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I would say that a lot of topology was discrete before it was continuous. The Euler characteristic was first observed (in 1752) as an invariant of polyhedra. Around 1900 Poincaré first calculated Bet …

Hilbert's 21st problem, on the existence of linear DEs with prescribed monodromy group, was for a long time thought to have been solved by Plemelj in 1908. In fact, Plemelj died in 1967 still believin …

Since no one has mentioned A.N. Kolmogorov (born 1903), I hope I may be forgiven for a second answer. The following is from Kolmogorov's Wikipedia biography. In classical mechanics, he is best known …

Here are a few examples from the 19th century. Unsolvability of the quintic equation. Abel (1826) proved this by algebraic ingenuity, but without clarifying the concepts involved. Galois (1830) gave …

P. S. Novikov was 54 when he gave the first proof (143 pages!) of the unsolvability of the word problem for groups in 1955, and 58 when he co-solved the Burnside problem with S. I. Adian.

Another example from logic is Gentzen's consistency proof for Peano arithmetic by transfinite induction up to $\varepsilon_0$, which I think was completely unexpected, and unprecedented.

Smale's eversion of the 2-sphere was first thought to be an "obvious counterexample" to a result he proved in his 1958 thesis. See the Wikipedia article "Smale's paradox" for further information.

I posted this earlier on the "narrowly-missed discoveries" thread, but I think the two paragraphs below address your three questions. For the most recent scholarly account of Post's work, see the arti …

Historians like nothing better than to debunk other historians, so there are plenty of papers shooting down one myth or another. I enjoy these papers as much as the next person, but sometimes the de …

Emil L. Post was very close to proving Gödel's incompleteness theorem, and the existence of algorithmically unsolvable problems in the early 1920s. He realized that one could enumerate all algorithms, …

Another unsolved problem from ancient Greek times is: which regular $n$-gons are constructible by ruler and compass? We know, since Gauss, that this problem reduces to finding all the Fermat primes, b …

Here are a few books on the history of recent mathematics that I recommend: A History of Algebraic and Differential Topology, 1900 - 1960 by Jean Dieudonne. History of Topology by I.M. James. Recip …

"Read the masters" should not be taken as blanket advice, because some masters are much easier to read, or more congenial to modern mathematicians, than others. Some 19th century works that I have lea …

Ludwig Schläfli discovered the regular polytopes in $\mathbb{R}^4$, including the 24-cell, 120-cell, and 600-cell, among many results of n-dimensional geometry, between 1850 and 1852. He wrote up his …

Here are a few: Girolamo Cardano: The Book of My Life. (trans. by Jean Stoner. New York: New York Review of Books, 2002) Norbert Wiener's two volumes Ex-Prodigy: My Childhood and Youth. (MIT Press 195 …