The influence of Abraham Robinson

Abraham Robinson

Abraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....

's theory of non-standard analysis

Non-standard analysis

Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...

 has been felt in a number of fields.

Probability theory

"Radically elementary probability theory" of Edward Nelson

Edward Nelson

Edward Nelson is a professor in the Mathematics Department at Princeton University. He is known for his work on mathematical physics and mathematical logic...

 combines the discrete and the continuous theory through the infinitesimal approach. Random infinitesimal steps can be used to model Brownian motion

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

, obviating the need for cumbersome measure-theoretic developments.

Economics

Economists have used non-standard analysis to model markets with large numbers of agents.

Education

The article by Michèle Artigue cited below concerns the teaching of analysis. Artigue devotes one page to "NSA and its weak impact on education", page 172. She writes:

The non-standard analysis revival and its weak impact on education. The publication in 1966 of Robinson's book NSA constituted in some sense a rehabilitation of infinitesimals which had fallen into disrepute [...] [Robinson's proposal] was met with suspicion, even hostility, by many mathematicians [...] Nevertheless, despite the obscurity of this first work, NSA developed rapidly [...] The attempts at simplification were often conducted with the aim of producing an elementary way of teaching NSA. This was the case with the work of Keisler and Henle-Kleinberg [...]

Artigue continues specifically with reference to the calculus textbook:

[Keisler's work] served as a reference text for a teaching experiment in the first year in university in the Chicago area in 1973-74. Sullivan used 2 questionnaires to evaluate the effects of the course, one for teachers, the other for students. The 11 teachers involved gave a very positive opinion of the experience. The student questionnaire revealed no significant difference in technical performance [...] but showed that those following the NSA course were better able to interpret the sense of the mathematical formalism of calculus [...] The appearance of the 2nd book of Keisler led to a virulent criticism by Bishop, accusing Keisler of seeking [...] to convince students that mathematics is only "an esoteric and meaningless exercise in technique", detached from any reality. These criticisms were in opposition to the declarations of the partisans of NSA who affirmed with great passion its simplicity and intuitive character. [...] However, it is necessary to emphasize the weak impact of NSA on contemporary education. The small number of reported instances of this approach are often accompanied with passionate advocacy, but this rarely rises above the level of personal conviction.