This paper analyses the hidden use of new axioms in set-theoretic practice with a focus on large cardinal axioms and presents a general overview of set-theoretic practices using large cardinal axioms. The hidden use of a new axiom provides extrinsic reasons in support of this axiom via the idea of verifiable consequences, which is especially relevant for set-theoretic practitioners with an absolutist view. Besides that, the hidden use has pragmatic significance for further important sub-groups of the set-theoretic (...) community---set-theoretic practitioners with a pluralist view and set-theoretic practitioners who aim for ZFC-proofs. By describing this, the paper gives a more complete picture of new axioms in set-theoretic practice. These observations, for instance, show that set-theoretic practitioners interested in ZFC-proofs use tools that go beyond ZFC. The analysis is based on empirical data that was collected in an extensive interview study with set-theoretic practitioners. (shrink)

The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the `maximal iterative concept', and the programme identi fies higher-order statements motivated by the maximal iterative concept. The satisfaction (...) of these statements (H-axioms) in countable transitive models, the collection of which constitutes the `hyperuniverse' (H), has remarkable 1st-order consequences, some of which we review in section 5. (shrink)

Discussion of new axioms for set theory has often focused on conceptions of maximality, and how these might relate to the iterative conception of set. This paper provides critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception. In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in particular (...) ways) face complementary problems. The latter view is unable to use maximality axioms that make use of extensions, where the former has to contend with the existence of extensions violating maximality axioms. An analysis of two kinds of multiversism, a Zermelian form and Skolemite form, leads to the conclusion that the kind of maximality captured by an axiom differs substantially according to background ontology. (shrink)

Although the concept of uncertainty is as old as Epicurus’s writings, and an excellent quantitative theory, with entropy as the measure of uncertainty having been developed in recent times, there has been little exploration of the qualitative theory. The purpose of the present paper is to give a qualitative axiomatization of uncertainty, in the spirit of the many studies of qualitative comparative probability. The qualitative axioms are fundamentally about the uncertainty of a partition of the probability space of events. (...) Of course, it is common to speak of the uncertainty, or randomness, of a random variable, but only the partition defined by the values of the random variable enter into the definition of uncertainty, not the actual values. It is straightforward to add axioms for decision making following the general line of Savage from the 1950s. Indeed, in the spirit of Epicurus, it is really our intuitive feeling about the uncertainty of the future that motivates much of our thinking about decisions. Here, the distinction between the concepts of probability and uncertainty can be made by citing many familiar examples. Without spelling out the technical details, the axiomatization of qualitative probability with uncertainty as the most important primitive concept, it is possible to raise a different kind of question about bounded rationality. This new question is whether or not one should bound the uncertainty in thinking and investigating any detailed framework of decision making. Discussion of this point is certainly different from the question of bounding rationality by not maximizing expected utility. In practice, we naturally bound uncertainty in our analysis of decision-making problems. As in the case of formulating an alternative for maximizing expected utility, so is the case of rational alternatives to maximizing uncertainty. There are several issues to consider. In the spirit of my other work in qualitative probability, I explore alternatives rather than attempt to give a definitive argument for one single solution. (shrink)

Spinoza's causal axiom is at the foundation of the Ethics. I motivate, develop and defend a new interpretation that I call the ‘causally restricted interpretation’. This interpretation solves several longstanding puzzles and helps us better understand Spinoza's arguments for some of his most famous doctrines, including his parallelism doctrine and his theory of sense perception. It also undermines a widespread view about the relationship between the three fundamental, undefined notions in Spinoza's metaphysics: causation, conception and inherence.

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us, all directions are the same to us and all units of length we use to create geometric figures are the same to us. On the other hand, through the process of algebraic simplification, this system of axioms directly provides (...) the Weyl’s system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: it supports the thesis that Euclidean geometry is a priori, it supports the thesis that in modern mathematics the Weyl’s system of axioms is dominant to the Euclid’s system because it reflects the a priori underlying symmetries, it gives a new and promising approach to learn geometry which, through the Weyl’s system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics. (shrink)

Ontology engineering is a hard and error-prone task, in which small changes may lead to errors, or even produce an inconsistent ontology. As ontologies grow in size, the need for automated methods for repairing inconsistencies while preserving as much of the original knowledge as possible increases. Most previous approaches to this task are based on removing a few axioms from the ontology to regain consistency. We propose a new method based on weakening these axioms to make them less (...) restrictive, employing the use of refinement operators. We introduce the theoretical framework for weakening DL ontologies, propose algorithms to repair ontologies based on the framework, and provide an analysis of the computational complexity. Through an empirical analysis made over real-life ontologies, we show that our approach preserves significantly more of the original knowledge of the ontology than removing axioms. (shrink)

A description of consciousness leads to a contradiction with the postulation from special relativity that there can be no connections between simultaneous event. This contradiction points to consciousness involving quantum level mechanisms. The Quantum level description of the universe is re- evaluated in the light of what is observed in consciousness namely 4 Dimensional objects. A new improved interpretation of Quantum level observations is introduced. From this vantage point the following axioms of consciousness is presented. Consciousness consists of two (...) distinct components, the observed U and the observer I. The observed U consist of all the events I is aware of. A vast majority of these occur simultaneously. Now if I were to be an entity within the space-time continuum, all of these events of U together with I would have to occur at one point in space-time. However, U is distributed over a definite region of space-time (region in brain). Thus, I is aware of a multitude of space-like separated events. It is seen that this awareness necessitates I to be an entity outside the space-time continuum. With I taken as such, a new concept called concept A is introduced. With the help of concept A a very important axiom of consciousness, namely Free Will is explained. Libet s Experiment which was originally seen to contradict Free will, in the light of Concept A is shown to support it. A variation to Libet s Experiment is suggested that will give conclusive proof for Concept A and Free Will. (shrink)

Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation. Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity. That is nothing else than a new reading at issue and comparative interpretation of Gödel’s papers (1930; 1931) meant here. Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of infinity. The most (...) utilized example of those generalizations is the complex Hilbert space. Any generalization of Peano arithmetic consistent to infinity, e.g. the complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself. (shrink)

A three dimensional hypercube representing all of the 4,096 dyadic computations in a standard bivalent system has been created. It has been constructed from the 16 functions arrayed in a table of functional completeness that can compute a dyadic relationship. Each component of the dyad is an operator as well as a function, such as “implication” being a result, as well as an operation. Every function in the hypercube has been color keyed to enhance the display of emerging patterns. At (...) the minimum, the hypercube is a “multiplication table” of dyadic computations and produces values in a way that shortens the time to do operations that normally would take longer using conventional truth table methods. It also can serve as a theorem prover and creator. With the hypercube comes a deductive system without the need for axioms. The main significance of the 3-D hypercube at this point is that it is the most fundamental way of displaying all dyadic computations in binary space, thus serving as a way of normalizing the rendition of uninterpreted, or raw, binary space. The hypercube is a dimensionless entity, a standard by which in binary spaces can be measured and classified, analogous to a meter stick. (shrink)

June 2022 A Revolutionary New Metaphysics, Based on Consciousness, and a Call to All Philosophers We are in a unique moment of our history unlike any previous moment ever. Virtually all human economies are based on the destruction of the Earth, and we are now at a place in our history where we can foresee if we continue on as we are, our own extinction. As I write, the planet is in deep trouble, heat, fires, great storms, and record flooding, (...) weather patterns drastic everywhere. Capitalism has been so successful in supplying us with an endless array of consumer items, that we have never noticed there is a cost. But there is a cost, the cost is the very life of the Earth herself, and now the day of reckoning is here. Days of reckoning are most often days of rath. But I am hoping that this is a day of major Awakening over the entire planet. We have caused a crisis so great for the Earth we can hardly get our minds around it. But I am proposing that deeply embedded in the crisis are ideas, about who and what the Earth is, Ourselves, and what we are here for. I am proposing that this is a call to all philosophy and philosophers to rethink ourselves from the ground up. The human race has never gotten the Earth right and now we must, for the Earth is the true foundation for civilization. We are in deep trouble with the Earth, and it owes to the fact that the great male thinkers never got the Earth right, they could never make the connection between the life of mind and the life of the Earth. I am proposing that deeply embedded in the crisis we are in are ideas. And that in fact the planet is in the grip of a false metaphysics, and scientific image of nature that really comes most recently from Descartes. His conviction that matter, and with it the Earth, are simply the weighable, the measurable the extended in space and time, utterly devoid of any inner life of mind thoughts or Consciousness. This view of matter and the Earth goes back to the very dawn of thought, Descartes is just its most recent and comprehensive version of things. This view of matter is really a false metaphysics that dominates most of the modern world and underpins the whole scientific image of nature itself. I am proposing this is a false metaphysics and offering a new version of things based on Consciousness. If there is a single idea fueling and driving the destruction of the planet, it is the conviction that the Earth is something without Consciousness. When you intend to ravage and savage the planet, turn it into consumer items to be converted into money in the bank, first pretend that all of the bright beings in the natural world are devoid of Consciousness, just “things” you do not have to have any conscious about destroying them. I am proposing that this term Consciousness, that has appeared as term and a problem for modern science, is the key to getting ourselves right with the Earth. I propose this new theory for science will ramify into every other disciplines as well, and the ways we think just about everything. This old paradigm based on a false theory of matter, has run its course, it has brought us to the brink of our own extinction. We need new ideas, and we need them now. You will find my recent work in the Scientific God Journal under the title: Consciousness as the Organizing Principle, and a very complete version of my arguments for this term Consciousness, along with radical reinterpretation of modern science itself, on the Galileo Commission site. It is a radical moment for all of the thinkers among us, take this as my small contribution to rethinking ourselves in new terms, based on my understanding that the true basis for the whole of Being, the Universe, the Cosmos, and with them Ourselves, is this term Consciousness. And unpack it. And I predict that with this term Consciousness every intractable human problem in both science and the modern world will go down like a line of dominos before it. And so, here are a few new ideas I would offer you. These are the times of the Reappearance of the Feminine, a great Awakening as of Springtime after a long dark and lonely Winter. We are leaving behind us an old civilization based on male dominance, fear lies and control, now in decay, disease, and death disintegrating all around us. We are moving into a whole new civilization based on Love and Truth. A new civilization requires new basic terms values and first principles. And so, I return to Descartes. 1. Just over 300 years ago a young man travelling with the army found himself stranded in a foreign town in a bare room lit by a candle with nothing to do but think. And in a couple of nights Descartes laid the entire thought foundations for modern science and most of the modern world. It isn’t clear that modern philosophers with their funding sources of grants, their graduate students, their reputation and all the perks and privileges of academia can do any better. Descartes claimed there are just two principles in the Universe, Matter and Mind. God was pure mind; human beings were mind and matter. And we are the only ones on the planet with mind. Everything else all of the bright array of beings who share the planet with us are simply matter. And matter? The weighable the measurable the extended in space and time utterly devoid of any inner life of thoughts, feelings, or Consciousness This classification was a boon for developing science. Scientists did not have to worry about any inner life in nature, ideas or Consciousness, they could just set about weighing and measuring, as they have until the present day. But it was a disaster for the Earth. Earth was once worshiped as a Goddess. With Descartes she became almost overnight “things.” Simply commodities and resources, an endless of supply of consumer items, to be turned into money in the bank. And now Descartes concept of matter is up for us. If there is a single idea that is enabling and driving the destruction of the Earth, it is the conviction that Earth is something without Consciousness. This whole scientific image of nature as something devoid of Consciousness is driving the destruction of the Earth, and driving us all insane, and it needs to change. Ever since Descartes formulated Dualism, scientists and philosophers have been trying to get rid of it. They have done this by attempting to make matter the fundamental term of reality, and to explain mind or Consciousness in terms of matter. And now, Consciousness has emerged as term and a problem for modern science. 2. The prevailing Universe picture has been mainly based on the belief that Consciousness appears only at the end of evolution in ourselves and some of the higher animals when matter attains “a certain state of complexity” and it is produced by neurons in the brain. But Consciousness is nothing real in itself, the neurons are doing the work, Consciousness is simply a byproduct of neural activity. As William James summed up this position, not his: Consciousness is to the brain as a shadow to the runner, it runs beside him but never influences his stride. And so, the great problem of neuroscience: How do neurons produce Consciousness? So far, no one has been able to say how. All we have is parallelism. Two processes are running in parallel a train of neural events and a train of Consciousness events. And so, which one is primary? The face of neuroscience and much of the modern world is that the neural events are what is really real, Consciousness is secondary. The conviction that Consciousness is something produced by neurons is so deeply held and has been so for so long, that it seems like an obvious fact. It is in fact an assumption and a theory, and it could all well be the other way around. The scientific image of nature as something devoid of Consciousness is now up for us. It has enabled the destruction of the Earth, and we are now at a unique place in our own evolution where we can foresee if we continue on as we are, our own extinction. And so, there are the best reasons for rethinking Descartes concepts of matter. Scientists and philosophers have been trying to escape dualism by making matter the fundamental term. And now, this has not worked. In a nutshell: The mind body problem, the “hard problem of Consciousness” ---how do neurons in the brain, atoms and molecules give rise to ideas thoughts and feelings and who what and where is the “I” who sees them? And then, the near-death experience. I remind you of the truth of the near-death experience. The body in the bed is brain dead eeg flat, but the patient, pure Consciousness is hovering at the ceiling watching the doctors work over the body below, able to read dials he could not have seen from the bed, sometimes able to events in the corridor beyond the room, all subsequently confirmed by the doctors. Evidence as good as it gets that the patient really was out of his body. And so, the patient is able to see without eyes, hear without ears and to think and remember without a brain. There is not a single scientific principle that can account for this. And I am proposing that the near-death experience is a threshold and a key for a much deeper understanding of the Universe. These three problems are so hard they cannot be solved by present scientific principles and therefore call all present principles and ways of thought into question. When your axioms cannot account for the data, it is time to drop them and find better fundamental terms. We have not been able to solve the problems of Consciousness with the assumption that matter is the most basic and fundamental term. And so, if Cartesian dualism is to be overcome at all, we need to try it the other way around. 3. We are in the position of Copernicus. Remember Copernicus? He was making his calculations on the assumption that the Earth was the center of the Universe until they became so unwieldy that he decided to try it all the other way around. To make the sun the center, the Earth revolving around it. And lo, all of his data fell neatly into place. I am proposing a whole new Universe picture. Evolution makes us continuous with every other being, Consciousness would not be in us if it were not in them also, and then, in the atoms and molecules and cells that make up living beings. I propose to make Consciousness the most basic term and see what follows. I am in fact, proposing a new Copernican Revolution, that Consciousness is the true basis for the Universe and the right fundamental term, for science itself all other disciplines as well, and the whole of the modern world. And I predict that every intractable problem in science or the modern world, will go down like a line of dominos before it. In a nutshell: Consciousness and not matter is “first and fundamental” in the Universe, it is there from the very “Beginning” everything has it and all of the true causalities, the explanatory principles belong to it. Not to matter. What then is matter? All matter is an Appearance of something much more real, Consciousness. All matter is an expression of Consciousness, even the least little bit of matter contains Consciousness, and is an expression of Consciousness. And so, we revise the scientific image of nature. Science at present is two termed, matter/energy. It needs to become three termed: Matter/energy/Consciousness, with the recognition that all of the basic principles, the true causalities of events, and of the Universe belong to Consciousness. And now physics. To accommodate this new Universe picture physics needs to alter just one term, energy. All energy contains Consciousness, that idea alone will change our world forever. The animals the trees the plants are just as conscious as we are, as is the Earth herself, a continuum of many forms of Consciousness of which our own is one. The Earth consists of interacting and intersecting forms of Consciousness, Consciousness within Consciousness within Consciousness within Consciousness. A remarkable work has just appeared entitled “Beyond Words What Animals Feel and Think.” Describing in detail how like ourselves all of the animals are. And the more we can see how like us everything else really is, the harder it will be for us to do them all in, in the interest of turning them into consumer items. And so, here is a new principle of understanding, that entails completely new forms of explanation, and a whole new Universe picture to be spelled out in terms of Consciousness and its various properties. This is a Universe that is aware awake and Enspirited throughout. And so, there is only one principle, not two. And so, how does Consciousness work? There is “no mind/body problem.” My Consciousness is aligning the Consciousness of neurons in the brain, that aligns the Consciousness of all other cells in the body, that aligns the Consciousness of molecules and atoms themselves. Mainly from the top down, but also interactive, and not reduction to the purely physical forces. 4. And so, here is a radical new Universe picture, and with it a new way to think about the Universe itself, the Earth, Ourselves, and Spirit. It requires a complete revision of both of the terms of modern science itself. The problem of the modern world owes in deep measure to Descartes concept of matter as something devoid of interior life. But he is only the most powerful and simple expression of a whole tradition of thought that predates him, that goes back to the very dawn of thought itself. It owes to the conviction of both Plato and Aristotle that the Earth was something devoid of reason, or mind, and therefore something to be held in complete contempt. We are emerging from a civilization based on contempt for the Earth, and down the road, it will become contempt for the body, sex, women, the Native people, and people of color everywhere. And so, this entire tradition is now up for us, it has brought us to the brink of our own extinction, and we need to get beyond it. I am proposing that this term Consciousness is the way out. The missing piece of the puzzle and the idea whose time has come. And it will enable us to move into a completely new civilization based on respect for the Earth, women, sex, people of color, that honors everyone and has a place for all. Revolutions in science occur when new data show up that cannot be accounted for by existing paradigms (Thomas Kuhns: The Structure of Scientific Revolutions). Consciousness, and all the properties and problems of Consciousness are just such data, and they compel a new scientific revolution. But we are a scientific age. A new Copernican revolution in science is such as revolution everywhere. And Consciousness is the key to getting every other reality at all levels and depths right as well. It will enable us finally to get the Earth right, get ourselves right with the Earth, and with one another. Archimedes: Give me a place to stand, and I will move the world. I am proposing a new synthesis among our fundamental terms in which this term Consciousness is the new lynchpin. And so, I return once more to Descartes. Descartes in the Mediations was a dualist, they were two realities interacting in the pineal gland mind and body. But Descartes age was high on “Cartesianisms,” on Descartes concept of matter. Animals were simply cleverly wired machines (Hobbes called them automata). Rumor had it that Descartes followers went about kicking dogs reasoning that their howls were not due to any feelings of pain, but simply the wiring of the machine. We have been doing this to the Earth ever since, and now the Earth is not going to stand for it any longer. It is time we all woke up. There is no matter in Descartes sense of it, or in Plato and Aristotle’s sense of it. This old concept of matter is embedded in all of the crisis and catastrophes---fires, drought, pandemics, over population, the disintegrating economies, our ways of life, capitalism and consumerism themselves. All matter is Enspirited, contains Consciousness, and is an expression of Consciousness. Now I tell you something about Descartes you may not know. In the Mediations was a dualist, there are minds and bodies. But by the end of his life, he knew the truth. Descartes guided his whole life by these vivid dreams, and toward the end of his life he had this dream: He saw the Universe as a giant machine and in the heart of the machine propelling the whole, was Consciousness. And so, out of the shades and shadows and into the light of day. Here is a renewed and a new science, a new cosmology. And with it, a revolutionary new metaphysics. The whole of the modern world is in the grip of a false metaphysics, based on Descartes theory of matter. Here is a new, and finally true metaphysics based on the single term, Consciousness. 5. And in all my works, I spell out this new metaphysics based on Consciousness, and what it means for how we think about virtually everything: The Universe, the Earth, Ourselves, and Spirit. Of who, what and where Spirit is, that we may connect with Spirit everywhere. There is only one term for the Universe, and not two, this single term Consciousness, that manifests itself in two modes: Consciousness in Form, Consciousness Formless. And so, here is a sketch of what this new term Consciousness really means. For four great aspects of the real, four new cornerstones for a whole new civilization on Earth, based on Love and Truth. A. For the Universe: New causalities, all matter is an expression of Consciousness, awake and aware, Enspirited throughout, which has a place for the causalities of love ideas intent plan and purpose, what Aristotle called final causality, that for the sake of which. And it has a place for subjectivity, the greatest mystery in the whole of the Universe, knowable only in ourselves, but there in everything else as well, bedrock in the chain of causalities. The buck stops here. Here are two very different Universe pictures. The Cartesian Universe consists of things, pushed, and pulled around by the physical forces that function blind. The true Universe consists of selves and not things. Selves at least in ourselves, are organized and focused about an I, an I am, or a sense of self in everything, ever other being dear to itself. Just one aspect of subjectivity. And so, it is Consciousness all the way down, and subjectivity all the way down. Science proceeds on the basis of the negligible of what it can leave out as unimportant. Subjectivity is the ultimate animating principle of the Universe. What Descartes really did was to strip from nature what is most essential about it, its animating principle Consciousness. And so, we come out of a long history of “clunk.” The great male thinkers have killed creation. First in concepts and now in the lived life. And so, after centuries of fruitless search, not all of it in vain, finally, and just in time the true basic principle for the Universe has shown up in virtually all disciplines, this term Consciousness. B. And so, the Earth: What does this term Consciousness really mean for the Earth? It means that all of the bright array of beings around us have Consciousness. And the more we can realize how like us they are, the harder it will be to do them all in. Is this what it really means to be “made in the image of Spirit?” To destroy ever other living being? The god of Genesis is a creator god who has created billions and billions of living forms. Man, taking himself to be made in the image of Spirit has destroyed virtually all of it. The Earth is not things, not commodities and resources, not an endless array of consumer goods. The Earth consists of conscious living spiritual beings like ourselves, who need to be honored and respected, nurtured and cared for, to be Loved. And if the human race can finally pull it off and bring Love here and start helping the Earth rather than destroying her, even our vast numbers now at 8 billion, may not matter. Deeply embedded in the modern world is a false cosmology based on the hitherto useful but ultimately false metaphysics of Descartes. In particular it owes to Descartes concepts of matter, and of the Earth, as something devoid of Consciousness. And now this concept is up for us. If there is a single idea enabling and driving the destruction of the planet, it is the conviction that the Earth is something devoid of Consciousness. And we all have a vested interest in keeping this false concept in place. When you intend to ravage and savage the Earth, to turn it into consumer items, and money in the bank, first pretend she has no Consciousness that way we need not have any conscious about it. We are in deep trouble with the planet. Virtually all human economies are based on the destruction of the Earth, and we are now at a place in our history where we can foresee if we continue on as we are, our own extinction. And so, here are a few new guiding principles for the planet, based on Consciousness. Consciousness has field properties. The Earth is Consciousness throughout, connected and interconnected throughout, intersecting and interacting fields of Consciousness of which our own is one. To harm one is to harm all. Only this understanding of the Earth can set our destructive technologies, and capitalism, their goals, and their limits. The real problem with capitalism, and with civilization itself is it has no true understanding of the Earth, all the exquisite networks and webworks that connect all beings with one another. Nor it is about to admit that the Earth is our “life support system,” reason gone insane in the modern world. We are destroying our life support system, capitalism is the full expression of the human death wish in the modern world, and it needs to change. And now, let the barriers between ourselves and all other beings go down. C. And so, Ourselves. This new Universe picture based on Consciousness opens into and has a place for new concepts of human identity. In particular, it has a place for reincarnation, a process whereby one Consciousness dons’ bodies again and again and lays them down. This is a process wholly impossible on materialistic theories. This new view of things puts a higher perspective on Freudian determinism, which is a partial knowing in a greater account. We are something more than the victims of genes drives and childhood. We are immortally eternal evolving spiritual beings who take on a cycle of earthly lives in order to achieve certain goals and abilities, and then, our earthly cycle complete, we depart the Earth plane for other areas of the Universe physical or purely spiritual, to continue our evolution elsewhere. The scientific evidence for reincarnation is increasing on all sides, see for example the work of Michael Newton, a mainstream scientist forced to confront the fact of reincarnation. A completely new reincarnational picture is now emerging everywhere. We choose our parents for abilities we could acquire through them. We all come here with reasons for being here, and we all have free will, an original property of Consciousness, and we work out our issues accordingly. We all come here with the same reason for being here. It is to experience the Earth plane and human society at all levels and depths, in every social role, every climate, both sexes male and female and in all races. We are all here to acquire a deep understanding of the Earth plane with all its challenges and all of its present problems. We are not here to devour and consume the Earth, but to honor her for the learning experience she makes available for us. And so, reincarnation is the great equalizer, the fundamental basis for any bill of rights. Many of the problems of the modern world owe to the fact that reincarnation has not been taught in the West. Reincarnation was the center piece of Jesus teachings. It was banned by a fourth century council that decided the populous could best be controlled through fear. They edited all mention of reincarnation from the scriptures, and inserted fear terms instead, a judgmental god, and so on. If reincarnation had been taught, there would have been no suppression of women, but the gift of both sexes would have been honored and developed. And there would have been no reason for contempt for people of color because who they are, we have been or will be. There is no reason to look down on anyone when we know that our own identity includes membership in theirs. The truth of reincarnation needs to be the foundation for getting ourselves right with the Earth, and with one another. D. And so, Spirit. This new Universe picture, based on Consciousness, has a place for Spirit as simply Infinite Consciousness. All universes all worlds are formed out of this Consciousness, divine and sacred throughout. And Spirit is everywhere in the Earth and can be connected everywhere. And the Native North American people had things right. Everything has its own Spirit, and without contradiction is also an expression of Spirit. And what I am really doing in terms of concepts of modern science is presenting the Native North American version of the Real. And this new Universe picture has a place for the essential insight of all religions almost smothered by church tradition: Love. Not an emotion but a great Oneness, that can be found everywhere, even in a busy city street. But especially in the Earth, that is levels and depths of Oneness everywhere. And so, for the first time in their long and bitter antagonistic history, science and religion now share a common base with this term Spirit, the lynchpin in a new conceptual framework. And I offer an interpretation of where we are in our history, this present moment of historical grace, and these tumultuous times in which we live, the return of the Goddess and the Divine Feminine, the Reappearance of the Feminine in history, the Advent of Women, this great Awakening, as of Springtime after a long dark Winter, the Awakening, the Second Coming. And so, stay tuned! (shrink)

Fundamental knowledge endures deep conceptual crisis manifested in total crisis of understanding, crisis of interpretation and representation, loss of certainty, troubles with physics, crisis of methodology. Crisis of understanding in fundamental science generates deep crisis of understanding in global society. What way should we choose for overcoming total crisis of understanding in fundamental science? It should be the way of metaphysical construction of new comprehensive model of ideality on the basis of the "modified ontology". Result of quarter-century wanderings: sum of (...) ideas, concepts and eidoses, new understanding of space, time, consciousness. (shrink)

The boundless nature of the natural numbers imposes paradoxically a high formal bound to the use of standard artificial computer programs for solving conceptually challenged problems in number theory. In the context of the new cognitive foundations for mathematics' and physics' program immersed in the setting of artificial mathematical intelligence, we proposed a refined numerical system, called the physical numbers, preserving most of the essential intuitions of the natural numbers. Even more, this new numerical structure additionally possesses the property of (...) being a bounded object allowing us to work with quite similar axioms like the classic Peano axioms, but in a finite environment and with an enriched physical dimension. Finally, we present several enlightening examples and we conclude that the physical numbers provide a natural formal setting for approaching classic problems in number theory from a more hybrid perspective, i.e. with a potential participation in the generation of solutions, not only of working mathematicians but also of more sophisticated artificial interactive (mathematical) intelligent agents (within the context of cognitive-computational metamathematics or artificial mathematical intelligence) and more controlled by physically-inspired principles. Finally, this paper addresses a highly new, bottom-up and paradigm-shifting approach to the foundations of physics: One should refine and improve the conceptual formal setting of the language that we use for understanding physical phenomena (e.g. the mathematical grounding concepts and theories) for being able to understand better more subtle and complex physical phenomena. (shrink)

While it is known that Euclid’s five axioms include a proposition that a line consists at least of two points, modern geometry avoid consistently any discussion on the precise definition of point, line, etc. It is our aim to clarify one of notorious question in Euclidean geometry: how many points are there in a line segment? – from discrete-cellular space (DCS) viewpoint. In retrospect, it may offer an alternative of quantum gravity, i.e. by exploring discrete gravitational theories. To elucidate (...) our propositions, in the last section we will discuss some implications of discrete cellular-space model in several areas of interest: (a) cell biology, (b) cellular computing, (c) Maxwell equations, (d) low energy fusion, and (e) cosmology modelling. (shrink)

In this article we challenge the notion that diversity serves as a good proxy for educational justice. First, we maintain that the story about how diversity might be accomplished and what it might do for students and society is internally inconsistent. Second, we argue that a disproportionate share of the benefits that might result from greater diversity often accrues to those already advantaged. Finally, we propose that many of the most promising and pragmatic remedies for educational injustice are often rejected (...) by liberal proponents of “diversity first” in favor of remedies that in most cases are practically impossible, and often problematic on their own terms. We argue that schools that are by geography and demography not ethnically or socioeconomically diverse still can successfully confront the obstacles that their students face in creating a life they have reason to value. (shrink)

In this paper, we offer a Leftist critique of standard liberal defenses of the public school. We suggest that the standard arguments employed by mainstream liberal defenders of the public school are generally inadequate because they fail to provide a credible representation of their historical object, let alone effective remedies to our current problems. Indeed, many of these narratives, in our view, are grounded in fantasies about what public schools, or teaching and learning, are or could be, as much as (...) they are grounded in the historical realities of public schools or the realities of so-called privatization. We speculate whether the self identification of the proponents of this cause as ‘progressive’ is not part of this ideological construction and if the underlying political agenda is not in fact more conservative. (shrink)

We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending ZFC by using his multiverse axioms MV and the ‘core hypothesis’. In the first part, we examine the evidential framework for MV, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of ZFC. In the second part, we address the existence and the possible features of the core of MV_T (where T is ZFC+Large (...) Cardinals). In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of ZFC. To this end, we take into account several strategies, and assess their prospects in the light of MV’s evidential framework. (shrink)

The contemporary versions of the ontological argument that originated from Charles Hartshorne are formalized proofs based on unique modal theories. The simplest well-known theory of this kind arises from the b system of modal logic by adding two extra-logical axioms: “If the perfect being exists, then it necessarily exists‘ and “It is possible that the perfect being exists‘. In the paper a similar argument is presented, however none of the systems of modal logic is relevant to it. Its only (...) premises are the axiom and, instead of, the new axiom : “If the perfect being doesn’t exist, it necessarily doesn’t‘. The main goal of the work is to prove that is no more controversial than and -- in consequence -- the whole strength of the modal ontological argument lies in the set of its extra-logical premises. In order to do that, three arguments are formulated: ontological, “cosmological‘ and metalogical. (shrink)

In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic justification relates to how `intuitively plausible' an axiom is, whereas extrinsic justification supports an axiom by identifying certain `desirable' consequences. This paper puts pressure on how this distinction is formulated and construed. In particular, we argue that the distinction as often presented is neither well-demarcated (...) nor sufficiently precise. Instead, we suggest that the process of justification in set theory should not be thought of as neatly divisible in this way, but should rather be understood as a conceptually indivisible notion linked to the goal of explanation. (shrink)

The motivating question of this paper is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by ‘intuitions’, as our scientific beliefs are justified by observations. On this view, axioms are analogous to laws of nature. They (...) are postulated to best systematize the data to be explained. We argue that there is a decisive difference between the cases. There is agreement on the data to be systematized in the scientific case that has no analog in the mathematical one. There is virtual consensus over observations, but conspicuous dispute over intuitions. In this respect, mathematics more closely resembles stereotypical philosophy. We conclude by distinguishing two ideas that have long been associated -- realism (the idea that there is an independent reality) and objectivity (the idea that in a disagreement, only one of us can be right). We argue that, while realism is true of mathematics and philosophy, these domains fail to be objective. One upshot of the discussion is that even questions of fundamental physics may fail to be objective in roughly the sense that the question, ‘Is the Parallel Postulate is true?’, understood as one of pure mathematics, fails to be. Another is a kind of pragmatism. Factual questions in mathematics, modality, logic, and evaluative areas go proxy for non-factual practical ones. -/- . (shrink)

Does individual desert matter for distributive justice? Is it relevant, for purposes of justice, that the pattern of distribution of justice’s “currency” (be it well-being, resources, preference-satisfaction, capabilities, or something else) is aligned in one or another way with the pattern of individual desert? -/- This paper examines the nexus between desert and distributive justice through the lens of individual claims. The concept of claims (specifically “claims across outcomes”) is a fruitful way to flesh out the content of distributive justice (...) so as to be grounded in the separateness of persons. A claim is a relation between a person and a pair of outcomes. If someone is better off in one outcome than a second, she has a claim in favor of the first. If she is equally well off in the two outcomes, she has a null claim between the two. In turn, whether one outcome is more just than a second depends upon the pattern of claims between them. -/- In prior work, I have elaborated the concept of claims across outcomes, and have used it to provide a unified defense of the Pareto and Pigou-Dalton axioms. Adding some further, plausible, axioms, we arrive at prioritarianism. -/- Here, I consider the possibility of desert-modulated claims—whereby the strength of an individual’s claim between two outcomes is determined not only by her well-being levels in the two outcomes, and her well-being difference between them, but also by her desert. This generalization of the notion of claims suggests a new axiom of justice: Priority for the More Deserving, requiring that, as between two individuals at the same well-being level, a given increment in well-being be allocated to the more deserving one. -/- If individual desert is intrapersonally fixed, this new axiom, together with a desert-modulated version of the Pigou-Dalton principle, and the Pareto axioms, yields a desert-modulated prioritarian account of distributive justice. Trouble arises, however, if an individual’s desert level can be different in different outcomes. In this case of intrapersonally variable desert, Priority for the More Deserving can conflict with the Pareto axioms (both Pareto indifference and strong Pareto). -/- This conflict, I believe, is sufficient reason to abandon the proposal to make claim strength a function of individual desert on top of well-being levels and differences. If distributive justice is truly sensitive to each individual’s separate perspective—if the justice ranking of outcomes is built up from the totality of individual rankings—we should embrace the Pareto axioms as axioms of justice and reject Priority for the More Deserving. In short: desert-modulated prioritarianism is a nonstarter. Rawls was right to sever distributive justice from desert. (shrink)

The naive theory of properties states that for every condition there is a property instantiated by exactly the things which satisfy that condition. The naive theory of properties is inconsistent in classical logic, but there are many ways to obtain consistent naive theories of properties in nonclassical logics. The naive theory of classes adds to the naive theory of properties an extensionality rule or axiom, which states roughly that if two classes have exactly the same members, they are identical. In (...) this paper we examine the prospects for obtaining a satisfactory naive theory of classes. We start from a result by Ross Brady, which demonstrates the consistency of something resembling a naive theory of classes. We generalize Brady’s result somewhat and extend it to a recent system developed by Andrew Bacon. All of the theories we prove consistent contain an extensionality rule or axiom. But we argue that given the background logics, the relevant extensionality principles are too weak. For example, in some of these theories, there are universal classes which are not declared coextensive. We elucidate some very modest demands on extensionality, designed to rule out this kind of pathology. But we close by proving that even these modest demands cannot be jointly satisfied. In light of this new impossibility result, the prospects for a naive theory of classes are bleak. (shrink)

A new non-Archimedean approach to interacted quantum fields is presented. In proposed approach, a field operator φ(x,t) no longer a standard tempered operator-valued distribution, but a non-classical operator-valued function. We prove using this novel approach that the quantum field theory with Hamiltonian P(φ)_4 exists and that the corresponding C^*­ algebra of bounded observables satisfies all the Haag-Kastler axioms except Lorentz covariance. We prove that the λ(φ^4 )_4 quantum field theory model is Lorentz covariant.

A description of consciousness leads to a contradiction with the postulation from special relativity that there can be no connections between simultaneous event. This contradiction points to consciousness involving quantum level mechanisms. The Quantum level description of the universe is re- evaluated in the light of what is observed in consciousness namely 4 Dimensional objects. A new improved interpretation of Quantum level observations is introduced. From this vantage point the following axioms of consciousness is presented. Consciousness consists of two (...) distinct components, the observed U and the observer I. The observed U consist of all the events I is aware of. A vast majority of these occur simultaneously. Now if I were to be an entity within the space-time continuum, all of these events of U together with I would have to occur at one point in space-time. However, U is distributed over a definite region of space-time (region in brain). Thus, I is aware of a multitude of space-like separated events. It is seen that this awareness necessitates I to be an entity outside the space-time continuum. With I taken as such, a new concept called concept A is introduced. With the help of concept A a very important axiom of consciousness, namely Free Will is explained. (shrink)

The cognition of quantum processes raises a series of questions about ordering and information connecting the states of one and the same system before and after measurement: Quantum measurement, quantum in-variance and the non-locality of quantum information are considered in the paper from an epistemological viewpoint. The adequate generalization of ‘measurement’ is discussed to involve the discrepancy, due to the fundamental Planck constant, between any quantum coherent state and its statistical representation as a statistical ensemble after measurement. Quantum in-variance designates (...) the relation of any quantum coherent state to the corresponding statistical ensemble of measured results. A set-theory corollary is the curious in-variance to the axiom of choice: Any coherent state excludes any well-ordering and thus excludes also the axiom of choice. However the above equivalence requires it to be equated to a well-ordered set after measurement and thus requires the axiom of choice for it to be able to be obtained. Quantum in-variance underlies quantum information and reveals it as the relation of an unordered quantum “much” (i.e. a coherent state) and a well-ordered “many” of the measured results (i.e. a statistical ensemble). It opens up to a new horizon, in which all physical processes and phenomena can be interpreted as quantum computations realizing relevant operations and algorithms on quantum information. All phenomena of entanglement can be described in terms of the so defined quantum information. Quantum in-variance elucidates the link between general relativity and quantum mechanics and thus, the problem of quantum gravity. The non-locality of quantum information unifies the exact position of any space-time point of a smooth trajectory and the common possibility of all space-time points due to a quantum leap. This is deduced from quantum in-variance. Epistemology involves the relation of ordering and thus a generalized kind of information, quantum one, to explain the special features of the cognition in quantum mechanics. (shrink)

L’Ipotesi del Continuo, formulata da Cantor nel 1878, è una delle congetture più note della teoria degli insiemi. Il Problema del Continuo, che ad essa è collegato, fu collocato da Hilbert, nel 1900, fra i principali problemi insoluti della matematica. A seguito della dimostrazione di indipendenza dell’Ipotesi del Continuo dagli assiomi della teoria degli insiemi, lo status attuale del problema è controverso. In anni più recenti, la ricerca di una soluzione del Problema del Continuo è stata anche una delle ragioni (...) fondamentali per la ricerca di nuovi assiomi in matematica. L’articolo fornisce un quadro generale dei risultati matematici fondamentali, e un’analisi di alcune delle questioni filosofiche connesse al Problema del Continuo. (shrink)

Quantum invariance designates the relation of any quantum coherent state to the corresponding statistical ensemble of measured results. The adequate generalization of ‘measurement’ is discussed to involve the discrepancy, due to the fundamental Planck constant, between any quantum coherent state and its statistical representation as a statistical ensemble after measurement. A set-theory corollary is the curious invariance to the axiom of choice: Any coherent state excludes any well-ordering and thus excludes also the axiom of choice. It should be equated to (...) a well-ordered set after measurement and thus requires the axiom of choice. Quantum invariance underlies quantum information and reveals it as the relation of an unordered quantum “much” (i.e. a coherent state) and a well-ordered “many” of the measured results (i.e. a statistical ensemble). It opens up to a new horizon, in which all physical processes and phenomena can be interpreted as quantum computations realizing relevant operations and algorithms on quantum information. All phenomena of entanglement can be described in terms of the so defined quantum information. Quantum invariance elucidates the link between general relativity and quantum mechanics and thus, the problem of quantum gravity. (shrink)

I prove both the mathematical conjectures P ≠ NP and the Continuum Hypothesis are eternally unprovable using the same fundamental idea. Starting with the Saunders Maclane idea that a proof is eternal or it is not a proof, I use the indeterminacy of human biological capabilities in the eternal future to show that since both conjectures are independent of Axioms and have definitions connected with human biological capabilities, it would be impossible to prove them eternally without the creation and (...) widespread acceptance of new axioms. I also show that the same fundamental concepts cannot be used to demonstrate the eternal unprovability of many other mathematical theorems and open conjectures. Finally I investigate the idea’s implications for the foundations of mathematics including its relation to Godel’s Incompleteness Theorem and Tarsky’s Undefinability Theorem. (shrink)

A principle, according to which any scientific theory can be mathematized, is investigated. Social science, liberal arts, history, and philosophy are meant first of all. That kind of theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term “theory” includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather (...) a metamathematical axiom about the relation of mathematics and reality. The main statement is formulated as follows: Any scientific theory admits isomorphism to some mathematical structure in a way constructive. Its investigation needs philosophical means. Husserl’s phenomenology is what is used, and then the conception of “bracketing reality” is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction. The sketch of the proof is organized in five steps: a generalization of epoché; involving transfinite induction in the transition between Peano arithmetic and set theory; discussing the finiteness of Peano arithmetic; applying transfinite induction to Peano arithmetic; discussing an arithmetical model of reality. Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom. The present paper follows a pathway grounded on Husserl’s phenomenology and “bracketing reality” to achieve the generalized arithmetic necessary for the principle to be founded in alternative ontology, in which there is no reality external to mathematics: reality is included within mathematics. That latter mathematics is able to self-found itself and can be called Hilbert mathematics in honour of Hilbert’s program for self-founding mathematics on the base of arithmetic. The principle of universal mathematizability is consistent to Hilbert mathematics, but not to Gödel mathematics. Consequently, its validity or rejection would resolve the problem which mathematics refers to our being; and vice versa: the choice between them for different reasons would confirm or refuse the principle as to the being. An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. The Schrödinger equation in quantum mechanics is involved to illustrate that ontology. Thus the problem which of the two mathematics is more relevant to our being is discussed again in a new way A few directions for future work can be: a rigorous formal proof of the principle as an independent axiom; the further development of information ontology consistent to both kinds of mathematics, but much more natural for Hilbert mathematics; the development of the information interpretation of quantum mechanics as a mathematical one for information ontology and thus Hilbert mathematics; the description of consciousness in terms of information ontology. (shrink)

Minimalism about truth is one of the main contenders for our best theory of truth, but minimalists face the charge of being unable to properly state their theory. Donald Davidson incisively pointed out that minimalists must generalize over occurrences of the same expression placed in two different contexts, which is futile. In order to meet the challenge, Paul Horwich argues that one can nevertheless characterize the axioms of the minimalist theory. Sten Lindström and Tim Button have independently argued that (...) Horwich’s attempt to formulate minimalism remains unsuccessful. We show how to properly state Horwich’s axioms by appealing to propositional functions that are given by definite descriptions. Both Lindström and Button discuss proposals similar to ours and conclude that they are unsuccessful. Our new suggestion avoids these objections. (shrink)

An axiom of medical research ethics is that a protocol is moral only if it has a “favorable risk-benefit ratio”. This axiom is usually interpreted in the following way: a medical research protocol is moral only if it has a positive expected value -- that is, if it is likely to do more good (to both subjects and society) than harm. I argue that, thus interpreted, the axiom has two problems. First, it is unusable, because it requires us to know (...) more about the potential outcomes of research than we ever could. Second, it is false, because it conflicts with the so-called “soft paternalist” principles of liberal democracy. In place of this flawed rule I propose a new way of making risk-benefit assessments, one that does comport with the principles of liberalism. I argue that a protocol is moral only if it would be entered into by competent subjects who are informed about the protocol. The new rule this eschews all pseudo-utilitarian calculation about the protocol’s likely harms and benefits. (shrink)

When people combine concepts these are often characterised as “hybrid”, “impossible”, or “humorous”. However, when simply considering them in terms of extensional logic, the novel concepts understood as a conjunctive concept will often lack meaning having an empty extension (consider “a tooth that is a chair”, “a pet flower”, etc.). Still, people use different strategies to produce new non-empty concepts: additive or integrative combination of features, alignment of features, instantiation, etc. All these strategies involve the ability to deal with conflicting (...) attributes and the creation of new (combinations of) properties. We here consider in particular the case where a Head concept has superior ‘asymmetric’ control over steering the resulting concept combination (or hybridisation) with a Modifier concept. Specifically, we propose a dialogical approach to concept combination and discuss an implementation based on axiom weakening, which models the cognitive and logical mechanics of this asymmetric form of hybridisation. (shrink)

A new non-Archimedean approach to interacted quantum fields is presented. In proposed approach, a field operator φ(x,t) no longer a standard tempered operator-valued distribution, but a non-classical operator-valued function. We prove using this novel approach that the quantum field theory with Hamiltonian P(φ)_4 exists and that the corresponding C^*­ algebra of bounded observables satisfies all the Haag-Kastler axioms except Lorentz covariance. We prove that the λ(φ^2n )_4,n≥2 quantum field theory models are Lorentz covariant.

Pettigrew offers new axiomatic constraints on legitimate measures of inaccuracy. His axiom called ‘Decomposition’ stipulates that legitimate measures of inaccuracy evaluate a credence function in part based on its level of calibration at a world. I argue that if calibration is valuable, as Pettigrew claims, then this fact is an explanandum for accuracy-rst epistemologists, not an explanans, for three reasons. First, the intuitive case for the importance of calibration isn’t as strong as Pettigrew believes. Second, calibration is a perniciously global (...) property that both contravenes Pettigrew’s own views about the nature of credence functions themselves and undercuts the achievements and ambitions of accuracy-rst epistemology. Finally, Decomposition introduces a new kind of value compatible with but separate from accuracy-proper in violation of Pettigrew’s alethic monism. introduction. (shrink)

The focus in this essay will be upon the paradoxes, and foremostly in set theory. A central result is that the librationist set theory £ extension \Pfund $\mathscr{HR}(\mathbf{D})$ of \pounds \ accounts for \textbf{Neumann-Bernays-Gödel} set theory with the \textbf{Axiom of Choice} and \textbf{Tarski's Axiom}. Moreover, \Pfund \ succeeds with defining an impredicative manifestation set $\mathbf{W}$, \emph{die Welt}, so that \Pfund$\mathscr{H}(\mathbf{W})$ %is a model accounts for Quine's \textbf{New Foundations}. Nevertheless, the points of view developed support the view that the truth-paradoxes and (...) the set-paradoxes have common origins, so that the librationist resolutions of the set theoretic paradoxes are at the same time resolutions of the truth theoretic paradoxes. Both the librationist resolutions of the set theoretic paradoxes and the truth theoretic paradoxes have non-trivial philosophical implications: librationist set theories have the consequence that there are no absolutely uncountable sets, and librationist truth theories allow the use of syntactical modalities in ways which circumvent limitations as those of \parencite{Montague1963}, and a truth predicate which is useful for more precise philosophical discourse. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter 2 argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter 3 provides an abstraction principle for epistemic (hyper-)intensions. Chapter 4 advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter 5 applies the fixed points of the modal μ-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal μ-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal μ-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter 6 advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter 7 provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters 2 and 4 are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters 8-12 provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter 8 examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of Ω-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter 9 examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter 10 examines the philosophical significance of hyperintensional Ω-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter 11 provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter 12 avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter 13 examines modal responses to the alethic paradoxes. Chapter 14 examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

We provide a new interpretation of Zeno’s Paradox of Measure that begins by giving a substantive account, drawn from Aristotle’s text, of the fact that points lack magnitude. The main elements of this account are (1) the Axiom of Archimedes which states that there are no infinitesimal magnitudes, and (2) the principle that all assignments of magnitude, or lack thereof, must be grounded in the magnitude of line segments, the primary objects to which the notion of linear magnitude applies. Armed (...) with this account, we are ineluctably driven to introduce a highly constructive notion of (outer) measure based exclusively on the total magnitude of potentially infinite collections of line segments. The Paradox of Measure then consists in the proof that every finite or potentially infinite collection of points lacks magnitude with respect to this notion of measure. We observe that the Paradox of Measure, thus understood, troubled analysts into the 1880’s, despite their knowledge that the linear continuum is uncountable. The Paradox was ultimately resolved by Borel in his thesis of 1893, as a corollary to his celebrated result that every countable open cover of a closed line segment has a finite sub-cover, a result he later called the “First Fundamental Theorem of Measure Theory.” This achievement of Borel has not been sufficiently appreciated. We conclude with a metamathematical analysis of the resolution of the paradox made possible by recent results in reverse mathematics. (shrink)

Arthur Clark and Michael Kube–McDowell (“The Triger”, 2000) suggested the sci-fi idea about the direct transformation from a chemical substance to another by the action of a newly physical, “Trigger” field. Karl Brohier, a Nobel Prize winner, who is a dramatic persona in the novel, elaborates a new theory, re-reading and re-writing Pauling’s “The Nature of the Chemical Bond”; according to Brohier: “Information organizes and differentiates energy. It regularizes and stabilizes matter. Information propagates through matter-energy and mediates the interactions of (...) matter-energy.” Dr Horton, his collaborator in the novel replies: “If the universe consists of energy and information, then the Trigger somehow alters the information envelope of certain substances –“. “Alters it, scrambles it, overwhelms it, destabilizes it” Brohier adds. There is a scientific debate whether or how far chemistry is fundamentally reducible to quantum mechanics. Nevertheless, the fact that many essential chemical properties and reactions are at least partly representable in terms of quantum mechanics is doubtless. For the quantum mechanics itself has been reformulated as a theory of a special kind of information, quantum information, chemistry might be in turn interpreted in the same terms. Wave function, the fundamental concept of quantum mechanics, can be equivalently defined as a series of qubits, eventually infinite. A qubit, being defined as the normed superposition of the two orthogonal subspaces of the complex Hilbert space, can be interpreted as a generalization of the standard bit of information as to infinite sets or series. All “forces” in the Standard model, which are furthermore essential for chemical transformations, are groups [U(1),SU(2),SU(3)] of the transformations of the complex Hilbert space and thus, of series of qubits. One can suggest that any chemical substances and changes are fundamentally representable as quantum information and its transformations. If entanglement is interpreted as a physical field, though any group above seems to be unattachable to it, it might be identified as the “Triger field”. It might cause a direct transformation of any chemical substance by from a remote distance. Is this possible in principle? (shrink)

This paper is an improvement of our paper “(t, i, f)-Neutrosophic Structures”, where we introduced for the first time a new type of structures, called (t, i, f)- Neutrosophic Structures, presented from a neutrosophic logic perspective, and we showed particular cases of such structures in geometry and in algebra.

Several philosophers have proposed Knowledge-Based Decision Theories (KDTs)—theories that require agents to maximize expected utility as yielded by utility and probability functions that depend on the agent’s knowledge. Proponents of KDTs argue that such theories are motivated by Knowledge-Reasons norms that require agents to act only on reasons that they know. However, no formal derivation of KDTs from Knowledge-Reasons norms has been suggested, and it is not clear how such norms justify the particular ways in which KDTs relate knowledge and (...) rational action. In this paper, I suggest a new axiomatic method for justifying KDTs and providing them with stronger normative foundations. I argue that such theories may be derived from constraints on the relation between knowledge and preference, and that these constraints may be evaluated relative to intuitions regarding practical reasoning. To demonstrate this, I offer a representation theorem for a KDT proposed by Hawthorne and Stanley (2008) and briefly evaluate it through its underlying axioms. -/- In section 1, I present knowledge-based decision theories generally and the versions of such theories suggested in the literature. In section 2, I argue that KDTs and theories of practical reasoning conjoined with Knowledge-Reasons norms generate constraints on the same relation—the relation between knowledge and preference. In section 3, I present a formal framework in which constraints on this relation may be expressed and from which KDTs may be derived. In section 4, I present a representation theorem for a specific KDT. In section 5, I briefly evaluate the axioms of the theory, and in section 6, I conclude. (shrink)

In this paper I present a new way of understanding Dutch Book Arguments: the idea is that an agent is shown to be incoherent iff he would accept as fair a set of bets that would result in a loss under any interpretation of the claims involved. This draws on a standard definition of logical inconsistency. On this new understanding, the Dutch Book Arguments for the probability axioms go through, but the Dutch Book Argument for Reflection fails. The question (...) of whether we have a Dutch Book Argument for Conditionalization is left open. (shrink)

We present a general framework for representing belief-revision rules and use it to characterize Bayes's rule as a classical example and Jeffrey's rule as a non-classical one. In Jeffrey's rule, the input to a belief revision is not simply the information that some event has occurred, as in Bayes's rule, but a new assignment of probabilities to some events. Despite their differences, Bayes's and Jeffrey's rules can be characterized in terms of the same axioms: "responsiveness", which requires that revised (...) beliefs incorporate what has been learnt, and "conservativeness", which requires that beliefs on which the learnt input is "silent" do not change. To illustrate the use of non-Bayesian belief revision in economic theory, we sketch a simple decision-theoretic application. (shrink)

In this essay I argue that childbearing and various kinds of organ donation are morally analogous activities. I argue, further, that the ethos of giftgiving ought to inform our analyses of both of these forms of bodily life support. This reframing of the abortion and organ donation debates yields new insights into two relatively neglected subtopics. First, though frequently asserted, few have demonstrated why bodily life support--especially in the form of childbearing--cannot be morally required. This comparison yields insights into the (...) reasons for such an axiom. Second, while the giving of bodily life support is sometimes exhorted and almost always respected and admired, its intelligibility and political meaningfulness as a moral choice is rarely explored. This analogical wager reveals why one ought to give another bodily life support. In summary, the analogy yields insights crucial to the development of cogent arguments regarding both the grounds for and limits of the responsibility to give bodily life support. Further, the analogy displays the disparity between what has been demanded traditionally of those who are pregnant and of those men (and women) who by virtue of tissue or blood type can offer other forms of bodily life support. The analogy enables reflection on abortion (and organ donation) to develop in a context free of sexist biases. Finally, efforts are made to assess this giftgiving ethos in light of the feminist "hermeneutics of suspicion" regarding arguments which have and can sacralize victimization. (shrink)

It is often supposed that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) (...) if a mathematical hypothesis is absolutely undecidable, then it is indeterminate. I shall argue that on no understanding of absolute undecidability could one hope to establish all of (a)–(c). However, I will identify one understanding of absolute undecidability on which one might hope to establish both (a) and (c) to the exclusion of (b). This suggests that a new style of mathematical antirealism deserves attention—one that does not depend on familiar epistemological or ontological concerns. The key idea behind this view is that typical mathematical hypotheses are indeterminate because they are relevantly similar to CH. (shrink)

We investigate the conflict between the ex ante and ex post criteria of social welfare in a new framework of individual and social decisions, which distinguishes between two sources of uncertainty, here interpreted as an objective and a subjective source respectively. This framework makes it possible to endow the individuals and society not only with ex ante and ex post preferences, as is usually done, but also with interim preferences of two kinds, and correspondingly, to introduce interim forms of the (...) Pareto principle. After characterizing the ex ante and ex post criteria, we present a first solution to their conflict that extends the former as much possible in the direction of the latter. Then, we present a second solution, which goes in the opposite direction, and is also maximally assertive. Both solutions translate the assumed Pareto conditions into weighted additive utility representations, and both attribute to the individuals common probability values on the objective source of uncertainty, and different probability values on the subjective source. We discuss these solutions in terms of two conceptual arguments, i.e., the by now classic spurious unanimity argument and a novel informational argument labelled complementary ignorance. The paper complies with the standard economic methodology of basing probability and utility representations on preference axioms, but for the sake of completeness, also considers a construal of objective uncertainty based on the assumption of an exogeneously given probability measure. JEL classification: D70; D81. (shrink)

Process philosophies tend to emphasise the value of continuous creation as the core of their discourse. For Bergson, Whitehead, Deleuze, and others the real is ultimately a creative becoming. Critics have argued that there is an irreducible element of (almost religious) belief in this re-evaluation of immanent creation. While I don’t think belief is necessarily a sign of philosophical and existential weakness, in this paper I will examine the possibility for the concept of universal creation to be a political and (...) ethical axiom, the result of a global social contract rather than of a new spirituality. I argue here that a coherent way to fight against potentially totalitarian absolutes is to replace them with a virtual absolute that cannot territorialise without deterritorialising at the same time: the Creal principle. (shrink)

This paper presents a range of new triviality proofs pertaining to naïve truth theory formulated in paraconsistent relevant logics. It is shown that excluded middle together with various permutation principles such as A → (B → C)⊩B → (A → C) trivialize naïve truth theory. The paper also provides some new triviality proofs which utilize the axioms ((A → B)∧ (B → C)) → (A → C) and (A → ¬A) → ¬A, the fusion connective and the Ackermann constant. (...) An overview over various ways to formulate Leibniz’s law in non-classical logics and two new triviality proofs for naïve set theory are also provided. (shrink)