The Pinocchio paradox, devised by Veronique Eldridge-Smith in February 2001, is a counter-example to solutions to the Liar that restrict the use or definition of semantic predicates. Pinocchio’s nose grows if and only if what he is stating is false, and Pinocchio says ‘My nose is growing’. In this statement, ‘is growing’ has its normal meaning and is not a semantic predicate. If Pinocchio’s nose is growing it is because he is saying something false; otherwise, (...) it is not growing. ‘Because’ stands here for a non-semantic relation; it might be supposed to be causal or of some other nature, but it is not semantic. The paradox is discussed in relation to Tarski’s and Kripke’s theories of truth. Although the paradox is not necessarily a counter-example to a theory of a truth predicate, it is a problem for a theory of truth of the kind preserved by validity. (shrink)

We develop an argument sketched by Luna (2011) based on the Pinocchio paradox, which was proposed by Eldridge-Smith and Eldridge- Smith (2010). We show that, upon plausible assumptions, the claim that mental states supervene on bodily states leads to the conclusion that some proposition is both paradoxical and not paradoxical. In order to show how the presence of paradoxes can be harnessed for philosophical argumentation, we present as well a couple of related arguments.

The Pinocchio paradox poses one dialetheia too many for semantic dialetheists (Eldridge-Smith 2011). However, Beall (2011) thinks that the Pinocchio scenario is merely an impossible story, like that of the village barber who shaves just those villagers who do not shave themselves. Meanwhile, Beall maintains that Liar paradoxes generate dialetheia. The Barber scenario is self-contradictory, yet the Pinocchio scenario requires a principle of truth for a contradiction. In this and other respects the Pinocchio paradox (...) is a version of the Liar, unlike the Barber. One wonders why some Liars would be impossible if others generate dialetheias. (shrink)

The Liar paradox is an obstacle to a theory of truth, but a Liar sentence need not contain a semantic predicate. The Pinocchio paradox, devised by Veronique Eldridge-Smith, was the first published paradox to show this. Pinocchio’s nose grows if, and only if, what Pinocchio is saying is untrue. What happens if Pinocchio says that his nose is growing? Eldridge-Smith and Eldridge-Smith : 212-5, 2010) posed the Pinocchio paradox against the Tarskian-Kripkean (...) solutions to the Liar paradox that use language hierarchies. Eldridge-Smith : 306-8, 2011) also set the Pinocchio paradox against semantic dialetheic solutions to the Liar. Beall argued the Pinocchio story was just an impossible story. Eldridge-Smith : 749-752, 2012b) responded that unless the T-schema is a necessary truth of some sort, the Pinocchio principle is possible. Luna : 77–86, 2016) argues that the Pinocchio contradiction proves the principle is false. D’Agostini & Ficara discuss a more plausible physical truth-tracking trait, the Blushing Liar, and argue that the Pinocchio contradiction is not a metaphysical dialetheia. I respond to Luna, and D’Agostini & Ficara, and prove that the Pinocchio paradox is a counterexample to hierarchical solutions to the Liar. (shrink)

Semantic dialetheists astutely dodge Explosion, the logical contagion of everything being true if a single contradiction is true. A dialetheia is contained in their semantics, and sustained by a paraconsistent logic. Graham Priest has shown that this is a solution to the Liar paradox. I use the Pinocchio paradox, devised by Veronique Eldridge-Smith, as a counter-example. The Pinocchio paradox turns on the truth of Pinocchio, whose nose grows if and only if what he is (...) saying is not true, saying ‘My nose is growing’. It is not just a matter of interpretation whether Pinocchio’s nose is and is not growing. (shrink)

This paper argues that, contrary to P. Eldridge-Smith, the so-called Pinocchio paradox affords no argument against ‘simply semantic dialetheism’.

A more provocative subject than “lying in politics” is difficult to imagine. Everybody, from the proverbial “Joe Sixpack” to ivory-tower philosophers, can wax eloquently on the subject, if only because easy-to-find, shocking examples abound. If moral outrage were judged an essential vitamin, then condemning dishonesty undoubtedly guarantees a daily megadose. Unfortunately, at least for those who crave self-indulgent outrage, the anti-lying case is less than 100 percent compelling. It is a quagmire of the first order, if only because those who (...) cherish frankness also usually confess to lying. Hannah Arendt once suggested that lies are a necessary and justifiable tool of the statesman's trade. Formulating damnation criteria invites mind-boggling paradoxes, and strident defenders of truth-telling, with scant exception, admit that falsehoods are “sometimes” permitted for “good” reasons. And what might be these “good reasons”? Who can say for sure? Centuries of erudite scholarship on this point might be encapsulated as “Lying to me is bad, but I can consent to deceiving others for noble purposes, as artfully decided by myself.”. (shrink)

Godel's and Tarski's theorems were inspired by paradoxes: the Richard paradox, the Liar. Godel, in the 1951 Gibbs lecture argued from his metatheoretical results for a metaphysical claim: the impossibility of reducing, both, mathematics to the knowable by the human mind and the human mind to a finite machine (e.g. the brain). So Godel reasoned indirectly from paradoxes for metaphysical theses. I present four metaphysical theses concerning mechanism, reductive physicalism and time for the only purpose of suggesting how it (...) could be argued for them directly from paradoxical sentences. (shrink)

Suppose a person blushes iff what she says is false and she says: ‘I am blushing’. If she blushes, she doesn’t, and if she doesn’t, she does. This Blushing Liar is a new paradox, similar in some respects to the Pinocchio Paradox : Pinocchio’s nose grows iff he says some falsity, and he says: ‘my nose is growing’. Both paradoxes involve physical properties, and both, supposedly, confirm the existence of metaphysical dialetheias. In the paper, we note (...) first that while PP relies on the rather implausible scenario of a wooden puppet whose nose grows iff he is lying, so it is debunked by the objection of fictionality, BL is more plausible. Many people in our world blush, and it is not difficult to imagine someone whose saying is causally related to blushing in a similar way. A second question is whether blushing can be directly connected to falsity, without deliberate conscience, on the part of the speaker, of saying a false statement. So we explore the consequences of intending ‘I am blushing’ as a ‘lie’, in the strict meaning of the term, and we claim there is no substantial difference, but for the fact that the paradoxical effect is even more plausible. Third, we check whether BL and PP do really release metaphysical dialetheias, and we argue they do not: they lack the fundamental requisite of Liar-like contradictions, i.e. the stratification of properties. (shrink)

In this book the main emphasis is twofold: on autonomous machine intelligence, and on mind uploading. This chapter shows that, while science fiction (SF) has depicted the extreme embrace of the “prosthetic impulse,” most notoriously in Star Trek's“Borg,” this is used as a warning of the potential Faustian consequences of such tendencies. The Star Trek franchise has also highlighted the converse, the Pinocchio syndrome, a reverse prosthetic impulse, most notably in the android Commander Data. Data is a sentient android (...) and a full member of the Enterprise starship crew in Star Trek: The Next Generation. The inverse of the Pinocchio syndrome, the prosthetic impulse, is arguably exemplified by Star Trek's Borg, who constitute a relentless inhuman tide that threatens to overwhelm every species violently, assimilating all individual beings into the Borg collective, stifling their lan vital and incorporating them as part of a hive mind, with an inexorable “insect mentality”. (shrink)

Philosophers often claim that forgiveness is a paradoxical phenomenon. I here examine two of the most widespread ways of dealing with the paradoxical nature of forgiveness. One of these ways, emblematized by Aurel Kolnai, seeks to resolve the paradox by appealing to the idea of repentance. Somehow, if a wrongdoer repents, then forgiving her is no longer paradoxical. I argue that this influential position faces more problems than it solves. The other way to approach the paradox, exemplified here (...) by the work of Jacques Derrida, is just too obscure to be by itself helpful. Yet, I argue that what I take to be its spirit is on the right track. I recommend distinguishing between the definition and the justification of forgiveness, and also between forgiveness understood as a mental phenomenon and an overt, communicative act. These distinctions are not given their due in the specialized literature, and I expose the nefarious consequences of this neglect. By focusing on forgiveness as a mental phenomenon I seek to analyze the root of the talk of paradoxes which surrounds the discussion of forgiveness. Finally, I present an analysis of forgiveness as a pure mental phenomenon, and argue that this analysis is the most important step in understanding forgiveness in any other sense. While my analysis reveals interesting aspects of forgiveness, it reveals, too, that forgiveness is not quite as paradoxical after all. (shrink)

Clark and Fischer analyze social robots as interactive depictions, presenting characters that people can interact with in social settings. Unlike other types of depictions, the props for social robot depictions depend on emerging interactive technologies. This raises questions about how such depictions depict: They conflate character and prop in ways that delight, confuse, mistreat, and may become ordinary human–technology interactions.

In this article, I imagine a scene in Plato’s cave where Gilles Deleuze and Félix Guattari encounter Pinocchio, the most famous puppet. I want to examine the forms of resistance created by the expressions of the Body Without Organs during the process of Pinocchio’s formation and bring together the elements that undermine the pedagogical causality established between telling a lie and the elongation of the nose to show that a different reading of The Adventures of Pinocchio is (...) possible. I hope to achieve this reading by demonstrating the constant conflict of the two logics in The Adventures of Pinocchio: the logic of punishment and the logic of symptoms. This research can also be read as preparing a possible strategic escape plan for Pinocchio, Deleuze and Guattari for a permanent escape from Plato’s cave. (shrink)

Logical paradoxes – like the Liar, Russell's, and the Sorites – are notorious. But in Paradoxes and Inconsistent Mathematics, it is argued that they are only the noisiest of many. Contradictions arise in the everyday, from the smallest points to the widest boundaries. In this book, Zach Weber uses “dialetheic paraconsistency” – a formal framework where some contradictions can be true without absurdity – as the basis for developing this idea rigorously, from mathematical foundations up. In doing so, Weber directly (...) addresses a longstanding open question: how much standard mathematics can paraconsistency capture? The guiding focus is on a more basic question, of why there are paradoxes. Details underscore a simple philosophical claim: that paradoxes are found in the ordinary, and that is what makes them so extraordinary. (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.

The paradox of knowability, derived from a proof by Frederic Fitch in 1963, is one of the deepest paradoxes concerning the nature of truth. Jonathan Kvanvig argues that the depth of the paradox has not been adequately appreciated. It has long been known that the paradox threatens antirealist conceptions of truth according to which truth is epistemic. If truth is epistemic, what better way to express that idea than to maintain that all truths are knowable? In the (...) face of the paradox, however, such a characterization threatens to undermine antirealism. If Fitch's proof is valid, then one can be an antirealist of this sort only by endorsing the conclusion of the proof that all truths are known.Realists about truth have tended to stand on the sidelines and cheer the difficulties faced by their opponents from Fitch's proof. Kvanvig argues that this perspective is wholly unwarranted. He argues that there are two problems raised by the paradox, one that threatens antirealism about truth and the other that threatens everybody's view about truth, realist or antirealist. The problem facing antirealism has had a number of proposed solutions over the past 40 years, and the results have not been especially promising with regard to the first problem. The second problem has not even been acknowledged, however, and the proposals regarding the first problem are irrelevant to the second problem. This book thus provides a thorough investigation of the literature on the paradox, and also proposes a solution to the deeper of the two problems raised by Fitch's proof. It provides a complete picture of the paradoxicality that results from Fitch's proof, and presents a solution to the paradox that claims to address both problems raised by the original proof. (shrink)

This essential guide to paradoxes takes the reader on a lively tour of puzzles that have taxed thinkers from Zeno to Galileo and Lewis Carroll to Bertrand Russell. Michael Clark uncovers an array of conundrums, such as Achilles and the Tortoise, Theseus' Ship, Hempel's Raven, and the Prisoners' Dilemma, taking in subjects as diverse as knowledge, ethics, science, art and politics. Clark discusses each paradox in non-technical terms, considering its significance and looking at likely solutions.

_Paradoxes from A to Z, Third edition_ is the essential guide to paradoxes, and takes the reader on a lively tour of puzzles that have taxed thinkers from Zeno to Galileo, and Lewis Carroll to Bertrand Russell. Michael Clark uncovers an array of conundrums, such as Achilles and the Tortoise, Theseus’ Ship, and the Prisoner’s Dilemma, taking in subjects as diverse as knowledge, science, art and politics. Clark discusses each paradox in non-technical terms, considering its significance and looking at (...) likely solutions. This third edition is revised throughout, and adds nine new paradoxes that have important bearings in areas such as law, logic, ethics and probability. Paradoxes from A to Z, Third edition is an ideal starting point for those interested not just in philosophical puzzles and conundrums, but anyone seeking to hone their thinking skills. (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)

We begin by showing that every theory of the value of uncertain prospects must have one of three unpalatable properties. _Reckless_ theories recommend giving up a sure thing, no matter how good, for an arbitrarily tiny chance of enormous gain; _timid_ theories permit passing up an arbitrarily large potential gain to prevent a tiny increase in risk; _non-transitive_ theories deny the principle that, if A is better than B and B is better than C, then A must be better than (...) C. Having set up this trilemma, we study its horns. Non-transitivity has been much discussed; we focus on drawing out the costs and benefits of recklessness and timidity when it comes to axiology, decision theory, and normative uncertainty. (shrink)

This short article pairs the realms of “Mathematics”, “Philosophy”, and “Poetry”, presenting some corners of intersection of this type of scientocreativity. Poetry have long been following mathematical patterns expressed by stern formal restrictions, as the strong metrical structure of ancient Greek heroic epic, or the consistent meter with standardized rhyme scheme and a “volta” of Italian sonnets. Poetry was always connected to Philosophy, and further on, notable mathematicians, like the inventor of quaternions, William Rowan Hamilton, or Ion Barbu, the creator (...) of the Barbilian spaces, have written appreciated poems. We will focus here on an avant-garde movement in literature, art, philosophy, and science, called Paradoxism, founded in Romania in 1980 by a mathematician, philosopher and poet, and on the laboured writing exercises of the Oulipo group, founded in Paris in 1960 by mathematicians and poets, both of them still in act. (shrink)

Benardete paradoxes involve infinite collections of Grim Reapers, assassins, demons, deafening peals, or even sentences. These paradoxes have recently been used in arguments for finitist metaphysical theses such as temporal finitism, causal finitism, and discrete views of time. Here we develop a new _finite_ Benardete-like paradox. We then use this paradox to defend a companions in guilt argument that challenges recent applications of patchwork principles on behalf of the aforementioned finitist arguments. Finally, we develop another problem for those (...) applications by examining the notion of exact duplication. (shrink)

Chinese Chan or Zen Buddhism is renowned for its improvisational, atypical, and perplexing use of words. In particular, the tradition’s encounter dialogues, which took place between Chan masters and their interlocutors, abound in puzzling, astonishing, and paradoxical ways of speaking. In this chapter, we are concerned with Chan’s use of paradoxical language. In philosophical parlance, a linguistic paradox comprises the confluence of opposite or incongruent concepts in a way that runs counter to our common sense and ordinary rational thinking. (...) One naturally wonders about Chan masters’ rationales for their use of paradox. There are also concerns about whether the use violates the logical principle of noncontradiction to the effect that nothing can be both P and not-P all over in the same way at the same time. Chan became a viable Chinese Buddhist tradition during the Tang dynasty (618−907) and continued to develop for several centuries. The tradition had produced a huge literature; consequently, our investigation of its use of paradox cannot but be limited and selective. In the second section, I first sketch key ideas of Chan that are pertinent to our investigation and then examine the use of paradox in the sermons associated with certain Tang masters of the southern Chan. In the third section, I analyze the presence of paradoxical language in post-Tang encounter dialogues. The fourth section concludes. (shrink)

It appears to be a feature of our ordinary understanding of morality that we ought not to act in certain ways at all. We ought not to kill, torture, deceive, break our promises (say)—exceptional circumstances apart. Many moral duties are thought of in this way. Killing another person would be wrong even if it achieved a great good, and even if it led to preventing the deaths of several others. This feature of moral thinking is at the core of deontological (...) ethics. But while it is also part and parcel of ordinary moral thought, it seems puzzling, because it requires an agent to act (or to abstain from acting) in certain ways, even if by violating the duty she could achieve a better outcome in terms of the very restriction that she is required to heed. If the explanation of the duty not to kill were that it preserves human life it would be hard to understand why killing is not morally permitted when it leads to preserving a greater number of lives. Moreover, why should we not violate a restriction if doing so leads to fewer violations of that very restriction? There are two questions here. (1) Why not violate a restriction whose point it is to preserve lives, if doing so leads to preserving a greater number of lives? (2) Why not violate a restriction if doing so minimizes the violation of that very restriction? The latter question concerns what is sometimes called ‘the paradox of deontology’ or the problem of ‘minimizing violations’. The former may concern the so-called trolley cases, for instance. After all, a trolley hurtling towards a number of innocent people is not about to violate the moral restriction on killing innocent people, as it is not the kind of thing that can be subject to such a restriction. In this paper I am concerned only with (2): the paradox of deontology. To put it in Nozick’s terms: ‘How can a concern for the nonviolation of C [i.e. some deontological constraint] lead to refusal to violate C even when this would prevent other more extensive violations of C? -/- Samuel Scheffler, puzzled by the same idea, put it in terms of ‘agent-centred restrictions’ (ACR), because the restriction requires an agent (not) to act in certain ways, contrasting thereby with a requirement that certain actions not be done. The restriction is thus an agent-relative rather than an agent-neutral one. Scheffler ended up rejecting the idea of agent-centred restrictions, as even on close investigation the puzzle would not dissolve. Scheffler maintains that ACRs have ‘an air of paradox’ to them, as they violate what he calls maximizing rationality. ACRs appear to clash with the idea of maximizing rationality, because they require an agent to choose the option that worse achieves her goal of not harming people. I call this, following common parlance, the paradox of deontology, even though it is not strictly speaking a paradox. Exploring whether there is really a problem here, and how it might be solved, is the main objective of this paper. I begin by explaining the puzzlement about agent-centred restrictions in more detail. I then discuss promising as an example of a kind of action which is thought to be subject to agent-centred restrictions. It turns out that they are surprisingly easy to explain. The difficult case is killing—and that should put us on guard. I argue that the problem is not with the very idea of deontological restrictions, but with puzzling aspects of the ethics of killing. (shrink)

Aside from bioethics, the main theme of Ronald Green's lifework has been an exploration of the relation between religion and morality, with special emphasis on the philosophies of Immanuel Kant and Søren Kierkegaard. This essay summarizes and assesses his work on this theme by examining, in turn, four of his relevant books. Religious Reason (1978) introduced a new method of comparative religion based on Kant's model of a rational religion. Religion and Moral Reason (1988) expanded on this project, clarifying that (...) religious traditions cannot be reduced to their moral grounding. Kierkegaard and Kant: The Hidden Debt (1992) offered bold new evidence that Kant, not Hegel, was the philosopher whose ideas primarily shaped Kierkegaard's overtly religious philosophy ; both philosophers focused on the problem of how to understand the relation between moral reasoning and historical religion. And Kant and Kierkegaard on Time and Eternity (2011) republished ten essays that explore various aspects of this theme in greater depth. I argue that throughout these works Green defends a " paradox of inwardness " : principles or ideals that are by their nature essentially inward end up requiring outward manifestation in order to be confirmed or fully justified as real. (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.

Schizophrenia as a pathology of self-awareness has attracted much attention from philosophical theorists and empirical scientists alike. I view schizophrenia as a basic self-disturbance leading to a lifeworld of solipsism adopted by the sufferer and explain how this adoption takes place, which then manifests in ways such as first-rank psychotic symptoms. I then discuss the relationships between these symptoms, not as isolated mental events, but as end-products of a loss of agency and ownership, and argue that symptoms like thought insertion (...) and other ego-boundary disorders are by nature a multitude of paradoxes created by a fragmented awareness. I argue that such fragmentation does not always require or lead to a delusional elaboration as the definitive feature of its phenomenology, and present reasons for the role of the first-person pronoun as a mere metaphor used to represent the patient’s bizarre experiences where sensory perception and thinking processes converge. Further, I discuss the initi... (shrink)

Does traditional Christianity involve paradoxical doctrines, that is, doctrines that present the appearance (at least) of logical inconsistency? If so, what is the nature of these paradoxes and why do they arise? What is the relationship between "paradox" and "mystery" in theological theorizing? And what are the implications for the rationality, or otherwise, of orthodox Christian beliefs? In Paradox in Christian Theology, James Anderson argues that the doctrines of the Trinity and the incarnation, as derived from Scripture and (...) formulated in the ecumenical creeds, are indeed paradoxical. But this conclusion, he contends, need not imply that Christians who believe these doctrines are irrational in doing so. In support of this claim, Anderson develops and defends a model of understanding paradoxical Christian doctrines according to which the presence of such doctrines is unsurprising and adherence to paradoxical doctrines cannot be considered as a serious intellectual obstacle to belief in Christianity. The case presented in this book has significant implications for the practice of systematic theology, biblical exegesis, and Christian apologetics. (shrink)

This article shows that there is a liar-like paradox that arises for rational credence that relies only on very weak logical and credal principles. The paradox depends on a weak rational reflection principle, logical principles governing conjunction, and principles governing the relationship between rational credence and proof. To respond to this paradox, we must either reject even very weak rational reflection principles or reject some highly plausible logical or credal principle.

Smith argues that, unlike other forms of evidence, naked statistical evidence fails to satisfy normic support. This is his solution to the puzzles of statistical evidence in legal proof. This paper focuses on Smith’s claim that DNA evidence in cold-hit cases does not satisfy normic support. I argue that if this claim is correct, virtually no other form of evidence used at trial can satisfy normic support. This is troublesome. I discuss a few ways in which Smith can respond.

Explores humanity's biological roots and astounding capacity for self-creation and self-alteration, examining freedom and choice, sexuality and love, conscience and justice, work and pleasure, and other issues.

Husserl’s first work formulated what proved to be an algorithmically complete arithmetic, lending mathematical clarity to Kronecker’s reduction of analysis to finite calculations with integers. Husserl’s critique of his nominalism led him to seek a philosophical justification of successful applications of symbolic arithmetic to nature, providing insight into the “wonderful affinity” between our mathematical thoughts and things without invoking a pre-established harmony. For this, Husserl develops a purely descriptive phenomenology for which he found inspiration in Mach’s proposal of a “universal (...) physical phenomenology.” To account for applications to any domain, Husserl envisages a theory of all possible deductive systems, which he develops extensively in his Göttingen lectures wherein he engages with Hilbert’s work on deductive systems for geometry, real arithmetic, and physics. This leads Husserl to formulate claims of decidability and proofs of completeness for various arithmetics that result from his analysis of Kronecker’s general arithmetic. Careful attention to these proofs seem to show that Husserl was not oblivious to problems that underlie our incompleteness theorems, namely that of showing that some inversions of his algorithmic arithmetic are undefined. His growing preoccupation with the issue of a pre-established harmony between mathematical thought and reality motivate his pursuit of a “supramathematics” of all possible complete theory forms to demystify such harmony, by having such a form on hand for describing any empirical domain. He soon decides that a transcendental idealism of nature will reveal the wonderful affinity of thoughts and things comprising such harmony, to be a wonderful “parallelism of objective unities and constituted manifolds of consciousness.” But the paradoxes of logic and set theory cloud the clarity of mathematics, which Weyl would restore with Brouwer’s intuitionism and Hilbert with his metamathematics. Husserl informed Weyl that his student Becker had formulated a phenomenological foundation not only for Weyl’s generalization of relativity theory but also for the Brouwer-Weyl continuum. But Weyl eventually rejected much of Becker’s work, especially when it became clear that his phenomenological intuitionism could not account for the success of Hilbert’s transfinite mathematics in quantum physics. Becker responded to this “crisis of phenomenological method” with his mantic phenomenology celebrating the magic of mathematical mysticism, which Husserl finally rejects in favor of a pluralistic phenomenology of mathematics and nature. (shrink)

The optimal-pessimal paradox (Bradley, 1975) has been criticized on bizarre grounds by Childs (1980). Assumptions that it never made were attributed to it and attacked. Empirical evidence for its existence (which occupied a large portion of the criticized article) was totally ignored, and it was treated as a mere theoretically based conjecture. Originally proposed solutions to the problems it presents were dismissed and replaced by ineffectual alternatives. In spite of Childs’ claim that “the paradox, although theoretically sound, is (...) grounded upon assumptions that are empirically untenable,” the paradox makes no such assumptions and is an empirical fact. (shrink)