A proof of Fermat’s last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat’s last theorem in the case of n = 3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat’s approach of infinite descent. (...) The infinite descent is linked to induction starting from n = 3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat’s last theorem. As far as Fermat had been proved the theorem for n = 4, one can suggest that the proof for n ≥ 4 was accessible to him. An idea for an elementary arithmetical proof of Fermat’s last theorem (FLT) by induction is suggested. It would be accessible to Fermat unlike Wiles’s proof (1995). (shrink)

In a previous paper, an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction has been demonstrated if the case for “n = 3” is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for (...) “n = 3” has been known for a long time. It needs “Hilbert mathematics”, which is inherently complete unlike the usual “Gödel mathematics”, and based on “Hilbert arithmetic” to generalize Peano arithmetic in a way to unify it with the qubit Hilbert space of quantum information. An “epoché to infinity” (similar to Husserl’s “epoché to reality”) is necessary to map Hilbert arithmetic into Peano arithmetic in order to be relevant to Fermat’s age. Furthermore, the two linked semigroups originating from addition and multiplication and from the Peano axioms in the final analysis can be postulated algebraically as independent of each other in a “Hamilton” modification of arithmetic supposedly equivalent to Peano arithmetic. The inductive proof of FLT can be deduced absolutely precisely in that Hamilton arithmetic and the pransfered as a corollary in the standard Peano arithmetic furthermore in a way accessible in Fermat’s epoch and thus, to himself in principle. A future, second part of the paper is outlined, getting directed to an eventual proof of the case “n=3” based on the qubit Hilbert space and the Kochen-Specker theorem inferable from it. (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)

The paper is a continuation of another paper published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The (...) relevant mathematical structure is Hilbert arithmetic in a wide sense, in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem. (shrink)

Conventional wisdom dictates that proofs of mathematical propositions should be treated as necessary, and sufficient, for entailing `significant' mathematical truths only if the proofs are expressed in a---minimally, deemed consistent---formal mathematical theory in terms of: * Axioms/Axiom schemas * Rules of Deduction * Definitions * Lemmas * Theorems * Corollaries. Whilst Andrew Wiles' proof of Fermat's Last Theorem FLT, which appeals essentially to geometrical properties of real and complex numbers, can be treated as meeting this criteria, it nevertheless (...) leaves two questions unanswered: (i) Why is x^n +y^n = z^n solvable only for n < 3 if x, y, z, n are natural numbers? (ii) What technique might Fermat have used that led him to, even if only briefly, believe he had `a truly marvellous demonstration' of FLT? Prevailing post-Wiles wisdom---leaving (i) essentially unaddressed---dismisses Fermat's claim as a conjecture without a plausible proof of FLT. -/- However, we posit that providing evidence-based answers to both queries is necessary not only for treating FLT as significant, but also for understanding why FLT can be treated as a true arithmetical proposition. We thus argue that proving a theorem formally from explicit, and implicit, premises/axioms using rules of deduction---as currently accepted---is a meaningless game, of little scientific value, in the absence of evidence that has already established---unambiguously---why the premises/axioms and rules of deduction can be treated, and categorically communicated, as pre-formal truths in Marcus Pantsar's sense. Consequently, only evidence-based, pre-formal, truth can entail formal provability; and the formal proof of any significant mathematical theorem cannot entail its pre-formal truth as evidence-based. It can only identify the explicit/implicit premises that have been used to evidence the, already established, pre-formal truth of a mathematical proposition. Hence visualising and understanding the evidence-based, pre-formal, truth of a mathematical proposition is the only raison d'etre for subsequently seeking a formal proof of the proposition within a formal mathematical language (whether first-order or second order set theory, arithmetic, geometry, etc.) By this yardstick Andrew Wiles' proof of FLT fails to meet the required, evidence-based, criteria for entailing a true arithmetical proposition. -/- Moreover, we offer two scenarios as to why/how Fermat could have laconically concluded in his recorded marginal noting that FLT is a true arithmetical proposition---even though he either did not (or could not to his own satisfaction) succeed in cogently evidencing, and recording, why FLT can be treated as an evidence-based, pre-formal, arithmetical truth (presumably without appeal to properties of real and complex numbers). It is primarily such a putative, unrecorded, evidence-based reasoning underlying Fermat's laconic assertion which this investigation seeks to reconstruct; and to justify by appeal to a plausible resolution of some philosophical ambiguities concerning the relation between evidence-based, pre-formal, truth and formal provability. (shrink)

The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of Hilbert (...) mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat’s last theorem proved by Andrew Wiles; Poincaré’s conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved “machine-likely” by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat’s last theorem is shown as a Gödel insoluble statement by means of Yablo’s paradox. Thus, Wiles’s proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in “Fermat arithmetic” introduced by “epoché to infinity” (following the pattern of Husserl’s original “epoché to reality”) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a “Wittgenstein ladder”. Poincaré’s conjecture can be reinterpreted physically by Minkowski space and thus reduced to the “nonstandard homeomorphism” of a bit of information mathematically. Perelman’s proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré’s conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions. (shrink)

Fermat’s Least Time Principle has a long history. World’s foremost academies of the day championed by their most prestigious philosophers competed for the glory and prestige that went with the solution of the refraction problem of light. The controversy, known as Descartes - Fermat controversy was due to the contradictory views held by Descartes and Fermat regarding the relative speeds of light in different media. Descartes with his mechanical philosophy insisted that every natural phenomenon must be explained by mechanical principles. (...) Fermat on the other hand insisted an end purpose for every motion. For example, least time of travel and not the least distance of travel is the end purpose for motion of light. This implied a thinking nature, which was rejected by Descartes. Surprisingly, with contradictory assumptions regarding the relative speeds of light in different media, both Descartes and Fermat came to the same result that the ratio of sines of angles of incidence and refraction is a constant. Fermat’s result came to be known as the ‘Fermat’s least time principle’. We show in this article that Fermat’s least time principle violates a fundamental theorem in geometry – the Ptolemy’s theorem. That leads to the invalidity of Fermat’s principle. -/- . (shrink)

Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum computer is not Turing machine – (...) Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 of their paper – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and psi-function – From a qualitative to a quantitative expression of relativity − psi-function, or the discrete by the random – Bartlett’s approach − psi-function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the as-sociated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Encoding a world line by psi-function – Spacetime and qubit − psi-function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = psi-function + self-adjoint operator – Reality and description – Does a „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The psi-function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocity-probability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − psi-function as physical quantity – Things in the world and instances in time – The generation of the physi-cal by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new view-point – The exception of two dimensions and Fermat’s last theorem. (shrink)

Andrew Wiles' analytic proof of Fermat's Last Theorem FLT, which appeals to geometrical properties of real and complex numbers, leaves two questions unanswered: (i) What technique might Fermat have used that led him to, even if only briefly, believe he had `a truly marvellous demonstration' of FLT? (ii) Why is x^n+y^n=z^n solvable only for n<3? In this inter-disciplinary perspective, we offer insight into, and answers to, both queries; yielding a pre-formal proof of why FLT can be treated as (...) a true arithmetical proposition (one which, moreover, might not be provable formally in the first-order Peano Arithmetic PA), where we admit only elementary (i.e., number-theoretic) reasoning, without appeal to analytic properties of real and complex numbers. We cogently argue, further, that any formal proof of FLT needs---as is implicitly suggested by Wiles' proof---to appeal essentially to formal geometrical properties of formal arithmetical propositions. (shrink)

Einstein wrote his famous sentence "God does not play dice with the universe" in a letter to Max Born in 1920. All experiments have confirmed that quantum mechanics is neither wrong nor “incomplete”. One can says that God does play dice with the universe. Let quantum mechanics be granted as the rules generalizing all results of playing some imaginary God’s dice. If that is the case, one can ask how God’s dice should look like. God’s dice turns out to be (...) a qubit and thus having the shape of a unit ball. Any item in the universe as well the universe itself is both infinitely many rolls and a single roll of that dice for it has infinitely many “sides”. Thus both the smooth motion of classical physics and the discrete motion introduced in addition by quantum mechanics can be described uniformly correspondingly as an infinite series converges to some limit and as a quantum jump directly into that limit. The second, imaginary dimension of God’s dice corresponds to energy, i.e. to the velocity of information change between two probabilities in both series and jump. (shrink)

The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present such as Fermat’s (...) last theorem, four-color theorem as well as its new-formulated generalization as “four-letter theorem”, Poincaré’s conjecture, “P vs NP” are considered over again, from and within the new-founding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested. (shrink)

Mathematics is everywhere and yet its objects are nowhere. There may be five apples on the table but the number five itself is not to be found in, on, beside or anywhere near the apples. So if not in space and time, where are numbers and other mathematical objects such as perfect circles and functions? And how do we humans discover facts about them, be it Pythagoras’ Theorem or Fermat’s Last Theorem? The metaphysical question of what numbers are and (...) the epistemological question of how we know about them are central to the philosophy of mathematics. These and related philosophical questions are of particular interest because of mathematics’ unusual status. Mathematics is exceptional in that, on the one hand, it appears unhesitatingly true—no one doubts that 2 + 3 = 5—but on the other, as just noted, it is not about the physical world. This ambivalent status is what gives the philosophy of mathematics its special interest. The philosophy of mathematics is also one of the oldest academic fields, more or less coeval with philosophy itself. But contemporary philosophy of mathematics is rather different from its pre-twentieth-century antecedents, largely for three reasons. The first is that since the seventeenth century, mathematics has become integral to science. Science has over the past few centuries become increasingly mathematical, and indeed the fundamental science of nature, physics, is today recognised as a branch of applied mathematics. The second is that mathematics underwent a transformation in the course of nineteenth century: having started the century as a rather traditional-looking science of quantity it emerged a hundred years later a radically transformed abstract theory of structure. The final factor in the transformation of the philosophy of mathematics is the rise of modern logic. Developed by Frege, Cantor and others in the late nineteenth century, modern logic pervades contemporary mathematics, philosophy and computer science, and has had an immeasurable effect on the philosophy of mathematics. These volumes will collect the major works in this major field, with a focus on the last few decades. The anthology will include technical work, which interprets philosophically significant mathematical results or subfields of mathematics, as well as purely philosophical writing, aimed at those without advanced mathematics. The collection should be of interest to both philosophers and mathematicians, as well as to anyone who is susceptible to wondering what the main intellectual tool used in science, economics and finance, and indeed everyday life is ultimately about. (shrink)

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. (...) The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines. (shrink)

My aim in this paper is to explain what Condorcet’s jury theorem is, and to examine its central assumptions, its significance to the epistemic theory of democracy and its connection with Rousseau’s theory of general will. In the first part of the paper I will analyze an epistemic theory of democracy and explain how its connection with Condorcet’s jury theorem is twofold: the theorem is at the same time a contributing historical source, and the model used by the authors to (...) this day. In the second part I will specify the purposes of the theorem itself, and examine its underlying assumptions. Third part will be about an interpretation of Rousseau’s theory, which is given by Grofman and Feld relying on Condorcet’s jury theorem, and about criticisms of such interpretation. In the fourth, and last, part I will focus on one particular assumption of Condorcet’s theorem, which proves to be especially problematic if we would like to apply the theorem under real-life conditions; namely, the assumption that voters choose between two options only. (shrink)

Leibniz proposed the ‘Most Determined Path Principle’ in seventeenth century. According to it, ‘ease’ of travel is the end purpose of motion. Using this principle and his calculus method he demonstrated Snell’s Laws of reflection and refraction. This method shows that light follows extremal (local minimum or maximum) time path in going from one point to another, either directly along a straight line path or along a broken line path when it undergoes reflection or refraction at plane or spherical (concave (...) or convex) surfaces. The extremal time path avoided the criticism that Fermat’s least time path was subjected to, by Cartesians who cited examples of reflections at spherical surfaces where light took the path of longest time. Thereby it became the standard method of demonstration of Snell’s Laws. Ptolemy’s theorem is a fundamental theorem in geometry. A special case of it offers a method of finding the minimum sum of the two distances of a point from two given fixed points. We show in this paper that Leibniz’s calculus proof of Snell’s Laws violates Ptolemy’s theorem, whereby Leibniz’s proof becomes invalid. (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)

Two of the most important ideas in the philosophy of law are the “Coase Theorem” and the “Prisoner’s Dilemma.” In this paper, the authors explore the relation between these two influential models through a creative thought-experiment. Specifically, the paper presents a pure Coasean version of the Prisoner’s Dilemma, one in which property rights are well-defined and transactions costs are zero (i.e. the prisoners are allowed to openly communicate and bargain with each other), in order to test the truth value of (...) the Coase Theorem. In addition, the paper explores what effect (a) uncertainty, (b) exponential discounting, (c) and elasticity have on the behavior of the prisoners in the Coasean version of the dilemma. Lastly, the paper considers the role of the prosecutor (and third-parties generally) in the Prisoner’s Dilemma and closes with some parting thoughts about the complexity of the dilemma. The authors then conclude by identifying the conditions under which the Prisoner’s Dilemma refutes the Coase Theorem. (shrink)

The study aimed to identify the reality of the use of social networks in the technical colleges in Palestine, where the variables of social networks were included. The analytical descriptive method was used in the study. A questionnaire consisting of (12) items was randomly distributed to college workers Technology in the Gaza Strip. The sample of the study consisted of (205) employees of these colleges. The response rate was 74.5%. The results showed a high degree of approval for the dimensions (...) of the social networks and a relative weight (74.15%) according to the perspective of the employees of the technical colleges in the Gaza Strip. The results of the study showed that there is a high level of social networking areas (site management and Website Content) in the technical colleges in the Gaza Strip. The field of site management ranked first with a relative weight of 74.91%, and second and last (Content Site) with a relative weight (73.38%). The results showed that there were differences between colleges in the use of social networks where the results showed that the most common colleges used these networks (UCAS) and the least used is (GTC). The results showed no differences between male and female employees in the use of social networks in technical colleges. The researchers suggest a number of recommendations, including: the need to raise awareness of the importance of Facebook and other social networking sites, through the holding of courses for employees in technical colleges, and to identify the ways to optimize the use of such sites, and the benefits of this use, and reflected positively on technical colleges. And the adoption of dealing with the various social networking sites as a reality, and the Palestinian and Arab technical colleges, use them in accordance with the objectives of technical colleges. Advise the Department of Technical Colleges to devote time to their presence on social networks to follow the public and respond to their queries. There is a need for the attention of decision-makers in technical colleges in social sites, because they are considered an important and effective means of communication, and the link between beneficiaries and decision-makers. There is a need to promote the use of modern electronic means of work and the need to increase the link of customers to the college through electronic services. (shrink)

We generalize and extend the class of Sahlqvist formulae in arbitrary polyadic modal languages, to the class of so called inductive formulae. To introduce them we use a representation of modal polyadic languages in a combinatorial style and thus, in particular, develop what we believe to be a better syntactic approach to elementary canonical formulae altogether. By generalizing the method of minimal valuations à la Sahlqvist–van Benthem and the topological approach of Sambin and Vaccaro we prove that all inductive formulae (...) are elementary canonical and thus extend Sahlqvist’s theorem over them. In particular, we give a simple example of an inductive formula which is not frame-equivalent to any Sahlqvist formula. Then, after a deeper analysis of the inductive formulae as set-theoretic operators in descriptive and Kripke frames, we establish a somewhat stronger model-theoretic characterization of these formulae in terms of a suitable equivalence to syntactically simpler formulae in the extension of the language with reversive modalities. Lastly, we study and characterize the elementary canonical formulae in reversive languages with nominals, where the relevant notion of persistence is with respect to discrete frames. (shrink)

This chapter sets forth an interpretation of public reason that appeals to our central capabilities as human beings. I argue that appealing to central human capabilities and to the related idea of respect for threshold capabilities is the best way to understand public reason. My defense of this position advances stepwise: first, I consider a central alternative to a capability account, which regards public reason as a matter of contracting; next, I describe central concerns with contract views and show how (...) a capability view can avoid them; lastly, I consider extending public reason beyond national borders and test the two views in this context. (shrink)

Commercial republicanism is the idea that a properly-structured commercial society can serve the republican end of minimizing the domination of citizens by states (imperium) and of citizens by other citizens (dominium). Much has been written about this idea in the last half-century, including analyses of individual commercial republicans (e.g., Adam Smith and Immanuel Kant) as well as discussions of national traditions of the same (e.g., in America, Britain, France, the Netherlands, and Italy). In this chapter, I review five kinds (...) of arguments that have been historically offered for how commercial society so understood advances the republican ideal of non-domination: first, by simply realizing non-dominating relationships in the marketplace itself (Instantiation); second, by encouraging the growth of a wide middle class that can offer a counterweight to other, dominating classes (Internal Check-and-Balance); third, by increasing the wealth and power of republics vis-à-vis other, dominating states in the international arena (External Check-and-Balance); fourth, by helpfully nurturing bourgeois-civic virtues in republican citizens (Cultivating Virtue); and finally, by transforming dangerous passions into more readily regulable, even socially beneficial material interests (Sublimating Vice). I end the chapter by briefly considering commercial republicanism’s prospects as a continuing research program. (shrink)

Collaborative learning emphasizes student-to-student interaction and the instructor’s role as a facilitator. Collaborative Online International Learning (COIL) was founded in 2005 by the State University of New York (SUNY) to help schools adapt their single classroom courses to an online, collaborative format and establish strong collaborations with professors with whom they would join classes and co-teach using SUNY COIL conferences and website, as well as pre-established partnerships between the institutions. However, as the globe becomes increasingly interconnected, educational challenges aimed at (...) cultivating intercultural competency become more important (Ceo-DiFrancesco & Bender-Slack, 2016). That is why this study aimed to (1) understand the lived experience of students who went under the COIL program in relation to SDG 12, (2) review the new knowledge students obtained when they took part in the COIL program in relation to SDG 12, and (3) discover the challenges students encountered while participating in the COIL program in relation to SDG 12. The researcher surveyed Filipino and Japanese students who participated in the COIL program concerning SDG12 from St. Patrick School of Quezon City, Kwansei Gakuin University, and Meiwa Senior High School. The study employed qualitative phenomenological research, and pertinent data were obtained through an open-ended survey questionnaire. The study was analyzed using Colaizzi’s descriptive phenomenological method. The results showed that the lived experiences of students who went under the COIL program in relation to SD12 have something to do with all their learnings and enjoyable experiences, intercultural and global interactions, and environmental discussions and action plans related to SDG 12. Moreover, students learned and understood better the negative impact of waste on the environment, environmental awareness and practices, the culture and norms in the Philippines and Japan, and the importance and benefits of collaboration. Lastly, the study also revealed the different challenges students encounter while participating in the COIL program in relation to SDG 12, such as cultural differences, language barriers, technical difficulties, and difficulties in experimentation. Thus, it was recommended to learn to adapt to everyone and adjust according to countries’ different cultures, practice and be proficient with the English language, have backup devices and platforms to use to ease technological problems, and improve implementations of experiments. (shrink)

In this paper, I use a comparative analysis of mysticism in Zen and the Abrahamic faiths to formulate a phenomenological account of mysticism “as such.” I argue that, while Zen Buddhism is distinct from other forms of mystical experience in important ways, it can still be fit into a general phenomenological category of mystical experience. First, I explicate the phenomenological accounts of mysticism provided by Anthony Steinbock and Angela Bello. Second, I offer an account of Zen mysticism which both coheres (...) with and problematizes these accounts, arguing that Zen demonstrates the inadequacy of these accounts as descriptions of mysticism as a universal religious category. Lastly, I use this investigation to propose that Zen mysticism does generally cohere with the mystical experiences of other religions, but only if we devise a new formula for speaking phenomenologically about mystical experience as such which captures this phenomenon in all of its manifestations. (shrink)

First paragraph: In the 1980s I went through a phase of writing limericks during idle moments when I lacked something to read. The result was a set of 27 limericks about cybernetics (Umpleby 1992). I occasionally use the limericks in class to restate a theoretical point. Limericks bring a smile and demonstrate that cybernetics can be approached in a variety of ways. Below are three limericks from this collection. The last was written by Ernst von Glasersfeld. It seems Ernst (...) believed that I was overly concerned with "reality" as opposed to perception, or at least that I had not captured the point that he was trying to make. (shrink)

This essay ties together some main strands of the author’s research spanning the last quarter-century. Because of its broad scope and space limitations, he prescinds from detailed arguments and instead intuitively motivates the general points which are supported more fully in other publications to which he provides references. After an initial delineation of several distinct notions of meaning, the author considers such a notion deriving from the evolutionary biology of communication that he terms ‘organic meaning’, and places it in (...) the context of evolutionary game theory. That provides a framework for a special type of organic meaning found in the phenomenon of expression, of which the author here offers an updated characterization while highlighting its wide philosophical interest. Expression in turn generalizes to a paradigmatic form of human communication—conversation—and section 4 provides a taxonomy of conversation-types while arguing that attention to such types helps to sharpen predictions of what speakers say rather than conversationally implicate. We close with a view of fictional discourse on which authors of fictional works are engaged in conversation with their readers, and can provide them with knowledge in spite of the fictional character of their conversation. Such knowledge includes knowledge of how an emotion feels and is thus a route to empathy. (shrink)

Hofstadter [1979, 2007] offered a novel Gödelian proposal which purported to reconcile the apparently contradictory theses that (1) we can talk, in a non-trivial way, of mental causation being a real phenomenon and that (2) mental activity is ultimately grounded in low-level rule-governed neural processes. In this paper, we critically investigate Hofstadter’s analogical appeals to Gödel’s [1931] First Incompleteness Theorem, whose “diagonal” proof supposedly contains the key ideas required for understanding both consciousness and mental causation. We maintain that bringing sophisticated (...) results from Mathematical Logic into play cannot furnish insights which would otherwise be unavailable. Lastly, we conclude that there are simply too many weighty details left unfilled in Hofstadter’s proposal. These really need to be fleshed out before we can even hope to say that our understanding of classical mind-body problems has been advanced through metamathematical parallels with Gödel’s work. (shrink)

Critics of the computational connectionism of the last decade suggest that it shares undesirable features with earlier empiricist or associationist approaches, and with behaviourist theories of learning. To assess the accuracy of this charge the works of earlier writers are examined for the presence of such features, and brief accounts of those found are given for Herbert Spencer, William James and the learning theorists Thorndike, Pavlov and Hull. The idea that cognition depends on associative connections among large networks of (...) neurons is indeed one with precedents, although the implications of this for psychological issues have been interpreted variously — not all versions of connectionism are alike. (shrink)

Can we find necessary and sufficient conditions for a mental state to be a pain state? That is, does pain have a nature? Or is the term ‘pain’ ambiguous? I argue here that our expression ‘pain’ lacks necessary use conditions if one considers a range of contexts. As use conditions constrain the reference class, I argue that ‘pain’ does not refer to a natural category, but binds together a bunch of loosely resembling phenomena. This leads to problems for scientific and (...) clinical discourse. To solve these, a method of explication is suggested, based on a discursive combination between analysis of first-person reports and theories of natural science. Lastly, I consider the ethical implications of this ambiguity that lead to a reformulation of the goal of pain science: Not alleviation of all pains ought to be our goal, but only manipulation of conscious and negatively emotionally charged pains. (shrink)

This article introduces the concept of ‘well-living’ as a transformative framework for reimagining quality of life in the face of current global socio-ecological challenges. Through a reflexive theoretical meta-analysis, it critically examines mainstream and reformist well-being discourses while drawing inspiration from transformative perspectives found in recent post-capitalist and indigenous movements. ‘Well-living’ is portrayed as both a civilizational endeavor and a multifaceted imperative, encompassing dimensions of creativity, liveability, conviviality, and alterity across various scales from individual to international contexts. Central to the (...) ‘well-living’ paradigm are nine key qualities, including harmonious coexistence, aspirational foresight and purposefulness, solidarity, autonomy, authenticity, and integrity, thereby promoting an integrated approach to living in balance with oneself, others, and the natural world. Embracing ‘well-living’ as a goal and process can empower individuals and communities to challenge prevailing global capitalist paradigms, re-establish connections with the interconnected web of life, and strive for a more just, regenerative, and diverse world, accommodating multiple perspectives. Lastly, employing a 'commonist' perspective, the article outlines essential institutional and legislative-policy changes required to actualize the vision of 'well-living.'. (shrink)

Abstract: In this paper, the definition of triple Aboodh transform of fractional order α, where α ϵ [0, 1], is introduced for functions which are fractional differentiable. We also present several properties of this transform. Furthermore, some main theorems and their proofs are discussed.

"The enterprise initiative is probably the most significant political and cultural influence to have affected Western and Eastern Europe in the last decade. In this book, academics from a range of disciplines debate Mary Douglas's distinctive Grid Group cultural theory and examine how it allows us to analyse the complex relation between the culture of enterprise and its institutions. Mary Douglas, Britain's leading cultural anthropologist, contributes several chapters."--BOOK JACKET.Title Summary field provided by Blackwell North America, Inc. All Rights Reserved.

Glucagon-like peptide-1 (GLP-1) receptor agonists, such as semaglutide, and dual GLP-1 and glucose-dependent insulinotropic polypeptide (GIP) receptor agonists, such as tirzepatide, have been found to be effective for treating obesity and diabetes, significantly reducing weight and the risk or predicted risk of adverse cardiovascular events. There is a global shortage of these medications that could last several years and raises questions about how limited supplies should be allocated. We propose a fair-allocation framework that enables evaluation of the ethics of (...) current practices and could guide governments, professional societies, and physicians in allocation decisions. (shrink)

Although they were never to meet and corresponded only briefly, Catharine Macaulay and Mary Wollstonecraft shared a mutual admiration and a strong intellectual bond. Macaulay’s work had a profound and lasting effect on Wollstonecraft, and she developed and expanded on many of Macaulay’s ideas. While she often took these in a different direction, there remains a great synergy between their ideas to the extent that we can understand Wollstonecraft’s own feminist arguments by approaching them through the frameworks and ideas that (...) Macaulay provided. These included the principles of classical republicanism, particularly in its understanding of the values of freedom, equality and virtue, and an understanding of reason as grounded in immutable principles that apply equally to both sexes. On the question of women’s freedom and social equality with men, I argue that though Macaulay sets up the problem in far richer and more detailed philosophical terms, in the end it is Wollstonecraft that has the more compelling account of its far-reaching social implications and of how this might be addressed. (shrink)

This paper has three main goals. First, to motivate a puzzle about how ignorance-expressing terms like maybe and if interact: they iterate, and when they do they exhibit scopelessness. Second, to argue that there is an ambiguity in our theoretical toolbox, and that exposing that opens the door to a solution to the puzzle. And third, to explore the reach of that solution. Along the way, the paper highlights a number of pleasing properties of two elegant semantic theories, explores some (...) meta-theoretic properties of dynamic notions of meaning, dips its toe into some hazardous waters, and offers characterization theorems for the space of meanings an indicative conditional can have. (shrink)

Techno-ethics is the area in the philosophy of technology which deals with emerging robotic and digital AI technologies. In the last decade, a new techno-ethical challenge has emerged: Autonomous Weapon Systems (AWS), defensive and offensive (the article deals only with the latter). Such AI-operated lethal machines of various forms (aerial, marine, continental) raise substantial ethical concerns. Interestingly, the topic of AWS was almost not treated in Jewish law and its research. This article thus proposes an introductory ethical-halakhic perspective on (...) AWS, in the Israeli context. The article has seven sections. Section 1 defines AWS and the main ethical concerns it evokes, while providing elementary definitions and distinctions. §2 locates AWS within the realm of Jewish laws of war (hilkhot-ẓava), which recognize the right for self-defense, as well as the status of universally accepted moral norms. §3 unfolds pragmatic ethical premises of a humane techno-ethics, which are required for the identifying AWS as a moral question: I. Relationality; II. Technology is not (completely) neutral; III. The fallaciousness of transhumanism. It is argued that these premises are compatible with halakhic tradition. §4 investigates the question of the morality of AWS, within the field of military AI ethics. It is clarified why the standard categories of war-ethics (ad bello, in bellum) do not capture the singular ethical problem of AWS, which pertains to the operation of military means, rather than their human targets. It is argued that reductive perception of the human mind is misleading about the feasibility of ‘ethical robots’, capable of independent moral discretion. To provide a thick examination of human agency from the perspective of Jewish tradition, §5 explores two stories from the biblical book of Samuel (the murder of Nob’s priests and of Uriah). The lessons about the significance of moral agency within the pubic-political military sphere are made explicit, as well as the possible costs resulting from the loss of human agency in the case of AWS. Given that realpolitik considerations are basic in halakhah, §6 considers some possible contemporary socio-political implications of the AWS, that may risk the sustainability of the democratic project. §7 concludes by pointing out humane contributions of Jewish law to contemporary techno-ethics. (shrink)

Abstract— In this paper, we introduce the definition of triple Aboodh transform, some properties for the transform are presented. Furthermore, several theorems dealing with the properties of the triple Aboodh transform are proved. In addition, we use this transform to solve partial differential equations with integer and non-integer orders.

Dmitri Nikulin is one of the few contemporary philosophers to have devoted books to the topic of dialogue and the dialogical self, especially in the last fifteen years. Yet his work on dialogue and the dialogical has received scant attention by philosophers, and this neglect has hurt the ongoing development of contemporary philosophical work on dialogicality. I want to address this lacuna in contemporary philosophical scholarship on dialogicality and suggest that, although Nikulin’s account is no doubt insightful and thought-provoking, (...) it is problematic for two main reasons: first, his account fails to recognize the proper relationship between dialogue and agency; and second, his enumeration of the necessary and sufficient conditions for dialogue contains conceptual inconsistencies. (shrink)

Education on authorship was delivered and evaluated by pre test and post test questionnairen on 30 post graduate medical students at the Department of Anestheology, Dhaka Medical College, Bangladesh between January and June 2019 to understand the knowledge, skill and attitude of post graduate medical students on authorship. Result: Before intervention, majority (60%) of the students felt that who perform the research work should be the author of the article. But 40% students were divided and felt that who advised the (...) design of the research (20%), who provided the grants (10%) and Chief/Head of the division (10%) should be the author of the article respectively. Maximum (70%) respondents did not know the order of authorship. Of 40% respondent felt that the PI should be always the first author and 40% don’t know the answer. Half of the students (50%) felt that keeping honorary author increased the opportunity of acceptance of publication. Of 36.7% and 13.3% of students felt that keeping honorary author increased the article’s value and made good relation to them to get some extra facility from them respectively. Of 20% participants were pressurized by lab head/head of department for inclusion of their name as an author. Half of the (56.7 %) respondents felt that the author’s contribution should be stated in the article. Only few 4 (13.3%) respondents said that their institute had guideline for authorship. However, after education 100% of students felt that who perform the research work should be only the author of the article. All (100%) respondents understood the order of authorship. Most of the students (86%) felt that PI should be always the first author. Of 100% respondent felt supervisor of the research should be the last author. All students (100%) felt author’s contribution should be mentioned in the article. All (100%) students did not want to include as author those who help in research design and secured grant; and they did not like to keep honorary author in their article. All (100%) students expressed that their institute had no guideline for authorship. After intervention, three groups of students were asked to write one page of article on Anesthesiology. Interestingly, they did not include any name in the author by line who were not participate or had any contribution in the writing. Conclusion: The comparative data between pre- and post-text have highlighted a general lack of understanding of the basic concept of authorship ethics which significantly improved after the intervention. The results also indicate that the education on authorship improved the awareness of postgraduate medical students in a particular centre. (shrink)

Open theists argue that God's relationship to time, as conceived in classical theism, is erroneous. They explain that it is contradictory for an atemporal being to act in a temporal universe, including experiencing its temporal successions. Contrary to the atemporalists, redemptive history has shown that God interacts with humans in time. This relational nature of God nullifies the classical notion of God as timelessly eternal. Therefore, it lacks a philosophical and theological basis. Because God is in time, He does not (...) know all future contingencies and, therefore, changes. This study examines open theism's appropriation of the A and B theories of time to the divine-human relationship. The study argues that divine temporality does not solve the tension of divine-human relationships, especially in relation to the future. Further, historical divine temporality does not negate the fact of divine atemporality. It mainly stems from God's choice to create temporal creatures and His relationship with them. Furthermore, if it is not logically and metaphysically contradictory for an omnipresent being to act in space, then it follows that an atemporal being can act in time. Whether time is understood from the metric or psychological point of view, it does not transcend God, and therefore, the limitation it places on human creatures with respect to the future does not apply to God. Lastly, although a few philosophers reject the notion of eternity as timelessly eternal, the doctrine has a philosophical and theological basis in the Scripture. (shrink)

In this paper, I offer an expansionist view of the Frickerian central case of testimonial injustice, citing examples from the South Asian context. To defend this expansionist position, I provide an argument in three parts. First, I argue that credibility deficit and credibility excess are entangled with each other in such a way that often, one produces the other. Secondly, I contend that we should not say that systematic testimonial injustice is a consequence of credibility deficit only because of the (...) entanglement between them. I also contend that for being the central case of testimonial injustice, identity prejudice should not be necessarily negative; it can be positive as well. Propounding a twofold condition of the status of a knower, the last part claims that testimonial injustice occurs when one of the two conditions remains unmet. (shrink)

The last twenty-five years or so have seen the emergence of exciting comparative work on Nietzsche and various philosophical traditions beyond the bounds of Europe. So far, however, the emphasis has been primarily on the cultures of India, China and Japan, with an almost exclusive focus on Buddhist, Hindu, Daoist, and Confucian traditions. Surprisingly, little work has been done on Nietzsche and the Islamic tradition. In this paper, I sketch out Nietzsche’s understanding of Islam, the ways in which he (...) uses it as a resource for his critique of Christianity and European modernity and the criticisms he has to make of it as one of the great monotheistic world religions. I then argue for the need to engage Nietzsche with specific Islamic falāsifa of the classical period (9-12th c.) rather than Islam itself, as some recent scholars have attempted to do. Although Nietzsche himself seems to have had no direct familiarity with any of the falāsifa, there are, I argue, many entry points for productive comparison and dialogue. This is at least in part because they share a significant common heritage: both were careful students of classical Greek and Hellenistic thought, and both put the insights they encountered there to work in bold new ways. Indeed, they appropriated, transformed and reanimated Greek ideas in new contexts and towards new ends that their progenitors would scarcely have recognized, and that were often radically challenging to their contemporaries. I focus here on a few select themes—the idea of philosophy as a “way of life,” the ideal of “becoming like God so far as it is possible” and the Platonic figure of the philosopher-ruler, all of which get taken up and re-imagined in radically novel and sometimes antipodal ways. (shrink)

Abstract: Lotteries have long been a source of fascination and intrigue, offering the tantalizing prospect of unexpected fortunes. In this research paper, we delve into the world of lottery predictions, employing cutting-edge AI techniques to unlock the secrets of lottery outcomes. Our dataset, obtained from Kaggle, comprises historical lottery draws, and our goal is to develop predictive models that can anticipate future winning numbers. This study explores the use of deep learning and time series analysis to achieve this elusive feat. (...) Through rigorous experimentation and data-driven approaches, we seek to determine the viability of AI in the realm of lottery predictions. Our findings reveal both the promise and limitations of AI in this context, shedding light on the complexities of lottery data and the potential need for quantum computing as a last resort. (shrink)

This paper sets mathematics among the sciences, despite not being empirical, because it studies relations of various sorts, like the sciences. Each empirical science studies the relations among objects, which relations determining which science. The mathematical science studies relations as such, regardless of what those relations may be or be among, how relations themselves are related. This places it at the extreme among the sciences with no objects of its own (A Subject with no Object, by J.P. Burgess and G. (...) Rosen). Examples are discussed. The historical development of written mathematics from algorithms on clay tablets to theorems with proofs is said to show that application of algorithms to specific problems and theorems to scientific objects is like allegorical interpretation. Mathematics fits into the modern scientific context because the sciences, beginning with Galileo, have been constructed in imitation of mathematics. Viewing mathematics this way does not solve any ontological problems, but it does show how mathematics avoids them. Epistemological problems, insuperable for object realism, are simplified. For example, we have access to relations among any objects that we can consider objectively, first physical objects and then mathematical objects of greater and greater abstraction. (shrink)

The use of language models in Web applications and other areas of computing and business have grown significantly over the last five years. One reason for this growth is the improvement in performance of language models on a number of benchmarks — but a side effect of these advances has been the adoption of a “bigger is always better” paradigm when it comes to the size of training, testing, and challenge datasets. Drawing on previous criticisms of this paradigm as (...) applied to large training datasets crawled from pre-existing text on the Web, we extend the critique to challenge datasets custom-created by crowdworkers. We present several sets of criticisms, where ethical and scientific issues in language model research reinforce each other: labour injustices in crowdwork, dataset quality and inscrutability, inequities in the research community, and centralized corporate control of the technology. We also present a new type of tool for researchers to use in examining large datasets when evaluating them for quality. (shrink)

We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the Milman-Pettis theorem that uniformly convex Banach spaces are reflexive.

Shows how, as a consequence of the Arrow Impossibility Theorem, objectivity in grading is chimerical, given a sufficiently knowledgeable teacher (of his students, not his subject) in a sufficiently small class. -/- PDF available from JStor only; permission to post full version previously granted by journal editors and publisher expired. -/- Unpublished reply posted gratis.

We extend the framework of Inductive Logic to Second Order languages and introduce Wilmers' Principle, a rational principle for probability functions on Second Order languages. We derive a representation theorem for functions satisfying this principle and investigate its relationship to the first order principles of Regularity and Super Regularity.

This article shows that underlying the Museyroom passage of James Joyce's Finnegans Wake is an account of the Battle of the Little Bighorn. Joyce's passage touches on what motivated the battle, its Irish participants, its commanders, Custer's disagreements with his Crow scouts, Sitting Bull's prediction of Custer's failure, Custer's (purely fanciful) joking reply to Sitting Bull's prediction, Custer's home near Bismarck, ND, and his signature song. Other passages scattered throughout Finnegans Wake treat of casualties, a straggler, Custer's trumpeter, Custer's from-the-grave (...) denunciations of his subordinates, and the burial of the dead. The famous "Three quarks for Muster Mark!" passage describes the desolation of Last Stand Hill. And Joyce's famous final paragraph contains an account of the legend of the narrow escape of Custer's Crow scout Ashishishe, who, at the battle's close, hides in the carcass of a horse before making his escape downriver. A brief look at Joyce's choice of names closes out the article. (shrink)