Infinitesimal

In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the 'infinity-th' item in a sequence.Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, 'infinitesimal' means 'extremely small'. To give it a meaning, it usually must be compared to another infinitesimal object in the same context (as in a derivative). Infinitely many infinitesimals are summed to produce an integral. I ( V , W ) = { f : V → W   |   f ( 0 ) = 0 , ( ∀ ϵ > 0 ) ( ∃ δ > 0 )   ∍   | | ξ | | < δ ⟹ | | f ( ξ ) | | < ϵ } {displaystyle {mathfrak {I}}(V,W)={f:V o W | f(0)=0,(forall epsilon >0)(exists delta >0) ackepsilon ||xi ||<delta implies ||f(xi )||<epsilon }} , O ( V , W ) = { f : V → W   |   f ( 0 ) = 0 ,   ( ∃ r > 0 , c > 0 )   ∍   | | ξ | | < r ⟹ | | f ( ξ ) | | ≤ c | | ξ | | } {displaystyle {mathfrak {O}}(V,W)={f:V o W | f(0)=0, (exists r>0,c>0) ackepsilon ||xi ||<rimplies ||f(xi )||leq c||xi ||}} , and o ( V , W ) = { f : V → W   |   f ( 0 ) = 0 ,   lim | | ξ | | → 0 | | f ( ξ ) | | / | | ξ | | = 0 } {displaystyle {mathfrak {o}}(V,W)={f:V o W | f(0)=0, lim _{||xi || o 0}||f(xi )||/||xi ||=0}} . f , g , h ∈ I ( R , R ) ,   g , h ∈ O ( R , R ) ,   h ∈ o ( R , R ) {displaystyle f,g,hin {mathfrak {I}}(mathbb {R} ,mathbb {R} ), g,hin {mathfrak {O}}(mathbb {R} ,mathbb {R} ), hin {mathfrak {o}}(mathbb {R} ,mathbb {R} )} but f , g ∉ o ( R , R ) {displaystyle f,g otin {mathfrak {o}}(mathbb {R} ,mathbb {R} )} and f ∉ O ( R , R ) {displaystyle f otin {mathfrak {O}}(mathbb {R} ,mathbb {R} )} . [ F ( α + ξ ) − F ( α ) ] − T ( ξ ) ∈ o ( V , W ) {displaystyle -T(xi )in {mathfrak {o}}(V,W)} In mathematics, infinitesimals are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the 'infinity-th' item in a sequence.Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, 'infinitesimal' means 'extremely small'. To give it a meaning, it usually must be compared to another infinitesimal object in the same context (as in a derivative). Infinitely many infinitesimals are summed to produce an integral. The concept of infinitesimals was originally introduced around 1670 by either Nicolaus Mercator or Gottfried Wilhelm Leibniz. Archimedes used what eventually came to be known as the method of indivisibles in his work The Method of Mechanical Theorems to find areas of regions and volumes of solids. In his formal published treatises, Archimedes solved the same problem using the method of exhaustion. The 15th century saw the work of Nicholas of Cusa, further developed in the 17th century by Johannes Kepler, in particular calculation of area of a circle by representing the latter as an infinite-sided polygon. Simon Stevin's work on decimal representation of all numbers in the 16th century prepared the ground for the real continuum. Bonaventura Cavalieri's method of indivisibles led to an extension of the results of the classical authors. The method of indivisibles related to geometrical figures as being composed of entities of codimension 1. John Wallis's infinitesimals differed from indivisibles in that he would decompose geometrical figures into infinitely thin building blocks of the same dimension as the figure, preparing the ground for general methods of the integral calculus. He exploited an infinitesimal denoted 1/∞ in area calculations. The use of infinitesimals by Leibniz relied upon heuristic principles, such as the law of continuity: what succeeds for the finite numbers succeeds also for the infinite numbers and vice versa; and the transcendental law of homogeneity that specifies procedures for replacing expressions involving inassignable quantities, by expressions involving only assignable ones. The 18th century saw routine use of infinitesimals by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange. Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his Cours d'Analyse, and in defining an early form of a Dirac delta function. As Cantor and Dedekind were developing more abstract versions of Stevin's continuum, Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions. Du Bois-Reymond's work inspired both Émile Borel and Thoralf Skolem. Borel explicitly linked du Bois-Reymond's work to Cauchy's work on rates of growth of infinitesimals. Skolem developed the first non-standard models of arithmetic in 1934. A mathematical implementation of both the law of continuity and infinitesimals was achieved by Abraham Robinson in 1961, who developed non-standard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955. The hyperreals implement an infinitesimal-enriched continuum and the transfer principle implements Leibniz's law of continuity. The standard part function implements Fermat's adequality. Vladimir Arnold wrote in 1990: The notion of infinitely small quantities was discussed by the Eleatic School. The Greek mathematician Archimedes (c.287 BC–c.212 BC), in The Method of Mechanical Theorems, was the first to propose a logically rigorous definition of infinitesimals. His Archimedean property defines a number x as infinite if it satisfies the conditions |x|>1, |x|>1+1, |x|>1+1+1, ..., and infinitesimal if x≠0 and a similar set of conditions holds for x and the reciprocals of the positive integers. A number system is said to be Archimedean if it contains no infinite or infinitesimal members. The English mathematician John Wallis introduced the expression 1/∞ in his 1655 book Treatise on the Conic Sections. The symbol, which denotes the reciprocal, or inverse, of ∞, is the symbolic representation of the mathematical concept of an infinitesimal. In his Treatise on the Conic Sections Wallis also discusses the concept of a relationship between the symbolic representation of infinitesimal 1/∞ that he introduced and the concept of infinity for which he introduced the symbol ∞. The concept suggests a thought experiment of adding an infinite number of parallelograms of infinitesimal width to form a finite area. This concept was the predecessor to the modern method of integration used in integral calculus. The conceptual origins of the concept of the infinitesimal 1/∞ can be traced as far back as the Greek philosopher Zeno of Elea, whose Zeno's dichotomy paradox was the first mathematical concept to consider the relationship between a finite interval and an interval approaching that of an infinitesimal-sized interval. Infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632. Prior to the invention of calculus mathematicians were able to calculate tangent lines using Pierre de Fermat's method of adequality and René Descartes' method of normals. There is debate among scholars as to whether the method was infinitesimal or algebraic in nature. When Newton and Leibniz invented the calculus, they made use of infinitesimals, Newton's fluxions and Leibniz' differential. The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work The Analyst. Mathematicians, scientists, and engineers continued to use infinitesimals to produce correct results. In the second half of the nineteenth century, the calculus was reformulated by Augustin-Louis Cauchy, Bernard Bolzano, Karl Weierstrass, Cantor, Dedekind, and others using the (ε, δ)-definition of limit and set theory.While the followers of Cantor, Dedekind, and Weierstrass sought to rid analysis of infinitesimals, and their philosophical allies like Bertrand Russell and Rudolf Carnap declared that infinitesimals are pseudoconcepts, Hermann Cohen and his Marburg school of neo-Kantianism sought to develop a working logic of infinitesimals. The mathematical study of systems containing infinitesimals continued through the work of Levi-Civita, Giuseppe Veronese, Paul du Bois-Reymond, and others, throughout the late nineteenth and the twentieth centuries, as documented by Philip Ehrlich (2006). In the 20th century, it was found that infinitesimals could serve as a basis for calculus and analysis; see hyperreal number. In extending the real numbers to include infinite and infinitesimal quantities, one typically wishes to be as conservative as possible by not changing any of their elementary properties. This guarantees that as many familiar results as possible are still available. Typically elementary means that there is no quantification over sets, but only over elements. This limitation allows statements of the form 'for any number x...' For example, the axiom that states 'for any number x, x + 0 = x' would still apply. The same is true for quantification over several numbers, e.g., 'for any numbers x and y, xy = yx.' However, statements of the form 'for any set S of numbers ...' may not carry over. Logic with this limitation on quantification is referred to as first-order logic.