Smooth infinitesimal analysis


Smooth infinitesimal analysis

Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete entities. As a theory, it is a subset of synthetic differential geometry.

The "nilsquare" or "nilpotent" infinitesimals are numbers "x" where "x"² = 0 is true, but "x" = 0 need not be true at the same time.

This approach departs from the classical logic used in conventional mathematics by denying the law of the excluded middle, i.e., "NOT" ("a" ≠ "b") does not have to mean "a" = "b". All functions whose domain is "R", the continuum, are continuous and infinitely differentiable. For example, one could attempt to define a discontinuous function "f"("x") with "f"("x") = 1 for "x" = 0, and "f"("x") = 0 for "x" ≠ 0. However, the domain of this function is not provably "R", since it is not provable that for any "x", either "x" = 0 or "x" ≠ 0 must hold.

In typical models of smooth infinitesimal analysis, the infinitesimals are not invertible, and therefore the theory does not contain infinite numbers. However, there are also models that include invertible infinitesimals.

Other mathematical systems exist which include infinitesimals, including nonstandard analysis and the surreal numbers. Smooth infinitesimal analysis is like nonstandard analysis in that (1) it is meant to serve as a foundation for analysis, and (2) the infinitesimal quantities do not have concrete sizes (as opposed to the surreals, in which a typical infinitesimal is 1/ω, where ω is the von Neumann ordinal). However, smooth infinitesimal analysis differs from nonstandard analysis in its use of nonclassical logic, and in lacking the transfer principle. Some theorems of standard and nonstandard analysis are false in smooth infinitesimal analysis, including the intermediate value theorem and the Banach-Tarski paradox. Statements in nonstandard analysis can be translated into statements about limits, but the same is not always true in smooth infinitesimal analysis.

Intuitively, smooth infinitesimal analysis can be interpreted as describing a world in which lines are made out of infinitesimally small segments, not out of points. These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved. The construction of discontinuous functions fails because a function is identified with a curve, and the curve cannot be constructed pointwise. We can imagine the intermediate value theorem's failure as resulting from the ability of an infinitesimal segment to straddle a line. Similarly, the Banach-Tarski paradox fails because a volume cannot be taken apart into points.

ee also

*Category theory
*Non-classical analysis
*Synthetic differential geometry

Further reading

*Bell, John L., [http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf Invitation to Smooth Infinitesimal Analysis] (PDF file)
*Bell, John L., "A Primer of Infinitesimal Analysis", Cambridge University Press, 1998.
*Moerdijk, I. and Reyes, G.E., "Models for Smooth Infinitesimal Analysis", Springer-Verlag, 1991.

External links

*Michael O'Connor, [http://arxiv.org/abs/0805.3307 An Introduction to Smooth Infinitesimal Analysis]


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Infinitesimal — Infinitesimals (from a 17th century Modern Latin coinage infinitesimus , originally referring to the infinite th member of a series) have been used to express the idea of objects so small that there is no way to see them or to measure them. For… …   Wikipedia

  • analysis — /euh nal euh sis/, n., pl. analyses / seez /. 1. the separating of any material or abstract entity into its constituent elements (opposed to synthesis). 2. this process as a method of studying the nature of something or of determining its… …   Universalium

  • Infinitesimal transformation — In mathematics, an infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three dimensional space. This is conventionally represented by a 3 times;3… …   Wikipedia

  • Differential (infinitesimal) — For other uses of differential in calculus, see differential (calculus), and for more general meanings, see differential. In calculus, a differential is traditionally an infinitesimally small change in a variable. For example, if x is a variable …   Wikipedia

  • Non-standard analysis — Abraham Robinson Gottfried Wilhelm Leibniz argued tha …   Wikipedia

  • Mathematical analysis — Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and… …   Wikipedia

  • Cálculo infinitesimal — Saltar a navegación, búsqueda El cálculo infinitesimal o cálculo de infinitesimales constituye una parte muy importante de la matemática moderna. Es normal en el contexto matemático, por simplificación, simplemente llamarlo cálculo. El cálculo,… …   Wikipedia Español

  • Constructive non-standard analysis — In mathematics, constructive nonstandard analysis is a version of Abraham Robinson s non standard analysis, developed by Moerdijk (1995), Palmgren (1998), Ruokolainen (2004). Ruokolainen wrote: The possibility of constructivization of nonstandard …   Wikipedia

  • Non-classical analysis — In mathematics, non classical analysis is any system of analysis, other than classical real analysis, and complex, vector, tensor, etc., analysis based upon it. Such systems include: Abstract Stone duality,[1] a programme to re axiomatise general …   Wikipedia

  • Differential of a function — For other uses of differential in mathematics, see differential (mathematics). In calculus, the differential represents the principal part of the change in a function y = ƒ(x) with respect to changes in the independent variable. The… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.