Foundations Of Mathematics
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Foundations Of Mathematics
Foundations of mathematics is the study of the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence, reason Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning ... and logical and/or algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...ic basis of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the natur ...
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Philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, language. Such questions are often posed as problems to be studied or resolved. Some sources claim the term was coined by Pythagoras (c. 570 – c. 495 BCE); others dispute this story, arguing that Pythagoreans merely claimed use of a preexisting term. Philosophical methodology, Philosophical methods include Socratic questioning, questioning, Socratic method, critical discussion, dialectic, rational argument, and systematic presentation.Quinton, Anthony. 1995. "The Ethics of Philosophical Practice." P. 666 in ''The Oxford Companion to Philosophy'', edited by Ted Honderich, T. Honderich. New York: Oxford University Press. . "Philosophy is rationally critical thinking, of a more or less systematic kind about the general nature of the world (metaphysics or theory ...
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Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was an Classical Athens, Athenian philosopher during the Classical Greece, Classical period in Ancient Greece, founder of the Platonist school of thought and the Platonic Academy, Academy, the first institution of higher learning in the Western world. He is widely considered a pivotal figure in the history of philosophy, history of Ancient Greek philosophy, Ancient Greek and Western philosophy, along with his teacher, Socrates, and his most famous student, Aristotle. Plato has also often been cited as one of the founders of Western religion and spirituality. The so-called neoplatonism of philosophers such as Plotinus and Porphyry (philosopher), Porphyry greatly influenced Neoplatonism and Christianity, Christianity through Church Fathers such as Augustine. Alfred North Whitehead once noted: "the safest general characterization of the European philosophical tradition is that it consists of a series of Note ...
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Metaphysics (Aristotle)
''Metaphysics'' (Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...: τὰ μετὰ τὰ φυσικά, "things after the ones about the natural world"; Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ...: ''Metaphysica'') is one of the principal works of Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest ..., in which he develops the doctrine that he refers to somet ...
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The Unreasonable Effectiveness Of Mathematics In The Natural Sciences
"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is a 1960 article by the physicist Eugene Wigner. In the paper, Wigner observes that a Theoretical physics, physical theory's Mathematics, mathematical structure often points the way to further advances in that theory and even to empirical predictions. Original paper and Wigner's observations Wigner begins his paper with the belief, common among those familiar with mathematics, that mathematical concepts have applicability far beyond the context in which they were originally developed. Based on his experience, he writes, "it is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena". He then invokes the fundamental gravity, law of gravitation as an example. Originally used to model freely falling bodies on the surface of the earth, this law was extended on the basi ...
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Platonism (mathematics)
The philosophy of mathematics is the Discipline (academia), branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methodology, methods of mathematics, and find out the place of mathematics in people's lives. The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts. The philosophy of mathematics has two major themes; Mathematical realism and Mathematic anti-realism. History The origin of mathematics is subject to arguments and disagreements. Whether the birth of mathematics was a random happening or induced by necessity during the development of other subjects, like physics, is still a matter of prolific debates. Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emp ...
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Meno
''Meno'' (; grc-gre, Μένων, ''Ménōn'') is a Socratic dialogue by Plato. Meno begins the dialogue by asking Socrates whether virtue is taught, acquired by practice, or comes by nature. In order to determine whether virtue is teachable or not, Socrates tells Meno that they first need to determine what virtue is. When the characters speak of virtue, or rather ''arete (excellence), arete'', they refer to virtue in general, rather than particular virtues, such as justice or temperance. The first part of the work showcases Socratic method, Socratic dialectical style; Meno (general), Meno, unable to adequately define virtue, is reduced to confusion or aporia. Socrates suggests that they seek an adequate definition for virtue together. In response, Meno suggests that it is impossible to seek what one does not know, because one will be unable to determine whether one has found it. Socrates challenges Meno's argument, often called "Meno's Paradox" or the "Learner's Paradox," by ...
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Syllogisms
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (both spellings are acceptable), intended to determine the degree of truth of another statement, the conclusion. The logical ... that applies deductive reasoning Deductive reasoning, also deductive logic, is the process of reasoning Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making ... to arrive at a conclusion based on two propositions In logic and linguistics, a proposition is the meaning of a declarative sentence (linguistics), sentence. In philosophy, "Meaning (philosophy), meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same mea ... that are asserted or assumed ...
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Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematics, mathematical treatise consisting of 13 books attributed to the ancient Greek mathematics, Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulates, propositions (theorems and Compass and straightedge constructions, constructions), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and Commensurability (mathematics), incommensurable lines. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science, and its logical rigor was not surpassed until the 19th century. Euclid's ''Elements'' has been referred to as the most successful and influential textbook ever written. It was one of the very earliest mathematical works to be printed after the movable type, invention of the printing press and ...
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Euclid
Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referred to as the "founder of geometry Geometr ... ; 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematics, Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I Soter, Ptolemy I (323–283 BC). His ''Euclid's Elements, Elements'' is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the ''Elements'', Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wr ...
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Axiomatic Method
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ... and logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ..., an axiomatic system is any set of axiom An axiom, postulate or assumption is a statement that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...s from which some or all axioms can be used in conjunction to logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to ...
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Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mi ... and polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a substantial number of subjects, known to draw on complex bodies of knowledge to solve specific prob ... during the Classical periodClassical period may refer to: *Classical Greece, specifically of the 5th and 4th centuries BC *Classical antiquity, in the Greco-Roman world *Classical India, an historic period of India (c. 322 BC - c. 550 CE) *Classical period (music), in music ... in Ancient Greece Ancient Greece ( el, Ἑλλάς, Hellás) was a civi ...
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Posterior Analytics
The ''Posterior Analytics'' ( grc-gre, Ἀναλυτικὰ Ὕστερα; la, Analytica Posteriora) is a text from Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest ...'s ''Organon The ''Organon'' ( grc, Ὄργανον, meaning "instrument, tool, organ") is the standard collection of Aristotle's six works on logic. The name ''Organon'' was given by Aristotle's followers, the Peripatetics. They are as follows: Constitut ...'' that deals with demonstration, definition A definition is a statement of the meaning of a term (a word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical ..., and scientific knowledge Science (from the ...
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