bluuebunny/crossref_metadata_2025_split huggingface dataset:
Logic in the early 20th century
Authors: Moore Gregory H. 
Published: September 2018

The creation of modern logic is one of the most stunning achievements of mathematics and philosophy in the twentieth century. Modern logic – sometimes called logistic, symbolic logic or mathematical logic – makes essential use of artificial symbolic languages.
Since Aristotle, logic has been a part of philosophy. Around 1850 the mathematician Boole began the modern development of symbolic logic. During the twentieth century, logic continued in philosophy departments, but it began to be seriously investigated and taught in mathematics departments as well. The most important examples of the latter were, from 1905 on, Hilbert at Göttingen and then, during the 1920s, Church at Princeton.
As the twentieth century began, there were several distinct logical traditions. Besides Aristotelian logic, there was an active tradition in algebraic logic initiated by Boole in the UK and continued by C.S. Peirce in the USA and Schröder in Germany. In Italy, Peano began in the Boolean tradition, but soon aimed higher: to express all major mathematical theorems in his symbolic logic. Finally, from 1879 to 1903, Frege consciously deviated from the Boolean tradition by creating a logic strong enough to construct the natural and real numbers. The Boole–Schröder tradition culminated in the work of Löwenheim (1915) and Skolem (1920) on the existence of a countable model for any first-order axiom system having a model.
Meanwhile, in 1900, Russell was strongly influenced by Peano’s logical symbolism. Russell used this as the basis for his own logic of relations, which led to his logicism: pure mathematics is a part of logic. But his discovery of Russell’s paradox in 1901 required him to build a new basis for logic. This culminated in his masterwork, Principia Mathematica, written with Whitehead, which offered the theory of types as a solution.
Hilbert came to logic from geometry, where models were used to prove consistency and independence results. He brought a strong concern with the axiomatic method and a rejection of the metaphysical goal of determining what numbers ‘really’ are. In his view, any objects that satisfied the axioms for numbers were numbers. He rejected the genetic method, favoured by Frege and Russell, which emphasized constructing numbers rather than giving axioms for them. In his 1917 lectures Hilbert was the first to introduce first-order logic as an explicit subsystem of all of logic (which, for him, was the theory of types) without the infinitely long formulas found in Löwenheim. In 1923 Skolem, directly influenced by Löwenheim, also abandoned those formulas, and argued that first-order logic is all of logic.
Influenced by Hilbert and Ackermann (1928), Gödel proved the completeness theorem for first-order logic (1929) as well as incompleteness theorems for arithmetic in first-order and higher-order logics (1931). These results were the true beginning of modern logic.
Quantum logic
Authors: Forrest Peter 
Published: September 2018

The topic of quantum logic was introduced by Birkhoff and von Neumann (1936), who described the formal properties of a certain algebraic system associated with quantum theory. To avoid begging questions, it is convenient to use the term ‘logic’ broadly enough to cover any algebraic system with formal characteristics similar to the standard sentential calculus. In that sense it is uncontroversial that there is a logic of experimental questions (for example, ‘Is the particle in region R?’ or ‘Do the particles have opposite spins?’) associated with any physical system. Having introduced this logic for quantum theory, we may ask how it differs from the standard sentential calculus, the logic for the experimental questions in classical mechanics. The most notable difference is that the distributive laws fail, being replaced by a weaker law known as orthomodularity.
All this can be discussed without deciding whether quantum logic is a genuine logic, in the sense of a system of deduction. Putnam argued that quantum logic was indeed a genuine logic, because taking it as such solved various problems, notably that of reconciling the wave-like character of a beam of, say, electrons, as it passes through two slits, with the thesis that the electrons in the beam go through one or other of the two slits. If Putnam’s argument succeeds this would be a remarkable case of the empirical defeat of logical intuitions. Subsequent discussion, however, seems to have undermined his claim.
Mereology
Authors: Forrest Peter 
Published: September 2018

Mereology is the theory of the part–whole relation and of derived operations such as the mereological sum. (The sum of several things is the smallest thing of which they are all parts.) It was introduced by Leśniewski to avoid Russell’s paradox.
Unlike the set-membership relation, the part–whole relation is transitive. This makes mereology much weaker than set theory, but gives the advantage of ontological parsimony. For example, mereology does not posit the proliferation of entities found in set theory, such as ∅⁣,{∅⁣},{{∅⁣}},….
Mereology has occasioned controversy: over whether many things really have a mereological sum if they are either scattered or, even worse, of different categories; over the uniqueness of sums; and over Lewis’ claim that the non-empty subsets of a set are literally parts of it.
Antirealism in the philosophy of mathematics
Authors: Moore A.W. 
Published: September 2018

Realism in the philosophy of mathematics is the position that takes mathematics at face value. According to realists, mathematics is the science of mathematical objects (numbers, sets, lines and so on); mathematicians, to use the old metaphor, are discoverers, not inventors. Moreover, just as there may be truths about physical reality which we can never know, so too, realists say, there may be truths about mathematical reality which we can never know.
It is this claim in particular which antirealists find unacceptable. Equating what can be known in mathematics with what can be proved, they insist that only what can be proved is true. (Only what can be proved: different accounts of what this ‘can’ means, facing different difficulties, generate different positions.) This leads antirealists to recoil not only from realism but also from the practice of mathematicians themselves. For the orthodox assumption that every mathematical statement is either true or false would be invalidated, on the antirealist view, by a statement that was neither provable nor disprovable. Not that antirealists themselves can see it in these terms. For if a statement were neither provable nor disprovable, that would itself be an unprovable truth about mathematical reality. Antirealists must learn how to be circumspect even in defence of their own circumspection.
Category theory, applications to the foundations of mathematics
Authors: McLarty Colin 
Published: September 2018

Since the 1960s Lawvere has distinguished two senses of the foundations of mathematics. Logical foundations use formal axioms to organize the subject. The other sense aims to survey ‘what is universal in mathematics’. The ontology of mathematics is a third, related issue.
Moderately categorical foundations use sets as axiomatized by the elementary theory of the category of sets (ETCS) rather than Zermelo–Fraenkel set theory (ZF). This claims to make set theory conceptually more like the rest of mathematics than ZF is. And it suggests that sets are not ‘made of’ anything determinate; they only have determinate functional relations to one another. The ZF and ETCS axioms both support classical mathematics.
Other categories have also been offered as logical foundations. The ‘category of categories’ takes categories and functors as fundamental. The ‘free topos’ (see Lambek and Couture 1991) stresses provability. These and others are certainly formally adequate. The question is how far they illuminate the most universal aspects of current mathematics.
Radically categorical foundations say mathematics has no one starting point; each mathematical structure exists in its own right and can be described intrinsically. The most flexible way to do this to date is categorically. From this point of view various structures have their own logic. Sets have classical logic, or rather the topos Set has classical logic. But differential manifolds, for instance, fit neatly into a topos Spaces with nonclassical logic. This view urges a broader practice of mathematics than classical.
This article assumes knowledge of category theory on the level of Category theory, introduction to §1.
Non-monotonic logic
Authors: Fuhrmann Andre 
Published: September 2018

A relation of inference is ‘monotonic’ if the addition of premises does not undermine previously reached conclusions; otherwise the relation is non-monotonic. Deductive inference, at least according to the canons of classical logic, is monotonic: if a conclusion is reached on the basis of a certain set of premises, then that conclusion still holds if more premises are added.
By contrast, everyday reasoning is mostly non-monotonic because it involves risk: we jump to conclusions from deductively insufficient premises. We know when it is worthwhile or even necessary (for example, in medical diagnosis) to take the risk. Yet we are also aware that such inference is ‘defeasible’ – that new information may undermine old conclusions. Various kinds of defeasible but remarkably successful inference have traditionally captured the attention of philosophers (theories of induction, Peirce’s theory of abduction, inference to the best explanation, and so on). More recently logicians have begun to approach the phenomenon from a formal point of view. The result is a large body of theories at the interface of philosophy, logic and artificial intelligence.
Boolean algebra
Authors: Bell J.L. 
Published: September 2018

Boolean algebra, or the algebra of logic, was devised by the English mathematician George Boole (1815–64) and embodies the first successful application of algebraic methods to logic.
Boole seems to have had several interpretations for his system in mind. In his earlier work he thinks of each of the basic symbols of his ‘algebra’ as standing for the mental operation of selecting just the objects possessing some given attribute or included in some given class; later he conceives of these symbols as standing for the attributes or classes themselves. In each of these interpretations the basic symbols are conceived as being capable of combination under certain operations: ‘multiplication’, corresponding to conjunction of attributes or intersection of classes; ‘addition’, corresponding to (exclusive) disjunction or (disjoint) union; and ‘subtraction’, corresponding to ‘excepting’ or difference. He also recognizes that the algebraic laws he proposes are satisfied if the basic symbols are interpreted as taking just the number values 0 and 1.
Boole’s ideas have since undergone extensive development, and the resulting concept of Boolean algebra now plays a central role in mathematical logic, probability theory and computer design.
Pan-Africanism
Authors: Appiah K. Anthony 
Published: September 2018

Pan-Africanism covers a wide range of intellectual positions which share the assumption of some common cultural or political projects for both Africans and people of African descent. The political project is the unification of all Africans into a single African state, sometimes thought of as providing a homeland for the return of those in the African diaspora. More vaguely, many self-identified pan-Africanists have aimed to pursue projects of solidarity – some political, some literary or artistic – in Africa or the African diaspora. The Pan-Africanist movement was founded in the nineteenth century by intellectuals of African descent in the Caribbean and North America, who saw themselves as belonging to a single negro race. As a result the Africa of pan-Africanism has sometimes been limited to those regions of sub-Saharan Africa largely inhabited by darker-skinned peoples, thus excluding those lighter-skinned north Africans, most of whom speak Arabic as a first language.
In the twentieth century this racialized understanding of African identity has been challenged by many of the African intellectuals who took over the movement’s leadership in the period after the Second World War. Founders of the Organization of African Unity, such as Gamal Abdel Nasser of Egypt and Kwame Nkrumah of Ghana had a notion of Africa that was continental. However, the movement’s intellectual roots lie firmly in the racial understanding of Africa in the thought of the African-American and Afro-Caribbean intellectuals who founded it.
Pan-Africanism began as a movement in the diaspora among the descendants of the slave populations of the New World and spread to Africa itself. As a result the forms of solidarity it articulated aimed to challenge anti-black racism on two fronts: racial domination in the diaspora and racialized colonial domination in the African continent. The movement’s fissures have occurred where these two clearly distinguishable projects have pulled it in different directions.
Latin America, colonial thought in
Authors: Redmond Walter B. 
Published: September 2018

Colonial refers to Spanish and Portuguese sovereignty in America from the arrival of Columbus in 1492 up to the emergence of modern Latin American states in the nineteenth century. The intellectual life of the colonies and their mother countries at that time falls into two phases: traditional and modern. The traditional phase includes the siglo de oro, or the Golden Age of the sixteenth and seventeenth centuries. This was a time when literature and the arts flourished, along with Scholastic philosophy, jurisprudence and theology. During the eighteenth century, traditional thought gradually gave way to modern movements, particularly from France.
The universities founded in the mid-sixteenth century, notably those of Mexico and Peru, as well as colleges and seminaries, were impressively productive in the area of philosophy. The pressure of events such as the clash between European and Native American cultures in the sixteenth century and the struggle for independence from Spain and Portugal in the nineteenth century brought about numerous nonacademic works with philosophic content. Authors wrote in both Latin and Spanish or Portuguese and often knew native languages, such as Nahuatl and Quechua as well. Many operated in several different areas, such as the nun, Sor Juana Inés de la Cruz, one of the greatest poets in the Spanish language, who wrote a book on logic in Latin, which has since been lost.
Students studied philosophy first, then specialized in medicine, law, or theology. The core philosophy curriculum was logic, natural philosophy or physics and metaphysics. In the eighteenth century Scholastic logic, similar to what has come to be known as formal logic, was weakened and natural philosophy began to incorporate experimental science. The bulk of philosophy was affected by modern thinkers such as René Descartes.
Eighteenth-century savants were critical of Scholasticism and later Latin American intellectuals tended to disavow the entire colonial past. However, historians since the 1940s have stressed the currency of modern scholarship, especially in science and since the 1960s have been rediscovering the sophisticated philosophy of the Golden Age.
Existentialist thought in Latin America
Authors: Bertelloni Maria Teresa 
Published: September 2018

In Latin America the thought and teaching of José Ortega y Gasset have been very influential. Their influence leaves an important mark on the substance of existentialism. The most effective aspect of Ortega y Gasset’s philosophical conception was his thesis that humans do not have a nature, only a history. It is this concept that encouraged Latin American thinkers to create their own original thought as a product of their concrete historical circumstances. This entry will deal with Latin American existence, unique in its historical concreteness, and from this general view will attempt to construct a metaphysical theory of Latin America’s historical endeavour.
Given that all historicist conception involves values that are objective in nature, it is not surprising that Latin American existentialism was profoundly influenced by Max Scheler and Nicolai Hartmann, together with the existential analysis of Martin Heidegger. Consequently, in opposition to the phenomenologists with a Husserlian orientation, implicit in Latin American existentialism, there is a phenomenological methodology in the interpretation of history as culture in accordance with the analysis of the dialectics of the structure of history. This opposes all possible perceptions of pure essences which might precede existence. The essence of existence is seen as progressive, constructing itself as it is bypassed by historical events.
Both in terms of the search for an original philosophy, which could be reduced to a philosophy of history (for example in the Orteguian philosophy of life) and in terms of a Heideggerian approach, Latin American philosophy applies a phenomenological method in its analysis. This would explain the fusion of phenomenology and existentialism in the works of Latin American philosophers. All Latin American phenomenological-existentialist philosophical effort is a struggle between the analysis and interpretation of the European currents and their search for the historical realization of the autonomous Latin American being.
Phenomenology in Latin America
Authors: Bertelloni Maria Teresa 
Published: September 2018

The Latin American struggle against the positivism of the nineteenth century was the primordial endeavour of the founders of Latin American thought, such as José Enrique Rodó (1872–1917), José Vasconcelos (1882–1959), Alejandro Korn (1860–1936), Carlos Vaz Ferreira (1871–1958), Alejandro Deústua (1849–1945), Enrique Molina (1871–1956) and Antonio Caso (1883–1946). These thinkers fought to win their philosophical freedom in a battle against continuing such European currents as Neo-Kantianism and the existential-phenomenological movement, on the one hand and on the other, developing a philosophy that was purely Latin American.
In all of this the influence of the Spanish philosopher José Ortega y Gasset (1883–1955) was fundamental. Ortega y Gasset was mainly interested in the problems of history and culture. His famous apothegm, ‘I am myself and my circumstance’ (1947: 322), was the justification for a phenomenological movement axiological in nature, as well as being the point of departure for the affirmation of Latin American circumstance.
Few have cultivated the Husserlian style of phenomenology in Latin America outside the classroom. Latin American thinkers have preferred to apply the phenomenological method to the different fields of knowledge. In particular, those dealing with sociocultural aspects, literary criticism, socio-economic structures and juridical axiology. In all works related to axiology the thought of the philosophers Nicolai Hartmann (1882–1950) and Max Scheler (1874–1928) were enormously influential.
Many of those originally inclined towards phenomenology drifted from the 1970s onwards, towards analytic philosophy, philosophy of science and logical neopositivism.
Positivist thought in Latin America
Authors: Marti Oscar R. 
Published: September 2018

Between 1850 and the 1920s European positivism became a major intellectual movement in Latin America. It asserted that all knowledge came from experience; that scientific thinking was the model for philosophizing and that the search for first causes or ultimate reason typical of religion and metaphysics, was an obsolete mode of thinking. Positivism set out to discover the most general features of experience. Its tasks were to use them to explain and predict phenomena, to develop a social science that would furnish objective grounds for moral choice and to help create the best society possible.
In Latin America positivism became a social philosophy which represented a cogent alternative to romanticism, eclecticism, Catholicism and traditional Hispanic values. It offered the prospect of a secular society in which the knowledge gained from science and industry would bring the benefits of order and progress. It assumed that the social sciences had the power to improve the human condition and it demanded political action. Three main currents were in evidence: autochthonous positivism indigenous to the region and concerned with local social and political issues, social positivism derived from Auguste Comte and stressing the historical nature of social change, and evolutionary positivism influenced by Herbert Spencer and asserting the biological nature of society.
Autochthonous positivism emerged in the 1830s from the influx of liberal ideas which followed the wars of independence fought on the US continent by those wishing to gain freedom from Spain. Urging an intellectual revolution, swift social change and material progress, autochthonous positivism paved the way for European positivism proper. Social positivism appeared around the 1850s and argued for the necessity of educational reforms to solve the continent’s problems. It required participation in political life and became a radical force in spite of opposition from supporters of the status quo. By the 1880s evolutionary positivism had steered the movement in a conservative direction in support of laissez-faire policies, individualism and gradual change.
Latin America, pre-Columbian and indigenous thought in
Authors: Schrenk Laura Mues De 
Published: September 2018

The term ‘pre-Columbian thought’ refers to the set of beliefs and ideas held by the civilizations existing in Latin America prior to the arrival of Columbus in 1492. Research in pre-Columbian thought poses several questions linked to language, interpretation, chronology and cultural diversity. They can roughly be organized according to the three main regions in which the indigenous cultures flourished upon the arrival of the Spanish invaders: Nahuatl/Aztec in central and southern Mexico, Quiché/Maya from Yucatan in southern Mexico to Honduras in central America and Quechua/Inca, from Ecuador to northern Chile. They each correspond to an empire into which previously many diverse, distinguishable peoples were assimilated. Since the Spanish invaders had destroyed most of their ‘heretical’ cultural objects by 1550, the question arises whether an accurate knowledge of their thought can be obtained.
Some ethnohistorians believe that each of the aforementioned cultures developed a hieroglyphic system of codification and documentation, called Codices, to preserve their theocosmogony, history and wisdom. Since the sixteenth century, however, it is known with certainty that only the Aztec and Mayan cultures developed such a system. According to some historians, the Incan culture did not use any kind of writing. They probably created ‘paintings’, as the Spaniards called their hieroglyphs, but these were totally destroyed.
It is known that all pre-Columbian religions worshipped the events and forces of nature. The term used to name them was translated into Spanish as ‘gods’ when they were acceptable, or ‘demons’ when they seemed heretical – the indigenous peoples were polytheistic. The gods did not dwell in a region beyond our world, but rather populated it and were actively intertwined with it. All pre-Columbian cultures believed the sun to be the highest deity. The universe was conceived as a holistic structure in which human life, society and the gods were parts of an interrelated universe. Beyond these, however, the three cultures believed in an intangible, abstract deity, or principle, which ruled above all others.
The sun, being the highest tangible deity, led the priests of these cultures to observe the skies. Based on a highly developed knowledge of astronomy and mathematics, they established an accurate solar calendar.