Since the period when Weierstrass pointed out the flaw in the proof of Riemann's mapping theorem is reported differently in different documents, I have doubts about the exact timing. When exactly did Weierstrass point out the flaw? If you have a clear source for this, I would appreciate it if you could attach it to your answer.


1 Answer 1


Yes, but there were other criticisms already during Riemann's lifetime, see Bottazzini, "Algebraic truths" vs "geometric fantasies" which gives a detailed story.

In 1864, when Riemann was in Pisa tending to his poor health, Italian mathematician Casorati traveled to Berlin to meet with Weierstrass and reported that "Riemann’s things are creating difficulties in Berlin." There was a rivalry involved, with Riemann upstaging some of Weierstrass's early work on analytic functions and doing so in a flawed manner (in Weierstrass's opinion).

"Riemann’s disciples are making the mistake of attributing everything to their master, while many [discoveries] had already been made by and are due to Cauchy, etc.; Riemann did nothing more than to dress them in his manner for his convenience."

In particular, Weierstrass called Riemann surfaces "geometric fantasies". More specifically, he objected to analytic continuation of functions along a path that avoids branch points and singularities, which Riemann treated as generally possible. "But this is not possible. It was precisely while searching for the proof of the general possibility that I realized it was in general impossible." Kronecker provided Casorati with an example of an analytic function given by a lacunary series that had essential boundary "entirely made of points where the function is not defined, it can take any value there", as Weierstrass pointed out. Later it became the basis for his nowhere differentiable continuous function.

But the criticisms of the Riemann mapping theorem more specifically, in the proof of which the Dirichlet principle was used, only appear in Weierstrass's correspondence with Schwarz from 1867 onward (i.e., after Riemann's death in 1866). Weierstrass encouraged Schwarz to find a "rigorous" proof and Schwarz succeeded in 1870. It is only then that Weierstrass presented his now famous counterexample. Here is from Bottazzini:

"In 1870 Schwarz discovered his alternating method. “With this method - he stated by presenting it in a lecture - all the theorems which Riemann has tried to prove in his papers by means of the Dirichlet principle, can be proved rigorously” ([8], vol. 2, 133). He submitted to Weierstrass an extended version of the paper, and in a letter of July 11, 1870 Schwarz asked him whether he had “objections to raise”. Apparently, Weierstrass’ answer has been lost. It is quite significant, however, that three days later, on July 14, 1870 Weierstrass presented to the Berlin Academy his celebrated counterexample to the Dirichlet principle ([10], vol. 2, 49-54), and then submitted Schwarz’s 1870 paper for publication in the Monatshefte of the Academy."

Later, in an 1875 letter to Schwarz, Weierstrass gave a general statement on the difference between his and Riemann's philosophies:

"The more I think about the principles of function theory - and I do it incessantly - the more I am convinced that this must be built on the basis of algebraic truths, and that it is consequently not correct when the ‘transcendental’, to express myself briefly, is taken as the basis of simple and fundamental algebraic propositions. This view seems so attractive at first sight, in that through it Riemann was able to discover so many of the important properties of algebraic functions."


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