“No one will be able to drive us out of the paradise that Cantor has created for us.” With these famous words, the great mathematician David Hilbert praised Georg Cantor’s insights into infinity and his invention of set theory, the language in which modern mathematics expresses itself.
The online magazine published on February 25 Quanta an article with the title The Man Who Stole Infinity (‘The Man Who Stole Infinity’), written by Joseph Howlett. The gist: a number of discoveries attributed to Cantor are actually those of his colleague Richard Dedekind. Howlett writes: “He [Cantor] carefully erased any trace of his associate’s contribution [Dedekind] including occasional use of terms that anyone who knew them would recognize as coming from Dedekind.” In short: plagiarism.
This soon followed a discussion on the platform Stack Exchange. “My opinion of Cantor and Dedekind has not changed,” writes one. Another thinks that the word ‘plagiarism’ is not at all out of place.
What’s going on?
In the summer of 1872, Georg Cantor (1845-1918) met Richard Dedekind (1831-1916) in the Swiss village of Gersau. During a long walk they discussed their ideas about mathematics. An intensive exchange of letters followed, which was later published.
In an article from 1874, Cantor proved that there are different orders of infinity: the real numbers – all points on the number line – have a higher order than the so-called algebraic numbers – solutions of polynomial equations. It has been known for some time that Cantor and Dedekind shared their ideas about this with each other. Cantor sent Dedekind a proof of the infinity order of the real numbers; Dedekind returned a sleeker version. Cantor published that version without mentioning Dedekind’s name.
The news in Howlett’s article concerns a until recently unknown letter from Dedekind. The German-Argentine mathematician and journalist Demian Goos found this letter, after lengthy research, in the archives of the Martin Luther University Halle-Wittenberg. Two things stand out: the subject—algebraic numbers—and the date—November 30, 1873, shortly before Cantor’s famous publication. The proof that the algebraic numbers are countable came from Dedekind, but Cantor claimed authorship for himself.
The division is over Stack Exchange – plagiarism or not? – arises from the feeling that, despite the discovery of the letter, there is little substantial news. Dedekind’s letter of November 30, 1873 is missing from the published correspondence, but a reference to it is included in a letter. Cantor wrote: “Isn’t it nice that, as you so aptly pointed out, one can speak of the nth algebraic number?” Moreover, Dedekind himself described the events in his diaries: “Cantor wrote it down and published it, with the proofs I suggested.” Nowhere does it appear that Dedekind felt passed over, although this does not alter the fact that Cantor did not present himself in an elegant manner.
Does this mean that ‘Cantor’s paradise’ should actually be ‘Dedekind’s paradise’? No, and Howlett also acknowledges this: “He [Cantor] was, after all, the first to prove that there are more real numbers than integers, which ultimately paved the way for research into infinity. […] But it remains important to recognize Dedekind’s role in one of the greatest discoveries in mathematics, as well as Cantor’s decision not to acknowledge him.”
