Infinitesimal

Infinitesimals (ε) and infinities (ω) on the hyperreal number line (ε = 1/ω)

Inmathematics,aninfinitesimal numberis a non-zero quantity that is closer to0than any non-zeroreal numberis. The wordinfinitesimalcomes from a 17th-centuryModern Latincoinageinfinitesimus,which originally referred to the "infinity-eth"item in asequence.

Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as thesurreal number systemand thehyperreal number system,which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are thereciprocalsof one another.

Infinitesimal numbers were introduced in thedevelopment of calculus,in which thederivativewas first conceived as a ratio of two infinitesimal quantities. This definition was notrigorously formalized.As calculus developed further, infinitesimals were replaced bylimits,which can be calculated using the standard real numbers.

Infinitesimals regained popularity in the 20th century withAbraham Robinson's development ofnonstandard analysisand thehyperreal numbers,which, after centuries of controversy, showed that a formal treatment of infinitesimal calculus was possible. Following this, mathematicians developed surreal numbers, a related formalization of infinite and infinitesimal numbers that include both hyperrealcardinalandordinal numbers,which is the largestordered field.

Vladimir Arnoldwrote in 1990:

Nowadays, when teaching analysis, it is not very popular to talk about infinitesimal quantities. Consequently, present-day students are not fully in command of this language. Nevertheless, it is still necessary to have command of it.

The crucial insightfor making infinitesimals feasible mathematical entities was that they could still retain certain properties such asangleorslope,even if these entities were infinitely small.

Infinitesimals are a basic ingredient incalculusas developed byLeibniz,including thelaw of continuityand thetranscendental 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 mathematics,infinitesimalmeans infinitely small, smaller than any standard real number. Infinitesimals are often compared to other infinitesimals of similar size, as in examining the derivative of a function. An infinite number of infinitesimals are summed to calculate anintegral.

The concept of infinitesimals was originally introduced around 1670 by eitherNicolaus MercatororGottfried Wilhelm Leibniz.Archimedesused what eventually came to be known as themethod of indivisiblesin his workThe Method of Mechanical Theoremsto find areas of regions and volumes of solids.In his formal published treatises, Archimedes solved the same problem using themethod of exhaustion.The 15th century saw the work ofNicholas of Cusa,further developed in the 17th century byJohannes Kepler,in particular, the calculation of the area of a circle by representing the latter as an infinite-sided polygon.Simon Stevin's work on the 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 ofcodimension1.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 denoted1/∞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 unassignable quantities, by expressions involving only assignable ones. The 18th century saw routine use of infinitesimals by mathematicians such asLeonhard EulerandJoseph-Louis Lagrange.Augustin-Louis Cauchyexploited infinitesimals both in definingcontinuityin hisCours d'Analyse,and in defining an early form of aDirac delta function.As Cantor andDedekindwere developing more abstract versions of Stevin's continuum,Paul du Bois-Reymondwrote a series of papers on infinitesimal-enriched continua based on growth rates of functions. Du Bois-Reymond's work inspired bothÉmile BorelandThoralf 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 byAbraham Robinsonin 1961, who developednonstandard analysisbased on earlier work byEdwin Hewittin 1948 andJerzy Łośin 1955. Thehyperrealsimplement an infinitesimal-enriched continuum and thetransfer principleimplements Leibniz's law of continuity. Thestandard part functionimplements Fermat'sadequality.

History of the infinitesimal

The notion of infinitely small quantities was discussed by theEleatic School.TheGreekmathematicianArchimedes(c. 287 BC – c. 212 BC), inThe Method of Mechanical Theorems,was the first to propose a logically rigorous definition of infinitesimals.HisArchimedean propertydefines a numberxas infinite if it satisfies the conditions|x| > 1,|x| > 1 + 1,|x| > 1 + 1 + 1,..., and infinitesimal ifx≠ 0and a similar set of conditions holds forxand the reciprocals of the positive integers. A number system is said to be Archimedean if it contains no infinite or infinitesimal members.

The English mathematicianJohn Wallisintroduced the expression 1/∞ in his 1655 bookTreatise 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 hisTreatise 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 athought experimentof adding an infinite number ofparallelogramsof infinitesimal width to form a finite area. This concept was the predecessor to the modern method of integration used inintegral calculus.The conceptual origins of the concept of the infinitesimal 1/∞ can be traced as far back as the Greek philosopherZeno of Elea,whoseZeno's dichotomy paradoxwas 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 usingPierre de Fermat's method ofadequalityandRené Descartes'method of normals.There is debate among scholars as to whether the method was infinitesimal or algebraic in nature. WhenNewtonandLeibnizinvented thecalculus,they made use of infinitesimals, Newton'sfluxionsand Leibniz'differential.The use of infinitesimals was attacked as incorrect byBishop Berkeleyin his workThe 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 byAugustin-Louis Cauchy,Bernard Bolzano,Karl Weierstrass,Cantor,Dedekind,and others using the(ε, δ)-definition of limitandset theory. While the followers of Cantor, Dedekind, and Weierstrass sought to rid analysis of infinitesimals, and their philosophical allies likeBertrand RussellandRudolf Carnapdeclared that infinitesimals arepseudoconcepts,Hermann Cohenand hisMarburg schoolofneo-Kantianismsought to develop a working logic of infinitesimals.The mathematical study of systems containing infinitesimals continued through the work ofLevi-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 (seehyperreal 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,elementarymeans that there is noquantificationoversets,but only over elements. This limitation allows statements of the form "for any number x..." For example, the axiom that states "for any numberx,x+ 0 =x"would still apply. The same is true for quantification over several numbers, e.g.," for any numbersxandy,xy=yx."However, statements of the form" for anysetSof numbers... "may not carry over. Logic with this limitation on quantification is referred to asfirst-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, thecompletenessproperty 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:

1. Anordered fieldobeys all the usual axioms of the real number system that can be stated in first-order logic. For example, thecommutativityaxiomx+y=y+xholds.
2. Areal closed fieldhas all the first-order properties of the real number system, regardless of whether they are usually taken as axiomatic, for statements involving the basic ordered-field relations +, ×, and ≤. This is a stronger condition than obeying the ordered-field axioms. More specifically, one includes additional first-order properties, such as the existence of a root for every odd-degree polynomial. For example, every number must have acube root.
3. The system could have all the first-order properties of the real number system for statements involvinganyrelations (regardless of whether those relations can be expressed using +, ×, and ≤). For example, there would have to be asinefunction that is well defined for infinite inputs; the same is true for every real function.

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, thetranscendental functionsare 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 ofLaurent serieswith 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 termxis thought of as the simplest infinitesimal, from which the other infinitesimals are constructed. Dictionary ordering is used, which is equivalent to considering higher powers ofxas negligible compared to lower powers.David O. Tallrefers to this system as the super-reals, not to be confused with thesuperreal numbersystem of Dales and Woodin. Since aTaylor seriesevaluated 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 infinitesimalxdoes not have a square root.

The Levi-Civita field

TheLevi-Civita fieldis 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 oftransseriesis larger than the Levi-Civita field.An example of a transseries is:

$e^{\sqrt {\ln \ln x}}+\ln \ln x+\sum _{j=0}^{\infty }e^{x}x^{-j},$

where for purposes of orderingxis considered infinite.

Surreal numbers

Conway'ssurreal numbersfall into category 2, except that the surreal numbers form aproper classand not a set.They are a system designed to be as rich as possible in different sizes of numbers, but not necessarily for convenience in doing analysis, in the sense that every ordered field is a subfield of the surreal numbers.There is a natural extension of the exponential function to the surreal numbers.

Hyperreals

Main article:Hyperreal number

The most widespread technique for handling infinitesimals is the hyperreals, developed byAbraham Robinsonin 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 thetransfer principle,proved byJerzy Łośin 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$\mathbb {N}$has a natural counterpart$^{*}\mathbb {N}$,which contains both finite and infinite integers. A proposition such as$\forall n\in \mathbb {N},\sin n\pi =0$carries over to the hyperreals as$\forall n\in {}^{*}\mathbb {N},{}^{*}\!\!\sin n\pi =0$.

Superreals

Main article:Superreal number

Thesuperreal numbersystem of Dales and Woodin is a generalization of the hyperreals. It is different from the super-real system defined byDavid Tall.

Dual numbers

Main article:Dual number

Inlinear algebra,thedual numbersextend the reals by adjoining one infinitesimal, the new element ε with the property ε²= 0 (that is, ε isnilpotent). Every dual number has the formz=a+bε withaandbbeing uniquely determined real numbers.

One application of dual numbers isautomatic differentiation.This application can be generalized to polynomials in n variables, using theExterior algebraof an n-dimensional vector space.

Smooth infinitesimal analysis

Main article:Smooth infinitesimal analysis

Synthetic differential geometryorsmooth infinitesimal analysishave roots incategory theory.This approach departs from the classical logic used in conventional mathematics by denying the general applicability of thelaw of excluded middle– i.e.,not(a≠b) does not have to meana=b.Anilsquareornilpotentinfinitesimal can then be defined. This is a numberxwherex²= 0 is true, butx= 0 need not be true at the same time. Since the background logic isintuitionistic 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

Cauchyused an infinitesimal$\alpha$to write down a unit impulse, infinitely tall and narrow Dirac-type delta function$\delta _{\alpha }$satisfying$\int F(x)\delta _{\alpha }(x)=F(0)$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 andLazare Carnot's terminology.

Modern set-theoretic approaches allow one to define infinitesimals via theultrapowerconstruction, where a null sequence becomes an infinitesimal in the sense of an equivalence class modulo a relation defined in terms of a suitableultrafilter.The article by Yamashita (2007) contains bibliography on modernDirac delta functionsin the context of an infinitesimal-enriched continuum provided by thehyperreals.

Logical properties

The method of constructing infinitesimals of the kind used in nonstandard analysis depends on themodeland which collection ofaxiomsare used. We consider here systems where infinitesimals can be shown to exist.

In 1936Maltsevproved thecompactness 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 integernthere is a positive numberxsuch that 0 <x< 1/n,then there exists an extension of that number system in which it is true that there exists a positive numberxsuch that for any positive integernwe 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 inZFCset theory:for any positive integernit is possible to find a real number between 1/nand zero, but this real number depends onn.Here, one choosesnfirst, then one finds the correspondingx.In the second expression, the statement says that there is anx(at least one), chosen first, which is between 0 and 1/nfor anyn.In this casexis 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 aone-dimensionallinearly orderedset of numbers, but fundamentally, there are two different approaches:

1. Extend the number system so that it contains more numbers than the real numbers.
2. Extend the axioms (or extend the language) so that the distinction between the infinitesimals and non-infinitesimals can be made in the real numbers themselves.

In 1960,Abraham Robinsonprovided an answer following the first approach. The extended set is called thehyperrealsand 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 callednonstandard.

In 1977Edward Nelsonprovided an answer following the second approach. The extended axioms are IST, which stands either forInternal set theoryor 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 at 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 classicCalculus Made EasybySilvanus P. Thompson(bearing the motto "What one fool can do another can") and the German textMathematik fur Mittlere Technische Fachschulen der Maschinenindustrieby R. Neuendorff.Pioneering works based onAbraham Robinson's infinitesimals include texts byStroyan(dating from 1972) andHoward Jerome Keisler(Elementary Calculus: An Infinitesimal Approach). 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 isInfinitesimal Calculusby 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 thehyperhyperreals, and demonstrate some applications for the extended model.

An elementary calculus text based on smooth infinitesimal analysis is Bell, John L. (2008). A Primer of Infinitesimal Analysis, 2nd Edition. Cambridge University Press. ISBN 9780521887182.

A more recent calculus text utilizing infinitesimals is Dawson, C. Bryan (2022), Calculus Set Free: Infinitesimals to the Rescue, Oxford University Press. ISBN 9780192895608.

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'sAdvanced Calculusdefines the function class of infinitesimals,${\mathfrak {I}}$,as a subset of functions$f:V\to W$between normed vector spaces by

${\mathfrak {I}}(V,W)=\{f:V\to W\ |\ f(0)=0,(\forall \epsilon >0)(\exists \delta >0)\ \backepsilon \ ||\xi ||<\delta \implies ||f(\xi )||<\epsilon \}$,

as well as two related classes${\mathfrak {O}},{\mathfrak {o}}$(seeBig-O notation) by

${\mathfrak {O}}(V,W)=\{f:V\to W\ |\ f(0)=0,\ (\exists r>0,c>0)\ \backepsilon \ ||\xi ||<r\implies ||f(\xi )||\leq c||\xi ||\}$,and

${\mathfrak {o}}(V,W)=\{f:V\to W\ |\ f(0)=0,\ \lim _{||\xi ||\to 0}||f(\xi )||/||\xi ||=0\}$.

The set inclusions${\mathfrak {o}}(V,W)\subsetneq {\mathfrak {O}}(V,W)\subsetneq {\mathfrak {I}}(V,W)$generally hold. That the inclusions are proper is demonstrated by the real-valued functions of a real variable$f:x\mapsto |x|^{1/2}$,$g:x\mapsto x$,and$h:x\mapsto x^{2}$:

$f,g,h\in {\mathfrak {I}}(\mathbb {R},\mathbb {R} ),\ g,h\in {\mathfrak {O}}(\mathbb {R},\mathbb {R} ),\ h\in {\mathfrak {o}}(\mathbb {R},\mathbb {R} )$but$f,g\notin {\mathfrak {o}}(\mathbb {R},\mathbb {R} )$and$f\notin {\mathfrak {O}}(\mathbb {R},\mathbb {R} )$.

As an application of these definitions, a mapping$F:V\to W$between normed vector spaces is defined to be differentiable at$\alpha \in V$if there is a$T\in \mathrm {Hom} (V,W)$[i.e, a bounded linear map$V\to W$] such that

$[F(\alpha +\xi )-F(\alpha )]-T(\xi )\in {\mathfrak {o}}(V,W)$

in a neighborhood of$\alpha$.If such a map exists, it is unique; this map is called thedifferentialand is denoted by$dF_{\alpha }$,coinciding with the traditional notation for the classical (though logically flawed) notion of a differential as an infinitely small "piece" ofF.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$(\Omega,{\mathcal {F}},\mathbb {P} )$be aprobability spaceand let$n\in \mathbb {N}$.An array$\{X_{n,k}:\Omega \to \mathbb {R} \mid 1\leq k\leq k_{n}\}$ofrandom variablesis called infinitesimal if for every$\epsilon >0$,we have:

$\max _{1\leq k\leq k_{n}}\mathbb {P} \{\omega \in \Omega \mid \vert X_{n,k}(\omega )\vert \geq \epsilon \}\to 0{\text{ as }}n\to \infty$

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 satisfyingLindeberg's conditionis infinitesimal, thus playing an important role inLindeberg's Central Limit Theorem(a generalization of thecentral limit theorem).

See also

• Cantor function
• Differential (infinitesimal)
• Indeterminate form
• Infinitesimal calculus
• Infinitesimal transformation
• Instant
• Nonstandard calculus
• Model theory

References

• B. Crowell, "Calculus" (2003)
• Dawson, C. Bryan, "Calculus Set Free: Infinitesimals to the Rescue" (2022) Oxford University Press
• Ehrlich, P. (2006) The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes. Arch. Hist. Exact Sci. 60, no. 1, 1–121.
• Malet, Antoni."Barrow, Wallis, and the remaking of seventeenth-century indivisibles".Centaurus39 (1997), no. 1, 67–92.
• J. Keisler, "Elementary Calculus" (2000) University of Wisconsin
• K. Stroyan "Foundations of Infinitesimal Calculus" (1993)
• Stroyan, K. D.;Luxemburg, W. A. J.Introduction to the theory of infinitesimals. Pure and Applied Mathematics, No. 72. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976.
• Robert Goldblatt(1998) "Lectures on the hyperreals" Springer.
• Cutlandet al. "Nonstandard Methods and Applications in Mathematics" (2007) Lecture Notes in Logic 25, Association for Symbolic Logic.
• "The Strength of Nonstandard Analysis" (2007) Springer.
• Laugwitz, D.(1989). "Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820".Archive for History of Exact Sciences.39(3): 195–245.doi:10.1007/BF00329867.S2CID120890300.
• Yamashita, H.: Comment on: "Pointwise analysis of scalar Fields: a nonstandard approach" [J. Math. Phys. 47 (2006), no. 9, 092301; 16 pp.]. J. Math. Phys. 48 (2007), no. 8, 084101, 1 page.