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Encyclopedia > Infinitesimal

Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. For everyday life, an infinitesimal object is an object which is smaller than any possible measure, whether we measure size, time, chemical concentration, etc. When used as an adjective in the vernacular, "infinitesimal" means extremely small. In chemistry, concentration is the measure of how much of a given substance there is mixed with another substance. ...


Before the nineteenth century none of the mathematical concepts as we know them today were formally defined, but many of these concepts were already there. The founders of calculus, Leibniz, Newton, Euler, Lagrange, the Bernoullis and many others, used infinitesimals in the way shown below and achieved essentially correct results even though no formal definition was available (similarly, there was no formal definition of real numbers at the time).

Contents

History of the infinitesimal

The first mathematician to make use of infinitesimals was Archimedes (c. 250 BC)[1], although he did not believe in the existence of physical infinitesimals.[citation needed] The Archimedean property is the property of an ordered algebraic structure of having no nonzero infinitesimals. Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... The ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse was the first mathematician to make explicit use of infinitesimals. ... Archimedes of Syracuse (Greek: c. ... Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 300s BC 290s BC 280s BC 270s BC 260s BC - 250s BC - 240s BC 230s BC 220s BC 210s BC 200s BC Years: 255 BC 254 BC 253 BC 252 BC 251 BC - 250 BC - 249 BC 248 BC... In mathematics (particularly abstract algebra), the Archimedean property is a property held by some ordered algebraic structures, and in particular by the ordered field of real numbers. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ...


In India from the 12th century until the 16th century, infinitesimals were discovered for use with differential calculus by Indian mathematician Bhaskara and various Keralese mathematicians. (11th century - 12th century - 13th century - other centuries) As a means of recording the passage of time, the 12th century was that century which lasted from 1101 to 1200. ... (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... This article is under construction. ... Bhaskara (1114-1185), also known as Bhaskara II and Bhaskara Achārya (Bhaskara the teacher), was an Indian mathematician-astronomer. ... The Kerala School was a school of mathematics and astronomy founded by Madhava in Kerala (in South India) which included as its prominent members Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ...


When Newton and Leibniz developed calculus, they made use of infinitesimals. A typical argument might go: Sir Isaac Newton FRS (4 January 1643 – 31 March 1728) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... “Leibniz” redirects here. ... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ...

To find the derivative f′(x) of the function f(x) = x2, let dx be an infinitesimal. Then,
f'(x), =frac{f(x + mathrm dx) - f(x)}{mathrm dx},
=frac{x^2 + 2x cdot mathrm dx + mathrm dx^2 -x^2}{mathrm dx},
=2x + mathrm dx,
=2x,
since dx is infinitely small.

This argument, while intuitively appealing, and producing the correct result, is not mathematically rigorous. The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work The Analyst.[2] The fundamental problem is that dx is first treated as non-zero (because we divide by it), but later discarded as if it were zero. For a non-technical overview of the subject, see Calculus. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... For the medical term see rigor (medicine) Rigour (American English: rigor) has a number of meanings in relation to intellectual life and discourse. ... George Berkeley (IPA: , Bark-Lee) (12 March 1685 – 14 January 1753), also known as Bishop Berkeley, was an influential Irish philosopher whose primary philosophical achievement is the advancement of a theory he called immaterialism (later referred to as subjective idealism by others). ... The Analyst, subtitled A DISCOURSE Addressed to an Infidel Mathematician, is a book published by George Berkeley in 1734. ...


When we consider numbers, the naive definition is clearly flawed: an infinitesimal is a number whose modulus is less than any non-zero positive number. Considering positive numbers, the only way for a number to be less than all numbers would be to be the least positive number. If h is such a number, then what is h/2? Or if h is indivisible, is it still a number? Also, intuitively, one would require that the reciprocal of an infinitesimal is infinitely large (in modulus) or unlimited, but this would make it the greatest number when clearly, there is no "last" biggest number.


It was not until the second half of the nineteenth century that the calculus was given a formal mathematical foundation by Karl Weierstrass and others using the notion of a limit. In the 20th century, it was found that infinitesimals could after all be treated rigorously. Neither formulation is wrong, and both give the same results if used correctly. Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as...


Modern uses of infinitesimals

Infinitesimal is necessarily a relative concept. If epsilon is infinitesimal with respect to a class of numbers it means that epsilon cannot belong to that class. This is the crucial point: infinitesimal must necessarily mean infinitesimal with respect to some other type of numbers. Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. ...


The path to formalisation

Proving or disproving the existence of infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used. We consider here systems where infinitesimals can be shown to exist. In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... This article is about a logical statement. ...


In 1936 Maltsev proved the compactness theorem. This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them. A consequence of this theorem is that if there is a number system in which it is true that for any positive integer n there is a positive number x such that 0< x < 1/n, then there exists an extension of that number system in which it is true that there exists a positive number x such that for any positive integer n we have 0 < x < 1/n. The possibility to switch "for any" and "there exists" is crucial. The first statement is true in the real numbers as given in ZFC set theory : for any positive integer n it is possible to find a real number between 1/n and zero, only 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? Anatoly Ivanovich Maltsev (Malcev) (Russian: Анато́лий Ива́нович Ма́льцев 27 November N.S./14 November O.S. 1909 - 7 June 1967) was born in Misheronsky, near Moscow, and died in Novosibirsk, USSR. He was a mathematician noted for his work on the decidability of various algebraic groups. ... The compactness theorem is a basic fact in symbolic logic and model theory and asserts that a set (possibly infinite) of first-order sentences is satisfiable, i. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ...


There are in fact many ways to construct such a one-dimensional linearly ordered set of numbers, but fundamentally, there are two different approaches: 2-dimensional renderings (ie. ... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...

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.

In 1960, Abraham Robinson provided an answer following the first approach. The extended set is called the hyperreals and contains numbers less in absolute value than any positive real number. The method may be considered relatively complex but it does prove that infinitesimals exist in the universe of ZFC set theory. The real numbers are called standard numbers and the new non-real hyperreals are called nonstandard. Abraham Robinson Abraham Robinson (October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics. ... The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ... In the most restricted sense, nonstandard analysis or non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of...


In 1977 Edward Nelson provided an answer following the second approach. The extended axioms are IST, which stands either for Internal Set Theory or for the initials of the three extra axioms: Idealization, Standardization, Transfer. In this system we consider that the language is extended in such a way that we can express facts about infinitesimals. The real numbers are either standard or nonstandard. An infinitesimal is a nonstandard real number which is less, in absolute value, than any positive standard real number. Reverend Edward Nelson was the father of British naval commander Horatio Nelson. ... Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson which provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. ...


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.


All of these approaches are mathematically rigorous.


This allows for a definition of infinitesimals which refers to these approaches:


A definition

An infinitesimal number is a nonstandard number whose modulus is less than any nonzero positive standard number.

What standard and nonstandard refer to depends on the chosen context.


Alternatively, we can have synthetic differential geometry or smooth infinitesimal analysis with its roots in category theory. This approach departs dramatically from the classical logic used in conventional mathematics by denying the law of excluded middle--i.e., not (ab) does not have to mean a = b. A nilsquare or nilpotent infinitesimal can then be defined. This is a number x where x2 = 0 is true, but x = 0 need not be true at the same time. With an infinitesimal such as this, algebraic proofs using infinitesimals are quite rigorous, including the one given above. Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. ... Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... “Excluded middle” redirects here. ... In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ...


References

  1. ^ Archimedes, The Method of Mechanical Theorems, see the Archimedes palimpsest
  2. ^ George Berkeley, The Analyst; or a discourse addressed to an infidel mathematician
  • J. Keisler "Elementary Calculus" (200) University of Wisconsin [1]
  • K. Stroyan "Foundations of Infinitesimal Calculus" (1993) [2]
  • Robert Goldblatt (1998) "Lectures on the hyperreals" Springer. [3]
  • "Nonstandard Methods and Applications in Mathematics" (2007) Lecture Notes in Logic 25, Association for Symbolic Logic. [4]
  • "The Strength of Nonstandard Analysis" (2007) Springer.[5]

The Archimedes Palimpsest is a palimpsest on parchment in the form of a codex which originally was a copy of an otherwise unknown work of the ancient mathematician, physicist, and engineer Archimedes of Syracuse and other authors. ...

See also


  Results from FactBites:
 
NationMaster - Encyclopedia: Infinitesimal (2705 words)
Proving or disproving the existence of infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used.
Infinitesimal calculus is an area of mathematics pioneered by Gottfried Leibniz based on the concept of infinitesimals, as opposed to the calculus of Isaac Newton, which is based upon the concept of the limit.
Infinitesimals are legitimate quantities in the non-standard analysis of Abraham Robinson.
Infinitesimal Calculus - LoveToKnow 1911 (17095 words)
The infinitesimal calculus is the body of rules and processes by means of which continuously varying magnitudes are dealt with in mathematical analysis.
The guise in which variable quantities presented themselves to the mathematicians of the 17th century was that of the lengths of variable lines.
Berkeley's criticism was levelled against all infinitesimals, that is to say, all quantities vaguely conceived as in some intermediate state between nullity and finiteness, as he took Newton's moments to be conceived.
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