Aristotle

Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical study of the relation of this framework with reality.

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.

Roger Bacon OFM (/ˈbeɪkən/; Latin: Rogerus or Rogerius Baconus, Baconis, also Frater Rogerus; c. 1219/20 – c. 1292), also known by the scholastic accolade Doctor Mirabilis, was a medieval English polymath, philosopher, scientist, theologian and Franciscan friar who placed considerable emphasis on the study of nature through empiricism. Intertwining his Catholic faith with scientific thinking, Roger Bacon is considered one of the greatest polymaths of the medieval period.

In the early modern era, he was regarded as a wizard and particularly famed for the story of his mechanical or necromantic brazen head. He is credited as one of the earliest European advocates of the modern scientific method, along with his teacher Robert Grosseteste. Bacon applied the empirical method of Ibn al-Haytham (Alhazen) to observations in texts attributed to Aristotle. Bacon discovered the importance of empirical testing when the results he obtained were different from those that would have been predicted by Aristotle.

Holism is the interdisciplinary idea that systems possess properties as wholes apart from the properties of their component parts.The aphorism "The whole is greater than the sum of its parts", typically attributed to Aristotle, is often given as a summary of this proposal. The concept of holism can inform the methodology for a broad array of scientific fields and lifestyle practices. When applications of holism are said to reveal properties of a whole system beyond those of its parts, these qualities are referred to as emergent properties of that system. Holism in all contexts is often placed in opposition to reductionism, a dominant notion in the philosophy of science that systems containing parts contain no unique properties beyond those parts. Proponents of holism consider the search for emergent properties within systems to be demonstrative of their perspective.

Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applications in many fields, including philosophy, theoretical computer science, artificial intelligence, economics, and linguistics. While philosophers since Aristotle have discussed modal logic, and Medieval philosophers such as Avicenna, Ockham, and Duns Scotus developed many of their observations, it was C. I. Lewis who created the first symbolic and systematic approach to the topic, in 1912. It continued to mature as a field, reaching its modern form in 1963 with the work of Saul Kripke.

The Organon (Ancient Greek: Ὄργανον, meaning "instrument, tool, organ") is the standard collection of Aristotle's six works on logical analysis and dialectic.

Sophistical Refutations (Greek: Σοφιστικοὶ Ἔλεγχοι, romanized: Sophistikoi Elenchoi; Latin: De Sophisticis Elenchis) is a text in Aristotle's Organon in which he identified thirteen fallacies. According to Aristotle, this is the first work to treat the subject of deductive reasoning in ancient Greece (Soph. Ref., 34, 183b34 ff.).
Cite error: There are <ref group=note> tags on this page, but the references will not show without a {{reflist|group=note}} template (see the help page).

Science in the medieval Islamic world was the science developed and practised during the Islamic Golden Age under the Abbasid Caliphate of Baghdad, the Umayyads of Córdoba, the Abbadids of Seville, the Samanids, the Ziyarids and the Buyids in Persia and beyond, spanning the period roughly between 786 and 1258. Islamic scientific achievements encompassed a wide range of subject areas, especially astronomy, mathematics, and medicine. Other subjects of scientific inquiry included alchemy and chemistry, botany and agronomy, geography and cartography, ophthalmology, pharmacology, physics, and zoology.

Medieval Islamic science had practical purposes as well as the goal of understanding. For example, astronomy was useful for determining the Qibla, the direction in which to pray, botany had practical application in agriculture, as in the works of Ibn Bassal and Ibn al-'Awwam, and geography enabled Abu Zayd al-Balkhi to make accurate maps. Islamic mathematicians such as Al-Khwarizmi, Avicenna and Jamshīd al-Kāshī made advances in algebra, trigonometry, geometry and Arabic numerals. Islamic doctors described diseases like smallpox and measles, and challenged classical Greek medical theory. Al-Biruni, Avicenna and others described the preparation of hundreds of drugs made from medicinal plants and chemical compounds. Islamic physicists such as Ibn Al-Haytham, Al-Bīrūnī and others studied optics and mechanics as well as astronomy, and criticised Aristotle's view of motion.

Latin translations of the 12th century were spurred by a major search by European scholars for new learning unavailable in western Europe at the time; their search led them to areas of southern Europe, particularly in central Spain and Sicily, which recently had come under Christian rule following their reconquest in the late 11th century. These areas had been under Muslim rule for a considerable time, and still had substantial Arabic-speaking populations to support their search. The combination of this accumulated knowledge and the substantial numbers of Arabic-speaking scholars there made these areas intellectually attractive, as well as culturally and politically accessible to Latin scholars. A typical story is that of Gerard of Cremona (c. 1114–87), who is said to have made his way to Toledo, well after its reconquest by Christians in 1085, because he:

Many Christian theologians were highly suspicious of ancient philosophies and especially of the attempts to synthesize them with Christian doctrines. St. Jerome, for example, was hostile to Aristotle, and St. Augustine had little interest in exploring philosophy, only applying logic to theology. For centuries, ancient Greek ideas in Western Europe were all but non-existent. Only a few monasteries had Greek works, and even fewer of them copied these works.

Christian philosophy includes philosophies carried out by Christians or in relation to the religion of Christianity. Christian philosophy emerged with the aim of reconciling science and faith, starting from natural rational explanations with the help of Christian revelation. Several thinkers such as Origen and Augustine of Hippo believed that there was a harmonious relationship between science and faith, others such as Tertullian claimed that there was contradiction; others tried to differentiate them.

There are scholars who question the existence of a Christian philosophy itself. These claim that there is no originality in Christian thought, and its concepts and ideas are inherited from Greek philosophy. Thus, Christian philosophy would protect philosophical thought, which would already be definitively elaborated by Greek philosophy. However, Catholic scholars Philotheus Boehner and Étienne Gilson claim that Christian philosophy is not a simple repetition of ancient philosophy, although they owe to Greek science the knowledge developed by Plato, Aristotle and the Neoplatonists. They even claim that in Christian philosophy, Greek culture survives in organic form.