Infinitesimal

In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such as the surreal numbers and hyperreal numbers, which can be thought of as the real numbers augmented with a system of infinitesimal quantities, as well as infinite quantities, which are the reciprocals of the infinitesimals.
They were famously introduced in the development of calculus, where the derivative was originally thought of as a ratio of two infinitesimal quantities. This definition, like much of the mathematics of the time, was not formalized in a perfectly rigorous way. As a result, subsequent formal treatments of calculus tended to drop the infinitesimal viewpoint in favor of limits, which can be performed using the standard reals.
Infinitesimals regained popularity in the 20th century with Abraham Robinson's development of nonstandard analysis and the hyperreal numbers, which showed that a formal treatment of infinitesimal calculus was possible, after a long controversy on this topic by centuries of mathematics. Following this was the development of the surreal numbers, a closely related formalization of infinite and infinitesimal numbers that includes both the hyperreal numbers and ordinal numbers, and which is the largest ordered field.
The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were infinitely 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 in mathematical use, "infinitesimal" means "infinitely small", or smaller than any standard real number. To give it a meaning, infinitesimals are often compared to other infinitesimals of similar size (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 nonstandard 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:
History of the infinitesimal.
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 numbers).
First-order properties.
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.
The resulting extended number system cannot agree with the reals on all properties that can be expressed by quantification over sets, because the goal is to construct a non-Archimedean system, and the Archimedean principle can be expressed by quantification over sets. One can conservatively extend any theory including reals, including set theory, to include infinitesimals, just by adding a countably infinite list of axioms that assert that a number is smaller than 1/2, 1/3, 1/4 and so on. Similarly, the completeness property cannot be expected to carry over, because the reals are the unique complete ordered field up to isomorphism.
We can distinguish three levels at which a non-Archimedean number system could have first-order properties compatible with those of the reals:
Systems in category 1, at the weak end of the spectrum, are relatively easy to construct, but do not allow a full treatment of classical analysis using infinitesimals in the spirit of Newton and Leibniz. For example, the transcendental functions are defined in terms of infinite limiting processes, and therefore there is typically no way to define them in first-order logic. Increasing the analytic strength of the system by passing to categories 2 and 3, we find that the flavor of the treatment tends to become less constructive, and it becomes more difficult to say anything concrete about the hierarchical structure of infinities and infinitesimals.
Number systems that include infinitesimals.
Formal series.
Laurent series.
An example from category 1 above is the field of Laurent series with a finite number of negative-power terms. For example, the Laurent series consisting only of the constant term 1 is identified with the real number 1, and the series with only the linear term "x" is thought of as the simplest infinitesimal, from which the other infinitesimals are constructed. Dictionary ordering is used, which is equivalent to considering higher powers of "x" as negligible compared to lower powers. David O. Tall refers to this system as the super-reals, not to be confused with the superreal number system of Dales and Woodin. Since a Taylor series evaluated with a Laurent series as its argument is still a Laurent series, the system can be used to do calculus on transcendental functions if they are analytic. These infinitesimals have different first-order properties than the reals because, for example, the basic infinitesimal "x" does not have a square root.
The Levi-Civita field.
The Levi-Civita field is similar to the Laurent series, but is algebraically closed. For example, the basic infinitesimal x has a square root. This field is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented in floating point.
Transseries.
The field of transseries is larger than the Levi-Civita field. An example of a transseries is:
where for purposes of ordering "x" is considered infinite.
Surreal numbers.
Conway's surreal numbers fall into category 2. They are a system designed to be as rich as possible in different sizes of numbers, but not necessarily for convenience in doing analysis. Certain transcendental functions can be carried over to the surreals, including logarithms and exponentials, but most, e.g., the sine function, cannot. The existence of any particular surreal number, even one that has a direct counterpart in the reals, is not known a priori, and must be proved.
Hyperreals.
The most widespread technique for handling infinitesimals is the hyperreals, developed by Abraham Robinson in the 1960s. They fall into category 3 above, having been designed that way so all of classical analysis can be carried over from the reals. This property of being able to carry over all relations in a natural way is known as the transfer principle, proved by Jerzy Łoś in 1955. For example, the transcendental function sin has a natural counterpart *sin that takes a hyperreal input and gives a hyperreal output, and similarly the set of natural numbers formula_2 has a natural counterpart formula_3, which contains both finite and infinite integers. A proposition such as formula_4 carries over to the hyperreals as formula_5 .
Superreals.
The superreal number system of Dales and Woodin is a generalization of the hyperreals. It is different from the super-real system defined by David Tall.
Dual numbers.
In linear algebra, the dual numbers extend the reals by adjoining one infinitesimal, the new element ε with the property ε2 = 0 (that is, ε is nilpotent). Every dual number has the form "z" = "a" + "b"ε with "a" and "b" being uniquely determined real numbers.
One application of dual numbers is automatic differentiation. This application can be generalized to polynomials in n variables, using the Exterior algebra of an n-dimensional vector space.
Smooth infinitesimal analysis.
Synthetic differential geometry or smooth infinitesimal analysis have roots in category theory. This approach departs from the classical logic used in conventional mathematics by denying the general applicability of the law of excluded middle – i.e., "not" ("a" ≠ "b") does not have to mean "a" = "b". A "nilsquare" or "nilpotent" infinitesimal can then be defined. This is a number "x" where "x"2 = 0 is true, but "x" = 0 need not be true at the same time. Since the background logic is intuitionistic logic, it is not immediately clear how to classify this system with regard to classes 1, 2, and 3. Intuitionistic analogues of these classes would have to be developed first.
Infinitesimal delta functions.
Cauchy used an infinitesimal formula_6 to write down a unit impulse, infinitely tall and narrow Dirac-type delta function formula_7 satisfying formula_8 in a number of articles in 1827, see Laugwitz (1989). Cauchy defined an infinitesimal in 1821 (Cours d'Analyse) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and Lazare Carnot's terminology.
Modern set-theoretic approaches allow one to define infinitesimals via the ultrapower construction, where a null sequence becomes an infinitesimal in the sense of an equivalence class modulo a relation defined in terms of a suitable ultrafilter. The article by Yamashita (2007) contains a bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the hyperreals.
Logical properties.
The method of constructing infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used. We consider here systems where infinitesimals can be shown to exist.
In 1936 Maltsev proved the compactness theorem. This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them. A consequence of this theorem is that if there is a number system in which it is true that for any positive integer "n" there is a positive number "x" such that 0 < "x" < 1/"n", then there exists an extension of that number system in which it is true that there exists a positive number "x" such that for any positive integer "n" we have 0 < "x" < 1/"n". The possibility to switch "for any" and "there exists" is crucial. The first statement is true in the real numbers as given in ZFC set theory : for any positive integer "n" it is possible to find a real number between 1/"n" and zero, but this real number depends on "n". Here, one chooses "n" first, then one finds the corresponding "x". In the second expression, the statement says that there is an "x" (at least one), chosen first, which is between 0 and 1/"n" for any "n". In this case "x" is infinitesimal. This is not true in the real numbers (R) given by ZFC. Nonetheless, the theorem proves that there is a model (a number system) in which this is true. The question is: what is this model? What are its properties? Is there only one such model?
There are in fact many ways to construct such a one-dimensional linearly ordered set of numbers, but fundamentally, there are two different approaches:
In 1960, Abraham Robinson provided an answer following the first approach. The extended set is called the hyperreals and contains numbers less in absolute value than any positive real number. The method may be considered relatively complex but it does prove that infinitesimals exist in the universe of ZFC set theory. The real numbers are called standard numbers and the new non-real hyperreals are called nonstandard.
In 1977 Edward Nelson provided an answer following the second approach. The extended axioms are IST, which stands either for Internal set theory or for the initials of the three extra axioms: Idealization, Standardization, Transfer. In this system we consider that the language is extended in such a way that we can express facts about infinitesimals. The real numbers are either standard or nonstandard. An infinitesimal is a nonstandard real number that is less, in absolute value, than any positive standard real number.
In 2006 Karel Hrbacek developed an extension of Nelson's approach in which the real numbers are stratified in (infinitely) many levels; i.e., in the coarsest level there are no infinitesimals nor unlimited numbers. Infinitesimals are in a finer level and there are also infinitesimals with respect to this new level and so on.
Infinitesimals in teaching.
Calculus textbooks based on infinitesimals include the classic "Calculus Made Easy" by Silvanus P. Thompson (bearing the motto "What one fool can do another can") and the German text "Mathematik fur Mittlere Technische Fachschulen der Maschinenindustrie" by R. Neuendorff. Pioneering works based on Abraham Robinson's infinitesimals include texts by Stroyan (dating from 1972) and Howard Jerome Keisler (). Students easily relate to the intuitive notion of an infinitesimal difference 1-"0.999...", where "0.999..." differs from its standard meaning as the real number 1, and is reinterpreted as an infinite terminating extended decimal that is strictly less than 1.
Another elementary calculus text that uses the theory of infinitesimals as developed by Robinson is "Infinitesimal Calculus" by Henle and Kleinberg, originally published in 1979. The authors introduce the language of first order logic, and demonstrate the construction of a first order model of the hyperreal numbers. The text provides an introduction to the basics of integral and differential calculus in one dimension, including sequences and series of functions. In an Appendix, they also treat the extension of their model to the "hyperhyper"reals, and demonstrate some applications for the extended model.
Functions tending to zero.
In a related but somewhat different sense, which evolved from the original definition of "infinitesimal" as an infinitely small quantity, the term has also been used to refer to a function tending to zero. More precisely, Loomis and Sternberg's "Advanced Calculus" defines the function class of infinitesimals, formula_9, as a subset of functions formula_10 between normed vector spaces by formula_11, as well as two related classes formula_12 (see Big-O notation) by formula_13, andformula_14.The set inclusions formula_15generally hold. That the inclusions are proper is demonstrated by the real-valued functions of a real variable formula_16, formula_17, and formula_18: formula_19 but formula_20 and formula_21.As an application of these definitions, a mapping formula_22 between normed vector spaces is defined to be differentiable at formula_23 if there is a formula_24 [i.e, a bounded linear map formula_25] such that formula_26in a neighborhood of formula_6. If such a map exists, it is unique; this map is called the "differential" and is denoted by formula_28, coinciding with the traditional notation for the classical (though logically flawed) notion of a differential as an infinitely small "piece" of "F". This definition represents a generalization of the usual definition of differentiability for vector-valued functions of (open subsets of) Euclidean spaces.
Array of random variables.
Let formula_29 be a probability space and let formula_30. An array formula_31 of random variables is called infinitesimal if for every formula_32, we have:
The notion of infinitesimal array is essential in some central limit theorems and it is easily seen by monotonicity of the expectation operator that any array satisfying Lindeberg's condition is infinitesimal, thus playing an important role in Lindeberg's Central Limit Theorem (a generalization of the central limit theorem).