From PBS:

It’s hard to imagine infinity: something that is, by definition, larger than everything you can imagine. Physicists have to deal with the unimaginable every day, and have the tools to do so. But does their math describe reality?

Mathematicians have found a way to pack infinity into manageable equations and theorems as part of a class of mathematical oddities called “singularities.” To a mathematician, a singularity is simply a point where a function breaks down, as 1/x does when x gets close to zero. The defining property of a singular point is that it’s impossible to predict what happens beyond it. But are the singularities in mathematicians’ equations just an abstract concept? Or do they occur in nature?

The word “singularity” was popularized in a 2005 book by Ray Kurzweil, who uses it to refer to an impending revolution in artificial intelligence (AI). According to Kurzweil, once artificial intelligences become smart enough to improve their own kin, a feedback loop will lead to a runaway process. After that, all bets are off: nobody knows what will happen. But Kurzweil’s technological singularity, if it comes to pass, is not a true singularity. There is no law of nature that limits our ability to predict what might happen once AI evolves past the “singularity” point; we’re confined instead by the limits of the human mind.

Though they sound exotic, mathematical singularities are actually common in solutions to all but the simplest equations in physics. The formation of shock waves and cracks, and even the motion of a billiard ball bouncing off a hard wall, can contain singularities. While these singularities fulfill the mathematical definition, they aren’t physically real either: they arise from idealized assumptions that physicists make to force the messy world of reality into the neat one of mathematics. In reality, no crack is perfectly sharp, no wall is perfectly hard, no shock wave is perfectly localized.

Here is another example: Turn down the water on your kitchen tap until it starts dripping. The hydrodynamical equation describing the surface of the drops has a singularity at the pinching point: you cannot from one drop predict where the next will be. But this singularity, too, can be avoided by applying a more suitable theory. Using atomic physics, you could, in principle, calculate exactly how the water stream breaks apart on the level of single atoms. All these singularities are thus artifacts of using a theory outside its range of applicability, on distances so short that a more precise theory would be needed.

The one type of singularity that might be real—that physicists don’t know how to resolve—is the one that appears in Einstein’s theory of General Relativity when matter collapses under the gravitational pull of its own weight. There is nothing in General Relativity that then stands in the way of this collapse. It will continue until all the matter is located at a single point of infinite matter density and infinite space-time curvature: a singularity.

The singularities that appear inside black…

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