Some random observations --

Taleb writes in a bombastic and broad-brush style.

He overgeneralizes.

He is writing for the consumers of statistics and the naïve practitioners, not for genuine experts.

He wants to stir things up.

On the whole, I think he's more right than wrong, in the sense that applying a dose of his thinking to the world as it is will tend to align thinking more closely with reality.

Every observational value is equally likely throughout the interval. Now do the same in two dimensions (uniform on the unit square), three, etc. You might think that in 12 dimensions, the observations are spread out evenly in the corresponding hypercube, but you'd be wrong. Once the CLT starts to work, the observations all lie geometrically on a thin spherical shell with spikes, like a hedgehog.

The high-dimensional cube looks like a horrid, spikey thing, but the observations are indeed spread evaently in the hypercube, and there is no geometrically thin shell in this problem. The distribution of distance from the center, on the other hand, does start to peak away from the center, and more sharply as the number of dimensions increases.

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