Smooth infinitesimal analysis
Smooth infinitesimal analysis is a modern reformulation of the
The nilsquare or nilpotent infinitesimals are numbers ε where ε² = 0 is true, but ε = 0 need not be true at the same time. Calculus Made Easy notably uses nilpotent infinitesimals.
Overview
This approach departs from the
- Every function whose 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 holds, so the function is not defined on the 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
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.
See also
- Category theory
- Non-standard analysis
- Synthetic differential geometry
- Dual number
References
- ^ Tao, Terrence (2012-04-03). "A cheap version of nonstandard analysis". What's new. Retrieved 2023-12-15.
- ISBN 9780521887182.
Further reading
- John Lane Bell, Invitation to Smooth Infinitesimal Analysis (PDF file)
- Ieke Moerdijk and Reyes, G.E., Models for Smooth Infinitesimal Analysis, Springer-Verlag, 1991.
External links
- Michael O'Connor, An Introduction to Smooth Infinitesimal Analysis