Smooth infinitesimal analysis
Encyclopedia
Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimal
s. 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 ε where ε² = 0 is true, but ε = 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 imply a = b. In particular, for all infinitesimals ε, neither ε = 0 nor NOT (ε = 0) is provable. One can see that the law of excluded middle cannot hold from the following basic theorem:
Despite this fact, one could attempt to define a discontinuous function f(x) by specifying that f(x) = 1 for x = 0, and f(x) = 0 for x ≠ 0. If the law of the excluded middle held, then this would be a fully defined, discontinuous function. However, there are plenty of x, namely the infinitesimals, such that neither x = 0 nor x ≠ 0 hold, so the function is not defined on the extended real numbers.
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 non-standard analysis
and the surreal number
s. Smooth infinitesimal analysis is like non-standard 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 non-standard analysis in its use of nonclassical logic, and in lacking the transfer principle
. Some theorems of standard and non-standard analysis are false in smooth infinitesimal analysis, including the intermediate value theorem
and the Banach–Tarski paradox
. Statements in non-standard 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.
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. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
s. Based on the ideas of F. W. Lawvere and employing the methods of category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, it views all functions as being continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
and incapable of being expressed in terms of discrete
Discrete mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...
entities. As a theory, it is a subset of synthetic differential geometry
Synthetic differential geometry
In mathematics, synthetic differential geometry is a reformulation of differential geometry in the language of topos theory, in the context of an intuitionistic logic characterized by a rejection of the law of excluded middle. There are several insights that allow for such a reformulation...
.
The nilsquare or nilpotent
Nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
infinitesimals are numbers ε where ε² = 0 is true, but ε = 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 imply a = b. In particular, for all infinitesimals ε, neither ε = 0 nor NOT (ε = 0) is provable. One can see that the law of excluded middle cannot hold from the following basic theorem:
- In smooth infinitesimal analysis, every function whose domain is R, the real numbers extended by infinitesimals, is continuous and infinitely differentiable.
Despite this fact, one could attempt to define a discontinuous function f(x) by specifying that f(x) = 1 for x = 0, and f(x) = 0 for x ≠ 0. If the law of the excluded middle held, then this would be a fully defined, discontinuous function. However, there are plenty of x, namely the infinitesimals, such that neither x = 0 nor x ≠ 0 hold, so the function is not defined on the extended real numbers.
In typical models
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
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 non-standard analysis
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
and the surreal number
Surreal number
In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number...
s. Smooth infinitesimal analysis is like non-standard 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 non-standard analysis in its use of nonclassical logic, and in lacking the transfer principle
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...
. Some theorems of standard and non-standard analysis are false in smooth infinitesimal analysis, including the intermediate value theorem
Intermediate value theorem
In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value....
and the Banach–Tarski paradox
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set theoretic geometry which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of non-overlapping pieces , which can then be put back together in a different way to yield two...
. Statements in non-standard analysis
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
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.
Further reading
- John Lane BellJohn Lane BellJohn Bell is Professor of Philosophy at the University of Western Ontario in Canada. He is an outstanding figure in mathematical logic and philosophy...
, Invitation to Smooth Infinitesimal Analysis (PDF file) - Bell, John L., A Primer of Infinitesimal Analysis, Cambridge University Press, 1998. Second edition, 2008.
- Ieke MoerdijkIeke MoerdijkIzak Moerdijk is professor of Mathematics at the Mathematisch Instituut of the Radboud University Nijmegen. He is the author of several influential books.-Selected works:...
and Reyes, G.E., Models for Smooth Infinitesimal Analysis, Springer-Verlag, 1991.
External links
- Michael O'Connor, An Introduction to Smooth Infinitesimal Analysis