Chosen notes on counterintuition

George Polya on intuition:

Intuition comes to us much earlier and with much less outside influence than formal arguments which we cannot really understand unless we have reached a relatively high level of logical experience and sophistication. In the first place, the beginner must be convinced that proofs deserve to be studied, that they have a purpose, that they are interesting.

– 0.999999.. = 1
– Banach-Tarski paradox (it isn’t possible to define a “size” of an arbitrary subset of the real numbers in a meaningful way)
– inscribed squares problem
– Goodstein’s theorem (and also: this theorem cannot be proved in Peano arithmetic, though it can be proven with transfinite induction)
– Monty Hall problem
– Brouwer’s fixed point theorem
– ham sandwitch theorem
– Smale’s paradox, Simpson’s paradox
– Leo Polovets: Something can have an infinite perimeter and a finite surface area (Koch snowflake), a finite volume and an infinite surface area (Gabriel’s Horn), or even zero volume and an infinite surface area (Menger sponge).
– Leo Polovets: If you take an infinitely long random walk in one or two dimensions, you will return to your starting point with probability 1. If you take a random walk in 3 or more dimensions, the probability of making it back to the starting point is significantly less than 1.
– Alon Amit: “a solid ball, a chair and a sponge full of holes are also curves. ”
– Alon Amit: Robertson-Seymour: ” if you have infinitely many graphs, one of them is guaranteed to be a minor of another. ”
– Alon Amit on differentiability: “in fact, most functions that are continuous everywhere are differentiable nowhere. ”
– Tarun Chitra: “Brownian Motion produces continuous everywhere, differentiable nowhere paths, almost surely”
– Tarun Chitra: Wiener Sphere theorem: “The Uniform Distribution on the n-sphere tends to Brownian Motion”
– Alexander Irpan: Galperin’s billiard computation
-Arshdeep Singh: Collatz Conjecture, Prozvilov’s Identity


+ Notes from “Monster vs Intuition” from Solomon Feferman!

– ubiquity of “intuition” in mathematics (e.g. cumulative hierarchy in set theory,
infinitesimals etc.)
– Weierstrass everywhere continuous and nowwhere differentiable function
– Peano’s space-filling curve, i.e. continuous function from [0,1] to R^2, whose
range is [0,1] x [0,1], violating the intuition that a curve is a 1-dim object
– Brouwer’s construct: three countries meeting each other at every point of their
– Sierpinski’s result: a curve which intersects itself at every point
– Peano-Hilbert: space-filling curve as a limit of curves that first go through every
quadrant of the unit square, then more quickly through every sub-quadrant, and so on
– Jordan curve theorem: every closed curve in the plane is the boundary of two open
connected regions, the “interior” one bounded and the “exterior” one unbounded,
– Alexander “horned” sphere: a subset S* of R^3 s.t. it is homeomorphic to the unit sphere
S^2 in R^3 (formed by successively growing pairs of honrs from S^2)
– Hausdorff’s paradox
– Banach and Tarski: “perhaps” paradoxical, the necessity of AC for results with “agree with
– arbitrarinesses in Bolayi-Gerwien (ignoring boundaries, notion of “piece”) or in Dougherty-
Foreman (notion of one set being dense in another)
– almost all (if not all) applicable mathematics can be formalized in a system W conservative over peano,
and therefore does not require ST notions


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Imagine a story that one can't believe. Hi. Life changes here. Small things only.
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