The 10 Hardest Math Problems That Remain Unsolved

A weekend post. Who knows might be able to solve one of these.

1. The Collatz Conjecture

Earlier this month, news broke of progress on this 82-year-old question, thanks to prolific mathematician Terence Tao. And while the story of Tao’s breakthrough is good news, the problem isn’t fully solved.

A refresher on the Collatz Conjecture: It’s all about that function f(n), shown above, which takes even numbers and cuts them in half, while odd numbers get tripled and then added to 1. Take any natural number, apply f, then apply f again and again. You eventually land on 1, for every number we’ve ever checked. The Conjecture is that this is true for all natural numbers.

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Dave Linkletter — Popular Mechanics

An Ode to Ugly Physics

The intuitive physicists were triumphant on many more occasions. Like Richard Feynman, who, from his wild imagination of virtual particles, wrote down outrageously ill-defined integral overall paths of virtual particles and fields through space and time.²⁰ At the level of perturbation theory, that is, pretending that the quantum was infinitesimal, Feynman’s path integral is nothing more than a mathematical trick that helps with organizing calculations. In fairness, it was a damn fine trick. It helped predict the magnetic dipole moment of the electron to eleven digits, while also helping to solve difficult mathematical problems from the topological invariants of knots to the deformation quantization of Poisson manifolds.²¹

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Xi Yin — Inference

Why mathematics has not been effective in economics

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I have always felt that mathematics could have been better used in economics.

Håvelmo argued also that if economics wanted to be taken as seriously as physics, chemistry and biology, it needed to employ probability because that was the way that opinions were expressed in science. He believed that if this was done, economics would make new insights, just as physicists and biologists had. He also observed that the natural sciences had found a perspective on nature that made it appear to follow stable laws. The goal of The Probability Approach in Econometrics was to present how this could be realised. Morgenstern began The Theory of Games, like Håvelmo, with an argument for the use of mathematics in economics and explained that what was required was the careful definition of terms, a pre-requisite of mathematics but lacking in economics. To this end, von Neumann started with the axioms of utility that had been at the core of Carl Menger’s, unmathematical, economics.

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Magic, maths and money — Tim Johnson

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