Draft for presentation at the 14th International Kant-Congress, September 2024. -/- Abstract: Kant claims we intuit infinite space. There’s a problem: Kant thinks full awareness of infinite space requires synthesis—the act of putting representations together and comprehending them as one. But our ability to synthesize is finite. Tobias Rosefeldt has argued in a recent paper that Kant’s notion of decomposing synthesis offers a solution. This talk criticizes Rosefeldt’s approach. First, Rosefeldt is committed to nonconceptual yet determinate (...) awareness of (potentially) infinite space, but I argue that such awareness is (1) prima facie implausible, as well as contravened by textual evidence from Kant’s (2a) definitions of ‘infinity’ and (2b) account of geometrical construction. Second, scrutiny of how Kant thinks awareness of potential infinity is afforded by geometrical construction (e.g., extending a line segment ad indefinitum) indicates no essential role for decomposing synthesis. Familiar synthesis from parts to wholes is sufficient. (shrink)

This chapter contains sections titled: Aristotle on the Infinite (to apeiron): From Cosmological Principle to Mathematical Operation Aristotle on Space: Magnitude (megethos) and Place (topos) Aristotle on Time: The “Number of Motion” and “Ever‐rolling Stream” Bibliography.

Donald Coxeter died in 2003, at a ripe old age of 96. Though I had regularly seen him at mathematics talks in Toronto for over twenty years, I never felt rushed to seek him out. It seemed he would go on forever. His death left me regretting my missed opportunity and Siobhan Robert's excellent book makes me regret it even more. Like any good biography of an intellectual, King of Infinite Space contains personal details and mathematical achievements in (...) some detail. Thus, we learn of the traumatic effects on Coxeter of his parents' divorce, his search for a spouse, his vegetarianism, and his progressive politics. We also learn a fair bit about the kaleidoscopes he made while in Cambridge during his student days in order to study the symmetry properties of polyhedra. These involved mirrors that Coxeter had specially constructed for this purpose. Along the way we are treated to interesting tidbits, such as G.H. Hardy's detestation of mirrors . There are brief accounts of the brilliant notation Coxeter invented, known as Coxeter diagrams, and of course, the now famous Coxeter groups. Short appendices fill in a bit more detail. It is all very well done and thoroughly engrossing.Wittgenstein befriended Coxeter, who was part of the very small group to whom …. (shrink)

In the Transcendental Aesthetic of the Critique of Pure Reason, Immanuel Kant famously claims that we have an a priori intuition of space as an ‘infinite given magnitude’. Later on, in the Transcendental Analytic, he seems to add that the intuition of space presupposes a synthetic activity of the transcendental imagination. Several authors have recently pointed out that these two claims taken together give rise to two problems. First, it is unclear how the transcendental imagination of a (...) finite mind could ever result in the intuition of an entity that is infinitely large. Second, Kant claims that our intuition of space has a ‘whole-prior-to-its-parts’ structure, such that its parts are given only as limitations of the whole, while synthesis is compositional and has a ‘parts-prior-to-their-whole’ structure, because it consists in first running through and then taking together the parts of a sensible manifold. I will solve these two problems by showing that Kant thought that synthesis does not always have a compositional structure but that there is also a form of ‘decomposing’ synthesis, which has a whole-prior-to-its-parts structure. Building on similarities between Kant, Edmund Husserl and G. W. F. Hegel, I will argue that infinite space is given to us in intuition by precisely such an activity of decomposition, one that allows us to differentiate between finite spatial objects and the unlimited phenomenal horizon in which they appear. (shrink)

Facing up to the head -- The secreting head -- Being my head -- The head comes to -- Airhead : breathing and its variations -- Communicating with air -- Enjoying and suffering my head -- Communicating without air -- Notes on the red-cheeked animal : the geology of a blush -- The watchtower -- The sensory room -- Having and using my head -- Head traffic : eating, vomiting and smoking -- Head on head : notes on kissing -- (...) Headgear -- Caretaking my head -- In the wars -- The dwindles -- Knowing (and not knowing) my head -- Head and world -- The thinking head -- Epilogue: Heading off. (shrink)

Abstractabstract:Kant's treatment of infinity seems to be plagued by two contradictions. First, the Transcendental Aesthetic claims that space is an infinite given magnitude, whereas the First Antinomy argues that the spatial world cannot be infinite. Second, the Transcendental Aesthetic claims that the representation of infinite space belongs to sensibility, but the third Critique seems to argue, instead, that infinity is an Idea of reason. This paper resolves these apparent contradictions by noting that Kant groups his (...) various conceptions of space into three kinds: (1) merely subjectively given space, (2) objectively given space, and (3) objective space as a mere Idea. Attending to these three conceptions of space illustrates that the Transcendental Aesthetic, First Antinomy, and third Critique refer to different conceptions of infinite space and thus do not contradict one another, illuminating the importance of Kant's various conceptions of space for his critical project. (shrink)

This essay explores the role of God’s omnipresence in Newton’s natural philosophy, with special emphasis placed on how God is related to space. Unlike Descartes’ conception, which denies the spatiality of God, or Gassendi and Charleton’s view, which regards God as completely whole in every part of space, it is argued that Newton accepts spatial extension as a basic aspect of God’s omnipresence. The historical background to Newton’s spatial ontology assumes a large part of our investigation, but with (...) attention also focused on the details of Newton’s unique approach to these traditional Scholastic conceptions. (shrink)

Aristotle argues that the world is populated by real and distinct physical substances; Spinoza that there must and can only be one physical substance. Aristotle’s view carries considerably intuitive appeal, but Spinoza’s logic can, under the right interpretation, seem awfully convincing. Andrew Burnside (2023) helps us to explore what occurs when Aristotle’s unstoppable intuitive appeal meets Spinoza’s impeccable logic. Burnside’s project, as I understand it, has two aims: to show that Spinoza’s argument for one extended substance is a better account (...) of physical reality than Aristotle’s arguments against an infinite body and, second, to support this claim by defending Yitzhak Melamed interpretation of Spinoza as genuinely dividing reality into substances and modes. I would like to press Burnside on the usefulness of comparing Aristotle and Spinoza with respect to an infinitely extended physical thing and then raise several question for his reading of Spinoza. (shrink)

Este artículo examina la concepción del espacio infinito y del tiempo en Hasdai Crescas, Gianfrancesco Pico della Mirandola y Giordano Bruno. Si la presencia de Crescas es explícita en el _Examen vanitatis_ (1520) de Pico, su recepción por Bruno, que nunca lo menciona, fue postulada por Harry A. Wolfson en 1929. Más recientemente, David Harari y Mauro Zonta han afirmado el papel intermediario de un autor judío desconocido. Sin embargo, una comparación de la crítica de Aristóteles efectuada por Crescas y (...) Bruno pone de manifiesto que, excepto dos puntos, Bruno pudo alcanzar sus posiciones con independencia de Crescas, a partir de su propia lectura crítica de Aristóteles y de su conocimiento de los desarrollos de la escolástica medieval y del concepto neoplatónico del tiempo. La significativa calificación por Crescas del espacio y del tiempo como atributos de Dios en un sentido puramente metafórico, una cuestión que Pico deja a un lado, deja paso en Bruno a la calificación del espacio infinito, de la duración y también de la materia como atributos reales de Dios, que es a la vez mente e intelecto, espacio y materia, en tanto que unidad indiferente de los opuestos. (shrink)

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers [Formula: see text] on [Formula: see text] as the prototype structures, we construct a class of continuum many topological Ramsey spaces [Formula: see text] which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projections. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces [Formula: see text], extending the Pudlák–Rödl theorem for barriers on the (...) Ellentuck space. The inspiration for these spaces comes from continuing the iterative construction of the forcings [Formula: see text] to the countable transfinite. The [Formula: see text]-closed partial order [Formula: see text] is forcing equivalent to [Formula: see text], which forces a non-p-point ultrafilter [Formula: see text]. This work forms the basis for further work classifying the Rudin–Keisler and Tukey structures for the hierarchy of the generic ultrafilters [Formula: see text]. (shrink)

I argue that Kant's claim in the “Transcendental Aesthetic” of the Critique of Pure Reason that space and time are immediately given in intuition as infinite magnitudes is undercut by his general theory of mathematical knowledge. On this general theory, pure intuition does not give objects of any determinate magnitude at all, but only forms of possible objects. Specifically, what pure intuition itself yields is the recognition that any determinate space or time is part of a larger (...) one, but it requires an inference of reason to go from that to the claim that space and time are infinite. I further argue that this result is consistent with Kant's claim in the second-edition “Transcendental Deduction” that the unity of space and time are the products of synthesis, but also means that the unity of space and time as objects cannot be used a premise in the Deduction but can only be regarded as a conclusion of the deduction and the following “System of Principles.”. (shrink)

We study theorems from Functional Analysis with regard to their relationship with various weak choice principles and prove several results about them: “Every infinite‐dimensional Banach space has a well‐orderable Hamel basis” is equivalent to ; “ can be well‐ordered” implies “no infinite‐dimensional Banach space has a Hamel basis of cardinality ”, thus the latter statement is true in every Fraenkel‐Mostowski model of ; “No infinite‐dimensional Banach space has a Hamel basis of cardinality ” is (...) not provable in ; “No infinite‐dimensional Banach space has a well‐orderable Hamel basis of cardinality ” is provable in ; (the Axiom of Choice for denumerable families of non‐empty finite sets) is equivalent to “no infinite‐dimensional Banach space has a Hamel basis which can be written as a denumerable union of finite sets”; Mazur's Lemma (“If X is an infinite‐dimensional Banach space, Y is a finite‐dimensional vector subspace of X, and, then there is a unit vector such that for all and all scalars α”) is provable in ; “A real normed vector space X is finite‐dimensional if and only if its closed unit ball is compact” is provable in ; (Principle of Dependent Choices) + “ can be well‐ordered” does not imply the Hahn‐Banach Theorem () in ; and “no infinite‐dimensional Banach space has a Hamel basis of cardinality ” are independent from each other in ; “No infinite‐dimensional Banach space can be written as a denumerable union of finite‐dimensional subspaces” lies in strength between (the Axiom of Countable Choice) and ; implies “No infinite‐dimensional Banach space can be written as a denumerable union of closed proper subspaces” which in turn implies ; “Every infinite‐dimensional Banach space has a denumerable linearly independent subset” is a theorem of, but not a theorem of ; and “Every infinite‐dimensional Banach space has a linearly independent subset of cardinality ” implies “every Dedekind‐finite set is finite”. (shrink)

It is well known that if the nonempty player of the Banach–Mazur game has a winning strategy on a space, then that space is Baire in all powers even in the box product topology. The converse of this implication may also be true: We know of no consistency result to the contrary. In this paper we establish the consistency of the converse relative to the consistency of the existence of a proper class of measurable cardinals.

This paper analyzes two different proposals, one by Ellis and Brundrit, based on classical relativistic cosmology, the other by Garriga and Vilenkin, based on the DH interpretation of quantum mechanics, both concluding that, in an infinite universe, planets and beings must be repeated an infinite number of times. We point to possible shortcomings in these arguments. We conclude that the idea of an infinite repetition of histories in space cannot be considered strictly speaking a consequence of (...) current physics and cosmology. Such ideas should be seen rather as examples of «ironic science» in the terminology of John Horgan. (shrink)

Kant's account of space as an infinite given magnitude in the Critique of Pure Reason is paradoxical, since infinite magnitudes go beyond the limits of possible experience. Michael Friedman's and Charles Parsons's accounts make sense of geometrical construction, but I argue that they do not resolve the paradox. I argue that metaphysical space is based on the ability of the subject to generate distinctly oriented spatial magnitudes of invariant scalar quantity through translation or rotation. The set (...) of determinately oriented, constructed geometrical spaces is a proper subset of metaphysical space, thus, metaphysical space is infinite. Kant's paradoxical doctrine of metaphysical space is necessary to reconcile his empiricism with his transcendental idealism. (shrink)

The theory of branching space-times is designed as a rigorous framework for modelling indeterminism in a relativistically sound way. In that framework there is room for "funny business", i.e., modal correlations such as occur through quantummechanical entanglement. This paper extends previous work by Belnap on notions of "funny business". We provide two generalized definitions of "funny business". Combinatorial funny business can be characterized as "absence of prima facie consistent scenarios", while explanatory funny business characterizes situations in which no localized (...) explanation of inconsistency can be given. These two definitions of funny business are proved to be equivalent, and we provide an example that shows them to be strictly more general than the previously available definitions of "funny business". (shrink)

We present a proof of a Ramsey-type theorem for infinite metric spaces due to Matoušek. Then we show that for every K>1 every uncountable Polish space has a perfect subset that K-bi-Lipschitz embeds into the real line. Finally we study decompositions of infinite separable metric spaces into subsets that, for some K>1, K-bi-Lipschitz embed into the real line.

This paper analyzes two different proposals, one by Ellis and Brundrit, based on classical relativistic cosmology, the other by Garriga and Vilenkin, based on the DH interpretation of quantum mechanics, both concluding that, in an infinite universe, planets and beings must be repeated an infinite number of times. We point to possible shortcomings in these arguments. We conclude that the idea of an infinite repetition of histories in space cannot be considered strictly speaking a consequence of (...) current physics and cosmology. Such ideas should be seen rather as examples of «ironic science» in the terminology of John Horgan. (shrink)

We classify the measures on the lattice ℒ of all closed subspaces of infinite-dimensional orthomodular spaces (E, Ψ) over fields of generalized power series with coefficients in ℝ. We prove that every σ-additive measure on ℒ can be obtained by lifting measures from the residual spaces of (E, Ψ). The measures being lifted are known, for the residual spaces are Euclidean. From the classification we deduce, among other things, that the set of all measures on ℒ is not separating.

A new construction of a certain conceptual space is presented. Elements of this conceptual space correspond to concept elements of reality, which potentially comprise an infinite number of qualities. This construction of a conceptual space solves a problem stated by Dietz and his co-authors in 2013 in the context of Voronoi diagrams. The fractal construction of the conceptual space is that this problem simply does not pose itself. The concept of convexity is discussed in this (...) new conceptual space. Moreover, the meaning of convexity is discussed in full generality, for example when space is deprived of it, its substitutes for concept domains are considered. (shrink)

The aesthetic techniques used in The Expanse are indicative of the infinite space that is an essential and ever‐present character in the show. The cinematography and set design of The Expanse make extensive use of chiaroscuro—a famous artistic technique in the history of painting. For some reason, the infinity of The Expanse attracts us. The look and design of the show indulges us in an experience of the sublime. The dynamically sublime is an experience of infinite power, (...) but not where our survival is threatened by it. The experience of the sublime is one proof of freedom. The absence of the ability to know the significance of what one does in the context of the infinite does not diminish the significance of what they do. The characters of The Expanse know this. They act because they know they must, and they have faith that what they do matters. (shrink)

Suppose you found that the universe around you was infinite—that it extended infinitely far in space or in time and, as a result, contained infinitely many persons. How should this change your moral decision-making? Radically, it seems, according to some philosophers. According to various recent arguments, any moral theory that is ’minimally aggregative’ will deliver absurd judgements in practice if the universe is (even remotely likely to be) infinite. This seems like sound justification for abandoning any such (...) theory. -/- My goal in this thesis is simple: to demonstrate that we need not abandon minimally aggregative theories, even if we happen to live in an infinite universe. I develop and motivate an extension of such theories, which delivers plausible judgements in a range of realistic cases. I show that this extended theory can overcome key objections—both old and new—and that it succeeds where other proposals do not. With this proposal in hand, we can indeed retain minimally aggregative theories and continue to make moral decisions based on what will promote the good. (shrink)

Almaatouq et al. argue that an “integrative experiment design” approach can help generating cumulative empirical and theoretical knowledge. Here, we discuss the novelty of their approach and scrutinize its promises and pitfalls. We argue that setting up a “design space” may turn out to be theoretically uninformative, inefficient, and even impossible. Designing truly diagnostic experiments provides a better alternative.

Infinity exists as a concept but has no existence in actuality. For infinity to have existence in actuality either time or space have to already be infinite. Unless something is already infinite, the only way to become infinite is by an 'infinity leap' in an infinitely small moment, and this is not possible. Neither does infinitely small have an existence since anything larger than zero is not infinitely small. Therefore infinity has no existence in actuality.

Many epistemologists have responded to the lottery paradox by proposing formal rules according to which high probability defeasibly warrants acceptance. Douven and Williamson present an ingenious argument purporting to show that such rules invariably trivialise, in that they reduce to the claim that a probability of 1 warrants acceptance. Douven and Williamson’s argument does, however, rest upon significant assumptions – amongst them a relatively strong structural assumption to the effect that the underlying probability space is both finite and uniform. (...) In this paper, I will show that something very like Douven and Williamson’s argument can in fact survive with much weaker structural assumptions – and, in particular, can apply to infinite probability spaces. (shrink)

Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.

How might we extend aggregative moral theories to compare infinite worlds? In particular, how might we extend them to compare worlds with infinite spatial volume, infinite temporal duration, and infinitely many morally valuable phenomena? When doing so, we face various impossibility results from the existing literature. For instance, the view we adopt can endorse the claim that worlds are made better if we increase the value in every region of space and time, or that they are (...) made better if we increase the value obtained by every person. But they cannot endorse both claims, so we must choose. In this paper I show that, if we choose the latter, our view will face serious problems such as generating incomparability in many realistic cases. Opting instead to endorse the first claim, I articulate and defend a spatiotemporal, expansionist view of infinite aggregation. Spatiotemporal views such as this do face some difficulties, but I show that these can be overcome. With modification, they can provide plausible comparisons in cases that we care about most. (shrink)

ABSTRACT This paper aims to show that intuitionistic Kripke models are a powerful tool for interpreting Kant’s ‘Critical Philosophy’. Part I reviews some old work of mine that applies these models to provide a reading of Kant’s second antinomy about the divisibility of matter and to answer several attacks on Kant’s antinomies. But it also points out three shortcomings of that original application. First, the reading fails to account for Kant’s second antinomy claim that matter is divisible ‘ad infinitum’ and (...) his first antinomy claim that empirical space extends only ‘ad indefinitum’. Secondly, in conformity with the ‘assertability’ heuristics for Kripke models, it attributes an assertability semantics to Kant. In fact, it attributes a pair of assertability semantics to Kant, but neither of these accords with modern assertabilism. And finally, one must wonder the propriety of using contemporary assertability and contemporary formal logic to save an eighteenth-century text. Part II goes historical to solve the problems about assertability. It outlines the Descartes–Leibniz dialectic about the infinite/indefinite distinction and demonstrates how each position reflects larger philosophical positions. It then demonstrates that Kant’s two versions of assertability and his version of the infinite/indefinite dichotomy emerge naturally from his attempts to resolve a tension in Leibniz’s philosophy. In light of this demonstration, Part III adapts the Kripke model interpretation so that it does reflect that dichotomy. It then refines the Kripke semantics in order to model some of Kant’s signature critical doctrines, and it shows how Kant’s critical version of the infinite/indefinite distinction differs from all those of his predecessors and from his own pre-critical version. It also addresses the question of using contemporary logical machinery to interpret Kant. Part IV turns the tables and uses what we have learned about Kant and modern semantics in order to address a pair of issues in the contemporary critical foundations of logic. (shrink)

Hume's discussion of the idea of space in his Treatise on Human Nature is fundamental to an understanding of his treatment of such central issues as the existence of external objects, the unity of the self, the relation between certainty and belief, and abstract ideas. Marina Frasca-Spada's rich and original study examines this difficult part of Hume's philosophical writings and connects it to eighteenth-century works in natural philosophy, mathematics and literature. Focusing on Hume's discussions of the infinite divisibility (...) of extension, the origin of the idea of space, geometry, and the notion of a vacuum, she shows that the central questions of Hume's 'science of human nature' - what does the 'science of human nature' reveal about the mind and its operations? what is experience? - underlie all of these discussions. Her analysis points the way to a reassessment of the central current interpretative problems in Hume studies. (shrink)

Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.

Hume’s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume’s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume’s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows (...) that, working within this epistemological model, we cannot attain complete certainty about the continuum but only at most about discrete quantity. Geometry, in contrast to arithmetic, cannot be a fully exact science. A number of more recent commentators have offered sympathetic interpretations of Hume’s discussion aiming to correct the older tendency to dismiss this part of the Treatise as weak and confused. Most of these commentators interpret Hume as anticipating the contemporary idea of a finite or discrete geometry. They view Hume’s conception that space is composed of simple indivisible minima as a forerunner of the conception that space is a discretely (rather than continuously) ordered set. This approach, in my view, is helpful as far as it goes, but there are several important features of Hume’s discussion that are not sufficiently appreciated. I go beyond these recent commentators by emphasizing three of Hume’s most original contributions. First, Hume’s epistemological model invokes the “confounding” of indivisible minima to explain the appearance of spatial continuity. Second, Hume’s sharp contrast between the perfect exactitude of arithmetic and the irremediable inexactitude of geometry reverses the more familiar conception of the early modern tradition in pure mathematics, according to which geometry (the science of continuous quantity) has its own standard of equality that is independent from and more exact than any corresponding standard supplied by algebra and arithmetic (the sciences of discrete quantity). Third, Hume has a developed explanation of how geometry (traditional Euclidean geometry) is nonetheless possible as an axiomatic demonstrative science possessing considerably more exactitude and certainty that the “loose judgements” of the vulgar. (shrink)

In her last known piece of work Lady Mary Shepherd’s Metaphysics (1832), Mary Shepherd writes that “mind, may inhere in definite portions of matter […] or of infinite space” (LMSM 699). Shepherd thus suggests that a mind – a “capacity for sensation in general” (e.g., EPEU 16) – may have a spatial location. This is prima facie surprising given that she is committed to the view that the mind is unextended. In this paper, we argue that Shepherd can (...) consistently honor both of these commitments. We argue, first, that even though finite minds, due to their unextendedness cannot, to use Shepherd’s term, ‘occupy’ space as material bodies do, such minds can nonetheless be said to have a spatial location in virtue of what we call their presence, or what Shepherd calls ‘inherence’, in a material body. Second, we argue that, in a similar way, the divine mind is present throughout the whole of space and is thus omnipresent. We then argue that, in contrast to finite minds, the divine mind does not require a body to be present in space. Rather, we contend, God’s omnipresence might be best understood in a ‘holenmerist’ sense, where the divine mind is wholly present in every part of creation. (shrink)

The paper investigates, in the framework of branching space–times, whether an infinite EPR-like correlation which does not involve finite EPR-like correlations is possible.

For infinite machines that are free from the classical Thomson’s lamp paradox, we show that they are not free from its inverted-in-time version. We provide a program for infinite machines and an infinite mechanism that demonstrate this paradox. While their finite analogs work predictably, the program and the infinite mechanism demonstrate an undefined behavior. As in the case of infinite Davies machines :671–682, 2001), our examples are free from infinite masses, infinite velocities, (...) class='Hi'>infinite forces, etc. Only infinite divisibility of space and time is assumed. Thus, the infinite devices considered are possible in a Newtonian Universe and they do not conflict with Newtonian mechanics. Note that the classical Thomson’s lamp paradox leads to infinite velocities which may not be producible in acceptable models of Newtonian mechanics. Finally, it is shown that the “paradox of predictability” is similar to the inverted Thomson’s lamp paradox. (shrink)

For thousands of years voyagers of inner space - spiritual seekers, shamans and mystics - have returned from their inner travels reporting another level of reality that is more real than the one we inhabit in 'waking life'. Others have claimed that under the influence of mysterious substances, known as entheogens, the everyday human mind can be given glimpses of this multidimensional realm of existence that is usually hidden from us by our five basic senses. Using information from the (...) leading edges of modern science, this book presents a startling new hypothesis that these 'inner worlds' are as real, or possibly even more real, than the 'reality' we experience in waking life. The Infinite Mindfield uses as its starting point the widespread historical belief that the pineal gland - the 'third eye' - is a profoundly important organ. It links this to the various myths, originating in ancient Sumer, that 'dragons' or 'serpents' have guided humanity and presents evidence that these beings are symbolic of DNA. It is now known that DNA gives off a form of light known as bioluminescence. This information-rich 'inner light' needs an organ of sight to process it - that organ is the pineal gland. It is through this small organ that we "perceive" the inner worlds of lucid dreaming, out-of-body experiences, hypnagogic imagery, near-death experiences, astral travel and the kundalini experience. The book ends with the mind-blowing proposition that all living beings are one unitary consciousness experiencing itself subjectively. This book is genuinely revolutionary and will cause both interest and controversy in equal measure. Its science is watertight as is its logic. Be ready to be stunned - you will never have read anything like this before. (shrink)

On Itamar Pitowsky’s subjective interpretation of quantum mechanics, “the Hilbert space formalism of quantum mechanics [QM] is just a new kind of probability theory”, one whose probabilities correspond to odds rational agents would accept on the outcomes of gambles concerning quantum event structures. Our aim here is to ask whether Pitowsky’s approach can be extended from its original context, of quantum theories for systems with an finite number of degrees of freedom, to systems with an infinite number of (...) degrees of freedom, such as quantum field theory and quantum statistical mechanics in the thermodynamic limit. An impediment to generalization is that Pitowsky adopts the framework of event structures encoded by atomic algebras, whereas the algebras typical of QM for infinitely many degrees of freedom are usually non-atomic. We describe challenges to Pitowsky’s approach deriving from this impediment, and sketch and assess strategies Pitowsky might use to meet those challenges. Although we offer no final verdict about the eventual success of those strategies, a testament to the worth of Pitowsky’s approach is that attempting to extend it engages us in provocative foundational issues. (shrink)

000000001. Introduction Call a theory of the good—be it moral or prudential—aggregative just in case (1) it recognizes local (or location-relative) goodness, and (2) the goodness of states of affairs is based on some aggregation of local goodness. The locations for local goodness might be points or regions in time, space, or space-time; or they might be people, or states of nature.1 Any method of aggregation is allowed: totaling, averaging, measuring the equality of the distribution, measuring the minimum, (...) etc.. Call a theory of the good finitely additive just in case it is aggregative, and for any finite set of locations it aggregates by adding together the goodness at those locations. Standard versions of total utilitarianism typically invoke finitely additive value theories (with people as locations). A puzzle can arise when finitely additive value theories are applied to cases involving an infinite number of locations (people, times, etc.). Suppose, for example, that temporal locations are the locus of value, and that time is discrete, and has no beginning or end.2 How would a finitely additive theory (e.g., a temporal version of total utilitarianism) judge the following two worlds? Goodness at Locations (e.g. times) w1:..., 2, 2, 2, 2, 2, 2, 2, 2, 2, ..... w2:..., 1, 1, 1, 1, 1, 1, 1, 1, 1, ..... Example 1 At each time w1 contains 2 units of goodness and w2 contains only 1. Intuitively, we claim, if the locations are the same in each world, finitely additive theorists will want to claim that w1 is better than w2. But it's not clear how they could coherently hold this view. For using standard mathematics the sum of each is the same infinity, and so there seems to be no basis for claiming that one is better than the other.3 (Appealing to Cantorian infinities is of no help here, since for any Cantorian infinite N, 2xN=1xN.). (shrink)