The n-Category Café

Wouldn’t this be more an example of a blending of algebraic structures, i.e., a movement towards yet richer algebraic structure? Tao seems to be interested in what escapes structural description.

Chris Hillman discusses something relevant in the last section (from p. 10) of What is Information? He gives a series of lessons, from

Lesson 1. The existence of “mathematical structure” in an object may mean that it can be specified more compactly than would otherwise be the case.

to

Lesson 8. The existence of symmetries in a subset $A\subset X$ usually means that we can describe $A$ much more efficiently than would otherwise be the case, by naming the group of symmetries and then describing how to pick out $A$ from among the other subsets invariant under the group.

Another tangential comment: you mention “what is pseudorandom” although I think there’s two levels here. Firstly there’s structures which are pseudorandom and those which we currently “probe” by pseudorandom/ensemble methods, and for which we don’t know where on the spectrum of pseudorandomness they lie. Eg, Shannon’s noisy channel coding theorem is proved in an ensemble-based way, but how much is that “a particular proof strategy” and how much is “pseudorandomness” an unavoidable part of coding. (Clearly creating an output stream full of “non-degenerate” correlations between elements of the input stream is a key, but is the pinacle of this pseudorandom or an intricate structure that appears random when you don’t look at it right? There’s also something I remember reading – but not where – that maybe most mathematical objects are pseudorandom but there’s a reverse “law of large numbers” that the problems humans can actually think about have so few “degrees of freedom” that they can’t not have structure.) There are numerous examples in computer science, particularly involving graph theory, where an object O of type T satisfying property P is desired and the overwhelming proportion of objects in T satisfy P yet all the imagined deterministic techniques for constructing such an object don’t work (presumably because they’re too simplistic to capture the desired property) and yet random generation is an easy, effective generation strategy.

The relevance to your point about learning is that you can clearly represent “this is a pseudorandom integer sequence s.t. its density is 1/2” much more compactly than an instance “0,8,9,…”, but how can you determine that this property is all that matters about the sequence, rather than say that it has density 1/2 and that it defines the topology of a multi-holed manifold that’s relevant because…? (This is clearly a non-sense example.) Of course, for finite learning examples you can perform cross-validation and argue that you want to learn what you can support from the data, but clearly as more general mathematical question is a better answer is desired.

Is there anything known about the pseudorandomness of these stable homotopy groups, or the unstable ones? Something like what remains after pieces of structure are removed, as with the ‘renormalization’ of the odd primes when mapped to (p - 1)/2, see p. 22 of The Dichotomy between structure and randomness.