![]() ![]() If you're asking for help learning/understanding something mathematical, post in the Quick Questions thread or /r/learnmath. ![]() Requests for calculation or estimation of real-world problems and values are best suited for the Quick Questions thread, /r/askmath or /r/theydidthemath. For example, if you think your question can be answered quickly, you should instead post it in the Quick Questions thread. Questions on /r/math should spark discussion. Rule 2: Questions should spark discussion Please avoid derailing such discussions into general political discussion, and report any comments that do so. In particular, any political discussion on /r/math should be directly related to mathematics - all threads and comments should be about concrete events and how they affect mathematics. Similarly, the Banach–Tarski paradox fails because a volume cannot be taken apart into points.All posts and comments should be directly related to mathematics, including topics related to the practice, profession and community of mathematics. We can imagine the intermediate value theorem's failure as resulting from the ability of an infinitesimal segment to straddle a line. The construction of discontinuous functions fails because a function is identified with a curve, and the curve cannot be constructed pointwise. These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved. 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. ![]() Statements in nonstandard analysis can be translated into statements about limits, but the same is not always true in smooth infinitesimal analysis. Some theorems of standard and nonstandard analysis are false in smooth infinitesimal analysis, including the intermediate value theorem and the Banach–Tarski paradox. However, smooth infinitesimal analysis differs from nonstandard analysis in its use of nonclassical logic, and in lacking the transfer principle. 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 a von Neumann ordinal). Other mathematical systems exist which include infinitesimals, including nonstandard analysis and the surreal numbers. However, there are also models that include invertible infinitesimals. In typical models of smooth infinitesimal analysis, the infinitesimals are not invertible, and therefore the theory does not contain infinite numbers. 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. If the law of the excluded middle held, then this would be a fully defined, discontinuous function. One can see that the law of excluded middle cannot hold from the following basic theorem (again, understood in the context of a theory of smooth infinitesimal analysis):Įvery function whose domain is R, the real numbers, is continuous and infinitely differentiable.ĭespite 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. In particular, in a theory of smooth infinitesimal analysis one can prove for all infinitesimals ε, NOT ( ε ≠ 0) yet it is provably false that all infinitesimals are equal to zero. This approach departs from the classical logic used in conventional mathematics by denying the law of the excluded middle, e.g., NOT ( a ≠ b) does not imply a = b. The nilsquare or nilpotent infinitesimals are numbers ε where ε² = 0 is true, but ε = 0 need not be true at the same time. As a theory, it is a subset of synthetic differential geometry. 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. Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Modern reformulation of the calculus in terms of infinitesimals ![]()
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