#impossibility-results

#impossibility-results

In the weeks leading up to September 1891, mathematician Georg Cantor prepared an ambush. For years he had sparred - philosophically, mathematically and emotionally - with his formidable rival Leopold Kronecker, one of Germany's most influential mathematicians. Kronecker thought that mathematics should deal only with whole numbers and proofs built from them and therefore rejected Cantor's study of infinity. "God made the integers," Kronecker once said. "All else is the work of man."

According to Einstein's General Relativity, for every black hole that exists within the Universe, there are only three properties that go into it that matter in any way: the black hole's total mass, the black hole's net electric charge, and the black hole's intrinsic angular momentum, and that's it. It doesn't matter what type of matter went into the black hole in order to form it; all that matters is its mass, charge, and angular momentum.

A drawn circle is at least something physical. You can see it, touch it, erase it. The skeptic can still say, "Circles are grounded in physical reality. Justice is different; it's just an idea in your head." So let's talk about the number two. Point to it. Not two apples, not two fingers, not a numeral on a page-that's just a symbol.

Walking through a field one day, a 17-year-old schoolteacher named George Boole had a vision. His head was full of abstract mathematics - ideas about how to use algebra to solve complex calculus problems. Suddenly, he was struck with a flash of insight: that thought itself might be expressed in algebraic form. Boole was born on November 2, 1815, at four o'clock in the afternoon, in Lincoln, England.

Recent integrative approaches suggest that physics cannot be adequately characterized by magnitude-based distinctions alone, such as those implied by Big-P, little-p, and mini-p physics. While these categories capture differences in scope and historical impact, they fail to address the heterogeneity of physical activity itself. To remedy this, I propose the Five Fs of physics: force, friction, flux, formulation, and foundational structure.

Star-formation will eventually end, and then the last shining stars will burn out. Galaxies will dissociate due to gravitational interactions, ejecting all masses and leaving only supermassive black holes behind. And then those black holes will decay via Hawking radiation, leaving only cold, stable, isolated bodies, from which no further energy can be extracted, all accelerating away from us within our dark energy-dominated Universe.