing as an incomplete mathematical calculus because ‘in mathematics, everything is algorithm [and syntax] and nothing is meaning [semantics]…” -/- I make some brief remarks which note the similarities of these ‘mathematical’ issues to economics, physics, game theory, and decision theory. -/- Those wishing further comments on philosophy and science from a Wittgensteinian two systems of thought viewpoint may consult my other writings -- Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle 2nd ed (2019), Suicide by Democracy 4th ed (2019), The Logical Structure of Human Behavior (2019), The Logical Structure of Consciousness (2019, Understanding the Connections between Science, Philosophy, Psychology, Religion, Politics, and Economics and Suicidal Utopian Delusions in the 21st Century 5th ed (2019), Remarks on Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason in Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych, Berto, Floyd, Moyal-Sharrock and Yanofsky (2019), and The Logical Structure of Philosophy, Psychology, Sociology, Anthropology, Religion, Politics, Economics, Literature and History (2019). (shrink)
When we sum up the results of Gödel's 1931 Incompleteness Theorem by formalizing Wittgenstein’s verbal specification such that this formalization meets Gödel's own sufficiency requirement: ”Every epistemological antinomy can likewise be used for a similar undecidability proof." then we can see that Gödel's famous logic sentence is only unprovable in PA because it is untrue in PA because it specifies the logical equivalence to self contradiction in PA.
This is what Daniel Simpson has to say of it: -/- An entertaining polemic that takes heartfelt swipes at Western scholars, accusing them of misreading Tantra. "Hinduism is Tantric in essence," the essay says, without proving that Tantra predates other influences, or that "Yoga in its various forms, arises out of Tantra". The latter seems at odds with the earliest descriptions of austerities, or the ascetic objective of bodily transcendence (which Tantric teachings later modified, as evinced by hatha (...) class='Hi'>yoga texts). Meanwhile, Patanjali is said to be Tantric because he describes a silent mind - despite not mentioning kundalini (as the author implies). And quoting Abhinavagupta does not mean that Vedanta is based on his framework. Yoga and experiential insight might be inseparable, but a history of ideas can still be written, however tangential it might seem to the practices it alludes to. If "that which is comprehensible is reductionist and is an exercise in structural scrutiny which is disastrous to Indology," then why compose an essay reducing Hinduism to Tantra, while dismissing all else as misguided archiving? Regardless, I enjoyed its invective. -/- . (shrink)
While elucidating the sense of ego-maker in classical Samkhya and Yoga philosophy I bear in mind several meanings of the word ‘sense’, or different levels of its understanding, namely: the semantic, ontological and epistemic as well as axiological sense. Thus, my aim is, firstly, to specify the semantic sense of the term ‘ahamkara’, that is to explain its contents or denotation. Secondly, when focusing on the ontological context I will try to define the nature and reason, or purpose (arthavattava), (...) of ahamkara. Thirdly, I shall also discuss the ego-maker in epistemic terms by displaying its function of the particular means or determinant of all experience. And finally, when concentrating on the axiological level I am going to consider the significance or value of ahamkara in the context of self-understanding and spiritual development. (shrink)
Spinoza distinguishes between causation that is external, as in A causing B where A is external to B, and causation that is internal, where C causes itself (causa sui), without any involvement of anything external to C. External causation is easy to understand, but self causation is not. This note explores an approach to self-causation based upon Gödelian undecidability and draws upon ideas from an earlier study of Gödel’s proof and the quantum measurement problem (Zwick, 1978).
In ‘Godel’s Way’ three eminent scientists discuss issues such as undecidability, incompleteness, randomness, computability and paraconsistency. I approach these issues from the Wittgensteinian viewpoint that there are two basic issues which have completely different solutions. There are the scientific or empirical issues, which are facts about the world that need to be investigated observationally and philosophical issues as to how language can be used intelligibly (which include certain questions in mathematics and logic), which need to be decided by looking (...) at how we actually use words in particular contexts. When we get clear about which language game we are playing, these topics are seen to be ordinary scientific and mathematical questions like any others. Wittgenstein’s insights have seldom been equaled and never surpassed and are as pertinent today as they were 80 years ago when he dictated the Blue and Brown Books. In spite of its failings—really a series of notes rather than a finished book—this is a unique source of the work of these three famous scholars who have been working at the bleeding edges of physics, math and philosophy for over half a century. Da Costa and Doria are cited by Wolpert (see below or my articles on Wolpert and my review of Yanofsky’s ‘The Outer Limits of Reason’) since they wrote on universal computation, and among his many accomplishments, Da Costa is a pioneer in paraconsistency. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019) . (shrink)
In ‘Godel’s Way’ three eminent scientists discuss issues such as undecidability, incompleteness, randomness, computability and paraconsistency. I approach these issues from the Wittgensteinian viewpoint that there are two basic issues which have completely different solutions. There are the scientific or empirical issues, which are facts about the world that need to be investigated observationally and philosophical issues as to how language can be used intelligibly (which include certain questions in mathematics and logic), which need to be decided by looking (...) at how we actually use words in particular contexts. When we get clear about which language game we are playing, these topics are seen to be ordinary scientific and mathematical questions like any others. Wittgenstein’s insights have seldom been equaled and never surpassed and are as pertinent today as they were 80 years ago when he dictated the Blue and Brown Books. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019). (shrink)
Section 1 reviews Strawson’s logic of presuppositions. Strawson’s justification is critiqued and a new justification proposed. Section 2 extends the logic of presuppositions to cases when the subject class is necessarily empty, such as (x)((Px & ~Px) → Qx) . The strong similarity of the resulting logic with Richard Diaz’s truth-relevant logic is pointed out. Section 3 further extends the logic of presuppositions to sentences with many variables, and a certain valuation is proposed. It is noted that, given this valuation, (...) Gödel’s sentence becomes neither true nor false. The similarity of this outcome with Goldstein and Gaifman’s solution of the Liar paradox, which is discussed in section 4, is emphasized. Section 5 returns to the definition of meaningfulness; the meaninglessness of certain sentences with empty subjects and of the Liar sentence is discussed. The objective of this paper is to show how all of the above-mentioned concepts are interrelated. (shrink)
Jung’s individuation process, the central process of human development, relies heavily on several core philosophical and psychological ideas including the unconscious, complexes, the archetype of the Self, and the religious function of the psyche. While working to find empirical evidence of the psyche’s religious function, Jung studied a variety of subjects including the Eastern liberatory traditions of Buddhism and Patañjali’s Classical Yoga. In these traditions, Jung found substantiation of his ideas on psychospiritual development. Although Jung’s career in soul work (...) was lengthy, throughout, he aimed to steer clear of metaphysics. Patañjali’s metaphysics, on the other hand, are straightforward, and his ontological commitments are evident. Because Jung’s ontological commitments were not explicit, his theories, when seen through Patañjali’s lens, confuse ontological questions with epistemic issues. As a result, when comparing the Jungian and Patañjalian notions of the Self, Jung’s insightful ideas seem to be constructed upon a considerably shaky foundation. Yet, utilizing the exceptionally consistent ontological and epistemological commitments of Patañjali Yoga, as well as the objective measures of affective neuroscience, brings credence to the innate aspects and instinctual nature of Jung’s archetype of the Self, and assists in answering the question of whether the archetype is innate or emergent. (shrink)
The relationship between the physical body and the conscious human mind has been a deeply problematic topic for centuries. Physicalism is the 'orthodox' metaphysical stance in contemporary Western thought, according to which reality is exclusively physical/material in nature. However, in the West, theoretical dissatisfaction with this type of approach has historically lead to Cartesian-style dualism, wherein mind and body are thought to belong to distinct metaphysical realms. In the current discussion I compare and contrast this standard Western approach with an (...) alternative form of dualism developed in the Sāṃkhya-Yoga philosophical tradition, where matter and pure consciousness are held to belong to distinct and independent realms, but where the mind is placed on the material side of the ontological divide. I argue that this model possesses a number of theoretical advantages over Cartesian-style dualism, and constitutes a compelling theoretical framework for re-conceptualizing the mind-body problem. (shrink)
On the heels of Franzén's fine technical exposition of Gödel's incompleteness theorems and related topics (Franzén 2004) comes this survey of the incompleteness theorems aimed at a general audience. Gödel's Theorem: An Incomplete Guide to its Use and Abuse is an extended and self-contained exposition of the incompleteness theorems and a discussion of what informal consequences can, and in particular cannot, be drawn from them.
Materialist and fundamentalist reductive ideologies obscure our capacity to directly experience the numinous. Thus, importantly, given the weight of the observable and measurable in orthodox science, and oftentimes a dismissal of both the soul and the subjective, a viable means of reconciling science and religious experience has continued to elude us. As a counter-measure to this obscuration, Jungian-oriented depth psychology has developed as an empirical science of the unconscious, researching both subject and object and offering theories and practices that foster (...) the psychospiritual development of the personality. Despite cultural and epochal differences, comparable evidence to Jung's process of psychospiritual development can be found in the Eastern liberatory tradition of Patañjali's Classical Yoga. However, given the elevated presence of neuroscience, no psychology, and especially no psychology that supports the soul, seems likely to survive much longer without finding an alliance with the objective measures of brain science. When considering the radically empirical measures of Jung and Patañjali, affective neuroscience may offer us a contemporary and objective means of languaging the bridge between the transcendent and immanent and fostering a contemporary science of the sacred. (shrink)
Modern science, based on the laws of physics, claims validity for all events in space and time. However, it also reveals its own limitations, such as the indeterminacy of quantum physics, the limits of decidability, and, presumably, limits of decodability of the mind-brain relationship. At the philosophical level, these intrinsic limitations allow for different interpretations of the relation between human cognition and the natural order. In particular, modern science may be logically consistent with religious as well as agnostic views of (...) humans and the universe. These points are exemplified through the transcript of a discussion between Kurt Gödel and Rudolf Carnap that took place in 1940. Gödel, discoverer of mathematical undecidability, took a proreligious view; Carnap, one of the founders of analytical philosophy, an antireligious view. By the time of the discussion, Carnap had liberalized his ideas on theoretical concepts of science: he believed that observational terms do not suffice for an exhaustive definition of theoretical concepts. Then, responded Gödel, one should formulate a theory or metatheory that is consistent with scientific rationality, yet also encompasses theology. Carnap considered such theories unproductive. The controversy remained unresolved, but its emphasis shifted from rationality to wisdom, not only in the Gödel-Carnap discussion but also in our time. -/- . (shrink)
Philosophers of language have drawn on metamathematical results in varied ways. Extensionalist philosophers have been particularly impressed with two, not unrelated, facts: the existence, due to Frege/Tarski, of a certain sort of semantics, and the seeming absence of intensional contexts from mathematical discourse. The philosophical import of these facts is at best murky. Extensionalists will emphasize the success and clarity of the model theoretic semantics; others will emphasize the relative poverty of the mathematical idiom; still others will question the aptness (...) of the standard extensional semantics for mathematics. In this paper I investigate some implications of the Gödel Second Incompleteness Theorem for these positions. I argue that the realm of mathematics, proof theory in particular, has been a breeding ground for intensionality and that satisfactory intensional semantic theories are implicit in certain rigorous technical accounts. (shrink)
The paper discusses the issue of psychophysical agency in the context of Indian philosophy, focusing on the oldest preserved texts of the classical tradition of Sāṃkhya–Yoga. The author raises three major questions: What is action in terms of Sāṃkhyakārikā (ca. fifth century CE) and Yogasūtra (ca. third century CE)? Whose action is it, or what makes one an agent? What is a right and morally good action? The first part of the paper reconsiders a general idea of action – (...) including actions that are deliberately done and those that ‘merely’ happen – identified by Patañjali and Ῑśvarakṛṣṇa as a permanent change or transformation (pariṇāma) determined by the universal principle of causation (satkārya). Then, a threefold categorization of actions according to their causes is presented, i.e. internal agency (ādhyātmika), external agency (ādhibhautika) and ‘divine’ agency (ādhidaivika). The second part of the paper undertakes the problem of the agent’s autonomy and the doer’s psychophysical integrity. The main issues that are exposed in this context include the relationship between an agent and the agent’s capacity for perception and cognition, as well as the crucial Sāṃkhya–Yoga distinction between ‘a doer’ and ‘the self’. The agent’s self-awareness and his or her moral self-esteem are also briefly examined. Moreover, the efficiency of action in present and future is discussed (i.e. karman, karmāśaya, saṃskāra, vāsanā), along with the criteria of a right act accomplished through meditative insight (samādhi) and moral discipline (yama). (shrink)
On a few occasions F.A. Hayek made reference to the famous Gödel theorems in mathematical logic in the context of expounding his cognitive and social theory. The exact meaning of the supposed relationship between Gödel's theorems and the essential proposition of Hayek's theory of mind remains subject to interpretation, however. The author of this article argues that the relationship between Hayek's thesis that the human brain can never fully explain itself and the essential insight provided by Gödel's theorems in mathematical (...) logic has the character of an analogy, or a metaphor. Furthermore the anti-mechanistic interpretation of Hayek's theory of mind is revealed as highly questionable. Implications for the Socialist Calculation Debate are highlighted. It is in particular concluded that Hayek's arguments for methodological dualism, when compared with those of Ludwig von Mises, actually amount to a strengthening of the case for methodological dualism. (shrink)
The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological, and logical character. This chapter focuses on two arguments from logic. First, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to which every proposition is either true or false, no matter whether the (...) proposition is about the past, present or future. In particular, the argument goes, whatever one does or does not do in the future is determined in the present by the truth or falsity of the corresponding proposition. The second argument coming from logic is much more modern and appeals to Gödel's incompleteness theorems to make the case against determinism and in favour of free will, insofar as that applies to the mathematical potentialities of human beings. The claim more precisely is that as a consequence of the incompleteness theorems, those potentialities cannot be exactly circumscribed by the output of any computing machine even allowing unlimited time and space for its work. The chapter concludes with some new considerations that may be in favour of a partial mechanist account of the mathematical mind. (shrink)
The aim of this essay is twofold. First, it outlines the concept of ontological frame. Secondly, two models are distinguished on this structure. The first one is connected to Kant’s concept of possible object and the second one relates to Leibniz’s. Leibniz maintains that the source of possibility is the mere logical consistency of the notions involved, so that possibility coincides with analytical possibility. Kant, instead, argues that consistency is only a necessary component of possibility. According to Kant, something is (...) possible if there is a cause capable of bringing it into existence; to this end consistency alone is not sufficient. Thus, while the Leibnizian notion of consistency is at the root of the concept of analytical possibility, the Kantian notion of possibility is the source of real possibility. This difference plays an important role in the discussion of Gödel’s ontological proof, which can be formally interpreted on the ontological frame of the pure perfections. While this proof, under some emendation condition, is conclusive in the context of Leibniz’s ontological model, it is not so within the Kantian one. This issue will be the subject of the second part of the present essay. (shrink)
Carl Ernst'ten çevirdiğimiz metinde Şettârî tarikati şeyhlerinden Muhammed Gavs'ın tercüme ettiği eser vasıtası ile Yogiler ile sûfîler arasındaki münâsebet ele alınmaktadır.
Hal ini sering berpikir bahwa kemustahilan, ketidaklengkapan, Paraconsistency, Undecidability, Randomness, komputasi, Paradox, ketidakpastian dan batas alasan yang berbeda ilmiah fisik atau matematika masalah memiliki sedikit atau tidak ada dalam Umum. Saya menyarankan bahwa mereka sebagian besar masalah filosofis standar (yaitu, Permainan bahasa) yang sebagian besar diselesaikan oleh Wittgenstein lebih dari 80years yang lalu. -/- "Apa yang kita ' tergoda untuk mengatakan ' dalam kasus seperti ini, tentu saja, bukan filsafat, tetapi bahan baku. Jadi, misalnya, apa yang seorang matematikawan cenderung mengatakan (...) tentang objektivitas dan realitas fakta matematika, bukan filsafat matematika, tetapi sesuatu untuk pengobatan filosofis. " Wittgenstein PI 234 -/- "Filsuf terus melihat metode ilmu di depan mata mereka dan tak tertahankan tergoda untuk bertanya dan menjawab pertanyaan dalam cara ilmu tidak. Kecenderungan ini adalah sumber nyata metafisika dan memimpin filsuf menjadi gelap gulita. " Wittgenstein -/- Aku memberikan ringkasan singkat dari beberapa temuan utama dari dua siswa yang paling terkemuka perilaku zaman modern, Ludwig Wittgenstein dan John Searle, pada struktur Logis intensionality (pikiran, bahasa, perilaku), mengambil sebagai titik awal Penemuan fundamental Wittgenstein – bahwa semua masalah ' filosofis ' adalah sama — kebingungan tentang bagaimana menggunakan bahasa dalam konteks tertentu, sehingga semua solusi sama — melihat bagaimana bahasa dapat digunakan dalam konteks yang menjadi masalah sehingga kebenaranNya kondisi (kondisi kepuasan atau COS) jelas. Masalah dasar adalah bahwa seseorang dapat mengatakan apa-apa, tetapi orang tidak dapat berarti (negara yang jelas cos untuk) sembarang ucapan dan makna hanya mungkin dalam konteks yang sangat spesifik. -/- Saya membedah beberapa tulisan dari beberapa komentator utama pada isu ini dari sudut pandang Wittgensteinian dalam kerangka perspektif modern dari dua sistem pemikiran (Dipopulerkan sebagai ' berpikir cepat, berpikir lambat '), mempekerjakan meja baru intensionality dan baru sistem ganda nomenklatur. Saya menunjukkan bahwa ini adalah heuristik yang kuat untuk menggambarkan sifat sebenarnya dari hal ini ilmiah, fisik atau matematika masalah yang benar-benar terbaik didekati sebagai masalah filosofis standar bagaimana bahasa yang akan digunakan (permainan bahasa di Wittgenstein's terminologi). -/- Ini adalah pendapat saya bahwa tabel intensionality (rasionalitas, pikiran, pikiran, bahasa, kepribadian dll) yang fitur mencolok di sini menggambarkan lebih atau kurang akurat, atau setidaknya berfungsi sebagai heuristic untuk, bagaimana kita berpikir dan berperilaku, dan sehingga mencakup tidak hanya filsafat dan psikologi, tetapi segala sesuatu yang lain (sejarah, sastra, matematika, politik dll). Perhatikan terutama bahwa intensionalitas dan rasionalitas sebagai I (bersama dengan Searle, Wittgenstein dan lain-lain) melihatnya, mencakup baik sistem linguistik pertimbangan sadar 2 dan tidak disadari otomatis sistem prelinguistik 1 tindakan atau refleks. (shrink)
On pense généralement que l'impossibilité, l'incomplétdulité, la paracohérence, l'indécidabilité, le hasard, la calcul, le paradoxe, l'incertitude et les limites de la raison sont des questions scientifiques physiques ou mathématiques disparates ayant peu ou rien dans terrain d'entente. Je suggère qu'ils sont en grande partie des problèmes philosophiques standard (c.-à-d., jeux de langue) qui ont été la plupart du temps résolus par Wittgenstein plus de 80 ans. Je fournis un bref résumé de quelques-unes des principales conclusions de deux des plus éminents (...) étudiants du comportement des temps modernes, Ludwig Wittgenstein et John Searle, sur la structure logique de l'intentionnalité (esprit, langue, comportement), en prenant comme point de départ La découverte fondamentale de Wittgenstein, à savoir que tous les problèmes véritablement « philosophiques » sont les mêmes, les confusions sur la façon d'utiliser la langue dans un contexte particulier, et donc toutes les solutions sont les mêmes— en regardant comment la langue peut être utilisée dans le contexte en cause afin que sa vérité (Conditions de satisfaction ou COS) sont claires. Le problème fondamental est que l'on peut dire n'importe quoi, mais on ne peut pas signifier (état clair COS pour) toute déclaration arbitraire et le sens n'est possible que dans un contexte très spécifique. Je dissé que quelques écrits de quelques-uns des principaux commentateurs sur ces questions d'un point de vue wittgensteinien dans le cadre de la perspective moderne des deux systèmes de pensée (popularisé comme «penser vite, penser lentement»), en utilisant une nouvelle table de intentionnalité et la nomenclature de nouveaux systèmes doubles. Je montre qu'il s'agit d'un puissant heuristique pour décrire la vraie nature de ces questions scientifiques, physiques ou mathématiques putatives qui sont vraiment mieux abordés comme des problèmes philosophiques standard de la façon dont la langue doit être utilisée (jeux de langue dans Wittgenstein terminologie). (shrink)
In this contribution an attempt is made to analyze an important mathematical discovery, the theorem of Gödel, and to explore the possible impact on the consistency of metaphysical systems. It is shown that mathematics is a pointer to a reality that is not exclusively subjected to physical laws. As the Gödel theorem deals with pure mathematics, the philosopher as such can not decide on the rightness of this theorem. What he, instead can do, is evaluating the general acceptance of this (...) mathematical finding and reflect on the consistency between consequences of the mathematical theorem with consequences of his metaphysical view. The findings of three mathematicians are involved in the argumentation: first Gödel himself, then the further elaboration by Turing and finally the consequences for the human mind as worked out by Penrose. As a result one is encouraged to distinguish two different types of intellectual activity in mathematics, which both can be carried out by humans. The astonishing thing is not the distinction between a formalized, logic approach on the one side and intuition, mathematical insight and meaning on the other. Philosophically challenging, however, is the claim that principally only one of these intellectual activities can be carried out by objects exclusively bound to the laws of physical reality. (shrink)
In his recent article Christopher Gauker (2001) has presented a thoughtprovoking argument against deﬂationist theories of truth. More exactly, he attacks what he calls ‘T-schema deﬂationism’, that is, the claim that a theory of truth can simply take the form of certain instances of the T-schema.
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