Non-standard Analysis

Non-standard Analysis, published in 1966, presents Abraham Robinson's groundbreaking mathematical framework for working with infinitesimals and infinite numbers. The book establishes rigorous foundations for concepts that were used informally by early calculus pioneers like Leibniz and Newton. Robinson employs model theory and mathematical logic to construct an extension of the real number system that includes infinitesimal and infinite quantities. The work provides formal justification for treating infinitesimals as actual numbers rather than as informal tools or limit concepts. The book progresses from basic principles through increasingly complex applications in calculus, topology, and mathematical analysis. Robinson demonstrates how his approach resolves historical controversies about infinitesimals while opening new avenues for research. This technical work represents a fusion of classical calculus methods with modern mathematical logic. The text bridges centuries of mathematical thought and offers an alternative foundation for core concepts in analysis.

Readers describe this as a dense, highly technical text requiring advanced mathematics knowledge. Several note it serves better as a reference than a learning tool. Likes: - Clear presentation of hyperreals and infinitesimals - Rigorous mathematical foundations - Historical background in Chapter 1 - Useful for research mathematicians Dislikes: - Not suitable for beginners - Heavy reliance on mathematical logic and model theory - Some sections are difficult to follow without extensive background - Language can be overly formal and abstract A mathematics professor on MathOverflow writes: "The book demands significant mathematical maturity. It's not meant as an introduction to NSA." Ratings: Goodreads: 4.0/5 (12 ratings) Amazon: 3.5/5 (4 ratings) Most reviews emphasize this is a foundational text for specialists rather than students learning non-standard analysis for the first time. Multiple readers recommend starting with Keisler's "Elementary Calculus" before attempting Robinson's work.

Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler This text develops calculus using hyperreal numbers and infinitesimals in a rigorous manner compatible with Robinson's framework.

A Primer of Infinitesimal Analysis by John L. Bell The book presents an alternative approach to infinitesimals through smooth infinitesimal analysis and synthetic differential geometry.

Model Theory by H. Jerome Keisler This text covers the mathematical foundations that underpin Robinson's construction of hyperreal numbers.

The Real Numbers and Real Analysis by Ethan D. Bloch The work bridges the gap between the intuitive calculus concepts and rigorous mathematical analysis through careful construction of the real number system.

Foundations of Infinitesimal Calculus by K.D. Stroyan This book develops the foundations of calculus using infinitesimals with focus on applications and computational aspects.

🔹 Published in 1966, this groundbreaking work finally provided a rigorous foundation for calculus using infinitesimals - a concept that Newton and Leibniz had used intuitively but mathematicians had struggled to formalize for 300 years. 🔹 Author Abraham Robinson fled Nazi Germany in 1933 at age 15, studied in Paris and Jerusalem, and went on to make significant contributions to both pure mathematics and aeronautical engineering, including work on wing design for aircraft. 🔹 The book introduced the concept of "hyperreal numbers" - a number system that includes infinitely large and infinitely small quantities while maintaining the familiar rules of arithmetic and algebra. 🔹 Though initially met with skepticism from the mathematical community, Non-standard Analysis has become an important tool in many areas of mathematics, including topology, probability theory, and mathematical economics. 🔹 Robinson's work helped resolve the apparent paradox of infinitesimals that had troubled philosophers and mathematicians since ancient Greece - Berkeley famously called them "the ghosts of departed quantities."