August Möbius [36] defended Libri, by presenting his former professor's reason for believing that 0^{0}=1 (basically a proof that lim_{x→0+}x^{x}=1). Möbius also went further and presented a supposed proof that lim_{x→0+}f(x)^{g(x)}=1 whenever lim_{x→0+}f(x)=lim_{x→0+}g(x)=0. Of course "S" then asked whether Möbius knew about functions such as f(x)=e^{−1/x} and g(x)=x. (And paper [36] was quietly omitted from the historical record when the collected works of Möbius were ultimately published.)
("His former professor" was Pfaff.)
In March of 1845, he published a paper in the Philosophical Magazine entitled “On Jacobi’s Elliptic Functions, in Reply to the Rev. B. Bronwin; and on Quaternions”. The bulk of this paper was an attempt to rebut an article pointing out mistakes in Cayley’s work on elliptic functions. Apparently as an afterthought, he tacked on a brief description of the octonions. In fact, this paper was so full of errors that it was omitted from his collected works—except for the part about octonions.
according to John Baez. The octonions. Bulletin of the American Mathematical Society 39:145–205, 2002, arXiv:math/0105155
Now although we know that Sylvester was in the habit of making guesses, and these guesses although often brilliant were not always so,* it would be next to impossible to find a generalisation of his which had no individual instances in support of it.
* See Crelle's Journal, lxxxix, pp. 82–85.
Borchardt was the editor of the journal. Sylvester has authorized Borchardt to withdraw his theorem in Sylvester's name. Final sentence:
Lorsque l'éminent géomètre qui a enrichi de si belles découvertes la théorie des déterminants et l'algèbre des fonctions entières en général, et dont les contributions forment un ornement bien precieux de ce Journal, reviendra sur sa théorie des déterminants composés et qu'il voudra bien destiner pour mon Journal la rectification dont sa formule générale est susceptible, il nous fera connaître, on peut en être sûr, un progrès nouveau que cette branche d'algèbre devra à son initiative.(I don't know if the French original sounds as weird as my attempt at translating it to English:)Berlin, 4 février 1880.
When the eminent geometer, who has enriched by such beautiful discoveries the theory of determinants and the algebra of entire(?) functions in general, and whose contributions form a very precious ornament of this Journal, will return to his theory of composite(?) determinants and that he will kindly direct to my Journal the correction to which his general formula is susceptible, he will certainly make known to us a new progress that this branch of algebra will owe to his initiative.
(Note the very polite tone: People did not say that "his formula is wrong", but "his formula is susceptible to a correction". Similarly, when Lhuilier discovered that Euler's polyhedral formula did not hold in all cases and required additional assumptions he phrased it as follows: [Die Formel] ... "erleidet Ausnahmen." ("suffers exceptions.")
Voevodsky lists several examples from his own experience:
A fast algorithm for embedding a graph on the torus was required. In 1978 Filotti had published a paper [8] presenting an algorithm for embedding 3-regular graphs on the torus. This was followed by a much expanded version [7] in 1980, which corrected a number of minor errors. This was then followed by papers by Filotti, Miller, and Reif (1979), Filotti and Mayer (1980), Miller (1978), and Djidjev and Reif (1991). These papers (except for Djidjev and Reif [1991]) all used Filotti’s techniques to address embedding and isomorphism problems for graphs of bounded genus.
Filotti’s algorithm is based on the planarity testing algorithm of Demoucron, Malgrange and Pertuiset 1964. We start by pointing out a misconception that Filotti [7] and also Gibbons [Algorithmic Graph Theory, Cambridge University Press, 1985, p. 89] had regarding this algorithm.
A fatal flaw in Filotti’s algorithm [7], which also appears in the algorithm of Filotti, Miller, and Reif (1979), is then described. The algorithm of Djidjev and Reif (1991) is also incorrect, and a fundamental error in it is presented. There appears to be no way to fix these problems without creating algorithms which take exponential time.
...
Juvan, Marinček and Mohar have created a linear time torus embedding algorithm [15] (1995). ... But these approaches are very complex, and very difficult to implement and to ensure the resulting code is correct. However, programming these might either point out further flaws in the reasoning or provide a better understanding of these results.
An attempt is being made by Mohar, Orbanic and Bonnington to create an implementation of [15], but as of July 2006, the program still has bugs. (There is a consensus that the two 24-vertex 3-regular obstructions mentioned earlier are torus obstructions, but the code as of July 2006 failed to recognize this).
Errors in reasoning can be subtle and it is a testament to the difficulties of the embedding problem that an algorithm like Filotti’s [7] can stand for 25 years before the errors are pointed out.
László Fejes Tóth, in his book Lagerungen in der Ebene, auf der Kugel und im Raum from 1953 [Section V.12, p. 151] asserts erroneously that a ruled surface such as a hyperboloid of one sheet can be triangulated with a Hausdorff error ε with only O(1/sqrt(ε)) triangles. This error was discovered by Mathijs Wintraecken, see Section 2.4 of his Ph.D. thesis, see also our manuscript Optimal triangulation of saddle surfaces.
D. P. Dobkin, L. Snyder, On a general method for maximizing and minimizing among certain geometric problems, in: 20th Annual Symposium on Foundations of Computer Science (FOCS 1979), 1979, pp. 9–17. doi:10.1109/SFCS.1979.28
From Wikipedia: In 1983, Gorenstein announced that the proof of the classification had been completed, but he had been misinformed about the status of the proof of classification of quasithin groups, which had a serious gap in it. A complete proof for this case was published by Aschbacher and Smith in 2004.
Carl Gustav Jacobi schrieb in einem Brief an Alexander von Humboldt am 21. Dezember 1846, über Dirichlet:
Er allein, nicht ich, nicht Cauchy, nicht Gauß weiß, was ein vollkommen strenger mathematischer Beweis ist, sondern wir kennen es erst von ihm. Wenn Gauß sagt, er habe etwas bewiesen, ist es mir sehr wahrscheinlich, wenn Cauchy es sagt, ist eben so viel pro als contra zu wetten, wenn Dirichlet es sagt, ist es gewiß, ich lasse mich auf diese Delicatessen lieber gar nicht ein.English translation attempt:
He alone, not I, not Cauchy, not Gauß, knows what a completely rigorous mathematical proof is, but we know it only from him. If Gauß says he has proved something, it is very likely to me, if Cauchy says it, the odds are one to one, if Dirichlet says it, it is certain, I'd rather not get into these delicate issues.Cited according to Briefwechsel zwischen Alexander von Humboldt und Carl Gustav Jacob Jacobi, Akademie-Verlag Berlin, ed. Heribert Pieper, 1987. Brief 22 (p.99). The original was at „Berlin, AAW, Nachlass Dirichlet Nr. 48“. AAW = Zentrales Archiv der Akademie der Wissenschaften der DDR. Now it is probably at the Berlin-Brandenburgische Akademie der Wissenschaften, or at the Staatsbibliothek.
The occasion of this letter was that Dirichlet, who was professor at the university and member of the Academy of Sciences in Berlin, had an offer of a position in Heidelberg. Humboldt and Jacobi were trying to persuade the king, Friedrich Wilhelm IV., to improve Dirichlet's salary in order to keep him in Berlin. Jacobi had already written a formal letter to the king on Dec 12 in support of Dirichlet, of which he enclosed a copy to Humboldt. Jacobi wanted to provide Humboldt, who was a high-ranking official at the Prussian court and had more possibilities to intervene personally, with arguments to underline the importance of Dirichlet. According to Note 3 in the Briefwechsel edition, p.100, Dirichlet still did not get the salary that he was supposed to receive. The surrounding passage concerning Dirichlet in the letter is:
Dirichlet hat in der Wissenschaft zwei Seiten, die seine Specialität ausmachen. Er allein … Delicatessen lieber gar nicht ein. Zweitens hat D. eine neue Disciplin der Mathematik geschaffen, die Anwendung derjenigen unendlichen Reihen, welche Fourier in die Wärmetheorie eingeführt hat, auf die Erforschung der Primzahlen. Dann hat er eine Menge Theoreme gefunden, welche in der Wissenschaft bleiben und die Grundpfeiler neuer Theorien bilden. D. hat es vorgezogen, sich hauptsächlich mit solchen Gegenständen zu beschäftigen, welche die größten Schwierigkeiten darbieten; darum liegen seine Arbeiten vielleicht nicht so auf der breiten Heerstraße der Wissenschaft und haben daher, wenn auch große Anerkennung, doch nicht alle die gefunden, die sie verdienden. Wäre er in Paris geblieben, so würde er dort jetzt ohne Nebenbuhler herrschen, und wie anders würde seine äußere Stellung sein!The efforts of Humboldt and Jacobi were successful, see letter 25 from Humboldt to Jacobi from Dec 27, 1846 in the Briefwechsel edition, p.108. Dirichlet remained in Berlin until 1855, when he moved to Göttingen to become the successor of Gauß. For the citation, see also Dieudonné (Hrsg.), Geschichte der Mathematik, Vieweg 1990, S. 389, and Wikipedia.