In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, a proof is a convincing demonstration (within the accepted standards of the field) that some mathematical statement A proposition is a sentence expressing something true or false. In philosophy, particularly in logic, a proposition is identified ontologically as an idea, concept, or abstraction whose token instances are patterns of symbols, marks, sounds, or strings of words. Propositions are considered to be syntactic entities and also truthbearers is necessarily true.[1][2] Proofs are obtained from deductive reasoning Deductive reasoning, also called Deductive logic, is reasoning which constructs or evaluates deductive arguments. In logic, an argument is deductive when its conclusion is a logical consequence of the premises. Deductive arguments are valid or invalid, never true or false. A deductive argument is valid if and only if the conclusion does follow, rather than from inductive Inductive reasoning, also known as induction or inductive logic, is a type of reasoning that involves moving from a set of specific facts to a general conclusion. It uses premises from objects that have been examined to establish a conclusion about an object that has not been examined. It can also be seen as a form of theory-building, in which or empirical The word empirical denotes information gained by means of observation, experience, or experiment. A central concept in science and the scientific method is that all evidence must be empirical, or empirically based, that is, dependent on evidence or consequences that are observable by the senses. It is usually differentiated from the philosophic arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproved proposition that is believed to be true is known as a conjecture A conjecture is a proposition that is unproven but appears correct and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds. In mathematics, a conjecture is an unproven proposition or.
The statement that is proved is often called a theorem In mathematics, a theorem is a statement which has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other.[1] Once a theorem is proved, it can be used as the basis to prove further statements. A theorem may also be referred to as a lemma In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself. A good stepping stone leads to many others, so some of the most powerful results in mathematics are known as lemmata, such as Bézout's lemma, Urysohn's lemma, Dehn's lemma, Fatou's lemma, Gauss's lemma,, especially if it is intended for use as a stepping stone in the proof of another theorem.
Proofs employ logic Logic is the study of arguments. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. It is one kind of critical thinking. In philosophy, the study of logic but usually include some amount of natural language In the philosophy of language, a natural language is any language which arises in an unpremeditated fashion as the result of the innate facility for language possessed by the human intellect. A natural language is typically used for communication, and may be spoken, signed, or written. Natural language is distinguished from constructed languages which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic The precise nature and definition of informal logic are matters of some dispute. Ralph H. Johnson and J. Anthony Blair define informal logic as "a branch of logic whose task is to develop non-formal standards, criteria, procedures for the analysis, interpretation, evaluation, criticism and construction of argumentation." This definition. Purely formal proofs A formal proof or derivation is a finite sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system. The notion of theorem is not in general effective, therefore there may be no method by which we can always find a, written in symbolic language instead of natural language, are considered in proof theory Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the. The distinction between formal and informal proofs Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the has led to much examination of current and historical mathematical practice Mathematical practice is used to distinguish the working practices of professional mathematicians from the end result of proven and published theorems, quasi-empiricism in mathematics Quasi-empiricism in mathematics is the attempt in the philosophy of mathematics to direct philosophers' attention to mathematical practice, in particular, relations with physics, social sciences, and computational mathematics, rather than solely to issues in the foundations of mathematics. Of concern to this discussion are several topics: the, and so-called folk mathematics As the term is understood by mathematicians, folk mathematics or mathematical folklore means theorems, definitions, proofs, or mathematical facts or techniques that are found by investigation and may circulate among mathematicians by word-of-mouth but have not appeared in print, either in books or in scholarly journals. Knowledge of folklore is (in both senses of that term). The philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of mathematics in people's lives. The logical and structural is concerned with the role of language and logic in proofs, and mathematics as a language Mathematical notation has assimilated symbols from many different alphabets and fonts. It also includes symbols that are specific to mathematics, such as.
History and etymology
See also: History of logic The history of logic is the study of the development of the science of valid inference . While many cultures have employed intricate systems of reasoning, and logical methods are evident in all human thought, an explicit analysis of the principles of reasoning was developed in only three traditions: those of India, of China, and of Greece. OfThe word Proof comes from the Latin probare meaning "to test". Related modern words are the English "probe", "proboscis”, "probation", and "probability", the Spanish "probar" (to smell or taste, or (lesser use) touch or test),[3] and the German "probieren" (to try). The early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony.[4]
Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof.[5] It is probable that the idea of demonstrating a conclusion first arose in connection with geometry Geometry "Earth-measuring" is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by, which originally meant the same as "land measurement".[6] The development of mathematical proof is primarily the product of ancient Greek mathematics Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language, and one of its greatest achievements. Thales Thales of Miletus was a pre-Socratic Greek philosopher from Miletus in Asia Minor, and one of the Seven Sages of Greece. Many, most notably Aristotle, regard him as the first philosopher in the Greek tradition. According to Bertrand Russell, "Western philosophy begins with Thales." Thales attempted to explain natural phenomena without (624–546 BCE) proved some theorems in geometry. Eudoxus Eudoxus of Cnidus was a Greek astronomer, mathematician, scholar and student of Plato. Since all his own works are lost, our knowledge of him is obtained from secondary sources, such as Aratus's poem on astronomy. Theodosius of Bithynia's Sphaerics may be based on a work of Eudoxus (408–355 BCE) and Theaetetus Theaetetus of Athens, son of Euphronius, of the Athenian deme Sunium, was a classical Greek mathematician. His principal contributions were on irrational lengths, which was included in Book X of Euclid's Elements, and proving that there are precisely five regular convex polyhedra (417–369 BCE) formulated theorems but did not prove them. Aristotle Aristotle (384 BC – 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, politics, government, ethics, biology, and zoology. Together with Plato and Socrates (Plato's teacher), Aristotle is one of the most (384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known. Mathematical proofs were revolutionized by Euclid Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry." He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (300 BCE), who introduced the axiomatic method In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system; usually though the effort towards still in use today, starting with undefined terms and axioms In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other truths (propositions regarding the undefined terms assumed to be self-evidently true from the Greek “axios” meaning “something worthy”), and used these to prove theorems using deductive logic Deductive reasoning, also called Deductive logic, is reasoning which constructs or evaluates deductive arguments. In logic, an argument is deductive when its conclusion is a logical consequence of the premises. Deductive arguments are valid or invalid, never true or false. A deductive argument is valid if and only if the conclusion does follow. His book, the Elements Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the, was known to all educated people in the West until the middle of the 20th century.[7] In addition to the familiar theorems of geometry, such as the Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle . In terms of areas, it states:, the Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.
Further advances took place in medieval Islamic mathematics In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics, is the mathematics developed in the Islamic world between 622 and 1600, during what is known as the Islamic Golden Age, in that part of the world where Islam was the dominant religion. Islamic science and mathematics flourished under the Islamic. While earlier Greek proofs were largely geometric demonstrations, the development of arithmetic Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers. In common usage, it refers to the simpler properties when and algebra Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi The Iraqi people or Mesopotamian people are natives or inhabitants of the country of Iraq, known since antiquity as Mesopotamia , and by virtue of a wide-ranging diaspora, throughout the Arab world, Europe, the Americas and Australasia. Before the arrival of Islam from the Arabian Peninsula, the population was mainly a non-Arabic speaking one but mathematician Al-Hashimi provided general proofs for numbers (rather than geometric demonstrations) as he considered multiplication, division, etc. for ”lines.” He used this method to provide the first proof for irrational numbers In mathematics, an irrational number is any real number which cannot be expressed as a fraction p/q, where p and q are integers, with q non-zero and is therefore not a rational number. Informally, this means that an irrational number cannot be represented as a simple fraction. It can be proven that irrational numbers are precisely those real.[8] The earliest inductive proof Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one for arithmetic sequences In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 3, 5, 7, 9, 11, 13, … is an arithmetic progression with common difference 2 was introduced in the Al-Fakhri (1000) by Al-Karaji Abū Bakr ibn Muḥammad ibn al Ḥusayn al-Karajī (c. 953 in Karaj or Karkh – c. 1029) was a 10th century Persian Muslim mathematician and engineer. His three major works are Al-Badi' fi'l-hisab (Wonderful on calculation), Al-Fakhri fi'l-jabr wa'l-muqabala (Glorious on algebra), and Al-Kafi fi'l-hisab (Sufficient on calculation), who used it to prove the binomial theorem In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power n into a sum involving terms of the form axbyc, where the coefficient of each term is a positive integer, and the sum of the exponents of x and y in each term is n. For example,, Pascal's triangle In mathematics, Pascal's triangle is a geometric arrangement of the binomial coefficients in a triangle. It is named after mathematician Blaise Pascal in much of the Western world, although other mathematicians studied it centuries before him in India, Persia, China, and Italy, and the sum formula for integral Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral cubes, a particular case of what is referred to as Waring's Problem.[9] His proof was the first to make use of the two basic components of an inductive proof: first, he notes the truth Truth can have a variety of meanings, from the state of being the case, being in accord with a particular fact or reality, being in accord with the body of real things, events, actuality, or fidelity to an original or to a standard, truth "behind" everything, the ontological truth. In archaic usage it could be fidelity, constancy or of the statement for n = 1; and secondly, he derives the truth for n = k from that of n = k − 1.[10][11] Alhazen Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham (Arabic: ابو علي، الحسن بن الحسن بن الهيثم, Persian: ابن هیثم, Latinized: Alhacen or Alhazen) (965 in Basra - c. 1039 in Cairo) was a Persian or Arab scientist and polymath. He made significant contributions to the principles of optics, as well as to physics, (965-1039) used an inductive proof to prove the sum of fourth powers, and by extension, the sum of any integral powers Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication; in other words, a product of n factors of a:.[12][13] Alhazen also developed the method of proof by contradiction In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the premise that the proposition is false implies a contradiction. Since by the law of bivalence a proposition must be either true or false, and its falsity has been shown impossible, the proposition must be true, as the first attempt at proving the Euclidean Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, whose Elements is the earliest known systematic discussion of geometry. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. Although many of Euclid's results had been parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that:.[14]
Modern proof theory Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the treats proofs as inductively defined data structures. There is no longer an assumption that axioms are "true" in any sense; this allows for parallel mathematical theories built on alternate sets of axioms (see Axiomatic set theory and Non-Euclidean geometry for examples).
Nature and purpose
There are two different conceptions of mathematical proof.[15] The first is an informal proof, a rigorous natural-language expression that is intended to convince the audience of the truth of a theorem. Because of their use of natural language, the standards of rigor for informal proofs will depend on the audience of the proof. In order to be considered a proof, however, the argument must be rigorous enough; a vague or incomplete argument is not a proof. Informal proofs are the type of proof typically encountered in published mathematics. They are sometimes called "formal proofs" because of their rigor, but logicians use the term "formal proof" to refer to a different type of proof entirely.
In logic, a formal proof is not written in a natural language, but instead uses a formal language consisting of certain strings of symbols from a fixed alphabet. This allows the definition of a formal proof to be precisely specified without any ambiguity. The field of proof theory studies formal proofs and their properties. Although each informal proof can, in theory, be converted into a formal proof, this is rarely done in practice. The study of formal proofs is used to determine properties of provability in general, and to show that certain undecidable statements are not provable.
A classic question in philosophy asks whether mathematical proofs are analytic or synthetic. Kant, who introduced the analytic-synthetic distinction, believed mathematical proofs are synthetic.
Proofs may be viewed as aesthetic objects, admired for their mathematical beauty. The mathematician Paul Erdős was known for describing proofs he found particularly elegant as coming from "The Book", a hypothetical tome containing the most beautiful method(s) of proving each theorem. The book Proofs from THE BOOK, published in 2003, is devoted to presenting 32 proofs its editors find particularly pleasing.
Methods of proof
Direct proof
Main article: Direct proofIn direct proof, the conclusion is established by logically combining the axioms, definitions, and earlier theorems.[16] For example, direct proof can be used to establish that the sum of two even integers is always even:
- Consider two even integers x and y. Since they are even, they can be written as x=2a and y=2b respectively for integers a and b. Then the sum x + y = 2a + 2b = 2(a + b). From this it is clear x+y has 2 as a factor and therefore is even, so the sum of any two even integers is even.
This proof uses definition of even integers, as well as distribution law.
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Q. Can we prove all mathematical statements? What does Godel's incompleteness theorem have to do with mathematical proof? Are there alternatives to rigorous proof? I hope to have links
Asked by Samuel Yee - Mon Jul 9 09:12:56 2007 - - 3 Answers - 0 Comments
A. Godel's incompleteness theorem doesn't really have anything to do with the notion of rigourous proof per se. In a nutshell, if you have a formal system of logic (e.g. mathematics), the system is considered complete if you can prove every true statement in it. However, Godel showed that every formal system will have true statements in it that cannot be proven by that particular formal system. In other words, the statement of the Incompleteness theorem is that in any formal system (such as mathematics) there are TRUE statements that cannot be proven within that system which makes these formal systems incomplete. For links you can start with this: 's_incompleteness_theorem There is also a good intuitive discussion of it in Roger… [cont.]
Answered by unmasked - Mon Jul 9 09:26:09 2007
