This is a re-look at the (Indian) Partition event through the lens of Advaita Vedanta.

It is now just over one hundred years since the beginning of the mathematical partnership between the Cambridge analyst G. H. Hardy and the Indian mathematical genius Srinivasa Ramanujan, one of the most celebrated collaborations in the history of mathematics. Indeed, the story of how Ramanujan was brought from India to Cambridge and feted by the British mathematical establishment now borders on legendary. But, in the context of this collection of articles, it provides an interesting case study of mathematical (...) exchange. This paper considers one particular product of the Hardy-Ramanujan creative partnership: their 1918 paper on partitions, and argues that the exchange of ideas prompted by their work on this paper was facilitated in part by their membership of the premier learned body in Britain for the advancement of mathematical research: the London Mathematical Society. (shrink)

It is now just over one hundred years since the beginning of the mathematical partnership between the Cambridge analyst G. H. Hardy and the Indian mathematical genius Srinivasa Ramanujan, one of the most celebrated collaborations in the history of mathematics. Indeed, the story of how Ramanujan was brought from India to Cambridge and feted by the British mathematical establishment now borders on legendary. But, in the context of this collection of articles, it provides an interesting case study of mathematical (...) exchange. This paper considers one particular product of the Hardy-Ramanujan creative partnership: their 1918 paper on partitions, and argues that the exchange of ideas prompted by their work on this paper was facilitated in part by their membership of the premier learned body in Britain for the advancement of mathematical research: the London Mathematical Society. (shrink)

This article examines Gandhi's writings, speeches, and correspondence, produced mainly from 1946 to the end of his life, on the subject of violence against women during the riots surrounding the Partition of the Indian subcontinent. Gandhi, the article demonstrates, persistently fails to address the gender pathology revealed at the heart of South Asian society by these violations, a pathology to which men were subject, but of which women were the victims. The article compiles a comprehensive overview of Gandhi's (...) shifting positions during this brief, though cataclysmic, period, in the belief that in so doing a certain core aspect of mainstream Indian nationalism's patriarchal underpinnings can be laid bare to critique. (shrink)

Presents the fundamental ideas of Indian thinkers that have shaped the mind of Indian from 1770 to the post-modern era in the middle of 20th century in India. Lists the most Indian influential figures in the field of philosophy, political theory, activicism such as Rabindranath Tagore, Ram Mohan Roy, Swami Vivekananda, and Mohandas Karamchand Gandhi.

This series of interviews, conducted in the form of exchanges between Roy Bhaskar and Mervyn Hartwig, tells a riveting story of the formation and development of critical realism. Three intersecting and interweaving narratives unfold in the course of this unfinished story: the personal narrative of Roy Bhaskar, born of an Indian father and English mother, a child of post-war Britain and Indian partition and independence; the intellectual narrative of the emergence and growth of critical realism; and a (...) world-historical story, itself theorized by critical realism in its discussion of the development of modernity. This book gives an invaluable account of the development of critical realism, and its consolidation as a leading philosophy of our times. It takes us through the major moments of its formation, the principal objections to and controversies within critical realism, the establishment of its institutions, and considers its limits and future development. Special features of the book include discussion of the genesis of critical realism, and the origins and nature of the so-called dialectical and spiritual turns. The informal dialogical style of The Formation of Critical Realism makes it compelling reading and an invaluable source for students of critical realism as well as all those interested in the intellectual story of our times. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Enlightenment in the Colony: The Jewish Question and the Crisis of Postcolonial CultureSpencer Hawkins (bio)Mufti, Aamir. Enlightenment in the Colony: The Jewish Question and the Crisis of Postcolonial Culture. Princeton UP, NJ: Princeton, 2007. xv + 325 pp.Mufti’s comparison of the Jewish question and the Indian Partition invites readers to join building projects that delineate and then endanger minorities within nations. Literature about minorities speaks a (...) language deliberately distant from the national one. Minority status has little to do with population numbers for Aamir Mufti; instead it signifies [End Page 61] tential victimization through disproportional rights, since “minority is always potentially exile” (Mufti 13). This definition of minority suits his two examples—Jews in Germany and Muslims in India—since in the twentieth century both fledgling nations adopted forms of nationalism that regarded the lives of these minorities as contaminants to the purity of the emergent national body. The prologue catalogues the brutality in India that prompted Mufti to write this book. Among the horrors listed were the destruction of Babri Mosque at Ayodhya in 1992 and the videotaped gang rapes of Muslim women that became a widespread form of entertainment in India. While Mufti reads violence toward minorities as correlative with the rise of nationstates, he considers similarities and differences between nation building in the West and in the subcontinent. In both cases, the new establishment of political equality for citizens came along with the desire to pare down the number of officially recognized citizens. Muslim and Jewish minorities migrated to “minority” nations in the collapsing British Empire, in present-day Pakistan and Israel, respectively, during the respective formations of India and Germany. By founding nations on formerly British-occupied territory, minorities potentially would enjoy the role of nationalist majorities. This solution strikes Mufti as spreading the nationalist contagion onto the minority.The whole book lets literature mark broad cultural attitudes and phenomena. The first chapter observes incipient Jewish nationalism in a work by a British author, Daniel Deronda by George Eliot, who spent time in Germany and grew fascinated with the Jewish question while abroad. Her Britishness reinforces a pattern in Mufti’s schema: British powers facilitate the rise of nationalism among minorities in India and in Germany. Daniel Deronda comes at the tail end of a long discourse on Jews’ relation to Europe. The first of Mufti’s cases within canonical literature is the hope for tolerance and a dialogue of cultures, which G. E. Lessing allegorizes through a drama about Saladin, the Muslim sultan of Egypt and Syria, who learns to trust the eponymous Jewish merchant in Nathan the Wise. “Muslim” stands in for “Christian” since Nathan’s alienation is allegorical for the alienation of Jews in Lutheran Germany. Heinrich Heine weighs in on the advantages to assimilation in this period with his fragmentary novel The Rabbi from Bacherach, where religiously raised Jewish children feel comforted when they hear the Tale of the Nibelungen from Father Rhein, the personification of a river that figures generally for German national myth and, in Heine’s story, as a possible source of comfort for partially assimilated minorities. If we read literature as a barometer of cultural pressures, then the appeal of assimilation for European Jews declines in the late colonial nineteenth century when George Eliot inscribed religion into the “secular” discourse of national citizenship.Since the Israeli state is outside of the range of his argument, we leave the Jewish Question with George Eliot’s philo-Semitism in Daniel Deronda, where the only hope for European Jews is to become a nation and to occupy the Palestinian territory. Mufti compares the concerns of early Zionism and Pakistani nationalism side by side in chapter 2. Pakistani issues dominate chapters 3 to 5. In chapter 2, he launches the comparison with the parallel between Hebrew’s development as a national language to identify [End Page 62] Jews secularly and culturally, not just religiously, and Urdu’s transition from a vernacular language to a national language with the secular goal of creating a Pakistani culture freed from Hindi’s erstwhile predominance. When he depicts language choice as mimetic of specific cultural trends, Mufti indicates the shape of... (shrink)

There is no safe place and no single locus of enunciation from where the uni-versal could be articulated for all and forever. Hindu nationalism and Western neo-liberalism are entangled in a long history of the logic of coloniality (domination, oppression, exploitation) hidden under the rhetoric of modernity (salvation, civilization, progress, development, freedom and democracy). There are, however, needs and possibilities for Indians and Western progressive intellectuals working together to undermine and supersede the assumptions that liberal thinkers in the West are (...) better placed to understand what is the common good better than Indian thinkers in post-partition India. Science, in the last analysis, doesn't carry in itself an ethics and politics. Therefore, it is doubtful to argue that science is beyond both, and only concerned with the advancement of the frontier of knowledge and understanding. Science could be (like Christianity, Hinduism, Liberalism or Marxism) both imperial and liberating. Knowledge and understanding, rather than science; gnoseology rather than epistemology, should be thought out as which the horizon toward a dialogical and critical cosmopolitanism (e.g., pluriversallity as universal project), could be envisioned beyond East and West, Hindu nationalism and Westertn (neo) liberalism. (shrink)

The study brings about the contribution of Islam to world civilization. It traces the rise of Muslim power in Asia, Europe and Africa over more than ten centuries combining political ascendancy with promotion of basic sciences, philosophy, literature and arts. It deals with the prevailing myths which have for long blocked the way to a correct understanding of the faith and the traditions of Islam. Vigorously debated issues, such as jihad, status of women, and national patriotism are discussed in their (...) historical context. The book provides lucid exposition of theological concepts, the schools oflaw, the Shariah, the on-going conflict between orthodoxy and modernism, and Sufism, on the basis of the Quran and the Hadith. It gives interesting account of the willing acceptance in medieval Europe of Islamic scholarship in sciences along with bitter hostility against Islam in the ecclesiastical establishment of Christian period. The nature of the Muslim rule and its relationship with non-Muslim communities in the Caliphates, in the Ottoman realm and in the Sultanate and Mughal empire in the Indian subcontinent has been evaluated. The events and the personalities contributing to India's partition in 1947 are examined to assess the truth of the so-called separatism in the mind of Indian Muslims. The book gives the background of the tension between Muslim countries and Western nations which intensified with the conclusion of World War I in 1918 and rose to be acute with the end of the Cold War in 1990. The continued political, economic, and military intervention by European powers and the U.S.A. has been discussed in the context of the present situation when Muslims tend to look upon the West as anti-Islam and in the Western world-view Islam is considered to be a menace to civilization itself. (shrink)

Preface -- Part I: Philosophical logic and philosophy of language -- Rules versus theorems : a new approach for mediation -- Between intuitionistic and two-valued logic -- On the relation between the partition of a whole into parts and the attribution of properties to an object -- Basic objectives of dialogic logic in historical perspective -- Pragmatic and semiotic prerequisites for predication : a dialogue model -- Pragmatics and semiotics : the peircean version of ontology and epistemology -- Intentionality (...) and its language-dependency -- Meaning postulates and rules of argumentation : remarks concerning the pragmatic tie between meaning (of terms) and truth (of propositions) -- What do language games measure? -- Features of Indian logic -- Part II: Methods in philosophy, in art, and in science -- The concept of science : some remarks on the methodological issue construction versus description in the philosophy of science -- Is and ought revisited -- Competition and cooperation : are they antagonistic or complementary? -- Another version of methodological dualism -- The pre-established harmony between the two Adams -- On the way to conceptual and perceptual knowledge -- Self and other : remarks on human nature and human culture -- On the concept of symmetry -- Procedural principles of the Eerlangen School : on the interrelation between the principles of method, of dialogue, and of reason. (shrink)

This paper is an attempt to make sense of the notion and ideal of nonviolence in these ultra-violent days. The paper is a dialogue with three “specialists” of violence, who nevertheless aspire to a different, brighter horizon: Ramchandra Gandhi, Saadat Hasan Manto and Mahasweta Devi. R. Gandhi is one of the most intriguing voices of twentieth-century Indian philosophy. Manto and Mahasweta are writers, the former known for his short partition stories in Urdu; the latter for her gut-wrenching literature (...) in Bengali. All three dare to look violence in the eye, implying that nonviolence can only emerge from deep reflection on violence as an inherent human tendency. Violence is part of me as much as of anyone else. R. Gandhi argues that partition, the cradle of violence, is in the eye, and suggests that we can train the human gaze, our gaze, to prioritize the common denominator between you and I, which hides under the obvious differences between us. For Manto, the remedy is to be found in language. He implies that an ethical dimension is concealed within language, waiting to be excavated. Mahasweta gives voice to those unheard. Acknowledging the unacknowledged, she and Manto show us, is an act of nonviolence. (shrink)

There is a long-standing myth that the history of modern India was foretold at the beginning of the nineteenth century by British liberals who predicted that the enlightened despotic rule of India's new conquerors would, by its beneficial effects, improve the native character and institutions sufficiently to prepare the people of that country one day to govern themselves. Lord William Bentinck, a disciple of Jeremy Bentham, while presenting as governor-general his case for the opening up of India to European settlers, (...) speculated on the possibility of “a vast change to have occurred in the frame of society . . . which would imply that the time had arrived when it would be wise for England to leave India to govern itself”, but added that such change “can scarcely be looked for in centuries to come”. The doctrinal basis within liberal theory for justifying a democratic country like Britain exercising despotic power in colonies such as Ireland and India was securely laid out by mid-century liberals such as John Stuart Mill. The project of “improvement” was revived at the end of the nineteenth century by Gladstonian liberals who inducted elite Indians into new representative institutions based on a very narrow franchise in preparation for some form of self-government. When power was ultimately transferred to the rulers of a partitioned subcontinent in 1947, the history of liberal progress in India was complete. The storyline was laid out, for instance, in Thompson and Garratt's Rise and Fulfilment of British Rule in India or in Percival Spear's revised edition of the hugely successful textbook by Vincent Smith. Even nationalist Indian scholars adopted at least a part of this story, nowhere more so than in the histories of constitutional law which traced the foundations of the postcolonial Indian republic to the progressive expansion of liberal state institutions under British rule. (shrink)

This Paper is divided into three sections. In the first section I provide a brief historical overview of Hindu-Muslim relations in India and of the condition of Indian Muslims today. I conclude by claiming that Indian Muslims are a marginalized minority who have been persistently underrepresented in political institutions, particularly in the Indian Parliament. This section is important for those who are less informed about these issues—and I assume that most readers fall in this category. In the (...) second section, I examine the case for political representation for Muslims. This was a much debated issue in pre-independent India. It was debated with subtlety and in considerable detail in the Constituent Assembly debates on the Indian constitution. However, with the partition of the country and the formation of the separate state of Pakistan, all debate on the political representation of Muslims ceased. I examine the merits and demerits of the case for the political representation of Indian Muslims. I also attempt a brief explanation of why this issue has virtually disappeared from the public arena in India. I conclude in the section that although political representation of Muslims qua Muslims is desirable, it is still unfeasible in the prevailing situation in India. In other words, I would support the recommendation to the Indian State that political rights not to be granted to any religious community. If political theory was to remain a handmaiden of state policy, then the matter ends right here. However, since I believe that political theory must think for the long run and design just institutions and policies for the future, and since, there is, I claim, no principled objection to the political representation of Muslims, in the third and final section I briefly outline which of the several electoral mechanisms are best suited to ensure fair political representation for Muslims in the future. In my view, the principle of fair political representation for Indian Muslims is best fulfilled by a complex mechanism consisting of preferential voting in multi-member constituencies with intra-party quotas in proportion to the overall population of Muslims in the country. (shrink)

The world witnessed a major historical event in 1947 when subcontinent, which was governed as a one unit from Khyber to Burma since almost last one thousand years, partitioned by the ruling British Empire resulting into two states namely India and Pakistan. The major reason behind partition of the subcontinent was the religious and cultural differences between the Hindus and Muslims. This difference made them hostile towards each other and India having superiority in all aspects, compelled Pakistan to become (...) a security state right after its inception. To expand its superiority over the whole region, the Indian nuclear program started in 1944, even before its independence. India conducted first nuclear tests in 1974 and continued expanding its nuclear program. This forced Pakistan to work seriously on its nuclear program with a fast pace. India announced its formal entry to the nuclear weapons club in May 1998 with 5 nuclear tests at Pokhran, Pakistan, having nuclear capability at that time, replied back in merely two weeks with 6 nuclear tests and became the 7th nation in the world and first Islamic country to join the nuclear countries club. The lives of over a billion people are at mercy of the two nuclear powers state heads, especially India having an extremist government poses a serious threat to the peace of subcontinent which needs great attention especially from the global community otherwise a nuclear Armageddon could be expected. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Between Two Worlds: East and West: An AutobiographyWilliam EdelglassBetween Two Worlds: East and West: An Autobiography. By J. N. Mohanty. Oxford: Oxford University Press, 2002. Pp. x + 134.The British philosopher Anthony Quinton once described J. N. Mohanty as "The one and only x who is a specialist in Navya-Nyāya, Husserl, and Frege." Between Two Worlds: East and West is the extraordinary story of Mohanty's career as a (...) student, teacher, and scholar of Indian and German philosophy. Told against the backdrop of the struggle for Indian Independence, the trauma of partition, the intellectual and cultural glory of Calcutta, postwar Germany, and the United States of the last generation, Mohanty's story is a page-turner filled with fascinating anecdotes from his own life as well as the lives of the many remarkable thinkers and public figures he has encountered. It is also a story of the concrete historical situations that enabled Mohanty's singular contribution to philosophy: the decades of study with some of India's most learned paṇḍits, the opportunities offered during his years as a student in Göttingen, and his close collegial relationships with Aron Gurwitsch, Hans Jonas, and Hannah Arendt. Though he set out to write an autobiography that was explicitly not a philosophical text, Mohanty observes that "since my life has been primarily dedicated to the pursuit of philosophical ideas, the connection of the story of my life to my philosophical interests was almost unavoidable" (p. ix). In addition to brief discussions of some of the philosophical questions he has pursued in his previous writings, Mohanty here reflects on cultural identity, the life of the mind, and religion and spirituality, as well as social and political questions.Born in 1928, Mohanty enjoyed a pleasant and comfortable childhood supported by the loving attention of an extended family. (In this story families are adaptable, ever incorporating others into the love and care of a network grounded in mutual support, shared food, and conversation.) The family home was in the small village of Nilakaṇṭhapur (the seat of the blue-necked god, i.e., Śiva), where Mohanty's [End Page 139] father had been raised. His father had left the village for a successful career as a judge under the British Raj, bringing both social prominence and moderate wealth. But his father's career required periodic moves, and it was the village to which they regularly returned that Mohanty considered home. It is a pleasure to read Mohanty's descriptions of the cycle of the seasons, village life, local customs, family characters, and the rambling house with its temple and many rooms. Mohanty still thinks of himself as a "village boy," and these pages are written with much tenderness and enough historical background to contextualize his descriptions.Unlike his father, Mohanty was not educated in the village school. He himself was born in the regional capital of Cuttack and educated at the English government school in town. Mohanty has fond recollections of his teachers and the special school holidays, including the sports festivals, the single religious holiday observed by the school—the worship of the goddess of learning, Saraswati—and the distribution of prizes for academic distinction. Mohanty would inevitably receive numerous prizes. Later, at his matriculation from high school, Mohanty achieved the highest scores in all of Orissa, and stood first in the university following his studies of mathematics, logic, and Sanskrit in Cuttack. Upon completion of his study of philosophy and law at Presidency College (Calcutta), Mohanty again stood first in the First Class, and then again for his graduate work at Calcutta University. Mohanty's schooling was augmented with lessons from private tutors at home that preceded or followed the regular school hours. "Every morning," Mohanty writes, he and his siblingswould recite conjugations and declensions of Sanskrit verbs and nouns loud enough to be heard outside the house; we would compete to do it first. The pandit [Nilakantha Misra, Mohanty's first Sanskrit teacher] taught me all the grammar that I learnt later. He also taught this ten-year-old boy the primer of logic Tarkasaṃgraha, and Kalidāsa's Raghuvaṃśaṃ. (p. 6)Mohanty... (shrink)

Nineteenth and twentieth century philosophies of science have consistently failed to identify any rational basis for the compelling character of scientific analogies. This failure is particularly worrisome in light of the fact that the development and diffusion of certain scientific analogies, e.g. Darwin’s analogy between domestic breeds and naturally occurring species, constitute paradigm cases of good science. It is argued that the interactivist model, through the notion of a partition epistemology, provides a way to understand the persuasive character of (...) compelling scientific analogies without consigning them to an irrational or arational context of discovery. (shrink)

There exist two main notions of typicality in computability theory, namely, Cohen genericity and randomness. In this article, we introduce a new notion of genericity, called partition genericity, which is at the intersection of these two notions of typicality, and show that many basis theorems apply to partition genericity. More precisely, we prove that every co-hyperimmune set and every Kurtz random is partition generic, and that every partition generic set admits weak infinite subsets, for various notions (...) of weakness. In particular, we answer a question of Kjos-Hanssen and Liu by showing that every Kurtz random admits an infinite subset which does not compute any set of positive effective Hausdorff dimension. Partition genericity is a partition regular notion, so these results imply many existing pigeonhole basis theorems. (shrink)

Lewis sender‐receiver games illustrate how a meaningful term language might evolve from initially meaningless random signals (Lewis 1969; Skyrms 2006). Here we consider how a meaningful language with a primitive grammar might evolve in a somewhat more subtle sort of game. The evolution of such a language involves the co‐evolution of partitions of the physical world into what may seem, at least from the perspective of someone using the language, to correspond to canonical natural kinds. While the evolved language may (...) allow for the sort of precise representation that is required for successful coordinated action and prediction, the apparent natural kinds reflected in its structure may be purely conventional. This has both positive and negative implications for the limits of naturalized metaphysics. (shrink)

There are some who defend a view of vagueness according to which there are intrinsically vague objects or attributes in reality. Here, in contrast, we defend a view of vagueness as a semantic property of names and predicates. All entities are crisp, on this view, but there are, for each vague name, multiple portions of reality that are equally good candidates for being its referent, and, for each vague predicate, multiple classes of objects that are equally good candidates for being (...) its extension. We provide a new formulation of these ideas in terms of a theory of granular partitions. We show that this theory provides a general framework within which we can understand the relation between vague terms and concepts on the one hand and correlated portions of reality on the other. We also sketch how it might be possible to formulate within this framework a theory of vagueness which dispenses with the notion of truth-value gaps and other artifacts of more familiar approaches. (shrink)

The connections between mathematical logic and combinatorics have a rich history. This paper focuses on one aspect of this relationship: understanding the strength, measured using the tools of computability theory and reverse mathematics, of various partition theorems. To set the stage, recall two of the most fundamental combinatorial principles, König's Lemma and Ramsey's Theorem. We denote the set of natural numbers by ω and the set of finite sequences of natural numbers by ω<ω. We also identify each n ∈ (...) ω with its set of predecessors, so n = {0, 1, 2, …, n − 1}. (shrink)

Given a regular infinite cardinal κ and a cardinal λ > κ, we study fine ideals H on Pκ that satisfy the square brackets partition relation equation image, where μ is a cardinal ≥2.

The paper examines Schopenhauer’s complex diagrams from the Berlin Lectures of the 1820 s, which show certain partitions of classes. Drawing upon ideas and techniques from logical geometry, we show that Schopenhauer’s partition diagrams systematically give rise to a special type of Aristotelian diagrams, viz. (strong) α -structures.

Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics (QM) is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets (or partitions) which are category-theoretically dual (...) to one another and which are developed in two mathematical logics, the usual Boolean logic of subsets and the more recent logic of partitions. Our sense-making strategy is "follow the math" by showing how the logic and mathematics of set partitions can be transported in a natural way to Hilbert spaces where it yields the mathematical machinery of QM--which shows that the mathematical framework of QM is a type of logical system over ℂ. And then we show how the machinery of QM can be transported the other way down to the set-like vector spaces over ℤ₂ showing how the classical logical finite probability calculus (in a "non-commutative" version) is a type of "quantum mechanics" over ℤ₂, i.e., over sets. In this way, we try to make sense out of objective indefiniteness and thus to interpret quantum mechanics. (shrink)

We show that Miller partition forcing preserves selective independent families and P-points, which implies the consistency of $\mbox {cof}(\mathcal {N})=\mathfrak {a}=\mathfrak {u}=\mathfrak {i}<\mathfrak {a}_T=\omega _2$. In addition, we show that Shelah’s poset for destroying the maximality of a given maximal ideal preserves tight mad families and so we establish the consistency of $\mbox {cof}(\mathcal {N})=\mathfrak {a}=\mathfrak {i}=\omega _1<\mathfrak {u}=\mathfrak {a}_T=\omega _2$.

The last section made it clear that an analysis which at first seems to fail is viable after all. It is viable if we let it depend on a partition function to be provided by the context of conversation. This analysis leaves certain traits of the partition function open. I have tried to show that this should be so. Specifying these traits as Pollock does leads to wrong predictions. And leaving them open endows counterfactuals with just the right (...) amount of variability and vagueness. (shrink)

We have a variety of different ways of dividing up, classifying, mapping, sorting and listing the objects in reality. The theory of granular partitions presented here seeks to provide a general and unified basis for understanding such phenomena in formal terms that is more realistic than existing alternatives. Our theory has two orthogonal parts: the first is a theory of classification; it provides an account of partitions as cells and subcells; the second is a theory of reference or intentionality; it (...) provides an account of how cells and subcells relate to objects in reality. We define a notion of well-formedness for partitions, and we give an account of what it means for a partition to project onto objects in reality. We continue by classifying partitions along three axes: (a) in terms of the degree of correspondence between partition cells and objects in reality; (b) in terms of the degree to which a partition represents the mereological structure of the domain it is projected onto; and (c) in terms of the degree of completeness with which a partition represents this domain. (shrink)

Partitions and their Afterlives engages with political partitions and how their aftermath affects the contemporary life of nations and their citizens.

It is shown that the notion of the partition of a set can be used to describe in a uniform way the meaning of the expression the same, in its basic uses in transitive and ditransitive sentences. Some formal properties of the function denoted by the same, which follow from such a description are indicated. These properties indicate similarities and differences between functions denoted by the same and generalised quantifiers.

A theorem on the partitioning of a randomly selected large population into stationary and non-stationary components by using a property of the stationary population identity is stated and proved. The methods of partitioning demonstrated are original and these are helpful in real-world situations where age-wise data is available. Applications of this theorem for practical purposes are summarized at the end.

Classical logic is usually interpreted as the logic of propositions. But from Boole's original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (nonempty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoretic duality--which is reflected in the duality between quotient objects and subobjects throughout algebra. Hence the idea arises of a dual (...) logic of partitions. That dual logic is described here. Partition logic is at the same mathematical level as subset logic since models for both are constructed from (partitions on or subsets of) arbitrary unstructured sets with no ordering relations, compatibility or accessibility relations, or topologies on the sets. Just as Boole developed logical finite probability theory as a quantitative treatment of subset logic, applying the analogous mathematical steps to partition logic yields a logical notion of entropy so that information theory can be refounded on partition logic. But the biggest application is that when partition logic and the accompanying logical information theory are "lifted" to complex vector spaces, then the mathematical framework of quantum mechanics is obtained. Partition logic models indefiniteness (i.e., numerical attributes on a set become more definite as the inverse-image partition becomes more refined) while subset logic models the definiteness of classical physics (an entity either definitely has a property or definitely does not). Hence partition logic provides the backstory so the old idea of "objective indefiniteness" in QM can be fleshed out to a full interpretation of quantum mechanics. (shrink)

This book brings together a diverse group of American Indian thinkers to discuss traditional and contemporary philosophies and philosophical issues. The essays presented here address philosophical questions pertaining to knowledge, time, place, history, science, law, religion, nationhood, ethics, and art, as understood from a variety of Native American standpoints. Unique in its approach, this volume represents several different tribes and nations and amplifies the voice of contemporary American Indian culture struggling for respect and autonomy. Taken together, the essays (...) collected here exemplify the way in which American Indian perspectives enrich contemporary philosophy. (shrink)

A desirable property of a diversity index is strict concavity. This implies that the pooled diversity of a given community sample is greater than or equal to but not less than the weighted mean of the diversity values of the constituting plots. For a strict concave diversity index, such as species richness S, Shannon''s entropy H or Simpson''s index 1-D, the pooled diversity of a given community sample can be partitioned into two non-negative, additive components: average within-plot diversity and between-plot (...) diversity. As a result, species diversity can be summarized at various scales measuring all diversity components in the same units. Conversely, violation of strict concavity would imply the non-interpretable result of a negative diversity among community plots. In this paper, I apply this additive partition model generally adopted for traditional diversity measures to Aczél and Daróczy''s generalized entropy of type . In this way, a parametric measure of -diversity is derived as the ratio between the pooled sample diversity and the average within-plot diversity that represents the parametric analogue of Whittaker''s -diversity for data on species relative abundances. (shrink)

We investigate the following weak Ramsey property of a cardinal κ: If χ is coloring of nodes of the tree κ <ω by countably many colors, call a tree ${T \subseteq \kappa^{ < \omega}}$ χ-homogeneous if the number of colors on each level of T is finite. Write ${\kappa \rightsquigarrow (\lambda)^{ < \omega}_{\omega}}$ to denote that for any such coloring there is a χ-homogeneous λ-branching tree of height ω. We prove, e.g., that if ${\kappa < \mathfrak{p}}$ or ${\kappa > \mathfrak{d}}$ (...) is regular, then ${{\kappa \rightsquigarrow (\kappa)^{ < \omega}_{\omega}}}$ and that ${\mathfrak{b}}$ ${(\mathfrak{b})^{ < \omega}_{\omega}}$ and ${\mathfrak{d}}$ ${(\mathfrak{d})^{ < \omega}_{\omega}}$ . The arrow is applied to prove a generalization of a theorem of Hurewicz: A Čech-analytic space is σ-locally compact iff it does not contain a closed homeomorphic copy of irrationals. (shrink)

Current theories make a distinction between two types of case, STRUCTURAL case and INHERENT (or LEXICAL) case (Chomsky 1981), similar to the older distinction between GRAMMATICAL and SEMANTIC case (Kuryłowicz 1964).1 Structural case is assumed to be assigned at S-structure in a purely configurational way, whereas inherent case is assigned at D-structure in possible dependence on the governing predicates’s lexical properties. It is well known that not all cases fall cleanly into this typology. In particular, there is a class of (...) cases that pattern syntactically with the structural cases, but are semantically conditioned. These cases however depend on different semantic conditions than inherent cases do: instead of being sensitive to the thematic relation that the NP bears to the verbal predicate, they are sensitive to the NP’s definiteness, animacy, or quantificational properties, or to the aspectual character of the VP, or to some combination of these factors. The Finnish partitive is a particularly clear instance of this apparently hybrid category of semantically conditioned structural case. (shrink)

For an infinite cardinal K a stronger version of K-distributivity for Boolean algebras, called k-partition completeness, is defined and investigated . It is shown that every k-partition complete Boolean algebra is K-weakly representable, and for strongly inaccessible K these concepts coincide. For regular K ≥ u, it is proved that an atomless K-partition complete Boolean algebra is an updirected union of basic K-tree algebras. Using K-partition completeness, the concept of γ-almost compactness is introduced for γ ≥ (...) K. For strongly inaccessible K we show that K is K-almost compact iff K is weakly compact, and if K is 2K-almost compact, then K is measurable. Further K is strongly compact iff it is γ-almost compact for all γ ≥ K. (shrink)

Two partition-theorems are proved for a particular causal decision theory. One is restricted to a certain kind of partition of circumstances, and analyzes the utility of an option in terms of its utilities in conjunction with circumstances in this partition. The other analyzes an option's utility in terms of its utilities conditional on circumstances and is quite unrestricted. While the first form seems more useful for applications, the second form may be of theoretical importance in foundational exercises. (...) Comparisons are made with theorems of Richard Jeffrey, Brad Armendt, and Peter Fishburn. (shrink)

In this paper we propose a formal theory of partitions (ways of dividing up or sorting or mapping reality) and we show how the theory can be applied in the geospatial domain. We characterize partitions at two levels: as systems of cells (theory A), and in terms of their projective relation to reality (theory B). We lay down conditions of well-formedness for partitions and we define what it means for partitions to project truly onto reality. We continue by classifying well-formed (...) partitions along three axes: (a) degree of correspondence between partition cells and objects in reality; (b) degree to which a partition represents the mereological structure of the domain it is projected onto; and (c) degree of completeness and exhaustiveness with which a partition represents reality. This classification is used to characterize three types of partitions that play an important role in spatial information science: cadastral partitions, categorical coverages, and the partitions involved in folk categorizations of the geospatial domain. (shrink)

Classical physics and quantum physics suggest two meta-physical types of reality: the classical notion of a objectively definite reality with properties "all the way down," and the quantum notion of an objectively indefinite type of reality. The problem of interpreting quantum mechanics is essentially the problem of making sense out of an objectively indefinite reality. These two types of reality can be respectively associated with the two mathematical concepts of subsets and quotient sets which are category-theoretically dual to one another (...) and which are developed in two dual mathematical logics, the usual Boolean logic of subsets and the more recent logic of partitions. Our sense-making strategy is "follow the math" by showing how the mathematics of set partitions can be transported in a natural way to complex vector spaces where it yields the mathematical machinery of QM. And then we show how the machinery of QM can be transported the other way down to set-like vector spaces over ℤ₂ yielding a rather fulsome "toy" or pedagogical model of "quantum mechanics over sets." In this way, we try to make sense out of objective indefiniteness and thus to interpret quantum mechanics. (shrink)