Is Smooth Infinitesimal Analysis the Solution for Calculi without Limits?

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SUMMARY

Smooth Infinitesimal Analysis (SIA) is a nonstandard calculus that allows for the definition of derivatives and integrals without relying on limits or the standard part function, as seen in traditional analysis. SIA introduces true infinitesimals and operates under a different logical framework where the law of excluded middle does not apply. This approach diverges from standard analysis, making it a unique alternative for those exploring calculus without conventional constraints.

PREREQUISITES
  • Understanding of nonstandard analysis concepts
  • Familiarity with derivatives and integrals
  • Knowledge of the law of excluded middle in logic
  • Basic principles of calculus
NEXT STEPS
  • Research the foundations of Smooth Infinitesimal Analysis
  • Explore the implications of nonstandard models in calculus
  • Study the differences between SIA and traditional calculus
  • Investigate applications of infinitesimals in mathematical logic
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Mathematicians, logicians, and students interested in alternative calculus frameworks, particularly those exploring nonstandard analysis and its implications in mathematical theory.

lolgarithms
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do you know of any nonstandard calculi that does not have to use limits or the standard part function? I don't really care how weird it is, as long as you can define the derivative and the integral.
 
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Smooth infinitesimal analysis might be what your looking for, a nonstandard model in the logicians sense of true infinitesimals along with derivatives and integrals, but without limits and without any standard part functions ala robinson. The price to pay is that the law of the excluded middle does not hold in SIA, and in general it is not equivalent to standard analysis.
 

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