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A curious footnote to the blocking of Arnold's Fields Medal by Pontryagin (if that is what it was) is the comment Arnold made following the award of medals to three French mathematicians (mainly for w …

I'll attempt an answer to question 1. Hausdorff was entitled to think that set theory was not yet mature, because his own 1914 book made considerable advances on what had been done previously (notably …

The article is probably referring to Dedekind's Was sind und was sollen die Zahlen of 1888, in which point 64 is Dedekind's definition of infinite. This of course is after Cantor had been investigatin …

An example from measure theory that might qualify is in the construction of non-measurable sets. The first example is well known, Vitali's 1905 construction obtained by choosing a member from each cos …

I'm pleased to hear that some MOers like my book, but I have to say that I think it has too much math for a class of non-science majors. At best, you might mine it for some homework problems because o …

Like many people, I am fascinated by the quote from Weyl (already listed here), that In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathem …

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.

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 …

An example of a slightly different kind -- not eliminating all calculation, but showing that "all calculations are easy" -- is Dehn's algorithm in combinatorial group theory. Dehn showed, using the co …

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 …

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 …

The invention of the Sierpinski carpet by Sierpinski in 1916. Who knew that cell phone antennas would later be based on this shape?

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 …

There is an English translation of the first 10 sections of Wessel's paper in the anthology edited by Henrietta Midonick, The Treasury of Mathematics, volume 2 (Penguin Books 1968) pp.321--329.

I don't think there is a decisive answer to this question, because some mathematicians accepted negative coordinates long before others did. However, here is another landmark from the 1690s: Huygens' …