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I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any mathematic …

The following story is a bit strange to be true, but we all believed it as students, and I think I still do believe that a somewhat weaker version of events must have indeed occurred. Michael Maschler …

Another urban legend, which I've heard told about various mathematicians, and which Misha Polyak self-effacingly tells about himself (and therefore might even be true), is the following: As a young p …

Calculus, particularly the ideas of derivation and integration, is surely the mathematical idea which has changed history most in the last 400 years. The ability to study and quantify change and rate …

This question is inspired by this Tim Gowers blogpost. I have some familiarity with the work of Jacob Lurie, which contains ideas which are simply astounding; but what I don't understand is which key …

The Four Colour Theorem might perhaps be a canonical example of a very hard proof of a major result which has improved, but is still very hard- no human-comprehensible proof exists, as far as I know, …

I've seen the following quote many times on the internet, and have used it myself. It is usually attributed to Grothendieck. It is better to have a good category with bad objects than a bad category …

There is a strong and growing trend to do mathematics via diagrammatic algebra, which involves constructing and manipulating equations whose elements are diagrams drawn in the plane. The manipulations …

As you mentioned, an often misapplied mathematical statement is Heisenberg's uncertainty principle, which for me, as a reader of Chriss-Ginzburg, is the purely mathematical statement that any subvarie …

A knot table, with the knots in it made out of a nice (pretty and pliable) material. It's aesthetic, and people might have fun playing with them. One might include also the Perko pair! They come with …

Perhaps the canonical example is Nikolai Ivanovich Lobachevsky? His work on hyperbolic geometry was subject to severe ridicule, stemming from a negative review of Ostrogradsky, the leading Russian mat …

In 1993, Pat Gilmer asserted as Theorem 1 of Classical knot and link concordance, that certain Casson-Gordon invariants vanish for all slice knots, which would be true if the kernel of the inclusion $ …

The Smale Conjecture. This was proven by Hatcher in 1983. It states that the diffeomorphism group $\mathrm{Diff}(S^3)$ of the $3$-sphere has the homotopy type of the orthogonal group $O(4)$, which …

Two widely believed conjectures: The proportion of hyperbolic knots amongst prime knots of $n$ or fewer crossings approaches $1$ as $n$ approaches infinity. The crossing number (the minimal number o …

Le and Murakami (HERE and HERE) discovered several previously unknown relations between multiple zeta values through the study of quantum invariants of knots. Further relations were later discovered t …