##
**Non-standard analysis.**
*(English)*
Zbl 0843.26012

Princeton, NJ: Princeton Univ. Press. xix, 293 p. (1996).

This reprint of the second edition (1974) of a pioneering classic is preceded by a foreword by W. A. J. Luxemburg, briefly recalling the historical development of the work, and by the author’s preface to the first edition (1966; Zbl 0151.00803).

Robinson had considered his new theory of infinitesimals as an application of model theory, eventually vindicating Leibnizian differential calculus. Though ultrapowers and axiomatic approaches have made nonstandard analysis both more easily accessible and more generally applicable, the text is still unsurpassed in the originality of its ideas, in the surprising range of applications, and in its philosophy of relevant mathematics. Also, Chapter 10, concerning the history of the calculus, has influenced historiography by initiating a debate on Cauchy and his infinitesimals. Though Gödel’s belief, that nonstandard analysis would be “the analysis of the future”, has not yet become true the topic is of lasting interest, and the use of the method in applications has steadily grown. Luxemburg has supplied a list of proceedings of some recent international conferences.

Robinson had considered his new theory of infinitesimals as an application of model theory, eventually vindicating Leibnizian differential calculus. Though ultrapowers and axiomatic approaches have made nonstandard analysis both more easily accessible and more generally applicable, the text is still unsurpassed in the originality of its ideas, in the surprising range of applications, and in its philosophy of relevant mathematics. Also, Chapter 10, concerning the history of the calculus, has influenced historiography by initiating a debate on Cauchy and his infinitesimals. Though Gödel’s belief, that nonstandard analysis would be “the analysis of the future”, has not yet become true the topic is of lasting interest, and the use of the method in applications has steadily grown. Luxemburg has supplied a list of proceedings of some recent international conferences.

Reviewer: D.Laugwitz (Darmstadt)

### MSC:

26E35 | Nonstandard analysis |

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

26-03 | History of real functions |

01A65 | Development of contemporary mathematics |

03H05 | Nonstandard models in mathematics |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |