|
Smooth Infinitesimal Analysis
Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete entities. As a theory, it is a subset of synthetic differential geometry. The ''nilsquare'' or ''nilpotent'' infinitesimals are numbers ''ε'' where ''ε''² = 0 is true, but ''ε'' = 0 need not be true at the same time. Overview This approach departs from the classical logic used in conventional mathematics by denying the law of the excluded middle, e.g., ''NOT'' (''a'' ≠ ''b'') does not imply ''a'' = ''b''. In particular, in a theory of smooth infinitesimal analysis one can prove for all infinitesimals ''ε'', ''NOT'' (''ε'' ≠ 0); yet it is provably false that all infinitesimals are equal to zero. One can see that the law of excluded middle cannot hold from the following basic theorem (again, unders ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous Rate of change (mathematics), rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence (mathematics), convergence of infinite sequences and Series (mathematics), infinite series to a well-defined limit (mathematics), limit. Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Later work, including (ε, δ)-definition of limit, codify ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ieke Moerdijk
Izak (Ieke) Moerdijk (; born 23 January 1958) is a Dutch mathematician, currently working at Utrecht University, who in 2012 won the Spinoza prize. Education and career Moerdijk studied mathematics, philosophy and general linguistics at the University of Amsterdam. He obtained his PhD ''cum laude'' in 1985 at the same institution. His thesis was entitled ''Topics in intuitionism and topos theory'' and was written under the supervision of Anne Sjerp Troelstra. After that, he worked as postdoctoral researcher at the University of Chicago and Cambridge. From 1988 to 2011 he was professor at Utrecht University. After working at the Mathematical Institute of the Radboud University Nijmegen for a few years, he returned to Utrecht University in 2016. In 2000 Moerdijk was an invited speaker to the 3rd European Congress of Mathematics. He was elected member of the Royal Netherlands Academy of Arts and Sciences in 2006 and of the Academia Europaea in 2014. Moerdijk received t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Lane Bell
John Lane Bell (born March 25, 1945) is an Anglo-Canadian philosopher, mathematician and logician. He is Professor Emeritus of Philosophy at the University of Western Ontario in Canada. His research includes such topics as set theory, model theory, lattice theory, modal logic, quantum logic, constructive mathematics, type theory, topos theory, infinitesimal analysis, spacetime theory, and the philosophy of mathematics. He is the author of more than 70 articles and of 13 books. In 2009, he was elected a Fellow of the Royal Society of Canada. Biography John Bell was awarded a scholarship to Oxford University at the age of 15, and graduated with a D.Phil. in Mathematics: his dissertation supervisor was John Crossley. During 1968–89 he was Lecturer in Mathematics and Reader in Mathematical Logic at the London School of Economics. Bell's students include Graham Priest Graham Priest (born 1948) is Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Number
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0. Dual numbers can be added component-wise, and multiplied by the formula : (a+b\varepsilon)(c+d\varepsilon) = ac + (ad+bc)\varepsilon, which follows from the property and the fact that multiplication is a bilinear operation. The dual numbers form a commutative algebra of dimension two over the reals, and also an Artinian local ring. They are one of the simplest examples of a ring that has nonzero nilpotent elements. History Dual numbers were introduced in 1873 by William Clifford, and were used at the beginning of the twentieth century by the German mathematician Eduard Study, who used them to represent the dual angle which measures the relative position of two skew lines in space. Study defined a dual angle as , where is the angle be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Synthetic Differential Geometry
In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulation. The first is that most of the analytic data for describing the class of smooth manifolds can be encoded into certain fibre bundles on manifolds: namely bundles of jets (see also jet bundle). The second insight is that the operation of assigning a bundle of jets to a smooth manifold is functorial in nature. The third insight is that over a certain category, these are representable functors. Furthermore, their representatives are related to the algebras of dual numbers, so that smooth infinitesimal analysis may be used. Synthetic differential geometry can serve as a platform for formulating certain otherwise obscure or confusing notions from differential geometry. For example, the meaning of what it means to be ''natural'' (or ''invariant'') has a particularly simple express ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-standard Analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Nonstandard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Nonstandard analysis originated in the early 1960s by the mathematician Abraham Robinson. He wrote: ... the idea of infinitely small or ''infinitesimal'' quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection ... that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit of a function is usually written as : \lim_ f(x) = L, (although a few authors may use "Lt" instead of "lim") and is read as "the limit of of as approaches equals ". The fact that a function approaches the limit as approaches is sometimes denoted by a right arrow (→ or \rightarrow), as in :f(x) \to L \text x \to c, which reads "f of x tends to L as x tends to c". History Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work ''Opus Geometricum'' (1647): "The ''terminus'' of a pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach–Tarski Paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces. An alternative form of the theorem states that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox". The theorem is called a paradox because it contradicts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intermediate Value Theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two important corollaries: # If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). # The image of a continuous function over an interval is itself an interval. Motivation This captures an intuitive property of continuous functions over the real numbers: given ''f'' continuous on ,2/math> with the known values f(1) = 3 and f(2) = 5, then the graph of y = f(x) must pass through the horizontal line y = 4 while x moves from 1 to 2. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper. Theorem The intermediate value theorem states the following: Consider an interval I = ,b/math> of real n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transfer Principle
In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first-order language of fields that is true for the complex numbers is also true for any algebraically closed field of characteristic 0. History An incipient form of a transfer principle was described by Leibniz under the name of "the Law of Continuity". Here infinitesimals are expected to have the "same" properties as appreciable numbers. The transfer principle can also be viewed as a rigorous formalization of the principle of permanence. Similar tendencies are found in Cauchy, who used infinitesimals to define both the continuity of functions (in Cours d'Analyse) and a form of the Dirac delta function. In 1955, Jerzy Łoś proved the transfer principle for any hyperreal number system. Its most common use is in Abraham Robinson's nonstanda ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
