A resolution to the Russell Paradox is presented that is similar to Russell's “theory of types” method but is instead based on the definition of why a thing exists as described in previous work by this author. In that work, it was proposed that a thing exists if it is a grouping tying "stuff" together into a new unit whole. In tying stuff together, this grouping defines what is contained within the new existent entity. A corollary is that a thing, (...) such as a set, does not exist until after the stuff is tied together, or said another way, until what is contained within is completely defined. A second corollary is that after a grouping defining what is contained within is present and the thing exists, if one then alters what is tied together (e.g., alters what is contained within), the first existent entity is destroyed and a different existent entity is created. A third corollary is that a thing exists only where and when its grouping exists. Based on this, the Russell Paradox's set R of all sets that aren't members of themselves does not even exist until after the list of the elements it contains (e.g. the list of all sets that aren't members of themselves) is defined. Once this list of elements is completely defined, R then springs into existence. Therefore, because it doesn't exist until after its list of elements is defined, R obviously can't be in this list of elements and, thus, cannot be a member of itself; so, the paradox is resolved. This same type of reasoning is then applied to Godel's first Incompleteness Theorem. Briefly, while writing a Godel Sentence, one makes reference to a future, not yet completed and not yet existent sentence, G, that claims its unprovability. However, only once the sentence is finished does it become a new unit whole and existent entity called sentence G. If one then goes back in and replaces the reference to the future sentence with the future sentence itself, a totally different sentence, G1, is created. This new sentence G1 does not assert its unprovability. An objection might be that all the possibly infinite number of possible G-type sentences or their corresponding Godel numbers already exist somehow, so one doesn't have to worry about references to future sentences and springing into existence. But, if so, where do they exist? If they exist in a Platonic realm, where is this realm? If they exist pre-formed in the mind, this would seem to require a possibly infinite-sized brain to hold all these sentences. This is not the case. What does exist in the mind is the system for creating G-type sentences and their corresponding numbers. This mental system for making a G-type sentence is not the same as the G-type sentence itself just as an assembly line is not the same as a finished car. In conclusion, a new resolution of the Russell Paradox and some issues with proofs of Godel's First Incompleteness Theorem are described. (shrink)

Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen’s influential account of monster culture and explore how well mathematical monsters fit each of his seven theses. The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. Firstly, late nineteenth-century mathematicians made numerous unsettling discoveries that threatened their understanding of their own discipline and challenged their intuitions. The great French mathematician (...) Henri Poincaré characterised these anomalies as ‘monsters’, a name that stuck. Secondly, the twentieth-century philosopher Imre Lakatos composed a seminal work on the nature of mathematical proof, in which monsters play a conspicuous role. Lakatos coined such terms as ‘monster-barring’ and ‘monster-adjusting’ to describe strategies for dealing with entities whose properties seem to falsify a conjecture. Thirdly, and most recently, mathematicians dubbed the largest of the sporadic groups ‘the Monster’, because of its vast size and uncanny properties, and because its existence was suspected long before it could be confirmed. (shrink)

This paper explores a special kind of loop of grounding: cosmic loops. A cosmic loop is a loop that intuitively requires us to go "around" the entire universe to come back to the original ground. After describing several kinds of cosmic loop scenarios, I will discuss what we can learn from these scenarios about constraints on grounding; the conceivability of cosmic loops; the possibility of cosmic loops; and the prospects for salvaging local reflexivity, asymmetry and transitivity of grounding in a (...) world containing a cosmic loop of ground. The considerations raised in this paper also bear on what we should think about relations that are meant to support grounding relations: in particular, revisions to theories of the part-whole relation are discussed. (shrink)

Philosophical questions about minds and computation need to focus squarely on the mathematical theory of Turing machines (TM's). Surrogate TM's such as computers or formal systems lack abilities that make Turing machines promising candidates for possessors of minds. Computers are only universal Turing machines (UTM's)—a conspicuous but unrepresentative subclass of TM. Formal systems are only static TM's, which do not receive inputs from external sources. The theory of TM computation clearly exposes the failings of two prominent critiques, Searle's Chinese room (...) (1980) and arguments from Gödel's Incompleteness theorems (e.g., Lucas, 1961; Penrose, 1989), both of which fall short of addressing the complete TM model. Both UTM-computers and formal systems provide an unsound basis for debate. In particular, their special natures easily foster the misconception that computation entails intrinsically meaningless symbol manipulation. This common view is incorrect with respect to full-fledged TM's, which can process inputs non-formally, i.e., in a subjective and dynamically evolving fashion. To avoid a distorted understanding of the theory of computation, philosophical judgments and discussions should be grounded firmly upon the complete Turing machine model, the proper model for real computers. (shrink)

Some theorists believe that there is a plurality of values, and that in many circumstances these values are incommensurable, or at least incomparable. Others believe that all values are reducible to a single super-value, or that even if there is a plurality of irreducible values these values are commensurable. But I will argue that both sides have got it wrong. Values are neither commensurable nor incommensurable, at least not in the way most people think. We are free to believe in (...) incommensurability or not, depending on what particular conception of morality we want to embrace. Incommensurability is accordingly not a theory about value. It is a presupposition that provides a necessary background condition for a certain kind of value to exist. It is therefore not the kind of view that can be morally true or false. As a presupposition, it can only be accepted or rejected on grounds that do not presuppose that morality already exists. Incommensurability is, like the rejection of hard determinism, one of the presuppositions on which morality as we know it happens to be based. (shrink)

My research into Virtual Reality technology and its central property of immersion has indicated that immersion in Virtual Reality (VR) electronic systems is a significant key to the understanding of contemporary culture as well as considerable aspects of previous culture as detected in the histories of philosophy and the visual arts. The fundamental change in aesthetic perception engendered by immersion, a perception which is connected to the ideal of total-immersion in virtual space, identifies certain shifts in ontology which are relevant (...) to a better understanding of the human being. This understanding was achieved through a broad inquiry into the histories of Virtual Reality, philosophy, and the visual arts and has lead to the formulation of an aesthetic theory of immersive consciousness indicative of immersive culture. The primary subject of this discourse is immersion then: an experience which will be identified within the dissertation as the indispensable characteristic of Virtual Reality. The understanding of immersion arrived at here will be used to fashion a synchronous theory of art particularly informed by encounters and concepts of immersion into virtuality. To sufficiently address this subject in a scholarly fashion, I have researched, found and accumulated aesthetic and philosophic examples of immersive tendencies, as found within the histories of art and philosophy, which subsequently contributed towards the articulation of what I have come to call immersive consciousness. As a result of formulating such an immersive consciousness, a good deal of the basis for the questioning of the Western ontological tradition has been found in the Western tradition itself when we look with new eyes and ask new uncertain questions. Moreover, this immersive consciousness will be used to propose some abstract questions encircling today's electronic-based culture. Through the structuring of the argument within the thesis - and overtly within the conclusion - I have articulated a non-teleological creative strategy which provides the basis for an unconstraining integration of noologies (ways of semblancing the thinking process). This strategy provides a means of exemplifying - and for honoring - various methods of thinking. This structuring strategy is consistent with the 'hacker ethic' as defined by Steven Levy, as a demand that access to computers - and anything which might teach us something about the way the world works - should be unlimited and total. To follow this strategy, this dissertation has set out to understand how topical conceptions of virtual immersion connect to pre-existing systems of thought as revealed in art as they have extended out of antecedent ontological self-understandings, historical human self-understandings which have evidenced themselves in the elaboration of technological objectives. To do this I have forged a certain rhizomatic paternity/maternity for Virtual Reality within this dissertation by joining choice immersive examples of simulacra technology into mental connections with the relevant examples culled from the histories of art, architecture, information-technology, sex, myth, space, consciousness and philosophy. (shrink)

The essay centers on Gödel's views on the place of our intuitive concept of time in philosophy and in physics. It presents my interpretation of his work on the theory of relativity, his observations on the relationship between Einstein's theory and Kantian philosophy, as well as some of the scattered remarks in his conversations with me in the seventies — namely, those on the philosophies of Leibniz, Hegel and Husserl — as a successor of Kant — in relation to their (...) conceptions of time. (shrink)

This study explores the transhuman from an East Asian perspective. In terms of cognitive science, mathematics, and theology, we define the transhuman system as characterized by transcendence, extension by compactification, and samtaegeuk. Compactification is conceptualized here in mathematical terms, as adding one or more elements so that a system becomes more complete—as one might join both ends of a line, and thereby create a circle. We assert that the East Asian transhuman could be defined as a three-point compactification: as an (...) extension of biophysical objects and events such as robots, cyborgs, and environments ; as an extension of culture, science, and art ; and as an extension of the interaction between the human and Cosmic Absolute such as in religions. Such a notion of the Transhuman might be associated with God, but any description of God, the Absolute Infinite, will apply to something less than God. (shrink)

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

We discuss the nature of reality in the ontological context of Penrose’s math-matter-mind triangle. The triangle suggests the circularity of the widespread view that math arises from the mind, the mind arises out of matter, and that matter can be explained in terms of math. Non-physicists should be wary of any claim that modern physics leads us to any particular resolution of this circularity, since even the sample of three theoretical physicists writing this paper hold three divergent views. Some physicists (...) believe that current physics has already found the basic framework for a complete description of reality, and only has to fill in the details. Others suspect that no single framework, from physics or other sources, will ever capture reality. Yet others guess that reality might be approached arbitrarily closely by some form of future physics, but probably based on completely different frameworks. We will designate these three approaches as the fundamentalist, secular and mystic views of the world, as seen by practicing physicists. We present and contrast each of these views, which arguably form broad categories capturing most if not all interpretations of physics. We argue that this diversity in the physics community is more useful than an ontological monoculture, since it motivates physicists to tackle unsolved problems with a wide variety of approaches. (shrink)

A Christian approach to scholarship, directed by the central biblical motive of creation, fall and redemption and guided by the theoretical idea that God subjected all of creation to His Law-Word, delimiting and determining the cohering diversity we experience within reality, in principle safe-guards those in the grip of this ultimate commitment and theoretical orientation from absolutizing or deifying anything within creation. In this article my over-all approach is focused on the one-sided legacy of mathematics, starting with Pythagorean arithmeticism (“everything (...) is number”), continuing with the geometrization of mathematics after the discovery of irrational numbers and once again, during the nineteenth century returning to an arithmeticistic position. The third option, never explored during the history of mathematics, guides our analysis: instead of reducing space to number or number to space it is argued that both the uniqueness of these two aspects and their mutual coherence ought to direct mathematics. The presence of different schools of thought is highlighted and then the argument proceeds by distinguishing numerical and spatial facts, while accounting for the strict correlation of operations on the law side of the numerical aspect and their correlated numerical subjects (numbers). Discussing the examples of 2 + 2 = 4 and the definition of a straight line as the shortest distance between two points provide the background for a brief sketch of the third alternative proposed (inter alia against the background of an assessment of infinity and continuity and the vicious circles present in contemporary mathematical arithmeticistic claims). (shrink)

The purpose of this paper is to discuss the various types of physical universe which could exist according to modern mathematical physics. The paper begins with an introduction that approaches the question from the viewpoint of ontic structural realism. Section 2 takes the case of the `multiverse' of spatially homogeneous universes, and analyses the famous Collins-Hawking argument, which purports to show that our own universe is a very special member of this collection. Section 3 considers the multiverse of all solutions (...) to the Einstein field equations, and continues the discussion of whether the notions of special and typical can be defined within such a collection. (shrink)

In this paper I investigate the relation between physics and metaphysics in Plato’s participation theory. I show that the logic shoring up Plato’s metaphysics in paraconsistent, as had been suggested already by Graham Priest. The transformation of the paradoxical One-and-Many of the pre-Socratics into a paraconsistent Great-and-Small bridges the abyss between archaic rationality and the world of classical logic based ultimately on the principle of contradiction. Indeed, language is an organ of perception, not simply a means of communication. J. Jaynes, (...) Origin of Consciousness. (shrink)

During the first phase of Greek mathematics a proof consisted in showing or making visible the truth of a statement. This was the epagogic method. This first phase was followed by an apagogic or deductive phase. During this phase visual evidence was rejected and Greek mathematics became a deductive system. Now epagoge and apagoge, apart from being distinguished, roughly according to the modern distinction between inductive and deductive procedures, were also identified on account of the conception of generality as continuity. (...) Epistemology of mathematics today only remembers the distinction, forgetting where they agreed, in this manner not only destroying the unity of the perceptual and conceptual but also forgetting what could be gained from Aristotelian demonstrative science. (shrink)

The emperor's new mind (hereafter Emperor) is an attempt to put forward a scientific alternative to the viewpoint of according to which mental activity is merely the acting out of some algorithmic procedure. John Searle and other thinkers have likewise argued that mere calculation does not, of itself, evoke conscious mental attributes, such as understanding or intentionality, but they are still prepared to accept the action the brain, like that of any other physical object, could in principle be simulated by (...) a computer. In Emperor I go further than this and suggest that the outward manifestations ofconscious mental activity cannot even be properly simulated by calculation. To support this view, I use various arguments to show that the results of mathematical insight, in particular, do not seem to be obtained algorithmically. The main thrust ofthis work, however, is to present an overview ofthe present state of physical understanding and to show that an important gap exists at the point where, quantum and classical physics meet, as well as to speculate on how the conscious brain might be taking advantage ofwhatever new physics is needed to fill this gap to achieve its nonalgorithmic effects. (shrink)

The alleged proof of the non-mechanical, or non-computational, character of the human mind based on Gödel’s incompleteness theorem is revisited. Its history is reviewed. The proof, also known as the Lucas argument and the Penrose argument, is refuted. It is claimed, following Gödel himself and other leading logicians, that antimechanism is not implied by Gödel’s theorems alone. The present paper sets out this refutation in its strongest form, demonstrating general theorems implying the inconsistency of Lucas’s arithmetic and the semantic inadequacy (...) of Penrose’s arithmetic. On the other hand, the limitations to our capacity for mechanizing or programming the mind are also indicated, together with two other corollaries of Gödel’s theorems: that we cannot prove that we are consistent, and that we cannot fully describe our notion of a natural number. (shrink)

Infinity and negation are in various relations and interdependencies one to another. The analysis of negation and infinity aims to better understanding them. Semantical, syntactical, and pragmatic issues will be considered.

I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, parallel universes and (...) Gödel incompleteness. I hypothesize that only computable and decidable (in Gödel’s sense) structures exist, which alleviates the cosmological measure problem and may help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems. (shrink)

This essay interprets the meaning of one of the cards in aTarot deck, "The Magician," in the context of process philosophy in the tradition of Alfred North Whitehead. It brings into the conversation the philosophical legacy of American semiotician Charles Sanders Peirce as well as French poststructuralist Gilles Deleuze. Some of their conceptualizations are explored herein for the purpose of explaining the symbolic function of the Magician in the world. From the perspective of the logic of explanation, the sign of (...) the Magician is an index of nonmechanistic, mutualist or circular, causality that enables self-organization embedded in coordination dynamics. Its action is such as to establish an unorthodox connection crossing over the dualistic gap between mind and matter, science and magic, process and structure, the world without and the world within, subject and object, and human experience and the natural world, thereby overcoming what Whitehead called the paradox of the connectedness of things. The Magician represents a certain quality that acts as a catalytic agent capable of eliciting transmutations, that is, the emergence of novelty. I present a model for process∼structure that uses mathematics on the complex plane and the rules of projective geometry. The corollary is such that the presence of the Magician in the world enables a particular organization of thought that makes pre-cognition possible. (shrink)

Gödel's Incompleteness Theorems have the same scientific status as Einstein's principle of relativity, Heisenberg's uncertainty principle, and Watson and Crick's double helix model of DNA. Our aim is to discuss some new faces of the incompleteness phenomenon unveiled by an information-theoretic approach to randomness and recent developments in quantum computing.

A version of the Church-Turing Thesis states that every effectively realizable physical system can be simulated by Turing Machines (‘Thesis P’). In this formulation the Thesis appears to be an empirical hypothesis, subject to physical falsification. We review the main approaches to computation beyond Turing definability (‘hypercomputation’): supertask, non-well-founded, analog, quantum, and retrocausal computation. The conclusions are that these models reduce to supertasks, i.e. infinite computation, and that even supertasks are no solution for recursive incomputability. This yields that the realization (...) of hypercomputing devices is implausible, and that Thesis P is not essentially different from the standard Church-Turing Thesis. (shrink)