http://www.ams.org/notices/201304/rnoti-p418.pdf
One popular example:
http://en.wikipedia.org/wiki/Four_color_theorem
-saige-
Every field of research has its horror stories about great ideas that were initially rejected by the research community. In mathematics, Fourier was ridiculed, Galois’ work was rejected, the Dirac delta function was laughed at, and Cantor’s work in set theory was highly controversial. All of these ideas are standard today. A classic case in control engineering is the development of the negative feedback amplifier, a spectacular technological breakthrough recounted in:http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.32.896&rep=rep1&type=pdf
H.S. Black, “Inventing the negative feedback amplifier,” IEEE Spectrum, vol. 14,
pp. 55-60, Dec. 1977
B. Friedland, “Introduction to ‘Stabilized feed-back amplifiers,’” Proc. IEEE, vol.
87, no. 2, 1999.
Unfortunately, most of the credit went to Nyquist, who expanded on Black’s insights in his stability theory.
... marked the birth of modern mathematical signal analysis, dates back to Joseph Fourier’s (1768–1830) investigations into the properties of heat transfer in the early 1800’s [14]. Fourier conjectured that an arbitrary periodic function, even with discontinuities, could be expressed by an infinite sum of pure harmonic terms. The idea was ridiculed among many great scholars, but a large portion of mathematical analysis in the 19th and 20th centuries was dedicated to making the statements of Fourier precise and the Fourier transformation eventually became a tool of utmost importance in signal analysis, quantum physics and the theory of partial differential equations.http://www.google.com/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=6&ved=0CEUQFjAF&url=http%3A%2F%2Fwww.researchgate.net%2Fprofile%2FDiederik_Aerts%2Fpublication%2F2195000_Necessity_of_Combining_Mutually_Incompatible_Perspectives_in_the_Construction_of_a_Global_View_Quantum_Probability_and_Signal_Analysis%2Flinks%2F0912f5093f71ec554f000000.pdf&ei=kS7JVLqHFqTlsAT7z4L4CQ&usg=AFQjCNE0RxDrKEc2DghUlAXjI9MCPSXU3A&sig2=2LceDn488yTUuah2Jmd7WA
After the announcement, Katz was appointed as one of the referees to review Wiles' manuscript. In the course of his review, he asked Wiles a series of clarifying questions that led Wiles to recognise that the proof contained a gap.Had the questions been asked differently, perhaps Wiles would not have seen the error, and perhaps once published, other mathematicians would not have seen the error for awhile.
was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy,[1] being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.Ref: http://en.wikipedia.org/wiki/Principia_Mathematica
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