In this article, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, Thomson’s lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Pirah ̃a, working with only three numerals (one, two, many) can help us to (...) change our perception of these paradoxes. A recently introduced methodology allowing one to work with finite, infinite, and infinitesimal numbers in a unique computational framework not only theoretically but also numerically is briefly described. This methodology is actively used nowadays in numerous applications in pure and applied mathematics and computer science as well as in teaching. It is shown in the article that this methodology also allows one to consider the paradoxes listed above in a new constructive light. (shrink)

Our evidence can be about different subject matters. In fact, necessarily equivalent pieces of evidence can be about different subject matters. Does the hyperintensionality of ‘aboutness’ engender any hyperintensionality at the level of rational credence? In this paper, I present a case which seems to suggest that the answer is ‘yes’. In particular, I argue that our intuitive notions of independent evidence and inadmissible evidence are sensitive to aboutness in a hyperintensional way. We are thus left with a paradox. While (...) there is strong reason to think that rational credence cannot make such hyperintensional distinctions, our intuitive judgements about certain cases seem to demand that it does. (shrink)

The purpose of this paper is to dissolve paradoxes of the infinite by correctly identifying the infinite natural numbers.

An analysis of the counter-intuitive properties of infinity as understood differently in mathematics, classical physics and quantum physics allows the consideration of various paradoxes under a new light (e.g. Zeno’s dichotomy, Torricelli’s trumpet, and the weirdness of quantum physics). It provides strong support for the reality of abstractness and mathematical Platonism, and a plausible reason why there is something rather than nothing in the concrete universe. The conclusions are far reaching for science and philosophy.

David Builes presents a paradox concerning how confident you should be that any given member of an infinite collection of fair coins landed heads, conditional on the information that they were all flipped and only finitely many of them landed heads. We argue that if you should have any conditional credence at all, it should be 1/2.

This work deals with problems involving infinities and infinitesimals. It explores the ideas behind zero, its relationship to ontological nothingness, finititude (such as finite numbers and quantities), and the infinite. The idea of infinity and zero are closely related, despite what many perceive as an intuitive inverse relationship. The symbol 0 generally refers to nothingness, whereas the symbol infinity refers to ``so much'' that it cannot be quantified or captured. The notion of finititude rests somewhere between complete nothingness (...) and something having no end. -/- My concern is that many of the philosophers arguing for or against the ontology (or possibility) of an actual infinite set are unaware or unfamiliar with the mathematical literature attempting to clearly and rigorously define these terms. I believe it is a mistake to leave mathematicians out of this conversation, as analysts in particular have defined (and used) infinity in a way that is relevant to the ongoing debate between philosophers. -/- For this paper, I have selected examples from Principles of Analysis by Walter Rudin, which has become a standard text for Real Analysis classes taken by pure mathematicians and engineers. I will also restrict the scope of examples to Euclidean spaces for simplicity. Finally, I will be using addition, + and multiplication * in their usual way. (shrink)

In this paper I present a novel supertask in a Newtonian universe that destroys and creates infinite masses and energies, showing thereby that we can have infinite indeterminism. Previous supertasks have managed only to destroy or create finite masses and energies, thereby giving cases of only finite indeterminism. In the Nothing from Infinity paradox we will see an infinitude of finite masses and an infinitude of energy disappear entirely, and do so despite the conservation of energy in all collisions. (...) I then show how this leads to the Infinity from Nothing paradox, in which we have the spontaneous eruption of infinite mass and energy out of nothing. I conclude by showing how our supertask models at least something of an old conundrum, the question of what happens when the immovable object meets the irresistible force. (shrink)

This article draws on several crucial and unpublished manuscripts from the Scholem Archive in exploration of Gershom Scholem's youthful statements on mathematics and its relation to extra-mathematical facts and, more broadly, to a concept of history that would prove to be consequential for Walter Benjamin's own thinking on "messianism" and a "futuristic politics." In context of critiquing the German Youth Movement's subsumption of active life to the nationalistic conditions of the "earth" during the First World War, Scholem turns to mathematics (...) for a genuine and self-consistent theory of action. In the concept of actual infinity (in Cantor and Bolzano) he finds an explanation of how mathematics relates to "the physical" without reducing the former to an "image" of the latter, and without relying on the concept of geometric intuition. This explanation, insofar as it relies on the notion of actual infinity, provides Scholem with a conception of mathematics (and the history of mathematics) that reconciles freedom and necessity—remarks on which he outlines in his diaries and communicates to Benjamin in early March 1916. (shrink)

The Nothing from Infinity paradox arises when the combination of two infinitudes of point particles meet in a supertask and disappear. Corral-Villate claims that my arguments for disappearance fail and concedes that this failure also produces an extreme kind of indeterminism, which I have called plenitudinous. So my supertask at least poses a dilemma of extreme indeterminism within Newtonian point particle mechanics. Plenitudinous indeterminism might be trivial, although easy attempts to prove it so seem to fail in the face (...) of plausible continuity principles. However, the question of its triviality is here moot, since I show that, except in one case, Corral-Villate’s disproofs fail, and with a correction, the original arguments are unrefuted. Consequently, of the two contenders for the outcome of my supertask, the Nothing from Infinity paradox has won out. (shrink)

We look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new logics and prove a completeness result for the first and last element of the chain stating that both logics are determined by a single finite logical (...) matrix. We show that the last logic of the chain is not finitely axiomatisable. (shrink)

Benardete paradoxes involve a beginningless set each member of which satisfies some predicate just in case no earlier member satisfies it. Such paradoxes have been wielded on behalf of arguments for the impossibility of an infinite past. These arguments often deploy patchwork principles in support of their key linking premise. Here I argue that patchwork principles fail to justify this key premise.

We look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new log- ics and prove a completeness result for the first and last element of the chain stating that both logics are determined by a single finite (...) logical matrix. We show that the last logic of the chain is not finitely axiomatisable. (shrink)

Two strategies to infinity are equally relevant for it is as universal and thus complete as open and thus incomplete. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, thеse phenomena can be elucidated as both complete and incomplete, after which choice is the border (...) between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or “observer”, or “user” to infinity implied by Henkin’s proposition as the only consistent ones as to infinity. (shrink)

(1st draft for review) -/- WARNING: -/- Reading this book comes with certain dangers to be mindful of; Please consider the following suggestions to avoid them: -/- 1: Do not try to perceive infinity; Any kind of success here leads to psychosis. 2: Do not try to resolve the paradoxes; To understand the greater truth of this book, paradoxes must be accepted as true. 3: Do not read this book if your faith is unstable and having it (...) challenged could potentially put you into existential crisis. (shrink)

Russell’s initial project in philosophy (1898) was to make mathematics rigorous reducing it to logic. Before August 1900, however, Russell’s logic was nothing but mereology. First, his acquaintance with Peano’s ideas in August 1900 led him to discard the part-whole logic and accept a kind of intensional predicate logic instead. Among other things, the predicate logic helped Russell embrace a technique of treating the paradox of infinite numbers with the help of a singular concept, which he called ‘denoting phrase’. Unfortunately, (...) a new paradox emerged soon: that of classes. The main contention of this paper is that Russell’s new conception only transferred the paradox of infinity from the realm of infinite numbers to that of class-inclusion. Russell’s long-elaborated solution to his paradox developed between 1905 and 1908 was nothing but to set aside of some of the ideas he adopted with his turn of August 1900: (i) With the Theory of Descriptions, he reintroduced the complexes we are acquainted with in logic. In this way, he partly restored the pre-August 1900 mereology of complexes and simples. (ii) The elimination of classes, with the help of the ‘substitutional theory’, and of propositions, by means of the Multiple Relation Theory of Judgment, completed this process. (shrink)

Bowin begins with an apparent paradox about Aristotelian infinity: Aristotle clearly says that infinity exists only potentially and not actually. However, Aristotle appears to say two different things about the nature of that potential existence. On the one hand, he seems to say that the potentiality is like that of a process that might occur but isn't right now. Aristotle uses the Olympics as an example: they might be occurring, but they aren't just now. On the other hand, (...) Aristotle says that infinity "exists in actuality as a process that is now occurring" (234). Bowin makes clear that Aristotle doesn't explicitly solve this problem, so we are left to work out the best reading we can. His proposed solution is that "infinity must be...a per se accident...of number and magnitude" (250). (Bryn Mawr Classical Review 2008.07.47). (shrink)

I propose a revision of Cantor’s account of set size that understands comparisons of set size fundamentally in terms of surjections rather than injections. This revised account is equivalent to Cantor's account if the Axiom of Choice is true, but its consequences differ from those of Cantor’s if the Axiom of Choice is false. I argue that the revised account is an intuitive generalization of Cantor’s account, blocks paradoxes—most notably, that a set can be partitioned into a set that (...) is bigger than it—that can arise from Cantor’s account if the Axiom of Choice is false, illuminates the debate over whether the Axiom of Choice is true, is a mathematically fruitful alternative to Cantor’s account, and sheds philosophical light on one of the oldest unsolved problems in set theory. (shrink)

This paper gives a definition of self-reference on the basis of the dependence relation given by Leitgeb (2005), and the dependence digraph by Beringer & Schindler (2015). Unlike the usual discussion about self-reference of paradoxes centering around Yablo's paradox and its variants, I focus on the paradoxes of finitary characteristic, which are given again by use of Leitgeb's dependence relation. They are called 'locally finite paradoxes', satisfying that any sentence in these paradoxes can depend on finitely (...) many sentences. I prove that all locally finite paradoxes are self-referential in the sense that there is a directed cycle in their dependence digraphs. This paper also studies the 'circularity dependence' of paradoxes, which was introduced by Hsiung (2014). I prove that the locally finite paradoxes have circularity dependence in the sense that they are paradoxical only in the digraph containing a proper cycle. The proofs of the two results are based directly on König's infinity lemma. In contrast, this paper also shows that Yablo's paradox and its nested variant are non-self-referential, and neither McGee's paradox nor the omega-cycle liar paradox has circularity dependence. (shrink)

When matter is falling into a black hole, the associated information becomes unavailable to the black hole's exterior. If the black hole disappears by Hawking evaporation, the information seems to be lost in the singularity, leading to Hawking's information paradox: the unitary evolution seems to be broken, because a pure separate quantum state can evolve into a mixed one.



This article proposes a new interpretation of the black hole singularities, which restores the information conservation. For the Schwarzschild black hole, it presents (...) new coordinates, which move the singularity at the future infinity (although it can still be reached in finite proper time). For the evaporating black holes, this article shows that we can still cure the apparently destructive effects of the singularity on the information conservation. For this, we propose to allow the metric to be degenerate at some points, and use the singular semiriemannian geometry. This view, which results naturally from Ashtekar's new variables formulation of Einstein's equation, repairs the incomplete geodesics.



The reinterpretation of singularities suggested here allows (in the context of standard General Relativity) the information conservation and unitary evolution to be restored, both for eternal and for evaporating black holes. (shrink)

When matter is falling into a black hole, the associated information becomes unavailable to the black hole's exterior. If the black hole disappears by Hawking evaporation, the information seems to be lost in the singularity, leading to Hawking's information paradox: the unitary evolution seems to be broken, because a pure separate quantum state can evolve into a mixed one.



This article proposes a new interpretation of the black hole singularities, which restores the information conservation. For the Schwarzschild black hole, it presents (...) new coordinates, which move the singularity at the future infinity (although it can still be reached in finite proper time). For the evaporating black holes, this article shows that we can still cure the apparently destructive effects of the singularity on the information conservation. For this, we propose to allow the metric to be degenerate at some points, and use the singular semiriemannian geometry. This view, which results naturally from the Cauchy problem, repairs the incomplete geodesics.



The reinterpretation of singularities suggested here allows (in the context of standard General Relativity) the information conservation and unitary evolution to be restored, both for eternal and for evaporating black holes.

. (shrink)

It is modernly debated whether application of the free will has potential to cause harm to nature. Power possessed to the discourse, sensory/perceptual, physical influences on life experience by the slow moving machinery of change is a viral element in the problems of civilization; failed resolution of historical paradox involving mind and matter is a recurring source of problems. Reference is taken from the writing of Euclid in which a oneness of nature as an indivisible point of thought is made (...) prerequisite in criteria of interpretation to demonstrate that contemporary scientific methodologies alternately ensue from the point of empirically centered induction. A qualification for conceptualizations is proposed that involves a physically describable form bound to energy in addition to contemporary notions of energy bound to form and a visually based mathematical-physical form is elaborated and discussed with respect to biological and natural processes. (shrink)

Georg Cantor's absolute infinity, the paradoxical Burali-Forti class Ω of all ordinals, is a monstrous non-entity for which being called a "class" is an undeserved dignity. This must be the ultimate vexation for mathematical philosophers who hold on to some residual sense of realism in set theory. By careful use of Ω, we can rescue Georg Cantor's 1899 "proof" sketch of the Well-Ordering Theorem––being generous, considering his declining health. We take the contrapositive of Cantor's suggestion and add Zermelo's choice (...) function. This results in a concise and uncomplicated proof of the Well-Ordering Theorem. (shrink)

The seventeenth century was an important period in the conceptual development of the notion of the infinite. In 1643, Evangelista Torricelli (1608-1647)—Galileo’s successor in the chair of mathematics in Florence—communicated his proof of a solid of infinite length but finite volume. Many of the leading metaphysicians of the time, notably Spinoza and Leibniz, came out in defense of actual infinity, rejecting the Aristotelian ban on it, which had been almost universally accepted for two millennia. Though it would be another (...) two centuries before the notion of the actually infinite was rehabilitated in mathematics by Dedekind and Cantor (Cauchy and Weierstrass still considered it mere paradox), their impenitent advocacy of the concept had significant reverberations in both philosophy and mathematics. In this essay, I will attempt to clarify one thread in the development of the notion of the infinite. In the first part, I study Spinoza’s discussion and endorsement, in the Letter on the Infinite (Ep. 12), of Hasdai Crescas’ (c. 1340-1410/11) crucial amendment to a traditional proof of the existence of God (“the cosmological proof” ), in which he insightfully points out that the proof does not require the Aristotelian ban on actual infinity. In the second and last part, I examine the claim, advanced by Crescas and Spinoza, that God has infinitely many attributes, and explore the reasoning that motivated both philosophers to make such a claim. Similarities between Spinoza and Crescas, which suggest the latter’s influence on the former, can be discerned in several other important issues, such as necessitarianism, the view that we are compelled to assert or reject a belief by its representational content, the enigmatic notion of amor Dei intellectualis, and the view of punishment as a natural consequent of sin. Here, I will restrict myself to the issue of the infinite, clearly a substantial topic in itself. (shrink)

Previous claims to have resolved the two-envelope paradox have been premature. The paradoxical argument has been exposed as manifestly fallacious if there is an upper limit to the amount of money that may be put in an envelope; but the paradoxical cases which can be described if this limitation is removed do not involve mathematical error, nor can they be explained away in terms of the strangeness of infinity. Only by taking account of the partial sums of the infinite (...) series of expected gains can the paradox be resolved. (shrink)

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.

This thesis attempts to present a new singular type of paradox called a “Monoletheia”, which itself creates a logical bridge to reconcile two subsequent paradoxes we call “Antinomies” and “Dialetheia”. The consequence follows therefrom, to state that the concept of infinity transcends its own regress of an existential construct to include a non-existent variable (representing void). This is done by applying the theoretical idea of co-existence between constructs and their own negation. -/- The working hypothesis can be summed (...) up in the following proposition: “Existence co-exists with non-existence“, and is expressed mathematically for convenience as “Ex + co-ex w/ X“. For the same purpose, the expression denoting an antinomy is “A*” and “Dt” is used for dialetheia. (shrink)

This paper is a response to David Oderberg's discussion of the Tristram Shandy paradox. I defend the claim that the Tristram Shandy paradox does not support the claim that it is impossible that the past is infinite.

The discrepancy between individual women and the stereotypes attributed to the group as a ‎whole has become progressively greater and more explicit over the course of history. The stereotypes remain the same age-old ‎allegations whilst the ‎developments in the occupations of women and the traits they have opportunity to express have increased the distance between women and those ascribed traits. Stereotypes’ abstention from revision in light of contrary evidence constitutes an epistemic paradox for it entails conflict between the stereotypical knowledge (...) and empirical evidence, as well as between the stereotypical knowledge and other empirical knowledges. This paradox of stubbornness also raises the question as to the epistemic source of stereotypes. For, people at young ages encounter evidence which should have formed a more complex knowledge regarding women rather than the pervasive stereotypical knowledge. It is this extremity of absurdness of stereotypes regarding women that is utile to understanding stereotypes as a whole. The epistemic paradox that is engrained in all stereotypes is exposed uncompromisingly by this group. The paradox cannot be dismissed with excuses of having a small basis of evidence, such as may be said regarding other social groups with which one may have had minimal or no contact. Thus, stereotypes of women provide a limit case to understanding the epistemology of this paradoxical form of knowledge. The traits and epistemic phenomena which render stereotypes epistemically puzzling and intriguing is the relationship between knowledges. Stereotypical knowledge regarding women as a group conflict with one’s experience-founded knowledges of particular women, deeming the corporate body of knowledge incoherent. Quine’s theory of knowledge, with its holistic empirical approach endorsing conceptual schemes, provides fertile ground for this analysis. The novelty in his account is his establishment of the criteria of coherency, which shifts the focal point of epistemic inquiry away from the adequacy of individual knowledges with empirical evidence to the intra-knowledge relationships themselves. In his empiricism, Quine rather examines the interactions between knowledges and the overarching coherence of the body of knowledge as a holistic unit. Quine creates place in his account for mechanism which provide for the epistemic incongruences of stubbornness and conflict found between stereotypical knowledges and empirical evidence. These very mechanisms are what this paper requests to examine. Quine, ultimately, formulates knowledge as a collectively held fabric consisting of relations and of posits – socially constructed mechanisms which are empirically-irreducible and implemented pragmatically for the organisation of empirical evidence. According to Quine, the fabric formation is a conceptual scheme sourced in social heritage. People are bestowed with an eclectic framework of ‎knowledge to pragmatically merge between an inherited, collective conceptual scheme and the personally experienced empirical evidence. This paper will delve into Quine’s fabric of knowledge, its relations, and inner-mechanisms. The finding of stereotypes to be an age-old posit entrenched pragmatically – thus stubbornly – in the collective conceptual scheme, will explain the epistemic paradoxes they entail and offer insight to their revision in light of empiric reality and its women. (shrink)

As with so many mystics, Alexis karpouzos intuitively know the oneness of cosmic creation and historic humanity as part of all that is and all there isn't. So, the originality of Alexis Karpouzos thought is that it crosses the most diverse fields, the most opposing philosophies, to unite them into an often contradictory and broken whole. Marx and Heidegger, Nietzsche, Freud and Heraclitus, poets and political theorists all come together in the same distance and the same unusual proximity. Alexis karpouzos (...) use Pre-Socratics philosophy and generally the ancient Greek philosophy, as well as the pre-philosophical thinking of The Upanishads, the Vedas and Buddhism in India, of Lao Tzu, of Zen Buddhism and the Taoist tradition in China, of the Arab mystics and poets, with their metaphysical religiosity as the metaphysical basis for the interpretation and understanding of the world and existence. At the same time the ancient metaphysics is connect with Hegel’s dialectical ontology and with the modern thinking of Marx, Nietzsche, Freud, Heidegger, and others as the interpretive context for understanding the central problems of the technical and scientific world during his time. Novalis, Hölderlin, Rimbaud, Whitman, Eliot and others show the dreamy nature of existence, the transcendence of the Cosmos, welcome the infinity. For Alexis karpouzos the combination of ancient and modern thought creates the holistic experience of universal space-time and the consciousness of the unity. The poetic thought of Alexis karpouzos is a expressions of soul's inner experiences, expression of universality. The inspiring visual images and the symbolic use of language offer a description of elevating experiences of consciousness, a glimpse of higher worlds. The philosophy of Alexis karpouzos speak to the human experience from a universal perspective, transcending all religions, cultural and national boundaries. Using vivid images and a direct language that speaks to the heart, his philosophy evokes a sense of deep communication with the collective unconscious, a sense of connection to all the creatures of the world, compassion for others, admiration for the beauty of nature, reverence for all life, and an abiding faith in the invisible touch of world. Alexis karpouzos thoughts are often terse and paradoxical, challenging us to to break out of the box of limiting beliefs and see things from a new perspective. Above all, Alexis karpouzos continually calls to us to wake up and explore the mysteries within our own selves, i.e the mysteries of universe. (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)

Supererogatory acts—good deeds “beyond the call of duty”—are a part of moral common sense, but conceptually puzzling. I propose a unified solution to three of the most infamous puzzles: the classic Paradox of Supererogation (if it’s so good, why isn’t it just obligatory?), Horton’s All or Nothing Problem, and Kamm’s Intransitivity Paradox. I conclude that supererogation makes sense if, and only if, the grounds of rightness are multi-dimensional and comparative.

Recently, the problem of national identity has been actualized, but here is a the question: identity to what? Traditions, religion, mentality? Ideals, perhaps? Inspired by the best examples of national historical thought, the author offers an an original approach to deciphering the Russian "cultural code". В последнее время актуализируется проблема национальной идентичности, но вот вопрос: идентичности чему? Традициям, религии, менталитету? Идеалам, может быть? Воодушевленный лучшими образцами отечественной исторической мысли, автор предлагает оригинальный подход к расшифровке русского «культурного кода».

Several interesting themes emerge from G. E. Moore’s previously unpub­lished review of _The Principles of Mathematics_. These include a worry concerning whether mathematical notions are identical to purely logical ones, even if coextensive logical ones exist. Another involves a conception of infinity based on endless series neglected in the Principles but arguably involved in Zeno’s paradox of Achilles and the Tortoise. Moore also questions the scope of Russell’s notion of material implication, and other aspects of Russell’s claim that mathematics (...) reduces to logic. (shrink)

In the late 1940s and early 1950s Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as the precursor to the more well-known dialogical logic and one could assumed that the same philosophical motivations were present in both works. However we want to show that this is not always the case. In particular, we claim, that Lorenzen’s well-known rejection of the actual infinite as stated in Lorenzen (1957) was not a major motivation (...) for operative logic and mathematics. In this article, we claim that this is in fact not the case. Rather, we argue for a shift that happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operativism is concerned with a critique of the Cantorian notion of set and related questions about the notion of countability and uncountability; only later, his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity. (shrink)

Illusionism is the view that conscious experience is some sort of introspective illusion. According to illusionism, there is no conscious experience, but it merely seems like there is conscious experience. This would suggest that much phenomenological enquiry, including work on phenomenological psychopathology, rests on a mistake. Some philosophers have argued that illusionism is obviously false, because seeming is itself an experiential state, and so necessarily presupposes the reality of conscious experience. In response, the illusionist could suggest that the relevant sort (...) of seeming here is not an experiential state, but is a cognitive state, such as a judgement or a belief, which is fully amenable to a physical or functionalist analysis. Herein, I argue that this response is unsuccessful and fails to undermine the reality of conscious experience. Nonetheless, the response does raise the problem of how a judgement or belief about the character of a conscious experience, even if it is true, can be justified if the conscious experience has no causal role in the formation of the judgement or belief. This is not a new problem, but is a reiteration of an old problem that is known in the philosophy of mind literature as the paradox of phenomenal judgement. I consider how the paradox of phenomenal judgement can be resolved and how the judgement or belief about conscious experience can be justified with appeal to the notion of acquaintance. (shrink)

This paper is concerned with the paradox of decrease. Its aim is to defend the answer to this puzzle that was propounded by its originator, namely, the Stoic philosopher Chrysippus. The main trouble with this answer to the paradox is that it has the seemingly problematic implication that a material thing could perish due merely to extrinsic change. It follows that in order to defend Chrysippus’ answer to the paradox, one has to explain how it could be that Theon is (...) destroyed by the amputation without changing intrinsically. In this paper, I shall answer this challenge by appealing to the broadly Aristotelian idea that at least some of the proper parts of a material substance are ontologically dependent on that substance. I will also appeal to this idea in order to offer a new solution to the structurally similar paradox of increase. In this way, we will end up with a unified solution to two structurally similar paradoxes. (shrink)

It is widely accepted that there is what has been called a non-hypocrisy norm on the appropriateness of moral blame; roughly, one has standing to blame only if one is not guilty of the very offence one seeks to criticize. Our acceptance of this norm is embodied in the common retort to criticism, “Who are you to blame me?”. But there is a paradox lurking behind this commonplace norm. If it is always inappropriate for x to blame y for a (...) wrong that x has committed, then all cases in which x blames x (i.e. cases of self-blame) are rendered inappropriate. But it seems to be ethical common-sense that we are often, sadly, in position (indeed, excellent, privileged position) to blame ourselves for our own moral failings. And thus we have a paradox: a conflict between the inappropriateness of hypocritical blame, and the appropriateness of self-blame. We consider several ways of resolving the paradox, and contend none is as defensible as a position that simply accepts it: we should never blame ourselves. In defending this startling position, we defend a crucial distinction between self-blame and guilt. (shrink)

Philosophers have long argued that duties to oneself are paradoxical, as they seem to entail an incoherent power to release oneself from obligations. I argue that self-release is possible, both as a matter of deontic logic and of metaethics.

The coordinated attack scenario and the electronic mail game are two paradoxes of common knowledge. In simple mathematical models of these scenarios, the agents represented by the models can coordinate only if they have common knowledge that they will. As a result, the models predict that the agents will not coordinate in situations where it would be rational to coordinate. I argue that we should resolve this conflict between the models and facts about what it would be rational to (...) do by rejecting common knowledge assumptions implicit in the models. I focus on the assumption that the agents have common knowledge that they are rational, and provide models to show that denying this assumption suffices for a resolution of the paradoxes. I describe how my resolution of the paradoxes fits into a general story about the relationship between rationality in situations involving a single agent and rationality in situations involving many agents. (shrink)

*As mentioned in Peter Coy's NYT essay "When Being Good Is Just a Matter of Being Lucky" (2023) -/- ----- -/- How is the problem of free will related to the problem of moral luck? In this essay, I answer that question and outline a new solution to the paradox of moral luck, the source-paradox solution. This solution both explains why the paradox arises and why moral luck does not exist. To make my case, I highlight a few key connections (...) between the paradox of moral luck and two related problems, namely the problem of free will and determinism and the paradox of self-creation. Piecing together intuitions, arguments, and insights from recent work on each of these three problems, I argue that the type of control necessary for moral responsibility can only be had by someone who is a genuine source of his own actions, but the relevant notion of sourcehood admits no coherent characterization. If our commonsense view of moral responsibility is incoherent, it is unsurprising that our commitment to the existence of morally responsible agents commits us to some paradoxical things—e.g. to both the existence and impossibility of moral luck. For more information about this article, please write to gmail account: kristin.mickelson.42. (shrink)

Most theories of suspense implicitly or explicitly have as a background assumption what I call suspense realism, i.e., that suspense is itself a genuine, distinct emotion. I claim that for a theory of suspense to entail suspense realism is for that theory to entail a contradiction, and so, we ought instead assume a background of suspense eliminativism, i.e., that there is no such genuine, distinct emotion that is the emotion of suspense. More precisely, I argue that i) any suspense realist (...) (...) theory must resolve the paradox of suspense, ii) if suspense is itself a genuine, distinct emotion, then in order to resolve the paradox of suspense it must be a radically sui generis genuine, distinct emotion, iii) according to any minimally adequate theory of the emotions, there can be no radically sui generis emotion, and so iv) there can be no genuine, distinct emotion that is the emotion of suspense. Quite simply, if a theory of suspense must entail suspense realism, then we ought to be eliminativists about suspense. This I call the Paradox of Suspense Realism, which I take to constitute a productive viability condition for any theory of suspense, i.e., any viable theory of suspense must be mutatis mutandis compatible with suspense eliminativism. (shrink)

Saying that x is ineffable seems to be paradoxical – either I cannot say anything about x, not even that it is ineffable – or I can say that it is ineffable, but then I can say something and it is not ineffable. In this article, I discuss Alston’s version of the paradox and a solution proposed by Hick which employs the concept of formal and substantial predicates. I reject Hick’s proposal and develop a different account based on some passages (...) from Pseudo-Dionysius’ Mystica Theologia. ‘God is ineffable’ is a metalinguistic statement concerning propositions about God: not all propositions about God are expressible in a human language. (shrink)

This paper deals with Veronica Rodriguez-Blanco’s answer to the paradox of the normativity of law: How can autonomous self-legislating persons act, without compromising their autonomy and their will, following legal rules? Regarding Rodriguez-Blanco’s answer, I offer two main critiques. The first one is based on Rodriguez-Blanco’s comments to David Enoch’s paper in which I argue against the idea that a descriptive theoretical account of law can, and should, give an answer to general problems of normativity due to the fact that (...) a theoretical approach that engages in questions about the normativity of law cannot be purely descriptive, therefore against Rodriguez-Blanco I conclude that Enoch’s answer is the only response a descriptive account is able to offer. The second criticism focuses on Rodriguez-Blanco’s response to the paradox and argues that her solution is not, in fact, a descriptive one, but one that relies on strong normative premises, which as it turns out defend a perfectionist perspective towards the law.Resumen:Este trabajo analiza la respuesta que Veronica Rodriguez-Blanco propone para resolver la paradoja de la normatividad del derecho: ¿Cómo es posible que una persona autónoma actúe siguiendo los mandatos de normas jurídicas sin comprometer su autonomía ni su voluntad? El autor ofrece dos críticas a la respuesta de Rodriguez-Blanco. La primera de ellas está basada en los comentarios que Rodriguez-Blanco ha hecho sobre la propuesta de David Enoch. En este punto, el autor argumenta en contra de la idea según la cual una perspectiva descriptiva del derecho puede, y debe, tratar de responder a los problemas generales de la normatividad del derecho. De acuerdo con el autor, una perspectiva que pretenda dar respuesta a los problemas de la normatividad del derecho no puede ser una perspectiva puramente descriptiva. En este sentido, el autor sostiene que la respuesta que ofrece Enoch es la única respuesta que una perspectiva descriptiva del derecho puede dar. La segunda crítica está dirigida a la respuesta que Rodriguez-Blanco da sobre la paradoja de la normatividad. En este segundo punto el autor argumenta que la perspectiva de Rodriguez-Blanco no es una perspectiva descriptiva como ella presume, sino que adopta una perspectiva normativa la cual, además, está comprometida con una idea perfeccionista del derecho. (shrink)

This paper resolves a paradox concerning colour constancy. On the one hand, our intuitive, pre-theoretical concept holds that colour constancy involves invariance in the perceived colours of surfaces under changes in illumination. On the other, there is a robust scientific consensus that colour constancy can persist in cerebral achromatopsia, a profound impairment in the ability to perceive colours. The first stage of the solution advocates pluralism about our colour constancy capacities. The second details the close relationship between colour constancy and (...) contrast. The third argues that achromatopsics retain a basic type of colour constancy associated with invariants in contrast processing. The fourth suggests that one person-level, conscious upshot of such processing is the visual awareness of chromatic contrasts ‘at’ the edges of surfaces, implicating the ‘colour for form’ perceptual function. This primitive type of constancy sheds new light on our most basic perceptual capacities, which mark the lower borders of representational mind. (shrink)

Conscientious objection in health care is a form of compromise whereby health care practitioners can refuse to take part in safe, legal, and beneficial medical procedures to which they have a moral opposition (for instance abortion). Arguments in defense of conscientious objection in medicine are usually based on the value of respect for the moral integrity of practitioners. I will show that philosophical arguments in defense of conscientious objection based on respect for such moral integrity are extremely weak and, if (...) taken seriously, lead to consequences that we would not (and should not) accept. I then propose that the best philosophical argument that defenders of conscientious objection in medicine can consistently deploy is one that appeals to (some form of) either moral relativism or subjectivism. I suggest that, unless either moral relativism or subjectivism is a valid theory—which is exactly what many defenders of conscientious objection (as well as many others) do not think—the role of moral integrity and conscientious objection in health care should be significantly downplayed and left out of the range of ethically relevant considerations. (shrink)

My aim in this paper is to develop a unified solution to two paradoxes of bounded rationality. The first is the regress problem that incorporating cognitive bounds into models of rational decisionmaking generates a regress of higher-order decision problems. The second is the problem of rational irrationality: it sometimes seems rational for bounded agents to act irrationally on the basis of rational deliberation. I review two strategies which have been brought to bear on these problems: the way of weakening (...) which responds by weakening rational norms, and the way of indirection which responds by letting the rationality of behavior be determined by the rationality of the deliberative processes which produced it. Then I propose and defend a third way to confront the paradoxes: the way of level separation. (shrink)

Some philosophers have argued that Aristotle’s view of habituation gives rise to a ‘paradox of moral education.’ The inculcation of habit, they contend, seems antithetical to the cultivation of virtue. I argue that this alleged paradox arises from significant misunderstandings of Aristotle’s view. Habit formation need not be at odds with the development of the kinds of intelligent, reflective capacities required for virtue. Indeed, Aristotle seems right to insist on an important role for habit in the cultivation of virtue. I (...) suggest that habit formation is part of the story of how the virtuous come to see the world aright. (shrink)

According to a widely held view epistemic justification is a normative notion. According to another widely held assumption, epistemic justification comes in degrees. Given that gradability requires a context-sensitivity that normativity seems to lack, these two assumptions stand in tension. Giving up the assumption of gradability of justification represents a lesser theoretical cost.

This book is my gift to Albert Einstein on the occasion of his 142nd birthday - and is also a gift to everybody in the world he helped to shape! -/- My book adopts the view that the universe is infinite and eternal - but scientifically created. This paradox of creating eternity depends on the advanced electronics developed by future humanity. Those humans will develop time travel, plus programs that use "imaginary" time and infinite numbers like pi. They'll also become (...) the El or Elohim (names used by various religions to mean "God" or "the gods"). As astronomer Carl Sagan wrote in "Pale Blue Dot", "Many religions teach that it is the goal of humans to become gods." (I think that Elohim would be termed supernatural today, though their infinite abilities are actually natural outcomes of progress.) -/- A look through the book will tell you that some ideas are frequently repeated. This is because each article is meant to be understood without reading the others … so the same ideas show up in more than one. I’ve tried to stay away from jargon and equations unless they’re necessary (I find that they often make a subject harder to understand, not easier). All objects and events on Earth, in space, and in time (including the inevitability of world peace and immortality) are just one thing – strings of electronics' binary digits 1 and 0. -/- -/- TABLE OF CONTENTS -/- CHAPTER 1 - HYPOTHESIS OF QUANTUM GRAVITY -/- CHAPTER 2 - THE PHYSICIST AND THE PHILOSOPHER -/- CHAPTER 3 - ATTENTION MEDICAL DOCTORS! PHYSICS THEORY IMPLIES ALL PATIENTS CAN BE NORMALISED USING GRAVITY -/- CHAPTER 4 - SETI, EVOLUTION, AND TIME -/- CHAPTER 5 - ANYONS - ONE KEY TO UNLOCK GRAVITATIONAL-ELECTROMAGNETIC UNION, THE TOPOLOGICAL UNIVERSE, SPACE-TIME TRAVEL, FUTURE COMPUTERS, DARK MATTER/DARK ENERGY -/- CHAPTER 6 - WHAT CAUSES THE PLANETS TO SPEED UP AND SLOW DOWN WHILE THEY ARE ORBITING THE SUN -/- CHAPTER 7 - A BRIEF OUTLINE OF THE POSSIBLE BASICS OF COSMOLOGY IN THE 22nd CENTURY, AND WHAT IT MEANS FOR RELIGION -/- CHAPTER 8 - THE WORLD'S REAL-LIFE EXPERIMENTS WITH COVID-19 AND PHYSICS ALTER SOCIETY AND GLOBAL ECONOMICS -/- CHAPTER 9 - UNITING TIME -/- CHAPTER 10 - EXTRACTING ENERGY FROM BLACK HOLES -/- CHAPTER 11 - TIME TREK: THE 25TH CENTURY'S ANSWER TO STAR TREK. (shrink)