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

boundaries

– 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

intuition”

– 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