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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. Terence Tao has referred to this concept under the name "cheap nonstandard analysis." 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. ''Calculus Made Easy'' notably uses nilpotent infinitesimals. 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 f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus
Calculus is the mathematics, 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. Originally called infinitesimal calculus or "the calculus of infinitesimals", 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. 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. It is the "mathematical backbone" for dealing with problems where variables change with time or another reference variable. Infinitesimal calculus was formulated separately ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surreal Number
In mathematics, the surreal number system is a total order, totally ordered proper class containing not only the real numbers but also Infinity, infinite and infinitesimal, infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book ''Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness''. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbe ... [...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 University of Cambridge, 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. Moerd ... [...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 a philosopher and logician who is distinguished professor of philosophy at the ... [...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 ... [...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 (ε, δ)-definition of limit, limits 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 inf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. 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 space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...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 argument (or index) approaches some value. Limits of functions 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. The limit inferior and limit superior provide generalizations of the concept of a limit which are particularly relevant when the limit at a point may not exist. Notation In formulas, a limit of a function is usually written as : \lim_ f(x) = L, and is read as "the limit of of as approaches equals ". This means that the value of the function can be made arbitrarily close to , by choosing sufficiently close to . Alternatively, the fact that a function approaches the limit as approaches is sometimes denoted by a right arrow (→ or ... [...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 original shape. However, the pieces themselves are not "solids" in the traditional 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 a veridical paradox: ... [...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 the closed interval I = ,b/math> ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
