Bertrand Russell in the context of Begriffsschrift

Alfred North Whitehead OM FRS FBA (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He created the philosophical school known as process philosophy, which has been applied in a wide variety of disciplines, including ecology, theology, education, physics, biology, economics, and psychology.

In his early career Whitehead wrote primarily on mathematics, logic, and physics. He wrote the three-volume Principia Mathematica (1910–1913), with his former student Bertrand Russell. Principia Mathematica is considered one of the twentieth century's most important works in mathematical logic, and placed 23rd in a list of the top 100 English-language nonfiction books of the twentieth century by Modern Library.

Bertrand Russell makes a distinction between two different kinds of knowledge: knowledge by acquaintance and knowledge by description. Whereas knowledge by description is something like ordinary propositional knowledge (e.g. "I know that snow is white"), knowledge by acquaintance is familiarity with a person, place, or thing, typically obtained through perceptual experience (e.g. "I know Sam", "I know the city of Bogotá", or "I know Russell's Problems of Philosophy"). According to Bertrand Russell's classic account of acquaintance knowledge, acquaintance is a direct causal interaction between a person and an object of experience.

The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1925–1927, it appeared in a second edition with an important Introduction to the Second Edition, an Appendix A that replaced ✱9 with a new Appendix B and Appendix C. PM was conceived as a sequel to Russell's 1903 The Principles of Mathematics, but as PM states, this became an unworkable suggestion for practical and philosophical reasons: "The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics ... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions."

PM, according to its introduction, had three aims: (1) to analyse to the greatest possible extent the ideas and methods of mathematical logic and to minimise the number of primitive notions, axioms, and inference rules; (2) to precisely express mathematical propositions in symbolic logic using the most convenient notation that precise expression allows; (3) to solve the paradoxes that plagued logic and set theory at the turn of the 20th century, like Russell's paradox.

Thomas Aquinas OP (/əˈkwaɪnəs/ ə-KWY-nəs; Italian: Tommaso d'Aquino, lit. 'Thomas of Aquino'; c. 1225 – 7 March 1274) was an Italian Dominican friar and priest, the foremost Scholastic thinker, as well as one of the most influential philosophers and theologians in the Western tradition. A Doctor of the Church, he was from the county of Aquino in the Kingdom of Sicily.

Thomas was a proponent of natural theology and the father of a school of thought (encompassing both theology and philosophy) known as Thomism. He argued that God is the source of the light of natural reason and the light of faith. He embraced several ideas put forward by Aristotle and attempted to synthesize Aristotelian philosophy with the principles of Christianity. He has been described as "the most influential thinker of the medieval period" and "the greatest of the medieval philosopher-theologians". Thomas Aquinas's philosophy influenced modern virtue ethics, aesthetics, and cognitive theory. He has been criticized, notably by Bertrand Russell, for seeking to justify conclusions already dictated by faith rather than follow reason independently.

Philosophy of language is the philosophical study of the nature of language. It investigates the relationship between language, language users, and the world. Investigations may include inquiry into the nature of meaning, intentionality, reference, the constitution of sentences, concepts, learning, and thought.

Gottlob Frege and Bertrand Russell were pivotal figures in analytic philosophy's "linguistic turn". These writers were followed by Ludwig Wittgenstein (Tractatus Logico-Philosophicus), the Vienna Circle, logical positivists, and Willard Van Orman Quine.

Analytic philosophy is a broad movement and methodology within contemporary Western philosophy, especially anglophone philosophy, focused on: analysis as a philosophical method; clarity of prose; rigor in arguments; and making use of formal logic, mathematics, and to a lesser degree the natural sciences. It is further characterized by the linguistic turn, or a concern with language and meaning. Analytic philosophy has developed several new branches of philosophy and logic, notably philosophy of language, philosophy of mathematics, philosophy of science, modern predicate logic and mathematical logic.

The proliferation of analysis in philosophy began around the turn of the 20th century and has been dominant since the latter half of the 20th century. Central figures in its historical development are Gottlob Frege, Bertrand Russell, G. E. Moore, and Ludwig Wittgenstein. Other important figures in its history include Franz Brentano, the logical positivists (especially Rudolf Carnap), the ordinary language philosophers, W. V. O. Quine, and Karl Popper. After the decline of logical positivism, Saul Kripke, David Lewis, and others led a revival in metaphysics.

Propositions are the meanings of declarative sentences, objects of beliefs, and bearers of truth values. They explain how different sentences, like the English "Snow is white" and the German "Schnee ist weiß", can have identical meaning by expressing the same proposition. Similarly, they ground the fact that different people can share a belief by being directed at the same content. True propositions describe the world as it is, while false ones fail to do so. Researchers distinguish types of propositions by their informational content and mode of assertion, such as the contrasts between affirmative and negative propositions, between universal and existential propositions, and between categorical and conditional propositions.

Many theories of the nature and roles of propositions have been proposed. Realists argue that propositions form part of reality, a view rejected by anti-realists. Non-reductive realists understand propositions as a unique kind of entity, whereas reductive realists analyze them in terms of other entities. One proposal sees them as sets of possible worlds, reflecting the idea that understanding a proposition involves grasping the circumstances under which it would be true. A different suggestion focuses on the individuals and concepts to which a proposition refers, defining propositions as structured entities composed of these constituents. Other accounts characterize propositions as specific kinds of properties, relations, or states of affairs. Philosophers also debate whether propositions are abstract objects outside space and time, psychological entities dependent on mental activity, or linguistic entities grounded in language. Paradoxes challenge the different theories of propositions, such as the liar's paradox. The study of propositions has its roots in ancient philosophy, with influential contributions from Aristotle and the Stoics, and later from William of Ockham, Gottlob Frege, and Bertrand Russell.

The Vienna Circle (German: Wiener Kreis) of logical empiricism was a group of elite philosophers and scientists drawn from the natural and social sciences, logic and mathematics who met regularly from 1924 to 1936 at the University of Vienna, chaired by Moritz Schlick. The Vienna Circle had a profound influence on 20th-century philosophy, especially philosophy of science and analytic philosophy.

The philosophical position of the Vienna Circle was called logical empiricism (German: logischer Empirismus), logical positivism or neopositivism. It was influenced by Ernst Mach, David Hilbert, French conventionalism (Henri Poincaré and Pierre Duhem), Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein and Albert Einstein. The Vienna Circle was pluralistic and committed to the ideals of the Enlightenment. It was unified by the aim of making philosophy scientific with the help of modern logic. Main topics were foundational debates in the natural and social sciences, logic and mathematics; the modernization of empiricism by modern logic; the search for an empiricist criterion of meaning; the critique of metaphysics and the unification of the sciences in the unity of science.

In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions.

According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let R be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) If R is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols:

The 1957 Nobel Prize in Literature was awarded the French writer Albert Camus (1913–1960) "for his important literary production, which with clear-sighted earnestness illuminates the problems of the human conscience in our times." He is the ninth French author to become a recipient of the prize after Catholic novelist François Mauriac in 1952, and the fourth philosopher after British analytic philosopher Bertrand Russell in 1950.

Aged 44 when he received the prize, Camus is the second youngest recipient of the Nobel Prize in Literature, after only Rudyard Kipling (41).

Charles Sanders Peirce (/pɜːrs/ PURSS; September 10, 1839 – April 19, 1914) was an American scientist, mathematician, logician, and philosopher who is sometimes known as "the father of pragmatism". According to philosopher Paul Weiss, Peirce was "the most original and versatile of America's philosophers and America's greatest logician". Bertrand Russell wrote "he was one of the most original minds of the later nineteenth century and certainly the greatest American thinker ever".

Educated as a chemist and employed as a scientist for thirty years, Peirce meanwhile made major contributions to logic, such as theories of relations and quantification. C. I. Lewis wrote, "The contributions of C. S. Peirce to symbolic logic are more numerous and varied than those of any other writer—at least in the nineteenth century." For Peirce, logic also encompassed much of what is now called epistemology and the philosophy of science. He saw logic as the formal branch of semiotics or study of signs, of which he is a founder, which foreshadowed the debate among logical positivists and proponents of philosophy of language that dominated 20th-century Western philosophy. Peirce's study of signs also included a tripartite theory of predication.

The theory of sense data is a view in the philosophy of perception, popularly held in the early 20th century by philosophers such as Bertrand Russell, C. D. Broad, H. H. Price, A. J. Ayer, and G. E. Moore. Sense data are taken to be mind-dependent objects whose existence and properties are known directly to us in perception. These objects are unanalyzed experiences inside the mind, which appear to subsequent more advanced mental operations exactly as they are.

Sense data are often placed in a time and/or causality series, such that they occur after the potential unreliability of our perceptual systems yet before the possibility of error during higher-level conceptual analysis and are thus incorrigible. They are thus distinct from the 'real' objects in the world outside the mind, about whose existence and properties we often can be mistaken.

In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after Cesare Burali-Forti, who, in 1897, published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Georg Cantor. Bertrand Russell subsequently noticed the contradiction, and when he published it in his 1903 book Principles of Mathematics, he stated that it had been suggested to him by Burali-Forti's paper, with the result that it came to be known by Burali-Forti's name.

Kurt Friedrich Gödel (/ˈɡɜːrdəl/ GUR-dəl; German: [ˈkʊʁt ˈɡøːdl̩] ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly influenced scientific and philosophical thinking in the 20th century (at a time when Bertrand Russell, Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics), building on earlier work by Frege, Richard Dedekind, and Georg Cantor.

Gödel's discoveries in the foundations of mathematics led to the proof of his completeness theorem in 1929 as part of his dissertation to earn a doctorate at the University of Vienna, and the publication of Gödel's incompleteness theorems two years later, in 1931. The incompleteness theorems address limitations of formal axiomatic systems. In particular, they imply that a formal axiomatic system satisfying certain technical conditions cannot decide the truth value of all statements about the natural numbers, and cannot prove that it is itself consistent. To prove this, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

The Principles of Mathematics (PoM) is a 1903 book by Bertrand Russell, in which the author presented his famous paradox and argued his thesis that mathematics and logic are identical.

The book presents a view of the foundations of mathematics and Meinongianism and has become a classic reference. It reported on developments by Giuseppe Peano, Mario Pieri, Richard Dedekind, Georg Cantor, and others.

Friedrich Ludwig Gottlob Frege (/ˈfreɪɡə/; German: [ˈɡɔtloːp ˈfreːɡə]; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was 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 one of the greatest logicians since Aristotle, and one of the most profound philosophers of mathematics ever.

His contributions include the development of modern logic in the Begriffsschrift and work in the foundations of mathematics. His book the Foundations of Arithmetic is the seminal text of the logicist project, and is cited by Michael Dummett as where to pinpoint the linguistic turn. His philosophical papers "On Sense and Reference" and "The Thought" are also widely cited. The former argues for two different types of meaning and descriptivism. In Foundations and "The Thought", Frege argues for Platonism against psychologism or formalism, concerning numbers and propositions respectively.