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'''Infinitesimals''' have been used to express the idea of objects so small that there is no way to see them or to measure them. The insight with exploiting infinitesimals was that objects could still retain certain specific properties, such as [[angle]] or [[slope]], even though these objects were quantitatively small.<ref>http://plato.stanford.edu/entries/continuity/#1</ref> The word ''infinitesimal'' comes from a 17th-century [[Modern Latin]] coinage ''infinitesimus'', which originally referred to the "[[Infinity|infinite]][[Ordinal number (linguistics)|-th]]" item in a sequence.  It was originally introduced around 1670 by either [[Nicolaus Mercator]] or [[Gottfried Wilhelm Leibniz]],<ref>*{{citation
 | last1 = Katz | first1 = Mikhail
 | author1-link = Mikhail Katz
 | last2 = Sherry | first2 = David
 | author2-link = 
 | arxiv = 1205.0174
 | doi = 10.1007/s10670-012-9370-y
 | issue = 
 | journal = [[Erkenntnis]]
 | pages = 
 | title = Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond
 | volume = 
 | year = 2012}}.</ref> and are a basic building block of [[infinitesimal calculus]].

In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, but not zero in size; or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small".  In order to give it a meaning it usually has to be compared to another infinitesimal object in the same context (as in a [[derivative]]).  Infinitely many infinitesimals are summed to produce an [[integral]].

[[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.<ref>Netz, Reviel; Saito, Ken; Tchernetska, Natalie: A new reading of Method Proposition 14: preliminary evidence from the Archimedes palimpsest. I. SCIAMVS 2 (2001), 9–29.</ref> 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 <math>\frac{1}{\infty}</math> in area calculations. 

The use of infinitesimals by [[Gottfried Wilhelm Leibniz|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 [[continuous function|continuity]] in his [[Cours d'Analyse]], and in defining an early form of a [[Dirac delta function]].  As Cantor and Dedekind were developing more abstract versions of Stevin's continuum, [[Paul du Bois-Reymond]] wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions.  Du Bois-Reymond's work inspired both [[Émile Borel]] and [[Thoralf Skolem]].  Borel explicitly linked du Bois-Reymond's work to Cauchy's work on rates of growth of infinitesimals.  Skolem developed the first non-standard models of arithmetic in 1934.  A mathematical implementation of both the law of continuity and infinitesimals was achieved by [[Abraham Robinson]] in 1961, who developed [[non-standard analysis]] based on earlier work by [[Edwin Hewitt]] in 1948 and [[Jerzy Łoś]] in 1955.  The [[hyperreal number|hyperreals]] implement an infinitesimal-enriched continuum and the [[transfer principle]] implements Leibniz's law of continuity. The [[standard part function]] implements Fermat's [[adequality]].

==History of the infinitesimal==
The notion of infinitely small quantities was discussed by the [[Eleatic School]]. The [[Greek mathematics|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.<ref>Archimedes, ''The Method of Mechanical Theorems''; see [[Archimedes Palimpsest]]</ref> 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 1/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 [[Indian mathematics|Indian]] mathematician [[Bhāskara II]] (1114–1185)<ref>{{cite journal  | last = '''Shukla'''  | first = Kripa Shankar  | authorlink =  | coauthors =  | title = Use of Calculus in Hindu Mathematics  | journal = Indian Journal of History of Science  | volume = 19  | issue =  | pages = 95–104  |year=1984  | url =  | doi =  | id =  | accessdate =  | postscript = .  }}</ref> described a geometric technique for expressing the change in  <math>\sin \theta</math> in terms of <math>\cos\theta</math> times a change in <math>\theta</math>.  Prior to the invention of calculus mathematicians were able to calculate tangent lines by the method [[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 [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]] invented the [[Infinitesimal calculus|calculus]], they made use of infinitesimals. The use of infinitesimals was attacked as incorrect by [[George Berkeley|Bishop Berkeley]] in his work ''[[The Analyst]]''.<ref>George Berkeley, ''The Analyst; or a discourse addressed to an infidel mathematician''</ref>  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]], [[Georg Cantor|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 infinitesimals from analysis and their philosophical allies like [[Bertrand Russell]] and [[Rudolf Carnap]] declared infinitesimals to be "pseudoconcepts", [[Hermann Cohen]] and his [[Marburg school]] of [[neo-Kantianism]] sought to develop a working logic of infinitesimals.<ref>[[Thomas Mormann]]; [[Mikhail Katz]]. Infinitesimals as an issue of neo-Kantian philosophy of science. [[International Society for the History of Philosophy of Science|HOPOS]]: The Journal of the International Society for the History of Philosophy of Science 3 (2013), no. 2, 236-280. See  http://www.jstor.org/stable/10.1086/671348 and http://arxiv.org/abs/1304.1027 </ref>  The mathematical study of systems containing infinitesimals continued through the work of [[Tullio Levi-Civita|Levi-Civita]], [[Paul du Bois-Reymond]], and others, throughout the late nineteenth and the twentieth centuries, as documented by [[Philip Ehrlich]] (2006).  In the 20th century, it was found that infinitesimals could serve as a basis for calculus and analysis; see [[hyperreal number]].

==First-order properties==

{{Expert-subject|Mathematics|section|talk=First-order properties|reason=exposition is muddled}}

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 will still be available. Typically ''elementary'' means that there is no [[quantifier|quantification]] over [[set (mathematics)|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&nbsp;''x'', ''x''&nbsp;+&nbsp;0&nbsp;=&nbsp;''x''" would still apply. The same is true for quantification over several numbers, e.g., "for any numbers&nbsp;''x'' and ''y'', ''xy''&nbsp;=&nbsp;''yx''."  However, statements of the form "for any ''set''&nbsp;''S''&nbsp;of numbers&nbsp;..." may not carry over. Logic with this limitation on quantification is referred to as [[first-order logic]].

It superficially seems clear that 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 nonarchimedean system, and the Archimedean principle can be expressed by quantification over sets, but this is incorrect. It is trivial to 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 [[Complete metric space|completeness]] property cannot be expected to carry over, because the reals are the unique complete ordered field up to isomorphism. This is also wrong, at least as a formal statement, since it presuming some underlying model of set theory.

We can distinguish three levels at which a nonarchimedean number system could have first-order properties compatible with those of the reals:

# An [[ordered field]] obeys all the usual axioms of the real number system that can be stated in first-order logic. For example, the [[commutativity]] axiom ''x''&nbsp;+&nbsp;''y''&nbsp;=&nbsp;''y''&nbsp;+&nbsp;''x'' holds.
# A [[real closed field]] 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 +,&nbsp;×, and&nbsp;≤. 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]].
# 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&nbsp;+,&nbsp;×, and&nbsp;≤). For example, there would have to be a [[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]] 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&nbsp;2 and&nbsp;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&nbsp;1, and the series with only the linear term&nbsp;''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&nbsp;''x'' as negligible compared to lower powers. [[David O. Tall]]<ref>{{cite web|url=http://www.jonhoyle.com/MAAseaway/Infinitesimals.html |title=Infinitesimals in Modern Mathematics |publisher=Jonhoyle.com |date= |accessdate=2011-03-11}}</ref> 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&nbsp;''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. <ref>Khodr Shamseddine, "Analysis on the Levi-Civia Field: A Brief Overview," http://www.uwec.edu/surepam/media/RS-Overview.pdf</ref>

====Transseries====
The field of [[transseries]] is larger than the Levi-Civita field.<ref>G. A. Edgar, "Transseries for Beginners," http://www.math.ohio-state.edu/~edgar/preprints/trans_begin/</ref> An example of a transseries is:

:<math>e^\sqrt{\ln\ln x}+\ln\ln x+\sum_{j=0}^\infty e^x x^{-j},</math>

where for purposes of ordering ''x'' is considered to be infinite.

===Surreal numbers===
Conway's [[surreal numbers]] fall into category 2. They are a system that was 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===
{{Main|Hyperreal number}}
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 in order to allow all of classical analysis to 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 <math>\mathbb{N}</math> has a natural counterpart <math>^*\mathbb{N}</math>, which contains both finite and infinite integers. A proposition such as <math>\forall n \in \mathbb{N}, \sin n\pi=0</math> carries over to the hyperreals as <math>\forall n \in {}^*\mathbb{N}, {}^*\!\!\sin n\pi=0</math> .

===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 O. Tall|David Tall]].

===Smooth infinitesimal analysis ===
{{Main|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]] &mdash; 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''<sup>2</sup> = 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 <math>\alpha</math> to write down a unit impulse, infinitely tall and narrow Dirac-type delta function <math>\delta_\alpha</math> satisfying <math>\int F(x)\delta_\alpha(x) = F(0)</math> 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 function]]s in the context of an infinitesimal-enriched continuum provided by the [[hyperreal number|hyperreals]].

==Logical properties==
The method of constructing infinitesimals of the kind used in nonstandard analysis depends on the [[Model theory|model]] and which collection of [[axiom]]s 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&nbsp;<&nbsp;''x''&nbsp;<&nbsp;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&nbsp;<&nbsp;''x''&nbsp;<&nbsp;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 will depend 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 will be 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 [[dimension|one-dimensional]] [[linear order|linearly ordered]] 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]] provided an answer following the first approach. The extended set is called the [[Hyperreal number|hyperreal]]s 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 [[Non-standard analysis|nonstandard]].

In 1977 [[Edward Nelson]] provided an answer following the second approach. The extended axioms are IST, which stands either for [[Internal set theory|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 which is less, in absolute value, than any positive standard real number.

In 2006 [http://math.sci.ccny.cuny.edu/people?name=Karel_Hrbacek 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]], and bearing the motto "What one fool can do another can".<ref>Available online at 
http://www.gutenberg.org/ebooks/33283</ref> Pioneering works based on [[Abraham Robinson]]'s infinitesimals include texts by [[Stroyan]] (dating from 1972) and [[Howard 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.<ref>*{{Cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=[[Journal for Research in Mathematics Education]] |volume=41 |issue=2 |pages=117–146}}</ref><ref>http://www.math.umt.edu/tmme/vol7no1/TMME_vol7no1_2010_article1_pp.3_30.pdf</ref>

==See also==
{{Portal|Mathematics}}
<div style="column-count:2;-moz-column-count:2;-webkit-column-count:2">
* [[Adequality]]
* [[Differential (mathematics)]]
* [[Dual number]]
* [[Hyperreal number]]
* [[Infinitesimal calculus]]
* [[Infinitesimal transformation]]
* [[Instant]]
* [[Levi-Civita field]]
* [[Non-standard calculus]]
* [[Non-standard analysis]]
* [[Surreal number]]
* [[Model theory]]
</div>

==Notes==
{{Reflist}}

==References==
{{refbegin}}
* B. Crowell, [http://www.lightandmatter.com/calc/ "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&ndash;121.
* J. Keisler, [http://www.math.wisc.edu/~keisler/calc.html "Elementary Calculus"] (2000) University of Wisconsin
* K. Stroyan [http://www.math.uiowa.edu/%7Estroyan/InfsmlCalculus/InfsmlCalc.htm "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) [http://www.springer.com/west/home/generic/order?SGWID=4-40110-22-1590889-0 "Lectures on the hyperreals"]  Springer.
* [[Nigel Cutland|Cutland]] et al. [http://www.aslonline.org/books-lnl_25.html "Nonstandard Methods and Applications in Mathematics"] (2007) Lecture Notes in Logic 25, Association for Symbolic Logic.
*  [http://www.springer.com/west/home/springerwiennewyork/mathematics?SGWID=4-40638-22-173705722-0 "The Strength of Nonstandard Analysis"] (2007) Springer.
*{{Cite journal|doi=10.1007/BF00329867|authorlink=Detlef Laugwitz|last=Laugwitz|first=D.|year=1989|title=Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820|journal=Arch. Hist. Exact Sci.|volume=39|issue=3|pages=195&ndash;245|postscript=<!--None-->}}.
* 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.
{{refend}}


{{Infinitesimals}}
{{Number systems}}

[[Category:Calculus]]
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