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Is there a formulation of calculus that uses infinitesimals rigorously without introducing an additional number system (non-standard analysis) and without deviating from classical logic?
How could this sentence possibly be satisfied? On the one hand, it's asking for new numbers (nonzero infinitessimals), but at the same time you reject introducing new numbers!Is there a formulation of calculus that uses infinitesimals rigorously without introducing an additional number system
Infinitesimals are mathematical objects that represent infinitely small quantities. They are used in calculus and other mathematical fields to describe continuous change and can be thought of as infinitely small numbers that are non-zero.
Infinitesimals are used in mathematics to describe and analyze continuously changing quantities, such as velocity, acceleration, and area under a curve. They are also used in the development of the calculus of variations, which is a branch of mathematics that deals with finding optimal solutions to problems with continuously varying parameters.
No, infinitesimals are not considered to be real numbers. They are a mathematical concept used to represent infinitely small quantities and are not part of the real number system. However, they are closely related to the concept of limits in calculus and can be used to approximate real numbers.
The controversy surrounding the use of infinitesimals stems from their historical development and the lack of a rigorous mathematical foundation. Some mathematicians and philosophers have argued that infinitesimals are not well-defined objects and cannot be used in a rigorous mathematical framework. However, modern developments in non-standard analysis have provided a more rigorous foundation for the use of infinitesimals.
Infinitesimals are used in physics to model and analyze continuously changing physical quantities, such as position, velocity, and acceleration. They are also used in the development of differential equations, which are essential in many areas of physics, including mechanics, electromagnetism, and thermodynamics.