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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, an infinitesimal or infinitesimal number is a quantity that is closer to
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

zero
than any standard
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, but that is not zero. The word ''infinitesimal'' comes from a 17th-century
Modern Latin New Latin (also called Neo-Latin or Modern Latin) is the List of revived languages, revival of Latin used in original, scholarly, and scientific works since about 1500. Modern scholarly and technical nomenclature, such as in zoological and botan ...
coinage ''infinitesimus'', which originally referred to the "
infinity Infinity is that which is boundless, endless, or larger than any number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything ...

infinity
- th" item in a
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

sequence
. Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another poly ...

reciprocal
s of one another. Infinitesimal numbers were introduced in the development of calculus, in which the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
was first conceived as a ratio of two infinitesimal quantities. This definition was not rigorously formalized. As calculus developed further, infinitesimals were replaced by
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
s, which can be calculated using the standard real numbers. Infinitesimals regained popularity in the 20th century with
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...
's development of
nonstandard 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)-definitio ...
and the
hyperreal numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, 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 includes both hyperreal numbers and
ordinal number In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ...
s, which is the largest
ordered field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
.
Vladimir Arnold Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–A ...

Vladimir Arnold
wrote in 1990: The crucial insight for making infinitesimals feasible mathematical entities was that they could still retain certain properties such as
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

angle
or
slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

slope
, even if these entities were infinitely small. Infinitesimals are a basic ingredient in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", 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 ...

calculus
as developed by
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, ...
, including the
law of continuity The law of continuity is a heuristic principle introduced by Gottfried Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. (; or ; – 14 November 1716) was a prominent German p ...
and the
transcendental law of homogeneity In mathematics, the transcendental law of homogeneity (TLH) is a heuristic principle enunciated by Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. (; or ; – ...
. 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, ''infinitesimal'' means 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 infinitesimals are summed to calculate an
integral In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

integral
. The concept of infinitesimals was originally introduced around 1670 by either
Nicolaus Mercator Nicholas (Nikolaus) Mercator (c. 1620, Holstein Holstein (; nds, label=Northern Low Saxon, Holsteen; da, Holsten; Latin and historical en, Holsatia, italic=yes) is the region between the rivers Elbe and Eider (river), Eider. It is the southern ...
or
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, " ...

Gottfried Wilhelm Leibniz
.
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

Archimedes
used what eventually came to be known as the
method of indivisibles In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...
in his work ''
The Method of Mechanical Theorems ''The Method of Mechanical Theorems'' ( el, Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as ''The Method'', is considered one of the major surviving works of the ancient Greek ...
'' 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 method of exhaustion (; ) is a method of finding the area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of ...

method of exhaustion
. The 15th century saw the work of
Nicholas of Cusa Nicholas of Cusa (1401 – 11 August 1464), also referred to as Nicholas of Kues and Nicolaus Cusanus (), was a German philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and f ...

Nicholas of Cusa
, further developed in the 17th century by
Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer An astronomer is a in the field of who focuses their studies on a specific question or field outside the scope of . They observe s such as s, s, , s and ...

Johannes Kepler
, in particular calculation of area of a circle by representing the latter as an infinite-sided polygon.
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (ar ...
's work on decimal representation of all numbers in the 16th century prepared the ground for the real continuum.
Bonaventura Cavalieri Bonaventura Francesco Cavalieri ( la, Bonaventura Cavalerius; 1598 – 30 November 1647) was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen ...

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
1.
John Wallis John Wallis (; la, Wallisius; ) was an English clergyman and mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), f ...

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 Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

Leonhard Euler
and
Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaAugustin-Louis Cauchy Baron Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord ...

Augustin-Louis Cauchy
exploited infinitesimals both in defining continuity in his ''
Cours d'Analyse ''Cours d'Analyse de l’École Royale Polytechnique; I.re Partie. Analyse algébrique'' is a seminal textbook in infinitesimal calculus published by Augustin-Louis Cauchy in 1821. The article follows the translation by Bradley and Sandifer in de ...
'', and in defining an early form of a
Dirac delta function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
. As Cantor and Dedekind were developing more abstract versions of Stevin's continuum,
Paul du Bois-Reymond Paul David Gustav du Bois-Reymond (2 December 1831 – 7 April 1889) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For ...
wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions. Du Bois-Reymond's work inspired both
Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such top ...
and
Thoralf Skolem Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set ...
. 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 Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...
in 1961, who developed
nonstandard 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)-definitio ...
based on earlier work by
Edwin Hewitt Edwin Hewitt (January 20, 1920, Everett, Washington Everett is the county seat of and the largest city in Snohomish County, Washington, United States. It is north of Seattle and is one of the main cities in the Seattle metropolitan area, me ...

Edwin Hewitt
in 1948 and Jerzy Łoś in 1955. The
hyperreals In mathematics, the system of hyperreal numbers is a way of treating Infinity, infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an Field extension, extension of the real numbe ...
implement an infinitesimal-enriched continuum and the
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 algebraic geometry and analytic geometry#The Lefschetz principle, the ...
implements Leibniz's law of continuity. The
standard part function In nonstandard 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 ...

standard part function
implements Fermat's
adequality Adequality is a technique developed by Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics ...
.


History of the infinitesimal

The notion of infinitely small quantities was discussed by the
Eleatic School The Eleatics were a pre-Socratic school of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of min ...
. The
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
mathematician
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

Archimedes
(c. 287 BC – c. 212 BC), in ''
The Method of Mechanical Theorems ''The Method of Mechanical Theorems'' ( el, Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as ''The Method'', is considered one of the major surviving works of the ancient Greek ...
'', was the first to propose a logically rigorous definition of infinitesimals. His
Archimedean property In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

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 John Wallis (; la, Wallisius; ) was an English clergyman and mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), f ...

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 A thought experiment is a hypothetical situation in which a hypothesis A hypothesis (plural hypotheses) is a proposed explanation An explanation is a set of statements usually constructed to describe a set of facts which clarifies the ...
of adding an infinite number of
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

parallelogram
s of infinitesimal width to form a finite area. This concept was the predecessor to the modern method of integration used in
integral calculus In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. The conceptual origins of the concept of the infinitesimal 1/∞ can be traced as far back as the Greek philosopher
Zeno of Elea Zeno of Elea (; grc, Ζήνων ὁ Ἐλεᾱ́της; ) was a pre-Socratic Pre-Socratic philosophy is ancient Greek philosophy Ancient Greek philosophy arose in the 6th century BC, at a time when the inhabitants of ancient Greece were st ...

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 Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of suc ...

Pierre de Fermat
's method of
adequality Adequality is a technique developed by Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics ...
and
René Descartes René Descartes ( or ; ; Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

René Descartes
'
method of normals In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", 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 gen ...
. There is debate among scholars as to whether the method was infinitesimal or algebraic in nature. When
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fi ...

Newton
and
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, ...
invented the
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", 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 ...
, they made use of infinitesimals, Newton's '' fluxions'' and Leibniz' '' differential''. The use of infinitesimals was attacked as incorrect by
Bishop Berkeley George Berkeley (; 12 March 168514 January 1753) – known as Bishop Berkeley (Bishop of Cloyne of the Anglican Church of Ireland) – was an Anglo-Irish people, Anglo-Irish philosopher whose primary achievement was the advancement of a theory ...

Bishop Berkeley
in his work ''
The Analyst ''The Analyst'' (subtitled ''A Discourse Addressed to an Infidel Mathematician: Wherein It Is Examined Whether the Object, Principles, and Inferences of the Modern Analysis Are More Distinctly Conceived, or More Evidently Deduced, Than Religious ...
''. 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 Baron Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord ...

Augustin-Louis Cauchy
,
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian A Bohemian () is a resident of Bohemia Bohemia ( ; cs, Čechy ; ; hsb, Čěska; szl, Czechy) is the westernmost a ...

Bernard Bolzano
,
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...

Karl Weierstrass
,
Cantor A cantor or chanter is a person who leads people in singing or sometimes in prayer Prayer is an invocation An invocation (from the Latin verb ''invocare'' "to call on, invoke, to give") may take the form of: * Supplication, prayer ...
, Dedekind, and others using the
(ε, δ)-definition of limit Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
and
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
. While the followers of Cantor, Dedekind, and Weierstrass sought to rid analysis of infinitesimals, and their philosophical allies like
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose know ...
and
Rudolf Carnap Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle The Vienna Circle (german: Wiener Krei ...
declared that infinitesimals are ''pseudoconcepts'',
Hermann Cohen Hermann Cohen (4 July 1842 – 4 April 1918) was a Germany, German Jewish philosophy, philosopher, one of the founders of the University of Marburg, Marburg school of neo-Kantianism, and he is often held to be "probably the most important ...

Hermann Cohen
and his
Marburg school In late modern continental philosophy Continental philosophy is a set of 19th- and 20th-century philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the abili ...
of
neo-Kantianism In late modern philosophy, late modern continental philosophy, neo-Kantianism (german: Neukantianismus) was a revival of the 18th-century philosophy of Immanuel Kant. More specifically, it was influenced by Arthur Schopenhauer's critique of the K ...
sought to develop a working logic of infinitesimals. The mathematical study of systems containing infinitesimals continued through the work of
Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italians, Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made s ...
,
Giuseppe Veronese Giuseppe Veronese (7 May 1854 – 17 July 1917) was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Italian language, ...

Giuseppe Veronese
,
Paul du Bois-Reymond Paul David Gustav du Bois-Reymond (2 December 1831 – 7 April 1889) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For ...
, 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 mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s).


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 First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rul ...
. 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: # An
ordered field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
obeys all the usual axioms of the real number system that can be stated in first-order logic. For example, the
commutativity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

commutativity
axiom ''x'' + ''y'' = ''y'' + ''x'' holds. # A
real closed field In mathematics, a real closed field is a Field (mathematics), field ''F'' that has the same first-order logic, first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, ...
has 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 a
cube root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

cube root
. # The system could have all the first-order properties of the real number system for statements involving ''any'' relations (regardless of whether those relations can be expressed using +, ×, and ≤). For example, there would have to be a
sine In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

sine
function 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, the
transcendental functions In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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 In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
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 In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. Each member a can be constructed as a formal series of the form : ...
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 In mathematics, the field \mathbb^ of logarithmic-exponential transseries is a Non-Archimedean ordered field, non-Archimedean ordered differential field which extends comparability of asymptotic analysis, asymptotic growth rates of elementary functi ...
is larger than the Levi-Civita field. An example of a transseries is: :e^\sqrt+\ln\ln x+\sum_^\infty e^x x^, where for purposes of ordering ''x'' is considered infinite.


Surreal numbers

Conway's
surreal number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s fall into category 2, except that the surreal numbers form a
proper class Proper may refer to: Mathematics * Proper map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and qu ...
and 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

The most widespread technique for handling infinitesimals is the hyperreals, developed by
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...
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 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 algebraic geometry and analytic geometry#The Lefschetz principle, the ...
, 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 \mathbb has a natural counterpart ^*\mathbb, which contains both finite and infinite integers. A proposition such as \forall n \in \mathbb, \sin n\pi=0 carries over to the hyperreals as \forall n \in ^*\mathbb, ^*\!\!\sin n\pi=0 .


Superreals

The
superreal number In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
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 Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
, the
dual number In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
s extend the reals by adjoining one infinitesimal, the new element ε with the property ε2 = 0 (that is, ε is
nilpotent In mathematics, an element ''x'' of a ring (mathematics), ring ''R'' is called nilpotent if there exists some positive integer ''n'', called the index (or sometimes the degree), such that ''x'n'' = 0. The term was introduced by Benj ...
). 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 differentiationIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. This application can be generalized to polynomials in n variables, using the
Exterior algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of an n-dimensional vector space.


Smooth infinitesimal analysis

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 ...
or
smooth infinitesimal analysis Smooth infinitesimal analysis is a modern reformulation of the calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that ge ...
have roots in
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
. This approach departs from the classical logic used in conventional mathematics by denying the general applicability of the
law of excluded middle In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
– i.e., ''not'' (''a'' ≠ ''b'') does not have to mean ''a'' = ''b''. A ''nilsquare'' or ''
nilpotent In mathematics, an element ''x'' of a ring (mathematics), ring ''R'' is called nilpotent if there exists some positive integer ''n'', called the index (or sometimes the degree), such that ''x'n'' = 0. The term was introduced by Benj ...
'' 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 Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic Mathematical logic, also called formal logic, is a subfield of mathematics exploring the formal applications of logic to mathematics. I ...
, 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 Baron Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was ...

Cauchy
used 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 and
Lazare Carnot Lazare Nicolas Marguerite, Count Carnot (13 May 1753 – 2 August 1823) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...

Lazare Carnot
's terminology. Modern set-theoretic approaches allow one to define infinitesimals via the
ultrapower The ultraproduct is a mathematics, mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structure ...
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 In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
. The article by Yamashita (2007) contains a bibliography on modern
Dirac delta function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
s in the context of an infinitesimal-enriched continuum provided by the
hyperreals In mathematics, the system of hyperreal numbers is a way of treating Infinity, infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an Field extension, extension of the real numbe ...
.


Logical properties

The method of constructing infinitesimals of the kind used in nonstandard analysis depends on the
model A model is an informative representation of an object, person or system. The term originally denoted the plan A plan is typically any diagram or list of steps with details of timing and resources, used to achieve an Goal, objective to do somet ...
and which collection of
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

axiom
s are used. We consider here systems where infinitesimals can be shown to exist. In 1936 Maltsev proved the
compactness theorem In mathematical logic, the compactness theorem states that a Set (mathematics), set of First-order predicate calculus, first-order Sentence (mathematical logic), sentences has a Model (model theory), model if and only if every Finite set, finite su ...
. 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
ZFC
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
: 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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

one-dimensional
linearly ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some Set (mathematics), set X, which satisfies the following for all a, b and c in X: # a ...
set 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 Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...
provided an answer following the first approach. The extended set is called the
hyperreals In mathematics, the system of hyperreal numbers is a way of treating Infinity, infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an Field extension, extension of the real numbe ...
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 Edward Nelson (May 4, 1932 – September 10, 2014) was an American mathematician. He was professor in the Mathematics Department at Princeton University Princeton University is a private Private or privates may refer to: Music * "In Priva ...
provided an answer following the second approach. The extended axioms are IST, which stands either for
Internal set theory Internal set theory (IST) is a mathematical theory of Set (mathematics), sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the ...
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 Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...
's infinitesimals include texts by Stroyan (dating from 1972) and
Howard Jerome Keisler Howard Jerome Keisler (born 3 December 1936) is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis. His Ph.D. advisor was Alfred Tarski a ...
( 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 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, \mathfrak, as a subset of functions f:V\to W between normed vector spaces by
\mathfrak(V,W) = \,
as well as two related classes \mathfrak,\mathfrak (see
Big-O notation Big O notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a member o ...
) by
\mathfrak(V,W) = \, and
\mathfrak(V,W) = \.
The set inclusions \mathfrak(V,W)\subsetneq\mathfrak(V,W)\subsetneq\mathfrak(V,W)generally hold. That the inclusions are proper is demonstrated by the real-valued functions of a real variable f:x\mapsto , x, ^, g:x\mapsto x , and h:x\mapsto x^2 :
f,g,h\in\mathfrak(\mathbb,\mathbb),\ g,h\in\mathfrak(\mathbb,\mathbb),\ h\in\mathfrak(\mathbb,\mathbb) but f,g\notin\mathfrak(\mathbb,\mathbb) and f\notin\mathfrak(\mathbb,\mathbb).
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(V,W) .e, a bounded linear map V\to Wsuch that
(\alpha+\xi)-F(\alpha)T(\xi)\in \mathfrak(V,W)
in a neighborhood of \alpha. If such a map exists, it is unique; this map is called the ''differential'' and 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" 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 (\Omega,\mathcal,\mathbb) be a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a space (mathematics), mathematical construct that provides a formal model of a randomness, random process or "experiment". For example, one can define a ...
and let n\in\mathbb. An array \ of
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
s is called infinitesimal if for every \epsilon>0, we have: :\max_\mathbb\\to 0\text 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 satisfying
Lindeberg's condition Lindeberg may refer to: Places * Lindeberg, Akershus, a village in Sørum municipality, Norway ** Lindeberg Station, an Oslo Commuter Rail station * Lindeberg, Oslo, an area of the borough Alna in Oslo, Norway ** Lindeberg (station), an Oslo Metro ...
is infinitesimal, thus playing an important role in Central limit theorem#Lindeberg CLT, Lindeberg's Central Limit Theorem (a generalization of the central limit theorem).


See also

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


Notes


References

* B. Crowell, "Calculus" (2003) *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. * Antoni Malet, Malet, Antoni. "Barrow, Wallis, and the remaking of seventeenth century indivisibles". ''Centaurus'' 39 (1997), no. 1, 67–92. * J. Keisler, "Elementary Calculus" (2000) University of Wisconsin * K. Stroyan "Foundations of Infinitesimal Calculus" (1993) *Keith Stroyan, Stroyan, K. D.; Wilhelmus Luxemburg, 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. * Nigel Cutland, Cutland et al. "Nonstandard Methods and Applications in Mathematics" (2007) Lecture Notes in Logic 25, Association for Symbolic Logic. * "The Strength of Nonstandard Analysis" (2007) Springer. * * 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. {{Authority control Calculus History of calculus Infinity Nonstandard analysis History of mathematics Mathematical logic Mathematics of infinitesimals