Sublime

out of which new law-like patterns at higher levels can—and

When he did so, he discovered a strange property of his arrays: multiplying one array, A, by another, B often gave a different answer from multiplying B by A.

This is unlike fluctuations in simpler physical systems, which are typically signs of disorder

to the lowest.

If you could figure out what patterns of neural activity in a 100-dimension population are fundamental to that population – and which are just recycled combinations of those fundamental patterns – you could explain that neural population with fewer than 100 dimensions.

On the morning of August 31, 2012, the Japanese mathematician Shinichi Mochizuki published four articles on his blog. Those six hundred pages contained a proof of one of the most important conjectures in number theory, known as a + b = c.

to produce branch effective physics and possibly higher levels of complexity.