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- Smooth_infinitesimal_analysis abstract "Smooth infinitesimal analysis is a modern 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 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, e.g., NOT (a ≠ b) does not imply a = b. 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. 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):Every function whose domain is R, the real numbers, 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 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 non-standard analysis and the surreal numbers. 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.".
- Smooth_infinitesimal_analysis wikiPageExternalLink 0805.3307.
- Smooth_infinitesimal_analysis wikiPageExternalLink invitation%20to%20SIA.pdf.
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- Smooth_infinitesimal_analysis wikiPageRevisionID "620933131".
- Smooth_infinitesimal_analysis wikiPageWikiLink Banach–Tarski_paradox.
- Smooth_infinitesimal_analysis wikiPageWikiLink Category:Mathematical_analysis.
- Smooth_infinitesimal_analysis wikiPageWikiLink Category:Mathematics_of_infinitesimals.
- Smooth_infinitesimal_analysis wikiPageWikiLink Category:Non-standard_analysis.
- Smooth_infinitesimal_analysis wikiPageWikiLink Category_theory.
- Smooth_infinitesimal_analysis wikiPageWikiLink Continuous_function.
- Smooth_infinitesimal_analysis wikiPageWikiLink Discrete_mathematics.
- Smooth_infinitesimal_analysis wikiPageWikiLink F._W._Lawvere.
- Smooth_infinitesimal_analysis wikiPageWikiLink Ieke_Moerdijk.
- Smooth_infinitesimal_analysis wikiPageWikiLink Infinitesimal.
- Smooth_infinitesimal_analysis wikiPageWikiLink Intermediate_value_theorem.
- Smooth_infinitesimal_analysis wikiPageWikiLink John_Lane_Bell.
- Smooth_infinitesimal_analysis wikiPageWikiLink Law_of_excluded_middle.
- Smooth_infinitesimal_analysis wikiPageWikiLink Law_of_the_excluded_middle.
- Smooth_infinitesimal_analysis wikiPageWikiLink Model_theory.
- Smooth_infinitesimal_analysis wikiPageWikiLink Nilpotent.
- Smooth_infinitesimal_analysis wikiPageWikiLink Non-classical_analysis.
- Smooth_infinitesimal_analysis wikiPageWikiLink Non-standard_analysis.
- Smooth_infinitesimal_analysis wikiPageWikiLink Surreal_number.
- Smooth_infinitesimal_analysis wikiPageWikiLink Synthetic_differential_geometry.
- Smooth_infinitesimal_analysis wikiPageWikiLink Transfer_principle.
- Smooth_infinitesimal_analysis wikiPageWikiLink William_Lawvere.
- Smooth_infinitesimal_analysis wikiPageWikiLinkText "Smooth infinitesimal analysis".
- Smooth_infinitesimal_analysis wikiPageWikiLinkText "smooth infinitesimal analysis".
- Smooth_infinitesimal_analysis hasPhotoCollection Smooth_infinitesimal_analysis.
- Smooth_infinitesimal_analysis wikiPageUsesTemplate Template:Reflist.
- Smooth_infinitesimal_analysis subject Category:Mathematical_analysis.
- Smooth_infinitesimal_analysis subject Category:Mathematics_of_infinitesimals.
- Smooth_infinitesimal_analysis subject Category:Non-standard_analysis.
- Smooth_infinitesimal_analysis type Field.
- Smooth_infinitesimal_analysis comment "Smooth infinitesimal analysis is a modern 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.".
- Smooth_infinitesimal_analysis label "Smooth infinitesimal analysis".
- Smooth_infinitesimal_analysis sameAs m.0b466d.
- Smooth_infinitesimal_analysis sameAs Гладкий_инфинитезимальный_анализ.
- Smooth_infinitesimal_analysis sameAs Q4139289.
- Smooth_infinitesimal_analysis sameAs Q4139289.
- Smooth_infinitesimal_analysis wasDerivedFrom Smooth_infinitesimal_analysis?oldid=620933131.
- Smooth_infinitesimal_analysis isPrimaryTopicOf Smooth_infinitesimal_analysis.