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Postulated
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning. In mathematics, an ''axiom'' may be a "logical axiom" or a " non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example ''a'' + 0 = ''a'' in integer arithmetic. Non-l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Realism
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathematical objects are purely abstract entities or are in some way concrete, and in what the relationship such objects have with physical reality consists. Major themes that are dealt with in philosophy of mathematics include: *''Reality'': The question is whether mathematics is a pure product of human mind or whether it has some reality by itself. *''Logic and rigor'' *''Relationship with physical reality'' *''Relationship with science'' *''Relationship with applications'' *''Mathematical truth'' *''Nature as human activity'' (science, art, game, or all together) Major themes Reality Logic and rigor Mathematical reasoning requires rigor. This means that the definitions must be absolutely unambiguous and the proofs must be reducible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Axioms
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning. In mathematics, an ''axiom'' may be a "logical axiom" or a " non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example ''a'' + 0 = ''a'' in integer arithmetic. Non- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid
Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics. Very little is known of Euclid's life, and most information comes from the scholars Proclus and Pappus of Alexandria many centuries later. Medieval Islamic mathematicians invented a fanciful biography, and medieval Byzantine and early Renaissance scholars mistook him for the earlier philo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. One of those is the parallel postulate which relates to parallel lines on a Euclidean plane. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logic, logical system in which each result is ''mathematical proof, proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-logical Axioms
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. In modern logic, an axiom is a premise or starting point for reasoning. In mathematics, an ''axiom'' may be a "logical axiom" or a " non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example ''a'' + 0 = ''a'' in integer arithmetic. Non- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statement (logic)
In logic and semantics, the term statement is variously understood to mean either: #a meaningful sentence (linguistics)#By_function_or_speech_act, declarative sentence that is Truth, true or false (logic), false, or #a proposition. Which is the ''Denotation, assertion'' that is made by (i.e., the Meaning (linguistics), meaning of) a true or false declarative sentence. "A statement is defined as that which is ''expressible'' by a ''sentence'', and is either true or false... A statement is a more abstract entity than even a sentence type. It is not identical with the sentence used to express it... [That is,] different sentences can be used to express the same statement." In the latter case, a (declarative) sentence is just one way of expressing an underlying statement. A statement is what a sentence means, it is the notion or idea that a sentence expresses, i.e., what it represents. For example, it could be said that "2 + 2 = 4" and "two plus two equals four" are two different sente ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proclus
Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of the most elaborate and fully developed systems of Neoplatonism and, through later interpreters and translators, exerted an influence on Byzantine philosophy, early Islamic philosophy, scholastic philosophy, and German idealism, especially G. W. F. Hegel, who called Proclus's ''Platonic Theology'' "the true turning point or transition from ancient to modern times, from ancient philosophy to Christianity." Biography The primary source for the life of Proclus is the eulogy ''Proclus'', ''or On Happiness'' that was written for him upon his death by his successor, Marinus, Marinus' biography set out to prove that Proclus reached the peak of virtue and attained eudaimonia. There are also a few details about the time in which he lived in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geminus
Geminus of Rhodes (), was a Greek astronomer and mathematician, who flourished in the 1st century BC. An astronomy work of his, the ''Introduction to the Phenomena'', still survives; it was intended as an introductory astronomy book for students. He also wrote a work on mathematics, of which only fragments quoted by later authors survive. Life Nothing is known about the life of Geminus. It is not even certain that he was born in Rhodes, but references to mountains on Rhodes in his astronomical works suggests that he worked there. His dates are not known with any certainty either. A passage in his works referring to the ''Annus Vagus'' (Wandering Year) of the Egyptian calendar of 120 years before his own time, has been used to imply a date of c. 70 BC for the time of writing, which would be consistent with the idea that he may have been a pupil of Posidonius, but a date as late as 50 AD has also been suggested. The crater Geminus on the Moon is named after him. Astronomy The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosopher
Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational and critical inquiry that reflects on its methods and assumptions. Historically, many of the individual sciences, such as physics and psychology, formed part of philosophy. However, they are considered separate academic disciplines in the modern sense of the term. Influential traditions in the history of philosophy include Western philosophy, Western, Islamic philosophy, Arabic–Persian, Indian philosophy, Indian, and Chinese philosophy. Western philosophy originated in Ancient Greece and covers a wide area of philosophical subfields. A central topic in Arabic–Persian philosophy is the relation between reason and revelation. Indian philosophy combines the Spirituality, spiritual problem of how to reach Enlightenment in Buddhism, enlighten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boethius
Anicius Manlius Severinus Boethius, commonly known simply as Boethius (; Latin: ''Boetius''; 480–524 AD), was a Roman Roman Senate, senator, Roman consul, consul, ''magister officiorum'', polymath, historian, and philosopher of the Early Middle Ages. He was a central figure in the translation of the Greek classics into Latin, a precursor to the Scholasticism, Scholastic movement, and, along with Cassiodorus, one of the two leading Christian scholars of the 6th century. The local cult of Boethius in the Diocese of Pavia was sanctioned by the Sacred Congregation of Rites in 1883, confirming the diocese's custom of honouring him on the 23 October. Boethius was born in Rome a few years after the forced abdication of the last Western Roman Empire, Western Roman emperor, Romulus Augustulus. A member of the Anicii family, he was orphaned following the family's sudden decline and was raised by Quintus Aurelius Memmius Symmachus, a later Roman consul, consul. After mastering both Latin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syllogisms
A syllogism (, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. In its earliest form (defined by Aristotle in his 350 BC book '' Prior Analytics''), a deductive syllogism arises when two true premises (propositions or statements) validly imply a conclusion, or the main point that the argument aims to get across. For example, knowing that all men are mortal (major premise), and that Socrates is a man (minor premise), we may validly conclude that Socrates is mortal. Syllogistic arguments are usually represented in a three-line form: All men are mortal. Socrates is a man. Therefore, Socrates is mortal.In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism. From the Middle Ages onwards, ''categorical syllogism'' and ''syllogism'' were usually used interchangeably. This article is concerne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
