Casuistry is a process of reasoning that seeks to resolve moral problems by extracting or extending abstract rules from a particular case, and reapplying those rules to new instances. This method occurs in applied ethics and jurisprudence. The term is also used pejoratively to criticise the use of clever but unsound reasoning, especially in relation to ethical questions. It has been defined as follows:
Study of cases of conscience and a method of solving conflicts of obligations by applying general principles of ethics, religion, and moral theology to particular and concrete cases of human conduct. This frequently demands an extensive knowledge of natural law and equity, civil law, ecclesiastical precepts, and an exceptional skill in interpreting these various norms of conduct....
George Edward Moore was an English philosopher, who with Bertrand Russell, Ludwig Wittgenstein and earlier Gottlob Frege was among the initiators of analytic philosophy. He and Russell began deemphasizing the idealism which was then prevalent among British philosophers and became known for advocating common-sense concepts and contributing to ethics, epistemology and metaphysics. He was said to have an "exceptional personality and moral character". Ray Monk later dubbed him "the most revered philosopher of his era".
Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory, algebra, geometry, analysis, and set theory.
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship with other human activities.
Hilary Whitehall Putnam was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science. Outside philosophy, Putnam contributed to mathematics and computer science. Together with Martin Davis he developed the Davis–Putnam algorithm for the Boolean satisfiability problem and he helped demonstrate the unsolvability of Hilbert's tenth problem.
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are lazy."
The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler, in 1736, laid the foundations of graph theory and prefigured the idea of topology.
Sir Michael Anthony Eardley Dummett was an English academic described as "among the most significant British philosophers of the last century and a leading campaigner for racial tolerance and equality." He was, until 1992, Wykeham Professor of Logic at the University of Oxford. He wrote on the history of analytic philosophy, notably as an interpreter of Frege, and made original contributions particularly in the philosophies of mathematics, logic, language and metaphysics.
Aristotelianism is a philosophical tradition inspired by the work of Aristotle, usually characterized by deductive logic and an analytic inductive method in the study of natural philosophy and metaphysics. It covers the treatment of the social sciences under a system of natural law. It answers why-questions by a scheme of four causes, including purpose or teleology, and emphasizes virtue ethics. Aristotle and his school wrote tractates on physics, biology, metaphysics, logic, ethics, aesthetics, poetry, theatre, music, rhetoric, psychology, linguistics, economics, politics, and government. Any school of thought that takes one of Aristotle's distinctive positions as its starting point can be considered "Aristotelian" in the widest sense. This means that different Aristotelian theories may not have much in common as far as their actual content is concerned besides their shared reference to Aristotle.
Ian MacDougall Hacking was a Canadian philosopher specializing in the philosophy of science. Throughout his career, he won numerous awards, such as the Killam Prize for the Humanities and the Balzan Prize, and was a member of many prestigious groups, including the Order of Canada, the Royal Society of Canada and the British Academy.
David Charles Stove was an Australian philosopher whose writings often challenged prevailing academic orthodoxy. He was known for his critiques of postmodernism, feminism, and multiculturalism.
Moral certainty is a concept of intuitive probability. It means a very high degree of probability, sufficient for action, but short of absolute or mathematical certainty.
David Papineau is a British academic philosopher, born in Como, Italy. He works as Professor of Philosophy of Science at King's College London and the City University of New York Graduate Center, and previously taught for several years at Cambridge University, where he was a fellow of Robinson College.
Patrick Joseph "Paddy" Ryan, invariably referred to as Dr P. J. Ryan, was an Australian Catholic priest and anti-communist organiser.
Australian realism, also called Australian materialism, is a school of philosophy that flourished in the first half of the 20th century in several universities in Australia including the Australian National University, the University of Adelaide, and the University of Sydney, and whose central claim, as stated by leading theorist John Anderson, was that "whatever exists … is real, that is to say it is a spatial and temporal situation or occurrence that is on the same level of reality as anything else that exists". Coupled with this was Anderson's idea that "every fact is a complex situation: there are no simples, no atomic facts, no objects which cannot be, as it were, expanded into facts." Prominent players included Anderson, David Malet Armstrong, J. L. Mackie, Ullin Place, J. J. C. Smart, and David Stove. The label "Australian realist" was conferred on acolytes of Anderson by A. J. Baker in 1986, to mixed approval from those realist philosophers who happened to be Australian. David Malet Armstrong "suggested, half-seriously, that 'the strong sunlight and harsh brown landscape of Australia force reality upon us'".
Structuralism is a theory in the philosophy of mathematics that holds that mathematical theories describe structures of mathematical objects. Mathematical objects are exhaustively defined by their place in such structures. Consequently, structuralism maintains that mathematical objects do not possess any intrinsic properties but are defined by their external relations in a system. For instance, structuralism holds that the number 1 is exhaustively defined by being the successor of 0 in the structure of the theory of natural numbers. By generalization of this example, any natural number is defined by its respective place in that theory. Other examples of mathematical objects might include lines and planes in geometry, or elements and operations in abstract algebra.
Australian philosophy refers to the philosophical tradition of the people of Australia and of its citizens abroad. Academic philosophy has been mostly pursued in universities. It has been broadly in the tradition of Anglo-American analytic philosophy, but has also had representatives of a diverse range of other schools, such as idealism, Catholic neo-scholasticism, Marxism, and continental, feminist and Asian philosophy.
Ethics in mathematics is an emerging field of applied ethics, the inquiry into ethical aspects of the practice and applications of mathematics. It deals with the professional responsibilities of mathematicians whose work influences decisions with major consequences, such as in law, finance, the military, and environmental science. When understood in its socio-economic context, the development of mathematical works can lead to ethical questions ranging from the handling and manipulation of big data to questions of responsible mathematisation and falsification of models, explainable and safe mathematics, as well as many issues related to communication and documentation. The usefulness of a Hippocratic oath for mathematicians is an issue of ongoing debate among scholars. As an emerging field of applied ethics, many of its foundations are still highly debated. The discourse remains in flux. Especially the notion that mathematics can do harm remains controversial.
Dale Jacquette was an American analytic philosopher. At the time of his death, he was Professor Ordinarius of Philosophy at the University of Bern. Jacquette had previously served on the faculty of Penn State University. He received his undergraduate degree in philosophy from Oberlin College in 1975, and his PhD in the same subject from Brown University in 1983, writing a dissertation on the logic of intention supervised by Roderick Chisholm. Jacquette had broad research interests in the philosophy of intentionality, logic, metaphysics, philosophy of mind, Wittgenstein, ethics, aesthetics, epistemology, and the history of philosophy. A prolific writer, Jacquette published books on Meinong, logic, cannabis, psychologism, and the ethics of capital punishment in the final decade of his life. He was a defender of Aristotelian realist philosophy of mathematics.
In the philosophy of mathematics, Aristotelian realism holds that mathematics studies properties such as symmetry, continuity and order that can be immanently realized in the physical world. It contrasts with Platonism in holding that the objects of mathematics, such as numbers, do not exist in an "abstract" world but can be physically realized. It contrasts with nominalism, fictionalism, and logicism in holding that mathematics is not about mere names or methods of inference or calculation but about certain real aspects of the world.
