Today there is no doubt that the empty set for the whole of mathematics is as reasonable and useful as zero for arithmetic. This however was not always the case, and surprisingly even Zermelo, who based his infinite set on the empty set, was unsure. The following quotes shed some light on this mostly forgotten development. My question is, when had these doubts become completely forgotten? I would also be interested in further quotes doubting the "justifyability" of the empty set.
Bernard Bolzano, the inventor of the notion set (Menge) in mathematics would not have named a nothing an empty set. In German the word set has the meaning of many or great quantity. Often we find in German texts the expression große (great or large) Menge, rarely the expression kleine (small) Menge. Therefore Bolzano apologizes for using this word in case of sets having only two elements: "Allow me to call also a collection containing only two parts a set." [J. Berg (ed.): B. Bolzano, Einleitung zur Grössenlehre, Friedrich Frommann Verlag, Stuttgart (1975) p. 152]
Also Richard Dedekind discarded the empty set. But he accepted the singleton, i.e., the non-empty set of less than two elements: "For the uniformity of the wording it is useful to permit also the special case that a system S consists of a single (of one and only one) element a, i.e., that the thing a is element of S but every thing different from a is not an element of S. The empty system, however, which does not contain any element, shall be excluded completely for certain reasons, although it might be convenient for other investigations to fabricate such." [R. Dedekind: "Was sind und was sollen die Zahlen?" Vieweg, Braunschweig (1887), 2nd ed. (1893) p. 2]
Bertrand Russell considered an empty class as not existing: "An existent class is a class having at least one member." [Bertrand Russell: "On some difficulties in the theory of transfinite numbers and order types", Proc. London Math. Soc. (2) 4 (1906) p. 47]
Georg Cantor mentioned the empty set with some reservations and only once in all his work: "Further it is useful to have a symbol expressing the absence of points. We choose for that sake the letter O. P = O means that the set P does not contain any single point. So it is, strictly speaking, not existing as such." [Cantor, p. 146]
And even Ernst Zermelo who made the "Axiom II There is an (improper) set, the 'null-set' 0 which does not contain any element" [E. Zermelo: "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen 65 (1908) p. 263], this same Zermelo himself said in private correspondence: "It is not a genuine set and was introduced by me only for formal reasons." [E. Zermelo, letter to A. Fraenkel (1 Mar 1921)] "I increasingly doubt the justifiability of the 'null set'. Perhaps one can dispense with it by restricting the axiom of separation in a suitable way. Indeed, it serves only the purpose of formal simplification." [E. Zermelo, letter to A. Fraenkel (9 May 1921)]
The answer and comments have shown me that it is hard if not impossible to answer this question. Zermelo for instance expressed his doubts only in private correspondence. What about McAtee or Peano or Young mentioned by Gerad Edgar? Therefore I will take the alternative: Who was the last recognized mathematician to utter doubts with respect to the existence of the empty set? Kanamori, mentioned by Dave L Renfro gives a hint to 1943.