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Georg Ferdinand Ludwig Philipp Cantor was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.
A paradigm shift, a concept brought into the common lexicon by the American physicist and philosopher Thomas Kuhn, is a fundamental change in the basic concepts and experimental practices of a scientific discipline. Even though Kuhn restricted the use of the term to the natural sciences, the concept of a paradigm shift has also been used in numerous non-scientific contexts to describe a profound change in a fundamental model or perception of events.
Imre Lakatos was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its "methodology of proofs and refutations" in its pre-axiomatic stages of development, and also for introducing the concept of the "research programme" in his methodology of scientific research programmes.
Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever.
The Structure of Scientific Revolutions is a book about the history of science by philosopher Thomas S. Kuhn. Its publication was a landmark event in the history, philosophy, and sociology of science. Kuhn challenged the then prevailing view of progress in science in which scientific progress was viewed as "development-by-accumulation" of accepted facts and theories. Kuhn argued for an episodic model in which periods of conceptual continuity where there is cumulative progress, which Kuhn referred to as periods of "normal science", were interrupted by periods of revolutionary science. The discovery of "anomalies" during revolutions in science leads to new paradigms. New paradigms then ask new questions of old data, move beyond the mere "puzzle-solving" of the previous paradigm, change the rules of the game and the "map" directing new research.
Pierre Maurice Marie Duhem was a French theoretical physicist who worked on thermodynamics, hydrodynamics, and the theory of elasticity. Duhem was also a historian of science, noted for his work on the European Middle Ages, which is regarded as having created the field of the history of medieval science. As a philosopher of science, he is remembered principally for his views on the indeterminacy of experimental criteria.
The historiography of science or the historiography of the history of science is the study of the history and methodology of the sub-discipline of history, known as the history of science, including its disciplinary aspects and practices and the study of its own historical development.
Commensurability is a concept in the philosophy of science whereby scientific theories are said to be "commensurable" if scientists can discuss the theories using a shared nomenclature that allows direct comparison of them to determine which one is more valid or useful. On the other hand, theories are incommensurable if they are embedded in starkly contrasting conceptual frameworks whose languages do not overlap sufficiently to permit scientists to directly compare the theories or to cite empirical evidence favoring one theory over the other. Discussed by Ludwik Fleck in the 1930s, and popularized by Thomas Kuhn in the 1960s, the problem of incommensurability results in scientists talking past each other, as it were, while comparison of theories is muddled by confusions about terms, contexts and consequences.
The Duhem–Quine thesis, also called the Duhem–Quine problem, after Pierre Duhem and Willard Van Orman Quine, is that in science it is impossible to experimentally test a scientific hypothesis in isolation, because an empirical test of the hypothesis requires one or more background assumptions : the thesis says that unambiguous scientific falsifications are impossible. In recent decades the set of associated assumptions supporting a thesis sometimes is called a bundle of hypotheses.
Rhetoric of science is a body of scholarly literature exploring the notion that the practice of science is a rhetorical activity. It emerged following a number of similarly-oriented disciplines during the late 20th century, including the disciplines of sociology of scientific knowledge, history of science, and philosophy of science, but it is practiced most fully by rhetoricians in departments of English, speech, and communication.
Objectivity in science is an attempt to uncover truths about the natural world by eliminating personal biases, emotions, and false beliefs. It is often linked to observation as part of the scientific method. It is thus intimately related to the aim of testability and reproducibility. To be considered objective, the results of measurement must be communicated from person to person, and then demonstrated for third parties, as an advance in a collective understanding of the world. Such demonstrable knowledge has ordinarily conferred demonstrable powers of prediction or technology.
Nonstandard analysis and its offshoot, nonstandard calculus, have been criticized by several authors, notably Errett Bishop, Paul Halmos, and Alain Connes. These criticisms are analyzed below.
Leslie Pearce Williams was a chaired professor at Cornell University's Department of History who also chaired the department for many years. He was also the founder, in the mid-1980s, of Cornell's program in the History and Philosophy of Science and Technology, which later became part of the Department of Science and Technology Studies.
Joseph Warren Dauben is a Herbert H. Lehman Distinguished Professor of History at the Graduate Center of the City University of New York. He obtained his PhD from Harvard University.
Donald A. Gillies is a British philosopher and historian of science and mathematics. He is an Emeritus Professor in the Department of Science and Technology Studies at University College London.
Thomas Samuel Kuhn was an American philosopher of science whose 1962 book The Structure of Scientific Revolutions was influential in both academic and popular circles, introducing the term paradigm shift, which has since become an English-language idiom.
An index list of articles about the philosophy of science.
Hendrik Jan Maarten "Henk" Bos is a Dutch historian of mathematics.
Hans-Ludwig Wußing was a German historian of mathematics and science.
References
- Pierre Kerszberg (1994, 2009) Review of Revolutions in Mathematics in Mathematical Reviews.
- Michael S. Mahoney (1994) "Review of Revolutions in Mathematics", American Mathematical Monthly 101(3):283–7.