axioms

Some of the problems and phenomena eventually addressed by renormalization actually appeared earlier in the classical electrodynamics of point particles in the 19th and early 20th century. The basic problem is that the observable properties of an interacting particle cannot be entirely separated from the field that mediates the interaction. The standard classical example is the energy of a charged particle. To cram a finite amount of charge into a single point requires an infinite amount of energy; this manifests itself as the infinite energy of the particle's electric field. The energy de ...

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Quantum field theory, Quantum field theory - Why quantum field theory, Quantum field theory - What QFT is, Quantum field theory - Technical statement, Quantum field theory - Quantizing a classical field theory, Quantum field theory - Canonical quantization, Quantum field theory - Path integral methods, Quantum field theory - The axiomatic approach, Quantum field theory - Renormalization, Quantum field theory - Gauge theories, Quantum field theory - Supersymmetry, Quantum field theory - Beyond local field theory, Quantum field theory - History, Quantum field theory - Suggested reading

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Reverse mathematics - Generalities. The principle of reverse mathematics is the following: one starts with a framework language and a base theory—a core (axiom) system—, which is too weak to prove most of the theorems one might be interested in, but still powerful enough to prove the equivalence of certain statements whose difference is deemed irrelevant or establish certain facts which are considered obvious enough (such as the fact that addition is commutative). Above that weak base theory there is a ful ...

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Reverse mathematics, Reverse mathematics - Principles, Reverse mathematics - Generalities, Reverse mathematics - Choice of the language and base system, Reverse mathematics - Second-order arithmetic, Reverse mathematics - The language, Reverse mathematics - Coding mathematics in second-order arithmetic, Reverse mathematics - Basic axioms, Reverse mathematics - Induction and comprehension axioms, Reverse mathematics - The full system, Reverse mathematics - Arithmetical comprehension, Reverse mathematics - The arithmetical hierarchy for formulas, Reverse mathematics - The base system, Reverse mathematics - Stronger systems, Reverse mathematics - Models of second-order arithmetic, Reverse mathematics - The main systems, Reverse mathematics - Recursive comprehension, Reverse mathematics - Weak König's lemma, Reverse mathematics - Arithmetical comprehension, Reverse mathematics - Arithmetical transfinite recursion, Reverse mathematics - Π¹₁-comprehension, Reverse mathematics - Some further systems, Reverse mathematics - An example of a reverse mathematical proof

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Gauge theory is a generalized version of the equivalence principle of general relativity. ...

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Quantum field theory, Quantum field theory - Why quantum field theory, Quantum field theory - What QFT is, Quantum field theory - Technical statement, Quantum field theory - Quantizing a classical field theory, Quantum field theory - Canonical quantization, Quantum field theory - Path integral methods, Quantum field theory - The axiomatic approach, Quantum field theory - Renormalization, Quantum field theory - Gauge theories, Quantum field theory - Supersymmetry, Quantum field theory - Beyond local field theory, Quantum field theory - History, Quantum field theory - Suggested reading

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More details can be found in the article on the history of quantum field theory. Quantum field theory was created by Dirac when he attempted to quantize the electromagnetic field in the late 1920s. The early development of the field involved Fock, Jordan, Pauli, Heisenberg, Bethe, Tomonaga, Schwinger, Feynman, and Dyson. This phase of development culminated with the construction of the ...

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Quantum field theory, Quantum field theory - Why quantum field theory, Quantum field theory - What QFT is, Quantum field theory - Technical statement, Quantum field theory - Quantizing a classical field theory, Quantum field theory - Canonical quantization, Quantum field theory - Path integral methods, Quantum field theory - The axiomatic approach, Quantum field theory - Renormalization, Quantum field theory - Gauge theories, Quantum field theory - Supersymmetry, Quantum field theory - Beyond local field theory, Quantum field theory - History, Quantum field theory - Suggested reading

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Quantum field theory originated in the problem of computing the power radiated by an atom when it dropped from one quantum state to another of lower energy. This problem was first examined by Max Born and Pascual Jordan in 1925. In 1926, Max Born, Werner Heisenberg and Pascual Jordan wrote down the quantum theory of the electromagnetic field neglecting polarization and sources to obtain what would today be called a free field theory. In order to quantize this theory, they used the canonical quantization procedure. In 1927, Paul Dirac gave th ...

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Quantum field theory, Quantum field theory - Why quantum field theory, Quantum field theory - What QFT is, Quantum field theory - Technical statement, Quantum field theory - Quantizing a classical field theory, Quantum field theory - Canonical quantization, Quantum field theory - Path integral methods, Quantum field theory - The axiomatic approach, Quantum field theory - Renormalization, Quantum field theory - Gauge theories, Quantum field theory - Supersymmetry, Quantum field theory - Beyond local field theory, Quantum field theory - History, Quantum field theory - Suggested reading

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Some of the problems and phenomena eventually addressed by renormalization actually appeared earlier in the classical electrodynamics of point particles in the 19th and early 20th century. A single particle state in quantum field theory incorporates within it multiparticle states. This is most simply demonstrated by examining the evolution of a single particle state in the interaction picture— Taking the overlap with the initial state, one retains the even powers of HI< ...

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Quantum field theory, Quantum field theory - Why quantum field theory, Quantum field theory - What QFT is, Quantum field theory - Technical statement, Quantum field theory - Quantizing a classical field theory, Quantum field theory - Canonical quantization, Quantum field theory - Path integral methods, Quantum field theory - The axiomatic approach, Quantum field theory - Renormalization, Quantum field theory - Gauge theories, Quantum field theory - Supersymmetry, Quantum field theory - Beyond local field theory, Quantum field theory - History, Quantum field theory - Suggested reading

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From the point of view of universal algebra, an algebra (or abstract algebra) is a set A together with a collection of operations on A. An n-ary operation on A is a function that takes n elements of A and returns a single element of A. Thus, a 0-ary operation (or nullary operation) is simply an element of A, or a constant, often denoted by a letter like a. A 1-ary operation (or unary operation) is simply a function from A to ASee also:

Universal algebra, Universal algebra - Basic idea, Universal algebra - Examples, Universal algebra - Groups, Universal algebra - Modules, Universal algebra - Further issues

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It is common to study vector spaces with certain additional structures. This is often necessary for recovering ordinary notions from geometry. Some of these additional structures include: A real or complex vector space with a defined length concept, i.e., a norm, is called a normed vector space. A real or complex vector space with a notion of both length and angle is called an inner product space. A vector space with a topology compatible with the operations (i.e., such that addition and scalar mu ...

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Vector space, Vector space - Formal definition, Vector space - Elementary properties, Vector space - Examples, Vector space - Subspaces and bases, Vector space - Linear transformations, Vector space - Generalizations and additional structures

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A well-known example is the Taniyama-Shimura conjecture, now the Taniyama-Shimura theorem, which proposed that each elliptic curve over the rational numbers can be translated into a modular form (in such a way as to preserve the associated L-function). There are difficulties in identifying this with an isomorphism, in any strict sense of the word. Certain curves had been known to be both elliptic curves (of genus 1) and modular curves, before the conjecture was formulated (about 1955). The surprising part of the conjecture was the extension ...

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Unifying theories in mathematics, Unifying theories in mathematics - Mathematical theories, Unifying theories in mathematics - Geometrical theories, Unifying theories in mathematics - Through-axiomatisation, Unifying theories in mathematics - Bourbaki, Unifying theories in mathematics - Category theory as a rival, Unifying theories in mathematics - Uniting theories, Unifying theories in mathematics - Reference list of major unifying concepts, Unifying theories in mathematics - Recent developments in relation with modular theory, Unifying theories in mathematics - Isomorphism conjectures in K-theory

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A short list of these theories might include: Cartesian geometry Calculus Complex analysis Galois theory Erlangen programme Lie group Set theory Hilbert space Recursive function Characteristic classes Homological algebra Homotopy theory Grothendieck's schemes Langlands philosophy Non-commutative g ...

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Unifying theories in mathematics, Unifying theories in mathematics - Mathematical theories, Unifying theories in mathematics - Geometrical theories, Unifying theories in mathematics - Through-axiomatisation, Unifying theories in mathematics - Bourbaki, Unifying theories in mathematics - Category theory as a rival, Unifying theories in mathematics - Uniting theories, Unifying theories in mathematics - Reference list of major unifying concepts, Unifying theories in mathematics - Recent developments in relation with modular theory, Unifying theories in mathematics - Isomorphism conjectures in K-theory

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On a less grandiose scale, there are frequent instances in which it appears that sets of results in two different branches of mathematics are similar, and one might ask whether there is a unifying framework which clarifies the connections. We have already noted the example of analytic geometry, and more generally the field of algebraic geometry thoroughly develops the connections between geometric objects (varieties, or more generally schemes) and algebraic ones (ideals); the touchstone result here is Hilbert's Nullstellensatz which roughly speaking shows that there is ...

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Unifying theories in mathematics, Unifying theories in mathematics - Mathematical theories, Unifying theories in mathematics - Geometrical theories, Unifying theories in mathematics - Through-axiomatisation, Unifying theories in mathematics - Bourbaki, Unifying theories in mathematics - Category theory as a rival, Unifying theories in mathematics - Uniting theories, Unifying theories in mathematics - Reference list of major unifying concepts, Unifying theories in mathematics - Recent developments in relation with modular theory, Unifying theories in mathematics - Isomorphism conjectures in K-theory

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Main article: Linear transformation Given two vector spaces V and W over the same field F, one can define linear transformations or “linear maps” from V to W. These are maps from V to W which are compatible with the relevant structure—i.e., they preserve sums and scalar products. The set of all linear maps from V to W, denoted L(V, W), is also a vector space over F. When bases for both V and W are given, linear ma ...

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Vector space, Vector space - Formal definition, Vector space - Elementary properties, Vector space - Examples, Vector space - Subspaces and bases, Vector space - Linear transformations, Vector space - Generalizations and additional structures

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Main articles: Linear subspace, Basis Given a vector space V, any nonempty subset W of V which is closed under addition and scalar multiplication is called a subspace of V. It is easy to see that subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set of vectors is called their span; if no vector can be removed without diminishing the span, the set is described as being linearly independent. A linearly independent set who ...

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Vector space, Vector space - Formal definition, Vector space - Elementary properties, Vector space - Examples, Vector space - Subspaces and bases, Vector space - Linear transformations, Vector space - Generalizations and additional structures

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Both of these seemingly contradictory statements are true using classical or two-valued logic - so long as the set of pink rhinoceros remains empty. Certainly, one would think it should be easy to avoid falling into the trap of employing vacuously true statements in rigorous proofs, but the history of mathematics contains many 'proofs' based on the negation of some accepted truth and subsequently demonstrating how this leads to a contradiction. One fundamental problem with such 'demonstrations' is the uncertainty of the truth-v ...

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Vacuous truth, Vacuous truth - Examples, Vacuous truth - Scope of the concept, Vacuous truth - Arguments of the semantic truth of vacuously true logical statements, Vacuous truth - Arguments that at least some vacuously true statements are true, Vacuous truth - Arguments for taking all vacuously true statements to be true, Vacuous truth - Arguments that only some vacuously true statements are true, Vacuous truth - Summary, Vacuous truth - Difficulties with the use of vacuous truth, Vacuous truth - Vacuous truths in mathematics

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The term "vacuously true" is generally applied to a statement S if S has a form similar to: P ⇒ Q, where P is false. ∀ x, P(x) ⇒ Q(x), where it is the case that ∀ x, ¬ P(x). ∀ x ∈ A, Q(x), where the set A is empty. ∀ ξ, Q(ξ), where the symbol ξ is restricted to a type that has no representatives. The first instance is the most basic one; the other three can ...

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Vacuous truth, Vacuous truth - Examples, Vacuous truth - Scope of the concept, Vacuous truth - Arguments of the semantic truth of vacuously true logical statements, Vacuous truth - Arguments that at least some vacuously true statements are true, Vacuous truth - Arguments for taking all vacuously true statements to be true, Vacuous truth - Arguments that only some vacuously true statements are true, Vacuous truth - Summary, Vacuous truth - Difficulties with the use of vacuous truth, Vacuous truth - Vacuous truths in mathematics

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The statement All elephants inside a loaf of bread are pink. is vacuously true since there are no elephants inside a loaf of bread; here property A is "being an elephant inside a loaf of bread", and property B is "being pink". Another example is If a prime number is even and bigger than two, then it must be divisible by three. There are no such prime numbers, so in a sense the ...

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Vacuous truth, Vacuous truth - Examples, Vacuous truth - Scope of the concept, Vacuous truth - Arguments of the semantic truth of vacuously true logical statements, Vacuous truth - Arguments that at least some vacuously true statements are true, Vacuous truth - Arguments for taking all vacuously true statements to be true, Vacuous truth - Arguments that only some vacuously true statements are true, Vacuous truth - Summary, Vacuous truth - Difficulties with the use of vacuous truth, Vacuous truth - Vacuous truths in mathematics

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An alternative (mostly complementary) to set theory but also serving to give a consistent approach to most of axiomatic mathematics is category theory, developed in the second half of the 20th century. A key theme from this point of view is that mathematics studies not only of certain kinds of objects (Lie groups, Banach spaces, etc.) but also of the mappings between them. In particular, this clarifies exactly what it means for the mathematical objects to be considered to be the same. (For example, are all equilateral triangles ...

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Unifying theories in mathematics, Unifying theories in mathematics - Mathematical theories, Unifying theories in mathematics - Geometrical theories, Unifying theories in mathematics - Through-axiomatisation, Unifying theories in mathematics - Bourbaki, Unifying theories in mathematics - Category theory as a rival, Unifying theories in mathematics - Uniting theories, Unifying theories in mathematics - Reference list of major unifying concepts, Unifying theories in mathematics - Recent developments in relation with modular theory, Unifying theories in mathematics - Isomorphism conjectures in K-theory

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The cause of axiomatic development was taken up in earnest by the Bourbaki group of mathematicians. Taken to its extreme this attitude demanded developing mathematics in its greatest generality, starting from the most general axioms and then specializing (e.g. introducing vector spaces over arbitrary fields and limiting to the real numbers only when absolutely necessary) even when the specializations were the theorems of primary interest. In particular, this perspective placed little value on fields of mathematics (such as combinatori ...

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Unifying theories in mathematics, Unifying theories in mathematics - Mathematical theories, Unifying theories in mathematics - Geometrical theories, Unifying theories in mathematics - Through-axiomatisation, Unifying theories in mathematics - Bourbaki, Unifying theories in mathematics - Category theory as a rival, Unifying theories in mathematics - Uniting theories, Unifying theories in mathematics - Reference list of major unifying concepts, Unifying theories in mathematics - Recent developments in relation with modular theory, Unifying theories in mathematics - Isomorphism conjectures in K-theory

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Topological spaces can be broadly classified, up to homeomorphism, by their topological properties. A topological property is a property of spaces that is invariant under homeomorphisms. To prove that two spaces are not homeomorphic it is sufficient to find a topological property which is not shared by them. Examples of such properties include connectedness, compactness, and various separation axioms. See t ...

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Topological space, Topological space - Definition, Topological space - Comparison of topologies, Topological space - Continuous functions, Topological space - Alternative definitions, Topological space - Examples of topological spaces, Topological space - Topological constructions, Topological space - Classification of topological spaces, Topological space - Topological spaces with algebraic structure, Topological space - Topological spaces with order structure

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The Socratic method is a negative method of hypotheses elimination, in that better hypotheses are found by steadily identifying and eliminating those which lead to contradictions. The method of Socrates is a search for the underlying hypotheses, assumptions, or axioms, which may subconsciously shape one's opinion, and to make them the subject of scrutiny, to determine their consistency with other beliefs. The basic form is a series of questions formulated as tests of logic and fact intended to help a person or group discover th ...

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Socratic method, Socratic method - Method, Socratic method - Practice, Socratic method - Application, Socratic method - Typical Application in Legal Education, Socratic method - Socratic Method in Psychotherapy

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The separation axioms are about the use of topological means to distinguish disjoint sets and distinct points. It's not enough for elements of a topological space to be distinct; we may want them to be topologically distinguishable. Similarly, it's not enough for subsets of a topological space to be disjoint; we may want them to be separated (in any of various ways). The separation axioms all say, in one way or another, that points or sets that are di ...

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Separation axiom, Separation axiom - Separated sets and topologically distinguishable points, Separation axiom - Definitions of the axioms, Separation axiom - Relationships between the axioms, Separation axiom - Other separation axioms, Separation axiom - Sources

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A function between topological spaces is said to be continuous if the inverse image of every open set is open. This is an attempt to capture the intuition that there are no "breaks" or "separations" in the function. A homeomorphism is a bijection that is continuous and whose inverse is also continuous. Two spaces are said to be homeomorphic if there exists a homeomorphism between them. From the standpoint o ...

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Topological space, Topological space - Definition, Topological space - Comparison of topologies, Topological space - Continuous functions, Topological space - Alternative definitions, Topological space - Examples of topological spaces, Topological space - Topological constructions, Topological space - Classification of topological spaces, Topological space - Topological spaces with algebraic structure, Topological space - Topological spaces with order structure

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