7
$\begingroup$

Looking at Wikipedia, I see that fat Cantor sets are also called Smith-Volterra-Cantor sets. Another name which is sometimes associated with these sets is Hermann Hankel.

I suppose that Cantor's name appears there simply because this construction is similar to the construction Cantor set. The construction of Volterra's function might explain the connection with Volterra.

Why is this construction named after Henry Smith and Vito Volterra? Did both of them discover this construction? Or are there also some other mathematicians who independently arrived at the same object? What is the history behind the discovery of fat Cantor sets?

I would be grateful both for references giving some historical background and also for answer giving brief account of this. (If it is short enough to fit a post on Stack Exchange.)


I have seen the following in Barry Simon: Real Analysis: A Comprehensive Course in Analysis, Part 1, page 202:

The nearly simultaneous discovery of the Cantor set is connected with some work of Hermann Hankel (1839–73) in 1870 [391] which attempted to prove that certain functions were Riemann integrable (see the Notes to Section 4.4 for the definition of Riemann integrability of discontinuous functions). Hankel made the error of thinking nowhere dense sets are thin; they can have positive size as seen in the following variant of the Cantor set, called the Smith–Volterra–Cantor set, S.

[391] H. Hankel, Untersuchungen über die unendlich oft oscillirenden und unstetigen Funktionen. Ein Beitrag zur Feststellung des Begriffes der Funktion überhaupt, Universitätspr. Tübingen, 1870.

I also found this chat message by t.b.

@robjohn I looked in one of my old history books (by Walter, no online version available): there the situation is presented as follows. Riemann's student Hankel fleshed out the passage I linked you to. Unfortunately he made some mistakes, which subsequently led Smith to find the fat Cantor sets (found independently by Volterra in his construction of the function named after him). The situation was clarified only Lebesgue, Vitali and Young who independently proved the "Lebesgue criterion".

The book then continues with description first of fat Cantor set and then Volterra's function.

I happened to find also chat message by robjohn which mentions the paper A Historical Gem from Vito Volterra by William Dunham; MAA website, link to pdf.

The context is somewhat different, but this part seems to be related to the topic of this question:

One such mathematician was Hermann Hankel, who in 1870 made a detailed examination of the "punktirt unstetigen," i.e., "pointwise discontinuous," functions. (This rather odd-sounding term was introduced as a contrast to the "total unstetige," i.e., "totally discontinuous," functions, whose points of continuity were not dense.) Hankel gave a proof [3, pp. 89-90] purporting to show that a function is (Riemann) integrable if and only if it is pointwise discontinuous. This conclusion certainly thrust pointwise discontinuous functions into the limelight, and, as Thomas Hawkins observes in his excellent book Lebesgue's Theory of Integration: Its Origins and Development [4, p. 301, seemed to identify them as precisely "the functions amenable to mathematical analysis."

However, Hankel's reasoning was flawed, and the error was exposed in 1875 by Oxford professor H. J. S. Smith [7, p. 150]. While Smith agreed that any integrable function must be pointwise discontinuous, his explicit example of a pointwise discontinuous but non-integrable function destroyed the "if and only if' nature of Hankel's proof. Smith had established, of course, that the integrable functions form a proper subset of the pointwise discontinuous ones, but his paper was not widely read and its impact was minimal. Consequently, pointwise discontinuous functions occupied an important, although not necessarily well-understood, position in the research of the day. (Hawkins gives a detailed account of the situation in [4, Ch. 21.)

This paper references the same book as already mentioned in Hans Lundmark's comment: T. Hawkins, Lebesgue's Theory of Integration: Its Origins and Development, Chelsea, 1975.

$\endgroup$
  • $\begingroup$ There's some information about this in the nice book Lebesgue's Theory of Integration: Its Origins and Development by Thomas Hawkins. $\endgroup$ – Hans Lundmark May 16 '18 at 7:23
5
$\begingroup$

See :

Cantor would prove in 1883 that given any set S, its derived set is a union of a countable set and a perfect set, a set that is its own derived set.

The first construction of a perfect, nowhere dense set was by the British mathematician Henry J. S. Smith (1826-1883) in 1875 ["On the integration of discontinuous functions", Proceedings of the London Mathematical Society].

In 1881, Vito Volterra showed how to construct such a set, but Volterra was still a graduate student, and he published in an Italian journal that was not widely read ["Alcune osservazioni sulle funzioni punteggiate discontinue", G. Mat]. Again, little notice was paid. Finally, in 1883, Cantor rediscovered this construction for himself, and suddenly everyone knew about it. Cantor's example is known as the Cantor ternary set. We shall use this term to refer to Cantor's specific example, but the family of examples of perfect, nowhere dense sets exemplified by the work of Smith, Volterra, and Cantor will be referred to as the Smith-Volterra-Cantor sets, or SVC sets.

$\endgroup$
  • $\begingroup$ +1 for mentioning this book. It is full of historical details and is an engaging read. Same goes for his other book A Radical Approach to Real Analysis. $\endgroup$ – Paramanand Singh May 16 '18 at 19:59
3
$\begingroup$

In his book The Fractal Geometry of Nature, Mandelbrot discusses the Cantor set. And says it is actually due to H.J.S. Smith (1875).

So I asked a graduate student (William Darnieder) to look into that. He read Smith's paper carefully, and did his Master's thesis on it.

So: Is the Cantor set due to Smith?

Well, maybe.

Smith has no good way to talk about "point sets", since that was pioneered later by Cantor. Smith discusses certain functions in view of the question of whether or not they are Riemann integrable. He has an example of a non-integrable function. Nowadays, we would say "The set of discontinuities of the function is a fat Cantor set". And Smith's arguments clearly show this, at least to us, since we know the later notion of "point set". And the main point: Smith shows that (in our language) contrary to Hankel, measure zero Cantor set compared fat Cantor set determine whether his examples are integrable.

A fat Cantor set: (It seems, from the quotes, that modern texts use the name "Smith-Volterra-Cantor set" for this: An innovation since my schooldays.) Of course none of these guys: Hankel, Smith, Volterra, or Cantor, had a grasp of Lebesgue measure, since that came later. But those of us who know Lebesgue measure can see that they were feeling their way along toward it. Surely Lebesgue was a great man; but he didn't start from nothing.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.