7
$\begingroup$

Abraham Robinson worked in applied mathematics for several decades. MathSciNet lists 12 articles by Robinson in wing theory. His production included the book

Robinson, A.; Laurmann, J. A. Wing theory. Cambridge, at the University Press, 1956.

The book is full of references to infinitesimals. Thus on page 36 one finds a reference to "infinitesimal horseshoe vortices of constant strength". Five years later Robinson published his first publication on infinitesimals in

Robinson, Abraham. Non-standard analysis. Nederl. Akad. Wetensch. Proc. Ser. A 64 = Indag. Math. 23 1961 432–440.

Are there any sources or references in the literature to a possible connection between Robinson's work exploiting infinitesimals in applied mathematics, on the one hand, and his eventual development of a rigorous mathematical theory thereof, on the other?

Note 1. I should add that Robinson's biographer J. Dauben was apparently unaware of the fact that Robinson's work in applied mathematics exploited "informal" infinitesimals extensively (at least I don't recall any mention of this in his book).

Improve this question
$\endgroup$

2 Answers 2

Reset to default
13
$\begingroup$

It turns out that Robinson himself answered my question. On page 238 of his book "Non-standard analysis" (Zbl 0151.00803), he explicitly connects infinitesimals in applied mathematics with hyperreal infinitesimals, in the following terms:

9.4 Sources and doublets. In the theory of continuous media, the language of infinitesimals has a long tradition, This tradition remained unbroken even though infinitesimals had been abandoned in pure Analysis. For example, the derivation of Euler's equations of motion is frequently based on the consideration of the balance of forces acting on a very small cube and several laws of Fluid Mechanics are expressed in terms of fluid particles whose dimensions are supposed to be infinitely small. Again, a doublet is defined by letting two sources of equal and opposite strength move infinitely close to each other while the strength of the sources varies in a suitable manner. It is usually taken for granted, and sometimes shown explicitly, that the use of infinitesimal language can be avoided. Naturally, this is not what we shall do here.

Having discussed supersonic flow in terms of infinitesimals in chapter 9, Robinson comments on page 259:

The reader is referred to SCHLICHTING [1955] or THWAITES [1960] for information on boundary layer theory; and to WARD [1955] or ROBINSON and LAURMANN [1956] for an introduction to the theory of linearized supersonic flow.

His 1956 book with Laurmann is the book "Wing theory" (Zbl 0073.41901) mentioned above.

The conclusion is that there was a clear connection in Robinson's mind between his work in applied mathematics using infinitesimals, on the one hand, and the rigorous theory he developed a few years later.

Improve this answer
$\endgroup$
4
$\begingroup$

A fascinating and well-organized book on the life and work of Abraham Robinson is "Abraham Robinson: The Creation of Nonstandard Analysis, A Personal and Mathematical Odyssey" by Joseph Warren Dauben http://press.princeton.edu/titles/5686.html If it does not mention a possible connection between Robinson's applied work and the development of non-standard analysis, hardly any other source would do it. Robinson himself in the preface of his 1966 book "Non-standard analysis" writes "The resulting subject was called by me Non-standard Analysis since it involves and was, in part, inspired by the so-called Non-standard models of Arithmetic whose existence was first pointed out by T. Skolem".

Improve this answer
$\endgroup$
6
  • $\begingroup$ I am familiar in detail with Dauben's book. With all due respect to its author, there are some serious mathematical errors in it. Also, Dauben does not actually claim that there was no connection. It is hard to see why you feel that "hardly any other source" would find such a connection. $\endgroup$
    – Mikhail Katz
    Commented Feb 7, 2014 at 11:04
  • $\begingroup$ Your quote from Robinson concerning Skolem's models is certainly appropriate. Note however that Robinson words himself carefully here: Robinson's theory was in part inspired by Skolem, implying there may have been other influences. In fact there definitely were other influences, such as the work of Hewitt (1948) and of Los (1955) both of which are cited in the book. $\endgroup$
    – Mikhail Katz
    Commented Feb 7, 2014 at 11:16
  • $\begingroup$ I admit that I was to hasty. Maybe there was such a connection, at least unconsciously. $\endgroup$
    – Zurab Silagadze
    Commented Feb 7, 2014 at 11:50
  • $\begingroup$ That's what I hope to find out. Notice that Dauben was apparently unaware of the fact that Robinson's work in applied mathematics exploited infinitesimals; at least I don't recall him mentioning this. $\endgroup$
    – Mikhail Katz
    Commented Feb 9, 2014 at 8:31
  • $\begingroup$ It seems Kutateladze in arxiv.org/abs/1306.4049 also does not mention this fact. $\endgroup$
    – Zurab Silagadze
    Commented Feb 9, 2014 at 13:27

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .