Infinitesimal

Números hiperreales

Infinitesimal refers to quantities that are closer to zero than any standard real number, but are not zero themselves. In the history of mathematics, the concept of infinitesimals has been a point of much debate and innovation, particularly in the development of calculus. The notion plays a crucial role in the formulation of concepts such as derivatives and integrals, which are foundational to many areas of mathematics and its applications.

History

The concept of infinitesimals dates back to the ancient Greek mathematicians, but it was not until the work of 17th century mathematicians such as Gottfried Wilhelm Leibniz and Isaac Newton that infinitesimals were formally used as a mathematical tool. Leibniz and Newton independently developed the foundations of calculus, with infinitesimals being central to Leibniz's approach. However, the lack of a rigorous mathematical foundation for infinitesimals led to significant criticism, most notably from George Berkeley.

In the 19th century, mathematicians moved away from infinitesimals in favor of the epsilon-delta approach to calculus, developed by Augustin-Louis Cauchy and formalized by Karl Weierstrass. This approach eliminated the need for infinitesimals by redefining the foundations of calculus in terms of limits.

The concept of infinitesimals saw a resurgence in the 20th century with the development of non-standard analysis by Abraham Robinson. Robinson's work provided a rigorous foundation for infinitesimals, using model theory, a branch of mathematical logic. This allowed mathematicians to use infinitesimals in a way that was both rigorous and consistent with classical mathematics.

Mathematical Definition

In modern mathematics, an infinitesimal is often defined in the context of non-standard analysis. An infinitesimal number is a number that is greater than 0 but less than any positive real number. More formally, if x is an infinitesimal, then for every positive real number ε, |x| < ε. In the hyperreal number system, which extends the real number system to include infinitesimals and their reciprocals (infinite numbers), infinitesimals are denoted as numbers that are not zero but whose absolute value is less than any positive real number.

Applications

Infinitesimals are used in various areas of mathematics and its applications. In calculus, they are used to provide intuitive explanations for the concepts of derivatives and integrals. In physics, infinitesimals are used in the mathematical modeling of continuous systems, such as in the study of motion and fields. The concept also finds application in engineering, economics, and other sciences where mathematical modeling of continuous processes is required.

Controversies and Philosophical Implications

The use of infinitesimals has been a subject of philosophical debate since their inception. Critics argue that infinitesimals are counterintuitive and lack a clear basis in reality. Supporters, however, see them as a useful and necessary abstraction for dealing with the infinitely small and the infinitely large. The development of non-standard analysis and the formalization of infinitesimals have addressed many of these criticisms, providing a rigorous foundation for their use in mathematics.

See Also

• Calculus
• Non-standard analysis
• Hyperreal number
• Limit (mathematics)