The article investigates a system of polymorphically typed combinatory logic which is equivalent to Gödel’s T. A notion of (strong) reduction is defined over terms of this system and it is proved that the class of well-formed terms is closed under both bracket abstraction and reduction. The main new result is that the number of contractions needed to reduce a term to normal form is computed by an ε0-recursive function. The ordinal assignments used to obtain this result (...) are also used to prove that the system under consideration is indeed equivalent to Gödel’s T. It is hoped that the methods used here can be extended so as to obtain similar results for stronger systems of polymorphically typed combinatory terms. An interesting corollary of such results is that they yield ordinally informative proofs of normalizability for sub-systems of second-order intuitionist logic, in natural deduction style. (shrink)

Abstract.This paper is the second in a series of three culminating in an ordinal analysis of Π12-comprehension. Its objective is to present an ordinal analysis for the subsystem of second order arithmetic with Δ12-comprehension, bar induction and Π12-comprehension for formulae without set parameters. Couched in terms of Kripke-Platek set theory, KP, the latter system corresponds to KPi augmented by the assertion that there exists a stable ordinal, where KPi is KP with an additional axiom stating that (...) every set is contained in an admissible set. (shrink)

Though the Random Utility Model (RUM) was conceived entirely in terms of ordinal utility, the apparatus through which it is widely practised exhibits properties of cardinal utility. The adoption of cardinal utility as a working operation of ordinal is perfectly valid, provided interpretations drawn from that operation remain faithful to ordinal utility. The article considers whether the latter requirement holds true for several measurements commonly derived from RUM. In particular it is found that measurements of consumer (...) surplus change may depart from ordinal utility, and exploit the cardinality inherent in the practical apparatus. (shrink)

In this paper, we explore the effects of certain forbidden substructure conditions on preordered sets. In particular, we characterize in terms of these conditions those preordered sets which can be represented as the supremum of a well-ordered ascending chain of lowersets whose members are constructed by means of alternating applications of disjoint union and ordinal sums with chains. These decompositions are examples of ordinal decompositions in relatively normal lattices as introduced by Snodgrass, Tsinakis, and Hart. We conclude (...) the paper with an application to information systems. (shrink)

In this article we investigate whether the following conjecture is true or not: does the addition-free theta functions form a canonical notation system for the linear versions of Friedman’s well-partial-orders with the so-called gap-condition over a finite set of n labels. Rather surprisingly, we can show this is the case for two labels, but not for more than two labels. To this end, we determine the order type of the notation systems for addition-free theta functions in terms of ordinals (...) less than ε0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _0$$\end{document}. We further show that the maximal order type of the Friedman ordering can be obtained by a certain ordinal notation system which is based on specific binary theta functions. (shrink)

Transfinite ordinal numbers enter mathematical practice mainly via the method of definition by transfinite recursion. Outside of axiomatic set theory, there is a significant mathematical tradition in works recasting proofs by transfinite recursion in other terms, mostly with the intention of eliminating the ordinals from the proofs. Leaving aside the different motivations which lead each specific case, we investigate the mathematics of this action of proof transforming and we address the problem of formalising the philosophical notion of elimination (...) which characterises this move. (shrink)

L’admission des femmes à l’ordination presbytérale ou au pastorat dans les Églises issues de la Réforme est liée à la situation historique particulière des États-Unis. Actuellement, aucun argument biblique ou théologique n’est jugé insurmontable pour l’ordination des femmes. Il existe certes de vives tensions, mais des ruptures effectives interviennent plutôt à la suite de l’ordination d’hommes et de femmes homosexuels. L’admission des femmes au ministère ordonné semble reposer sur l’hypothèse qu’un clergé mixte à tous les degrés du sacrement de l’ordre (...) assure à une Église la stabilité et la paix interne. Les débats au sein de ces Églises ont montré la faiblesse d’une interprétation de l’Écriture en termes de complémentarité entre l’homme et la femme. Ils les ont aussi conduites à une redécouverte de la grande variété de la tradition ecclésiale en la matière. L’article pose la question des enseignements que l’Église catholique romaine devrait à l’avenir tirer de l’expérience des Églises chrétiennes des États-Unis. (shrink)

Contrary to appearances, Machiavelli's Art of War is a carefully ordered whole. The order of treatment of topics in the body of the work deviates from the order initially proposed by the dialogue's main interlocutor, yet it strictly follows another order, inscribed in the text itself, which is neither accidental nor dependent on any ancient source. A clear understanding of this new order, and of the dramatic event that precipitates it, illuminates key questions, including the work's dialogic character, its relation (...) to its ancient sources, Machiavelli's apparent deprecation of cavalry as opposed to infantry, and, most importantly, Machiavelli's own long-term impresa to blaze an altogether new path in political thought. More specifically, attention to the order of the military section suggests that the dialogue's main speaker and his young interlocutors become the commanders of an army-in-speech on the march against an entire way of life whose origins are found in Christianity. (shrink)

There is a received view on measurement scales. It includes both a classification of scales and a set of prescriptions regarding measurement inferences. This article casts doubt on the adequacy of this received view. To do this, I propose an epistemic characterization of the ordinal/interval distinction, that is, one in terms of researchers’ beliefs. This novel characterization reveals the ordinal/interval distinction as too coarse grained and thus the received view as too restrictive of a framework for measurement (...) research. (shrink)

Quando si usa il termine laico, occorre sempre rispondere ad alcune domande implicite: laico rispetto a chi? Quando? Un percorso tra le varie classificazioni che si possono trovare nei testi medievali. Nella convinzione che la struttura stessa della classificazione sia importante, l'articolo procede sulla base di un criterio quantitativo, facendo riferimento al numero di classi o generi identificati, a partire dalla divisione più semplice. Una passaggio dal semplice al complesso alla luce della centralità del binomio concettuale di unità-differenza.When we use (...) the term laico, we must ever answer to some implicit questions: laico compared to whom? When? A path among the various classifications that we can find in medieval texts. Persuaded that the very structure of the classification is important, the paper goes on, according to a quantitative criterion, by reference to the number of identified classes or genres, starting from the simplest partition. A transition from the simple to the complex in the light of the centrality of two-faced concept of unit-difference. (shrink)

This paper is intended to give for a general mathematical audience a survey of intriguing connections between analytic combinatorics and logic. We define the ordinals below ε0 in non-logical terms and we survey a selection of recent results about the analytic combinatorics of these ordinals. Using a versatile and flexible compression technique we give applications to phase transitions for independence results, Hilbert’s basis theorem, local number theory, Ramsey theory, Hydra games, and Goodstein sequences. We discuss briefly universality and renormalization (...) issues in this context. Finally, we indicate how regularity properties of ordinal count functions can be used to prove logical limit laws. (shrink)

How do ordinals measure the strength and computational power of formal theories? This paper is concerned with the connection between ordinal representation systems and theories established in ordinal analyses. It focusses on results which explain the nature of this connection in terms of semantical and computational notions from model theory, set theory, and generalized recursion theory.

This study aimed to investigate whether the ordinal judgments of high math-anxious and low math-anxious individuals differ. Two groups of 20 participants with extreme scores on the Shortened Mathematics Anxiety Rating Scale had to decide whether a triplet of numbers was presented in ascending order. Triplets could contain one-digit or two-digit numbers and be formed by consecutive numbers, numbers with a constant distance of two or three or numbers with variable distances between them. All these triplets were also presented (...) unordered: sequence order in these trials could be broken at the second or third number. A reverse distance effect of equal size was found in both anxiety groups. However, HMA participants made more judgment errors than their LMA peers when they judged one-digit counting ordered triplets. This effect was related to worse performance of HMA individuals on a symmetry span test and might be related to group differences on working memory. Importantly, HMAs were less accurate than LMA participants at rejecting unordered D2 sequences. This result is interpreted in terms of worse cognitive flexibility in HMA individuals. (shrink)

We introduce the appropriate iterated version of the system of ordinal notations from [G1] whose order type is the familiar Howard ordinal. As in [G1], our ordinal notations are partly inspired by the ideas from [P] where certain crucial properties of the traditional Munich' ordinal notations are isolated and used in the cut-elimination proofs. As compared to the corresponding “impredicative” Munich' ordinal notations (see e.g. [B1, B2, J, Sch1, Sch2, BSch]), our ordinal notations arearbitrary (...) terms in the appropriate simple term algebra based on the notion of collapsing functions (which we would rather identify as projective functions). In Sect. 1 below we define the systems of ordinal notationsPRJ( ), for any primitive recursive limit wellordering . In Sect. 2 we prove the crucial well-foundness property by using the appropriate well-quasi-ordering property of the corresponding binary labeled trees [G3]. In Sect. 3 we interprete inPRJ( ) the familiar Veblen-Bachmann hierarchy of ordinal functions, and in Sect. 4 we show that the corresponding Buchholz's system of ordinal notationsOT( ) is a proper subsystem ofPRJ( ), although it has the same order type according to [G3] together with the interpretation from Sect. 2 in the terms of labeled trees. In Sect. 5 we use Friedman's approach in order to obtain an appropriate purely combinatorial statement which is not provable in the theory of iterated inductive definitions ID< λ, for arbitrarily large limit ordinalλ. Formal theories, axioms, etc. used below are familiar in the proof theory of subsystems of analysis (see [BFPS, T, BSch]). (shrink)

This text unites the logical and philosophical aspects of set theory in a manner intelligible both to mathematicians without training in formal logic and to logicians without a mathematical background. It combines an elementary level of treatment with the highest possible degree of logical rigor and precision. Starting with an explanation of all the basic logical terms and related operations, the text progresses through a stage-by-stage elaboration that proves the fundamental theorems of finite sets. It focuses on the Bernays (...) theory of finite classes and finite sets, exploring the system's basis and development, including Stage I and Stage II theorems, the theory of finite ordinals, and the theory of finite classes and finite sets. This volume represents an excellent text for undergraduates studying intermediate or advanced logic as well as a fine reference for professional mathematicians. (shrink)

This book is an introduction to the philosophy of Justus Buchler--so far as he has developed it to date. Singer carefully defines for us what Buchler means by his key terms, how he uses them, how he has developed them, and how his thought is related to the other philosophers of his time who are generally called "naturalists" of one sort or another.

Within the research field of computer-mediated communication, extensive attention has been paid to the differences between CMC and spoken conversation, particularly in terms of sequential structure. In this study, the aim is to analyse how participants maintain continuity and handle discontinuities in institutionally arranged, computer-mediated communication. The empirical material consists of chat log files from in-service training courses for professionals in the food production industry. In the chat sessions we analysed, participants initially had some problems in co-ordinating their postings, (...) that is, keeping track of the discussion’s threading.We also found, however, that methods for co-ordinating postings rather quickly were established among the participants. Two methods for making recognizable how postings were to be read were common in the material: 1) the use of personal names to address a specific participant, and 2) recycling part of the content of a previous posting or reformulating the content in the posting addressed in recognizable ways. We also found an emerging norm in their interaction: that of waiting for one’s turn. (shrink)

This note defends the reduction of ordinals to pure sets against an argument put forward by Beau Madison Mount. In the first part I will defend the claim that dependence simpliciter can be reduced to immediate dependence and define a notion of predecessor dependence. In the second part I will provide and defend a way to model the dependence profile of ordinals akin to Mount’s proposal in terms of immediate dependence and predecessor dependence. I furthermore show that my alternative (...) dependence profile allows us to single out the reduction of ordinals to von Neumann ordinals as the only viable set-theoretic reduction. (shrink)

Each of us has a story which we can think of as a set of beliefs, hopefully consistent. We make our decisions in view of our beliefs which may be probabilistic, in the general case, but simple yes or no as in this paper. Our beliefs are our envelope just as the shell of a tortoise is its envelope. Decision theory—or single agent game theory tells us when to make the best choice in a game of us against nature. But (...) nature has no desire to further or frustrate our efforts. Nature is mysterious but not malign. Things change when there are other agents involved. Then the best thing for us to do will depend on what they do. And they will think the same. And we predict their actions in terms of what we think their beliefs are. But how do we coordinate with others whose beliefs are different? This paper addresses the issue of working together despite different beliefs. (shrink)

We show the following results on Wainer's notation for a minimal subrecursive inaccessible ordinal τ: First, we give a constructive proof of the collapsing theorem. Secondly, we prove that the slow-growing hierarchy and the fast-growing hierarchy up to τ have elementary properties on increase and domination, which completes Wainer's proof that τ is a minimal subrecursive inaccessible. Our results are obtained by showing a strong normalization theorem for the term structure of the notation. MSC: 03D20, 03F15.

Despite many reservations, Transition to an Ordinal Metaphysics is interesting because it continues a tradition which takes the metaphysical enterprise seriously, which attempts to examine the nature and function of metaphysics, and which attempts to communicate the insights and conceptual power of what Ross terms the ordinal approach to that enterprise.

In this paper we study with proof-theoretic methods the function(al) s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functional provably recursive from $RCA_0 + COH + \Pi _1^0 - CP$ are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that $WKL_0 + \Pi _1^0 - (...) CP + COH$ is $\Pi _1^0 $ -conservative over PRA. Recent work of the first author showed that $\Pi _1^0 + CP + COH$ is equivalent to a weak variant of the Bolzano-Weierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs. For Ramsey's theorem for pairs and two colors $\left( {RT_2^2 } \right)$ we obtain the upper bounded that the type 2 functionals provable recursive relative to $RCA_0 + \sum\nolimits_2^0 {IA + RT_2^2 } $ are in T₁. This is the fragment of Gödel's system T containing only type 1 recursion—roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction of T₁-bounds from proofs that use RT2..... Moreover, this yields a new proof of the fact that $WKL_0 + \sum\nolimits_2^0 {IA} + RT_2^2 $ is $\Pi _1^0 $ -conservative over $RCA_0 + \sum\nolimits_2^0 { - IA} $ . The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel's functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using $\Pi _1^0 $ -comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard's ordinal analysis of bar recursion (for $RT_2^2 $. (shrink)

The symbolic argument against women's ordination supposes that the theological significance of Christ's sex is his saving relationship to the Church, which takes the form of that of a bridegroom and his bride. It infers that a male priest alone is fit to represent Christ in his capacity as the Saviour of the Church, and thus that only men should be ordained. Since the emergence of the symbolic argument, however, scholars have rediscovered a long tradition of understanding Christ's saving relationship (...) to the Church in maternal terms. While remaining neutral on whether women ultimately ought to be ordained or not, I argue that the kind of reasoning in the symbolic argument, if updated with the Jesus as Mother tradition, would suggest that it is fitting for a female priest to represent Jesus as Mother, just as it is fitting for a male priest to represent Christ the Bridegroom. Other critics of the symbolic argument tend to contest what is seen as its ‘literalist’ or ‘essentialist’ or overly gendered-valanced assumptions about priestly representation. I show that even if we grant to the symbolic argument more gendered and ‘essentialist’ views on each of the major points of disagreement in these debates, women might still be fit to represent Christ because of the maternal ways that Christ has traditionally been thought to relate to the Church. As a result, the symbolic argument may be repurposed to support women's ordination rather than undermine it. (shrink)

The Byzantine rite of ordination to the presbyterate culminates in the conferral of a portion of the Eucharist upon the newly-ordained by the bishop. After reviewing the historical roots of this rite, this article examines the philological and biblical background of the term. Parakatathêkê coincides with the notion of paradosis but goes beyond it in personal and eschatological senses. Three simultaneous but distinct acts of personal transmission are highlighted: the personal participation of the newly-ordained in the apostolic ministry of the (...) bishop; the handing over of Christ personally present in the Eucharist to the newly-ordained; and the corresponding personal surrender of the new priest to Christ, to whom he will render an account of his own realization of Christ's priesthood in the service of His Mystical Body, the Church. (shrink)

The standard method of generating countable ordinals from uncountable ordinals can be replaced by a use of fixed point extractors available in the term calculus of Howard’s system. This gives a notion of the intrinsic complexity of an ordinal analogous to the intrinsic complexity of a function described in Gödel’s T.

There are two ways of thinking about the natural numbers: as ordinal numbers or as cardinal numbers. It is, moreover, well-known that the cardinal numbers can be defined in terms of the ordinal numbers. Some philosophies of mathematics have taken this as a reason to hold the ordinal numbers as (metaphysically) fundamental. By discussing structuralism and neo-logicism we argue that one can empirically distinguish between accounts that endorse this fundamentality claim and those that do not. In (...) particular, we argue that if the ordinal numbers are metaphysically fundamental then it follows that one cannot acquire cardinal number concepts without appeal to ordinal notions. On the other hand, without this fundamentality thesis that would be possible. This allows for an empirical test to see which account best describes our actual mathematical practices. We then, finally, discuss some empirical data that suggests that we can acquire cardinal number concepts without using ordinal notions. However, there are some important gaps left open by this data that we point to as areas for future empirical research. (shrink)

This paper proposes a revised Theory of Moves (TOM) to analyze matrix games between two players when payoffs are given as ordinals. The games are analyzed when a given player i must make the first move, when there is a finite limit n on the total number of moves, and when the game starts at a given initial state S. Games end when either both players pass in succession or else a total of n moves have been made. Studies are (...) made of the influence of i, n, and S on the outcomes. It is proved that these outcomes ultimately become periodic in n and therefore exhibit long-term predictable behavior. Efficient algorithms are given to calculate these ultimate outcomes by computer. Frequently the ultimate outcomes are independent of i, S, and n when n is sufficiently large; in this situation this common ultimate outcome is proved to be Pareto-optimal. The use of ultimate outcomes gives rise to a concept of stable points, which are analogous to Nash equilibria but consider long-term effects. If the initial state is a stable point, then no player has an incentive to move from that state, under the assumption that any initial move could be followed by a long series of moves and countermoves. The concept may be broadened to that of a stable set. It is proved that every game has a minimal stable set, and any two distinct minimal stable sets are disjoint. Comparisons are made with the results of standard TOM. (shrink)

The termination of rewrite systems for parameter recursion, simple nested recursion and unnested multiple recursion is shown by using monotone interpretations both on the ordinals below the first primitive recursively closed ordinal and on the natural numbers. We show that the resulting derivation lengths are primitive recursive. As a corollary we obtain transparent and illuminating proofs of the facts that the schemata of parameter recursion, simple nested recursion and unnested multiple recursion lead from primitive recursive functions to primitive recursive (...) functions. (shrink)

In 1986 David Gauthier proposed an arbitration scheme for two player cardinal bargaining games based on interpersonal comparisons of players’ relative concessions. In Gauthier’s original arbitration scheme, players’ relative concessions are defined in terms of Raiffa-normalized cardinal utility gains, and so it cannot be directly applied to ordinal bargaining problems. In this paper I propose a relative benefit equilibrating bargaining solution for two and n-player ordinal and quasiconvex ordinal bargaining problems with finite sets of feasible basic (...) agreements based on the measure of players’ ordinal relative individual advantage gains. I provide an axiomatic characterization of this bargaining solution and discuss the conceptual relationship between RBEBS and ordinal egalitarian bargaining solution proposed by Conley and Wilkie. I show the relationship between the measurement procedure for ordinal relative individual advantage gains and the measurement procedure for players’ ordinal relative concessions, and argue that the proposed arbitration scheme for ordinal games can be interpreted as an ordinal version of Gauthier’s arbitration scheme. (shrink)

This article sets out to analyse some of the most basic elements of our number concept - of our awareness of the one and the many in their coherence with multiplicity, succession and equinumerosity. On the basis of the definition given by Cantor and the set theoretical definition of cardinal numbers and ordinal numbers provided by Ebbinghaus, a critical appraisal is given of Frege’s objection that abstraction and noticing (or disregarding) differences between entities do not produce the concept of (...) number. By introducing the notion of subject functions, an account is advanced of the (nominalistic) reason why Frege accepted physical, kinematic and spatial properties (subject functions) of entities, but denied the ontic status of their quantitative properties (their quantitative subject function). With reference to intuitionistic mathematics (Brouwer, Weyl, Troelstra, Kreisel, Van Dalen) the primitive meaning of succession is acknowledged and connected to an analysis of what is entailed in the term ‘Gleichzahligkeit’ (‘equinumerosity’). This expression enables an analysis of the connections between ordinality and cardinality on the one hand and succession and wholeness (totality) on the other. The conceptions of mathematicians such as Frege, Cantor, Dedekind, Zermelo, Brouwer, Skolem, Fraenkel, Von Neumann, Hilbert, Bernays and Weyl, as well as the views of the philosopher Cassirer, are discussed in order to arrive at an assessment of the relation between ordinality and cardinality (also taking into account the relation between logic and arithmetic) - and on the basis of this evaluation, attention is briefly given to the definition of an ordered pair in axiomatic set theory (with reference to the set theory of Zermelo-Fraenkel) and to the defmition of an ordered pair advanced by Wiener and Kuratowski. (shrink)

La dimensione teorica della musica occidentale, nella sua evoluzione storica, ha inevitabilmente fatto uso di concetti essenziali alla definizione di come il suono è musicalmente organizzato o organizzabile. Fra questi, risultano imprescindibili le nozioni di materia, forma e ordine, che implicano rispettivamente, nei pur diversi ambiti linguistici e contesti storico-filosofici di riferimento, ciò di cui è fatta la musica, ciò a cui la materia sonora tende, e come tale tensione è realizzata. Scopo di questo contributo è una valutazione d’insieme sulla (...) valenza filosofica e l’uso di tali concetti nel pensiero musicale occidentale altomedievale e in quello del Novecento, che nonostante la distanza cronologica presentano convergenze significative. Nel Medioevo, un lungo processo di elaborazione teorico-pratica permise la matematizzazione dello spazio musicale, attraverso l’adattamento di moduli melodici, trasmessi oralmente, alle strutture scalari desunte dalla teoria greco-boeziana. La forma e l’ordo musicali furono quindi riferiti ai criteri fisico-matematici atti a distinguere nella materia sonora, in sé disorganica e indefinita, ciò che è musicale da ciò che non lo è. Tale concezione sparisce con l’affermarsi del sistema tonale, ove il suono fisico è concepito come “pre-formato” e “pre-ordinato” nella gamma scalare e nei vincoli attrattivi “naturali” degli armonici. In conseguenza, i concetti di forma e ordine abbandonano il contesto acustico, andando a definire i criteri teorici della composizione artistica, in linea con l’idea filosofica del loro esprimere la coerenza logico-sintattica dell’“insieme organico” . Nell’ambito del pensiero musicale del Novecento, la crisi del sistema tonale e la messa a punto di nuove modalità e tecniche di produzione del suono hanno riorientato la valenza significante e l’uso di questi tre concetti, implicando il loro tornare a imporsi sugli elementi primari della fisica del suono, non solo l’altezza, ma soprattutto il timbro, il volume e la durata. Già Schönberg, nel 1911, affermava come “l’ordine che chiamiamo forma artistica” è “un necessario espediente” elaborato “a causa della nostra incapacità a comprendere l’indistinto e il non-ordinato”, che è la condizione propria della materia sonora. In its historical development, the Western theory of music has inevitably made use of specific concepts to define how the sound is musically organized or organizable. Among these notions, those of matter, form and order signify – in spite of the different linguistic areas and the various historical and philosophical backgrounds – that from which music is made, that to which it tends and how such tendency is realized. This paper aims to evaluate the philosophical implications and the use of these three concepts in the musical theories of the high Middle Ages and of the twentieth century, since, in spite of their chronological distance, they present significant similarities. A long process of theoretical and practical adjustments of the melodic structures of liturgical chants to the scale-systems of the Greek and Boethian theory contributed to developing what would become a familiar notion throughout the Middle Ages: that musical intervals can be expressed in mathematical terms. According to this view, the words forma and ordo signify the mathematical measures that are applied to the “matter” of sound, which is considered as inorganic and indefinite in itself. This conception gradually disappears when the tonal system begins to emerge, since this new modality of composition implies that the physical sound is “pre-formed” and “pre-ordered” within the musical scale, according to the “natural” links of the harmonics. Consequently, the concepts of “form” and “order” do not refer any more to the context of sound generation, but are now used to define the theoretical criteria of the artistic composition, so to accomplish the new philosophic idea of their expressing the logical-syntactic coherence of the “organic-whole” . In the early twentieth century, when the tonal system starts to be replaced by new methods of composition and when new modalities and techniques of sound production emerge, those three concepts are used, again, to express the basic elements of the physics of sound: not only its height, but also its timbre, volume, and duration. In 1911, Schönberg argued that the order that we call “artistic form” is a necessary device, elaborated “because of our incapacity to comprehend the indistinct and unordered thing”, that is the own condition of the matter of sound. (shrink)

Since the late 1980s, in defiance of Sri Lanka’s major monastic fraternities and the government, Buddhist women and men have begun to organize across distinctions of national boundary and Buddhist tradition to reinstate a defunct bhikkhun? ordination lineage for renunciant women. Drawing on fieldwork from the winter of 2015–16, this article considers some of the strategies by which Sri Lanka’s bhikkhun?s and their supporters constitute the burgeoning lineage as both legitimate and necessary for the continued health and vitality of an (...) otherwise ailing Buddhist s?sana. I argue that Sri Lanka’s bhikkhun?s engage in highly-visible forms of adherence to vinaya rules and social expectations for ideal monastic behavior set against a popular discourse about the laxity of male renunciants. Such engagement is both political and soteriological; while it is aimed at fulfilling legitimizing gendered expectations of women’s piety, it is expressed primarily in terms of the eradication of personal and societal suffering through forms of practice that accord with the ideal of a pious monastic. Thus, in contrast to discourses which locate bhikkhun?s as subjects whose presence weakens the s?sana’s duration and strength, in this new discourse Sri Lanka’s bhikkhun?s become virtuous agents of social service and moral restoration. The article concludes by identifying emerging connections between this discourse and an alreadygendered xenophobic Buddhist nationalism. (shrink)

In this essay, I explore Justus Buchler’s ordinal naturalism with the goal of establishing how his phenomenological approach extends the range of human inquiry to include the many and varied traits of natural phenomena that are not “simply” the result of sensate experience or material functions. To achieve this goal I critically assess Buchler’s notion of “ontological parity”–the idea that abstract phenomena such as values, relations, ideals, and other mental contents are just as relevant as sense-data when one attempts (...) to provide an adequate description of the world in naturalistic terms. I argue that certain phenomena, subsisting within what Buchler calls the “proceptive domain,” are legitimate objects of knowledge as they are par t of a larger domain of phenomenological analysis: nature more broadly and justly understood. It is my view that in the attempt to describe the natural world Buchler’s ordinal naturalism succeeds where other forms of naturalism fail because his form of naturalism offers a more capacious view of nature that attempts to describe whatever is in any way, not just focus on what is readily apparent to speci?c forms of observation that may privilege one domain of analysis over another. I draw the conclusion that because Buchler’s ordinal naturalism contains within it a working principle of ontological parity, his approach to nature ful?lls the criteria of the phenomenological method, and so I title his ordinal naturalism an ordinal phenomenology (Corrington 1992, 1-6, 9-14). Ultimately it is my aim to bring Buchler’s thought into closer connection with continental phenomenology, as well as to illustrate a more just and open understanding of nature through an analysis of his unique variety of philosophical naturalism. (shrink)

Despite the introduction of female bishops, women do not hold offices on equal terms with men in the Church of England, where conservative evangelical male clergy often reject the validity of women’s ordination. This article explores the gender values of such clergy, investigating how they are expressed and the factors that shape them. Data is drawn from semi-structured interviews and is interpreted with thematic narrative analysis. The themes were analyzed with theories on postfeminism, engaged orthodoxy and group schism. It (...) is argued that participants’ gender values are best understood as postfeminist and that the wider evangelical tradition, as well as a perceived change in Anglican identity with the onset of women’s ordination, shape their postfeminism. Moreover, whilst evangelical gender values possess the potential to foster greater gender equality within the Church of England, gender differentiation limits this possibility, a limitation that could be addressed by increasing participants’ engagement beyond the Church. (shrink)

“A little masterpiece of originality and clarity.”—George Steiner “A necessary book.”—Roberto Saviano “A wonderful little book that will delight you.”—François Busnel International Best Seller / Now in English for the First Time In this thought-provoking and extremely timely work, Nuccio Ordine convincingly argues for the utility of useless knowledge and against the contemporary fixation on utilitarianism—for the fundamental importance of the liberal arts and against the damage caused by their neglect. Inspired by the reflections of great philosophers and writers (e.g., (...) Plato, Dante, Montaigne, Shakespeare, Borges, and Calvino), Ordine reveals how the obsession for material goods and the cult of utility ultimately wither the spirit, jeopardizing not only schools and universities, art, and creativity, but also our most fundamental values—human dignity, love, and truth. Also included is Abraham Flexner’s 1939 essay “The Usefulness of Useless Knowledge,” which originally prompted Ordine to write this book. Flexner—a founder and the first director of the Institute for Advanced Study in Princeton—offers an impassioned defense of curiosity-driven research and learning. (shrink)

Buchholz presented a method to build notation systems for infinite sequent-style derivations, analogous to well-known systems of notation for ordinals. The essential feature is that from a notation one can read off by a primitive recursive function its n th predecessor and, e.g. the last rule applied. Here we extend the method to the more general setting of infinite terms, in order to make it applicable in other proof-theoretic contexts as well as in recursion theory. As examples, we use (...) the method to 1. give a new proof of a well-known trade-off theorem , which says that detours through higher types can be eliminated by the use of transfinite recursion along higher ordinals, and 2. construct a continuous normalization operator with an explicit modulus of continuity. (shrink)

The Commission for Religious Relations with the Jews produced The Gifts and Calling of God Are Irrevocable, in which supersessionism is firmly rejected. In this document, the term fulfillment occurs frequently to describe the relationship between the Old and New Covenant. It implies an evolutionary development from old to new, or from promise to fulfillment. Therefore, the use of this term may lead one to suspect that it is merely a synonym for supersession or a progression from good to better. (...) To avoid this connotation, I redefine this term by locating it within the Israelite cult. Through a study of Aaron’s ordination to the high priesthood in Leviticus and the claims for Jesus’s high priesthood in the Epistle to the Hebrews, I show that fulfillment already occurs in the Old Covenant by relating the historical, earthly cult to the eternal, heavenly cult. (shrink)

In this highly original study, Nuccio Ordine uses the figure of the ass as a lens through which to focus on the thought and writings of the great Renaissance humanist philosopher Giordano Bruno.

The Bachmann-Howard structure, that is the segment of ordinal numbers below the proof theoretic ordinal of Kripke-Platek set theory with infinity, is fully characterized in terms of CARLSON’s approach to ordinal notation systems based on the notion of Σ1-elementarity.