A thread on de Rham's Theorem from Differential Topology:
"Euclid missed a great opportunity here: if he had stated the principle , 'The extremity of an extremity is empty' ,he could be considered as the discoverer of the BASIC EQUATION OF HOMOLOGICAL ALGEBRA: d•d = 0." 1/n
This basic equation of Homological Algebra is just an abstract incarnation of something we encounter at different places in Mathematics. Like, in Topology given a space X with boundary, one can not construct boundary of that boundary 2/n
Sep 20, 2020 • 5 tweets • 1 min read
An excerpt taken from Mike Shulman's Homotopy Type Theory - A Synthetic Theory of higher equalities-
"It is important to understand that rules of Type theory are not the same sort of thing as the axioms of a theory like ZFC or ETCS." 1/n
"Axioms are statements inside an ambient superstructure of (usually first-order) logic, whereas
the rules of type theory exist at the same level as the deductive system of the logic itself." 2/n
Sep 19, 2020 • 6 tweets • 2 min read
The hole argument was invented by Albert Einstein late in 1913 as part of his quest for the general theory of relativity. plato.stanford.edu/entries/spacet….
"What is space? What is time? Do they exist independently of the things and processes in them? Or is their existence parasitic on these things and processes? Are they like a canvas onto which an artist paints; they exist whether or not the artist paints on them?" 1/n
