(About "gray swans" - that is, black swans that occur as a result of fractal or Mandelbrotian randomness.)

Benoit Mandelbrot

Benoit Mandelbrot was born in Warsaw in 1924 into a Jewish family that went to France in 1936 to escape the Nazis. In 1958 he moved to the US and joined IBM as a researcher. His work was founded on two ideas - that the world is loaded with self-similar phenomena and that much reality does not fit neatly under a Gaussin bell curve. The former belief led him in 1975 to coin the term fractal, which made him famous. The later belief led him to Nassim Taleb - or the other way around. (Seeing them together on TV is pretty weird, the placid old mathematician and the bouncing trader/philosopher.)

Fractals

Fractals refer to aspects of nature - physical stuff, events, processes - that are similar across all scales, big and small. Look at the basic shape of a mountain. It is pretty much the same viewed from a distance or close up. It's the same with branches of a fern - as well as crystals, snowflakes, coastlines, broccoli, lightning, etc. All are fractals.  They are the result of fairly simple rules applied recursively - that is the same rule applied over and over to the same thing.

In a fractal system, the ratios of differences between large and small, tend to the same or close. Also there is no obvious limit to the size of the system. (There might be a limit, some point at which the system becomes predictable, or Gaussian; however, there is no way to determine what that limit might be.)

Fractals can be applied to visual arts, music, poetry, literature, any activity where there is a repeating theme or pattern.

Non-Linear Chaos

Fractals are examples of non-linear, chaotic systems. These are systems that start out simple then over time, as rules are recursively applied, evolve into something much more complex. Such systems are sensitive to initial conditions. A small change at the outset results in major changes later. A simple fractal shape can quickly become a complex shape (that still contains the seed of the starting shape). A butterfly flapping its wings and creating tiny wind patterns in Africa can result in a hurricane in Florida.

Chaotic systems may appear to be stable for some number of iterations, then start to rapidly change. This is called the crossover point, when things start to get weird, when the system becomes "chaotic". The crossover point is related to exponent or power attributes of chaotic systems - which determines the difference between each iteration.

Taleb's point in all this is that you can't get inside a chaotic system to see what these values and attributes are. Although such a system may in fact be deterministic, operating according to certain rules, there is no practical way to predict where it is going. Scientists who study complex systems have discovered certain patterns that might be universal across totally different external realities. Cool, Taleb might say, but you still can't predict what's going to happen.  A system can start out Gaussian, become Mandelbrotian, then become Gaussian again.

Gray Swans and Black Swans

Gray swans are the results of Mandelbrotian randomness. They are theoretically predictable but practically not. Many black swans are actually gray swans. We don't know when they will happen, just that they will. True black swans are completely unknown. They blindside us every time.