For more than fifty years, taxonomists have proposed numerous alternative definitions of species while they searched for a unique, comprehensive, and persuasive definition. This monograph shows that these efforts have been unnecessary, and indeed have provably been a pursuit of a will o’ the wisp because they have failed to recognize the theoretical impossibility of what they seek to accomplish. A clear and rigorous understanding of the logic underlying species definition leads both to a recognition of the inescapable ambiguity that (...) affects the definition of species, and to a framework-relative approach to species definition that is logically compelling, i.e., cannot not be accepted without inconsistency. An appendix reflects upon the conclusions reached, applying them in an intellectually whimsical taxonomic thought experiment that conjectures the possibility of an emerging new human species. (shrink)

The elimination of ambiguity and redundancy are unquestioned goals in the exact sciences, and yet, as this paper shows, there are inescapable lower bounds that constrain our wish to eliminate them. The author discusses contributions by Richard Hamming (inventor of the Hamming code) and Satosi Watanabe (originator of the Theorems of the Ugly Duckling). Utilizing certain of their results, the author leads readers to recognize the unavoidable, central roles in effective communication, of redundancy, and of ambiguity of meaning, (...) reference, and identification. (shrink)

According to Satosi Watanabe's "theorem of the ugly duckling", the number of predicates satisfied by any two different particulars is a constant, which does not depend on the choice of the two particulars. If the number of predicates satisfied by two particulars is their number of properties in common, and the degree of resemblance between two particulars is a function of their number of properties in common, then it follows that the degree of resemblance between any two (...) different particulars is also constant, which is absurd. Avoiding this absurd conclusion requires questioning assumptions about infinity in the proof or interpretation of the theorem, adopting a sparse conception of properties, or denying degree of resemblance is a function of number of properties in common. After arguing against both the first two options, this paper argues for a version of the third which analyses degree of resemblance as a function of properties in common, but weighted by their degree of naturalness or importance. (shrink)

Kant’s theory of taste, as expounded in the Critique of Judgment, deals exhaustively with judgments of beauty. Rarely does Kant mention ugliness. This omission has led to a debate among commentators about how judgments of ugliness should be explained in a Kantian framework. I argue that the judgment of ugliness originates in the disharmonious play between the faculties of imagination and understanding. Such disharmony occurs when the understanding finds that it cannot in principle form any concept suitable to a representation (...) as it is presented by the imagination. (shrink)

The alt-right claims it responsibly advocates for its positions while the Ku Klux Klan was “ad-hoc.” This allows them to accept the philosophy of white nationalism while rejecting comparisons with prior white nationalist organizations. I confront this by comparing the methodologies of alt-right trolls and the KKK. After studying each movement’s genesis in pranks done for amusement, I demonstrate that each uses threats to police behavior, encourages comradery around ethnic heritage, and manipulates politics to exclude voices from public deliberation. Differences (...) between alt-right trolls and the KKK originate in the technologies they use, not out of a concern for responsible advocacy. (shrink)

Representation theorems are often taken to provide the foundations for decision theory. First, they are taken to characterize degrees of belief and utilities. Second, they are taken to justify two fundamental rules of rationality: that we should have probabilistic degrees of belief and that we should act as expected utility maximizers. We argue that representation theorems cannot serve either of these foundational purposes, and that recent attempts to defend the foundational importance of representation theorems are unsuccessful. As a result, we (...) should reject these claims, and lay the foundations of decision theory on firmer ground. (shrink)

I offer the first sustained defence of the claim that ugliness is constituted by the disposition to disgust. I advance three main lines of argument in support of this thesis. First, ugliness and disgustingness tend to lie in the same kinds of things and properties (the argument from ostensions). Second, the thesis is better placed than all existing accounts to accommodate the following facts: ugliness is narrowly and systematically distributed in a heterogenous set of things, ugliness is sometimes enjoyed, and (...) ugliness sits opposed to beauty across a neutral midpoint (the argument from proposed intensions). And third, ugliness and disgustingness function in the same way in both giving rise to representations of contamination (the argument from the law of contagion). In making these arguments, I show why prominent objections to the thesis do not succeed, cast light on some of the artistic functions of ugliness, and, in addition, demonstrate why a dispositional account of disgustingness is correct, and present a novel problem for warrant-based accounts of disgustingness (the ‘too many reasons’ problem). (shrink)

In this paper, we present two proofs of the reciprocal butterfly theorem. The statement of the butterfly theorem is: Let us consider a chord PQ of midpoint M in the circle Ω(O). Through M, two other chords AB and CD are drawn, such that A and C are on the same side of PQ. We denote by X and U the intersection of AD respectively CB with PQ. Consequently, XM = YM. For the proof of this theorem, (...) see [1]. The reciprocal of the butterfly theorem has the following statement: In the circle Ω(O), let us consider the chords PQ, AB and CD which are concurrent in the point M≠O, such as the points A and C are on the same side of the line PQ. Let X and Y respectively be the intersections of the chord PQ with AD and BC respectively. If XM = YM, then M is the middle of the chord PQ. (shrink)

In arguing for his theory of pure reflective judgments of taste Kant extensively analyses beauty, but almost wholly disregards ugliness. We commonly take ugliness as paradigmatic when we reflect on our negative aesthetic judgments, and so does Kant. Consequently, there ought to be a more explicit story explaining how Kantian judgments of ugliness are possible. In this paper I argue that a disharmony is the key to understanding Kantian ugliness. This way, an answer to the question of ugliness in Kant (...) can be given in terms of a disharmonious free play of the faculties of imagination and understanding. (shrink)

The problem of multiple-computations discovered by Hilary Putnam presents a deep difficulty for functionalism (of all sorts, computational and causal). We describe in out- line why Putnam’s result, and likewise the more restricted result we call the Multiple- Computations Theorem, are in fact theorems of statistical mechanics. We show why the mere interaction of a computing system with its environment cannot single out a computation as the preferred one amongst the many computations implemented by the system. We explain why (...) nonreductive approaches to solving the multiple- computations problem, and in particular why computational externalism, are dualistic in the sense that they imply that nonphysical facts in the environment of a computing system single out the computation. We discuss certain attempts to dissolve Putnam’s unrestricted result by appealing to systems with certain kinds of input and output states as a special case of computational externalism, and show why this approach is not workable without collapsing to behaviorism. We conclude with some remarks about the nonphysical nature of mainstream approaches to both statistical mechanics and the quantum theory of measurement with respect to the singling out of partitions and observables. (shrink)

Following a long-standing philosophical tradition, impartiality is a distinctive and determining feature of moral judgments, especially in matters of distributive justice. This broad ethical tradition was revived in welfare economics by Vickrey, and above all, Harsanyi, under the form of the so-called Impartial Observer Theorem. The paper offers an analytical reconstruction of this argument and a step-wise philosophical critique of its premisses. It eventually provides a new formal version of the theorem based on subjective probability.

A fundamental problem in science is how to make logical inferences from scientific data. Mere data does not suffice since additional information is necessary to select a domain of models or hypotheses and thus determine the likelihood of each model or hypothesis. Thomas Bayes’ Theorem relates the data and prior information to posterior probabilities associated with differing models or hypotheses and thus is useful in identifying the roles played by the known data and the assumed prior information when making (...) inferences. Scientists, philosophers, and theologians accumulate knowledge when analyzing different aspects of reality and search for particular hypotheses or models to fit their respective subject matters. Of course, a main goal is then to integrate all kinds of knowledge into an all-encompassing worldview that would describe the whole of reality. A generous description of the whole of reality would span, in the order of complexity, from the purely physical to the supernatural. These two extreme aspects of reality are bridged by a nonphysical realm, which would include elements of life, man, consciousness, rationality, mental and mathematical abstractions, etc. An urgent problem in the theory of knowledge is what science is and what it is not. Albert Einstein’s notion of science in terms of sense perception is refined by defining operationally the data that makes up the subject matter of science. It is shown, for instance, that theological considerations included in the prior information assumed by Isaac Newton is irrelevant in relating the data logically to the model or hypothesis. In addition, the concepts of naturalism, intelligent design, and evolutionary theory are critically analyzed. Finally, Eugene P. Wigner’s suggestions concerning the nature of human consciousness, life, and the success of mathematics in the natural sciences is considered in the context of the creative power endowed in humans by God. (shrink)

Political theorists study the attributes of desirable social-moral states of affairs. Schaefer (forthcoming) aims to show that "static political theory" of this kind rests on shaky foundations. His argument revolves around an application of an abstruse mathematical theorem -- Kakutani's fixed point theorem -- to the social-moral domain. We show that Schaefer has misunderstood the implications of this theorem for political theory. Theorists who wish to study the attributes of social-moral states of affairs should carry on, safe (...) in the knowledge that Kakutani's theorem poses no threat to this enterprise. (shrink)

Central limit theorem for the functional of jump Markov process. Nguyễn Văn Hữu, Vương Quân Hoàng và Trần Minh Ngọc. Báo cáo: Hội nghị toàn quốc lần thứ III “Xác suất - Thống kê: Nghiên cứu, ứng dụng và giảng dạy” (tr. 34). Ba Vì, Hà Tây, ngày 12-14 tháng 05 năm 2005. Viện Toán học / Trường Đại học Khoa học tự nhiên / Đại học Quốc gia Hà Nội.

We show that the respective oversights in the von Neumann's general theorem against all hidden variable theories and Bell's theorem against their local-realistic counterparts are homologous. When latter oversight is rectified, the bounds on the CHSH correlator work out to be ±2√2 instead of ±2.

In this paper, we will argue that, while Big Data enthusiasts imply that the analysis of massive data sets can produce serendipitous (that is, unexpected and fortunate) discoveries, the way those models are currently designed not only does not create serendipity so easily but also frequently generates zemblanitous (that is, expected and unfortunate) findings.

Can an ugly experiment be a good experiment? Philosophers have identified many beautiful experiments and explored ways in which their beauty might be connected to their epistemic value. In contrast, the present chapter seeks out (and celebrates) ugly experiments. Among the ugliest are those being designed by AI algorithms. Interestingly, in the contexts where such experiments tend to be deployed, low aesthetic value correlates with high epistemic value. In other words, ugly experiments can be good. Given this, (...) we should conclude that beauty is not generally necessary or sufficient for epistemic value, and increasing beauty will not generally tend to increase epistemic value. (shrink)

All acknowledged proofs of the Four Colour Theorem (4CT) are computerdependent. They appeal to the existence, and manual identification, of an ‘unavoidable’ set containing a sufficient number of explicitly defined configurations—each evidenced only by a computer as ‘reducible’—such that at least one of the configurations must occur in any chromatically distinguished, putatively minimal, planar map. For instance, Appel and Haken ‘identified’ 1,482 such configurations in their 1977, computer-dependent, proof of 4CT; whilst Neil Robertson et al ‘identified’ 633 configurations as (...) sufficient in their 1997, also computer-dependent, proof of 4CT. However, treating any specific number of ‘reducible’ configurations in an ‘unavoidable’ set as sufficient entails a minimum number as necessary and sufficient. We now show that the minimum number of such configurations can only be the one corresponding to the ‘unavoidable’ set of the single, ‘reducible’, 4-sided configuration identified by Alfred Kempe in his, seemingly fatally flawed, 1879 ‘proof’ of 4CT. We shall further show that although Kempe fallaciously concluded that a 5-sided configuration was also in the ‘unavoidable’ set, and appealed to unproven properties of ‘Kempe’ chains in a graphical representation to then argue for its ‘reducibility’, neither flaw in his ‘proof’ is fatal when the argument is expressed geometrically; and that, essentially, Kempe correctly argued that any planar map which admits a chromatic differentiation with a five-sided area C that shares non-zero boundaries with four, all differently coloured, neighbours can be 4-coloured. (shrink)

According to conciliatory views about the epistemology of disagreement, when epistemic peers have conflicting doxastic attitudes toward a proposition and fully disclose to one another the reasons for their attitudes toward that proposition (and neither has independent reason to believe the other to be mistaken), each peer should always change his attitude toward that proposition to one that is closer to the attitudes of those peers with which there is disagreement. According to pure higher-order evidence views, higher-order evidence for a (...) proposition always suffices to determine the proper rational response to disagreement about that proposition within a group of epistemic peers. Using an analogue of Arrow's Impossibility Theorem, I shall argue that no conciliatory and pure higher-order evidence view about the epistemology of disagreement can provide a true and general answer to the question of what disagreeing epistemic peers should do after fully disclosing to each other the (first-order) reasons for their conflicting doxastic attitudes. (shrink)

It is a widespread intuition that the coherence of independent reports provides a powerful reason to believe that the reports are true. Formal results by Huemer, M. 1997. “Probability and Coherence Justification.” Southern Journal of Philosophy 35: 463–72, Olsson, E. 2002. “What is the Problem of Coherence and Truth?” Journal of Philosophy XCIX : 246–72, Olsson, E. 2005. Against Coherence: Truth, Probability, and Justification. Oxford University Press., Bovens, L., and S. Hartmann. 2003. Bayesian Epistemology. Oxford University Press, prove that, under (...) certain conditions, coherence cannot increase the probability of the target claim. These formal results, known as ‘the impossibility theorems’ have been widely discussed in the literature. They are taken to have significant epistemic upshot. In particular, they are taken to show that reports must first individually confirm the target claim before the coherence of multiple reports offers any positive confirmation. In this paper, I dispute this epistemic interpretation. The impossibility theorems are consistent with the idea that the coherence of independent reports provides a powerful reason to believe that the reports are true even if the reports do not individually confirm prior to coherence. Once we see that the formal discoveries do not have this implication, we can recover a model of coherence justification consistent with Bayesianism and these results. This paper, thus, seeks to turn the tide of the negative findings for coherence reasoning by defending coherence as a unique source of confirmation. (shrink)

In this contribution an attempt is made to analyze an important mathematical discovery, the theorem of Gödel, and to explore the possible impact on the consistency of metaphysical systems. It is shown that mathematics is a pointer to a reality that is not exclusively subjected to physical laws. As the Gödel theorem deals with pure mathematics, the philosopher as such can not decide on the rightness of this theorem. What he, instead can do, is evaluating the general (...) acceptance of this mathematical finding and reflect on the consistency between consequences of the mathematical theorem with consequences of his metaphysical view. The findings of three mathematicians are involved in the argumentation: first Gödel himself, then the further elaboration by Turing and finally the consequences for the human mind as worked out by Penrose. As a result one is encouraged to distinguish two different types of intellectual activity in mathematics, which both can be carried out by humans. The astonishing thing is not the distinction between a formalized, logic approach on the one side and intuition, mathematical insight and meaning on the other. Philosophically challenging, however, is the claim that principally only one of these intellectual activities can be carried out by objects exclusively bound to the laws of physical reality. (shrink)

This work describes a seminal framework of law by one of the founders of the field of law and economics, Judge Guido Calabresi. It broadens what is known as the framework of law among legal scholars, and posits a calabresi theorem which is developed and explained, in part, in comparison to the coase theorem. The framework provides policymakers a tool for creating balanced policies.

All accepted proofs of the Four Colour Theorem (4CT) are computer-dependent; and appeal to the existence, and manual identification, of an ‘unavoidable’ set containing a sufficient number of explicitly defined configurations—each evidenced only by a computer as ‘reducible’—such that at least one of the configurations must occur in any chromatically distinguished, minimal, planar map. For instance, Appel and Haken ‘identified’ 1,482 such configurations in their 1977, computer-dependent, proof of 4CT; whilst Neil Robertson et al ‘identified’ 633 configurations as sufficient (...) in their 1997, also computer-dependent, proof of 4CT. However, treating any specific number of ‘reducible’ configurations in an ‘unavoidable’ set as sufficient entails a minimum number as necessary and sufficient. We now show that the minimum number of such configurations can only be the one corresponding to the ‘unavoidable’ set of a ‘reducible’, 4-sided configuration identified by Alfred Kempe in his, seemingly fatally flawed, 1879 ‘proof’ of 4CT. Although Kempe appealed to putative properties of ‘Kempe’ chains in a graphical representation to fallaciously argue that a 5-sided configuration was also in the ‘unavoidable’ set, and ‘reducible’, we shall show why the flaw in his ‘proof’ is not fatal when the argument is expressed geometrically; and that, essentially, Kempe correctly argued that any planar map which admits a chromatic differentiation with a five-sided area C that shares non-zero boundaries with four, all differently coloured, neighbours can be 4-coloured. (shrink)

It seems like we care about at least two features of our credence function: gradational-accuracy and verisimilitude. Accuracy-first epistemology requires that we care about one feature of our credence function: gradational-accuracy. So if you want to be a verisimilitude-valuing accuracy-firster, you must be able to think of the value of verisimilitude as somehow built into the value of gradational-accuracy. Can this be done? In a recent article, Oddie has argued that it cannot, at least if we want the accuracy measure (...) to be proper. I argue that it can. 1Introduction2Some Nuts and Bolts3First Attempts4Oddie’s Constraint5The Good5.1Proximity over the disagreement metric 5.2Proximity over the magnitude metric 6The Bad and the Ugly 7Some More Good: The Role of Evenness of Distribution 8Some More Bad: Which Propositions to Privilege? 9Concluding Thoughts: Accuracy and Practical Value. (shrink)

It has been known for long time that the cosmic sound wave was there since the early epoch of the Universe. Signatures of its existence are abound. However, such an acoustic model of cosmology is rarely developed fully into a complete framework from the notion of space, cancer therapy up to the sky. This paper may be the first attempt towards such a complete description of the Universe based on classical wave equation of sound. It is argued that one can (...) arrive at a consistent description of space, elementary particles, Sachs-Wolfe acoustic theorem, up to a novel approach for cancer therapy, starting from this simple classical wave equation of sound. We also discuss a plausible extension of Acoustic Sachs-Wolfe theorem based on its analogue with Klein-Gordon equation to become Acoustic Sachs-Wolfe-Christianto-Smarandache-Umniyati (ASWoCSU) equation. It is our hope that the new proposed equation can be verified with observation data. But we admit that our model is still in its infancy, more researches are needed to fill all the missing details. (shrink)

If the concept of “free will” is reduced to that of “choice” all physical world share the latter quality. Anyway the “free will” can be distinguished from the “choice”: The “free will” involves implicitly certain preliminary goal, and the choice is only the mean, by which it can be achieved or not by the one who determines the goal. Thus, for example, an electron has always a choice but not free will unlike a human possessing both. Consequently, and paradoxically, the (...) determinism of classical physics is more subjective and more anthropomorphic than the indeterminism of quantum mechanics for the former presupposes certain deterministic goal implicitly following the model of human freewill behavior. The choice is usually linked to very complicated systems such as human brain or society and even often associated with consciousness. In its background, the material world is deterministic and absolutely devoid of choice. However, quantum mechanics introduces the choice in the fundament of physical world, in the only way, in which it can exist: All exists in the “phase transition” of the present between the uncertain future and the well-ordered past. Thus the present is forced to choose in order to be able to transform the coherent state of future into the well-ordering of past. The concept of choice as if suggests that there is one who chooses. However quantum mechanics involves a generalized case of choice, which can be called “subjectless”: There is certain choice, which originates from the transition of the future into the past. Thus that kind of choice is shared of all existing and does not need any subject: It can be considered as a low of nature. There are a few theorems in quantum mechanics directly relevant to the topic: two of them are called “free will theorems” by their authors, Conway and Kochen, and according to them: “Do we really have free will, or, as a few determined folk maintain, is it all an illusion? We don’t know, but will prove in this paper that if indeed there exist any experimenters with a modicum of free will, then elementary particles must have their own share of this valuable commodity” “The import of the free will theorem is that it is not only current quantum theory, but the world itself that is non-deterministic, so that no future theory can return us to a clockwork universe”. Those theorems can be considered as a continuation of the so-called theorems about the absence of “hidden variables” in quantum mechanics. (shrink)

I demonstrate that Bell's theorem is based on circular reasoning and thus a fundamentally flawed argument. It unjustifiably assumes the additivity of expectation values for dispersion-free states of contextual hidden variable theories for non-commuting observables involved in Bell-test experiments, which is tautologous to assuming the bounds of ±2 on the Bell-CHSH sum of expectation values. Its premises thus assume in a different guise the bounds of ±2 it sets out to prove. Once this oversight is ameliorated from Bell's argument (...) by identifying the impediment that leads to it and local realism is implemented correctly, the bounds on the Bell-CHSH sum of expectation values work out to be ±2√2 instead of ±2, thereby mitigating the conclusion of Bell's theorem. Consequently, what is ruled out by any of the Bell-test experiments is not local realism but the linear additivity of expectation values, which does not hold for non-commuting observables in any hidden variable theories to begin with. (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)

In the present paper, we prove the normalization theorem and the consistency of the first-order classical logic with disjunctive syllogism. First, we propose the natural deduction system SCD for classical propositional logic having rules for conjunction, implication, negation, and disjunction. The rules for disjunctive syllogism are regarded as the rules for disjunction. After we prove the normalization theorem and the consistency of SCD, we extend SCD to the system SPCD for the first-order classical logic with disjunctive syllogism. It (...) can be shown that SPCD is conservative extension to SCD. Then, the normalization theorem and the consistency of SPCD are given. (shrink)

This paper critically engages Philip Mirowki's essay, "The scientific dimensions of social knowledge and their distant echoes in 20th-century American philosophy of science." It argues that although the cold war context of anti-democratic elitism best suited for making decisions about engaging in nuclear war may seem to be politically and ideologically motivated, in fact we need to carefully consider the arguments underlying the new rational choice based political philosophies of the post-WWII era typified by Arrow's impossibility theorem. A distrust (...) of democratic decision-making principles may be developed by social scientists whose leanings may be toward the left or right side of the spectrum of political practices. (shrink)

Even with Brexit, the European Union will remain a market of more than 450 million people and a prominent promoter of Western values shared by Australia. Given that the EU has been Australia’s largest investor and economic partner for the past 25 years, it is pertinent to reflect on what an EU without Britain might look like.

Plato seems to countenance both positive and negative Forms, that is to say, both good and bad ones. He may not say so outright, but he invokes both and rejects neither. The apparent finality of this impression creates a lack of direct interest in the subject: Plato scholars do not give negative Forms much thought except as the prospect relates to something else they happen to be doing. Yet when they do give the matter any thought, typically for the sake (...) of a prior concern, they try either to support the textual evidence or to contradict it, indicating that the evidence does not stand on its own. The purpose of this paper is to determine why they tend to affirm or deny the obvious, how they try to confirm or dispute it, and what this says about Plato’s position. The strategic vehicle is a comparative case study. The confirmation comes from Debra Nails (2013), who needs to embrace negative Forms to demonstrate that the unhypothetical first principle of the all is not identical to the Form of the good, something she cannot do unless Plato recognizes negative Forms. The contradiction comes from Holger Thesleff (2013), who needs to reject negative (Ideal) Forms because the defining feature of his (Ideal) Forms is the possession of positive intrinsic value, which cannot be predicated of anything negative. Despite defending opposite views, or perhaps because of this, they jointly make up for any lack of interest in the scholarly community. I appreciate both yet side with Thesleff. (shrink)

Retractions play an important role in research communication by highlighting and explaining how research projects have failed and thereby preventing these mistakes from being repeated. However, the process of retraction and the data it produces is often sparse or incomplete. Drawing on evidence from 2046 retraction records, Quan-Hoang Vuong discusses the emerging trends this data highlights and argues for the need to enforce reporting standards for retractions, as a means of de-stigmatising retraction and rewarding practising integrity in the scholarly record.

This note clarifies an error in the proof of the main theorem of “The Ricean Objection: An Analogue of Rice’s Theorem for First-Order Theories”, Logic Journal of the IGPL, 16(6): 585–590(2008).

In the recent literature on the issue, a number of commentators have argued that Kant’s aesthetic theory commits him to the position that nothing is ugly. For instance, in ‘Why Kant finds nothing ugly’, Shier argues that ‘within Kant’s aesthetics, there cannot be any negative judgments of taste’ (Shier (1998): 413). And in ‘Kant’s problems with ugliness’, Thomson claims that ‘Kant’s aesthetic theory precludes […] ugliness’ (Thomson (1992): 107). In other words, as it is presented in some of (...) the literature, Kant’s account of aesthetic judgment seems to preclude him from accounting for the possibility of ugliness. A number of reasons have been put forward to explain why there can be no ugliness for Kant. Some have to do with his account of experience in general. Others have to do with the very nature of the free play between imagination and understanding. Of course, the real source of the problem is that the Critique of Judgment does not actually discuss ugliness, so commentators are left with the task of filling in the gaps. This paper aims to argue that Kant’s account of aesthetic judgment commits him not only to the existence of the ugly, but to the distinction between two kinds of ugliness. To support this claim, I will start from the premise that if there is to be room for ugliness, it should be defined as the contrary of beauty. However, I will suggest that Kant’s account of beauty can be negated in two ways, one that gives rise to impure ugliness and the other to pure ugliness. I will examine them in turn, first by showing that impure ugliness is the contrary of beauty insofar as it contravenes our interests and produces a desire to dispose of it, whereas beauty is necessarily disinterested for Kant. In the second section, I will use Kant’s definition of judgments of pure beauty to carve a space for an account of pure ugliness. For, it should present all the characteristics that make it “pure” whilst replacing the characteristics that make it beautiful with the ones that make it ugly. On this basis, I will define pure ugliness as the disinterested displeasure caused by the experience of what I will call “foul play” between the imagination and the understanding. (shrink)

Condorcet's famous jury theorem reaches an optimistic conclusion on the correctness of majority decisions, based on two controversial premises about voters: they are competent and vote independently, in a technical sense. I carefully analyse these premises and show that: whether a premise is justi…ed depends on the notion of probability considered; none of the notions renders both premises simultaneously justi…ed. Under the perhaps most interesting notions, the independence assumption should be weakened.

This paper makes a novel linkage between the multiple-computations theorem in philosophy of mind and Landauer’s principle in physics. The multiple-computations theorem implies that certain physical systems implement simultaneously more than one computation. Landauer’s principle implies that the physical implementation of “logically irreversible” functions is accompanied by minimal entropy increase. We show that the multiple-computations theorem is incompatible with, or at least challenges, the universal validity of Landauer’s principle. To this end we provide accounts of both ideas (...) in terms of low-level fundamental concepts in statistical mechanics, thus providing a deeper understanding of these ideas than their standard formulations given in the high-level terms of thermodynamics and cognitive science. Since Landauer’s principle is pivotal in the attempts to derive the universal validity of the second law of thermodynamics in statistical mechanics, our result entails that the multiple-computations theorem has crucial implications with respect to the second law. Finally, our analysis contributes to the understanding of notions, such as “logical irreversibility,” “entropy increase,” “implementing a computation,” in terms of fundamental physics, and to resolving open questions in the literature of both fields, such as: what could it possibly mean that a certain physical process implements a certain computation. (shrink)

Classical interpretations of Goedels formal reasoning, and of his conclusions, implicitly imply that mathematical languages are essentially incomplete, in the sense that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is, both, non-algorithmic, and essentially unverifiable. However, a language of general, scientific, discourse, which intends to mathematically express, and unambiguously communicate, intuitive concepts that correspond to scientific investigations, cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth (...) verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Goedel's reasoning, under which any formal system of Peano Arithmetic---classically accepted as the foundation of all our mathematical Languages---is verifiably complete in the above sense. We show how some paradoxical concepts of Quantum mechanics can, then, be expressed, and interpreted, naturally under a constructive definition of mathematical truth. (shrink)

Agents are often assumed to have degrees of belief (“credences”) and also binary beliefs (“beliefs simpliciter”). How are these related to each other? A much-discussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the “lottery paradox”: the set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed that this problem (...) generalizes: there exists no local function from degrees of belief to binary beliefs that satisfies some minimal conditions of rationality and non-triviality. “Locality” means that the binary belief in each proposition depends only on the degree of belief in that proposition, not on the degrees of belief in others. One might think that the impossibility can be avoided by dropping the assumption that binary beliefs are a function of degrees of belief. We prove that, even if we drop the “functionality” restriction, there still exists no local relation between degrees of belief and binary beliefs that satisfies some minimal conditions. Thus functionality is not the source of the impossibility; its source is the condition of locality. If there is any non-trivial relation between degrees of belief and binary beliefs at all, it must be a “holistic” one. We explore several concrete forms this “holistic” relation could take. (shrink)

Tennenbaum's Theorem yields an elegant characterisation of the standard model of arithmetic. Several authors have recently claimed that this result has important philosophical consequences: in particular, it offers us a way of responding to model-theoretic worries about how we manage to grasp the standard model. We disagree. If there ever was such a problem about how we come to grasp the standard model, then Tennenbaum's Theorem does not help. We show this by examining a parallel argument, from a (...) simpler model-theoretic result. (shrink)

Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case (...) on which to focus in analyzing mathematical depth. After introducing the theorem, four accounts of mathematical depth will be considered. (shrink)

A survey of more philosophical applications of Gödel's incompleteness results.

Those incompleteness theorems mean the relation of (Peano) arithmetic and (ZFC) set theory, or philosophically, the relation of arithmetical finiteness and actual infinity. The same is managed in the framework of set theory by the axiom of choice (respectively, by the equivalent well-ordering "theorem'). One may discuss that incompleteness form the viewpoint of set theory by the axiom of choice rather than the usual viewpoint meant in the proof of theorems. The logical corollaries from that "nonstandard" viewpoint the relation (...) of set theory and arithmetic are demonstrated. (shrink)

The aim of this paper is twofold. First, to explain the distinction between Kant’s notions of the sublime and ugliness, and to answer an important question that has been left unnoticed in contemporary studies, namely why it is the case that even though both sublime and ugliness are contrapurposive for the power of judgment, occasioning the feeling of displeasure, yet that after all we should feel pleasure in the former, while not in the latter. Second, to apply my interpretation of (...) the sublime and ugliness to contemporary art, and to resolve certain issues that have been raised in accounting for the possibility of artistic sublimity. I argue that an experience of a genuine artistic sublimity is an uncommon occurrence. I propose that the value of contemporary art can be best explained by referring to Kant’s notion of ugliness and his theory of aesthetic ideas. (shrink)

Jury theorems are mathematical theorems about the ability of collectives to make correct decisions. Several jury theorems carry the optimistic message that, in suitable circumstances, ‘crowds are wise’: many individuals together (using, for instance, majority voting) tend to make good decisions, outperforming fewer or just one individual. Jury theorems form the technical core of epistemic arguments for democracy, and provide probabilistic tools for reasoning about the epistemic quality of collective decisions. The popularity of jury theorems spans across various disciplines such (...) as economics, political science, philosophy, and computer science. This entry reviews and critically assesses a variety of jury theorems. It first discusses Condorcet's initial jury theorem, and then progressively introduces jury theorems with more appropriate premises and conclusions. It explains the philosophical foundations, and relates jury theorems to diversity, deliberation, shared evidence, shared perspectives, and other phenomena. It finally connects jury theorems to their historical background and to democratic theory, social epistemology, and social choice theory. (shrink)

Democratic decision-making is often defended on grounds of the ‘wisdom of crowds’: decisions are more likely to be correct if they are based on many independent opinions, so a typical argument in social epistemology. But what does it mean to have independent opinions? Opinions can be probabilistically dependent even if individuals form their opinion in causal isolation from each other. We distinguish four probabilistic notions of opinion independence. Which of them holds depends on how individuals are causally affected by environmental (...) factors such as commonly perceived evidence. In a general theorem, we identify causal conditions guaranteeing each kind of opinion independence. These results have implications for whether and how ‘wisdom of crowds’ arguments are possible, and how truth-conducive institutions can be designed. (shrink)

The article examines the reorientations of the appreciation of ugliness within different national contexts in a comparative and relational frame, juxtaposing the Australian, American, British and Italian milieus. It also explores the ways in which the transformation of the urban fabric and the effect of suburbanization were perceived in the aforementioned national contexts. Special attention is paid to the production and dissemination of how the city’s uglification was conceptualized between the 1950s and 1970s. Pivotal for the issues that this article (...) addresses are Ian Nairn’s Outrage: On the Disfigurement of Town and Countryside, Robin Boyd’s The Australian Ugliness, Donald Gazzard’s Australian Outrage: The Decay of a Visual Environment, and the way the phenomenon of urban expansion is treated in these books in comparison with other books from the four national contexts under study, such as Ludovico Quaroni’s La torre di Babele and Reyner Banham’s The New Brutalism: Ethic or Aesthetic?. Particular emphasis is placed on Boyd’s articles in The Architectural Review between 1951 and 1970. At the core of the article is the analysis of the debates around ugliness between the 1950s and 1970s within the British, Italian, American and Australian contexts. (shrink)

The previous two parts of the paper demonstrate that the interpretation of Fermat’s last theorem (FLT) in Hilbert arithmetic meant both in a narrow sense and in a wide sense can suggest a proof by induction in Part I and by means of the Kochen - Specker theorem in Part II. The same interpretation can serve also for a proof FLT based on Gleason’s theorem and partly similar to that in Part II. The concept of (probabilistic) measure (...) of a subspace of Hilbert space and especially its uniqueness can be unambiguously linked to that of partial algebra or incommensurability, or interpreted as a relation of the two dual branches of Hilbert arithmetic in a wide sense. The investigation of the last relation allows for FLT and Gleason’s theorem to be equated in a sense, as two dual counterparts, and the former to be inferred from the latter, as well as vice versa under an additional condition relevant to the Gödel incompleteness of arithmetic to set theory. The qubit Hilbert space itself in turn can be interpreted by the unity of FLT and Gleason’s theorem. The proof of such a fundamental result in number theory as FLT by means of Hilbert arithmetic in a wide sense can be generalized to an idea about “quantum number theory”. It is able to research mathematically the origin of Peano arithmetic from Hilbert arithmetic by mediation of the “nonstandard bijection” and its two dual branches inherently linking it to information theory. Then, infinitesimal analysis and its revolutionary application to physics can be also re-realized in that wider context, for example, as an exploration of the way for physical quantity of time (respectively, for time derivative in any temporal process considered in physics) to appear at all. Finally, the result admits a philosophical reflection of how any hierarchy arises or changes itself only thanks to its dual and idempotent counterpart. (shrink)