02/15/2017, 10:10 AM

Some remarks

I didn't mean that. The paper does not talk about ranks but only about complexity, dimension and entropy.

About the analogies... I guess that it's easy to see! There is a subtle and fascinating network of analogies between all these concepts.

And if it is not, start by what is probably the most easy-to-spot analogy. (remember that I'm not making the domains precise, I'm just trying to be suggestive)

The ackermann function and all the recursive(but not-primitive) functions that are "beyond" the hierarchy (finite rank) are a bit like the transcendental functions (or smooth, or analytic) that are beyond the polynomial (finite degree): in both cases we have a procedure going back "through dimensions" (differences and anti-recursion) and in the other direction a mutivalued procedure (indefinite sum and not based-recursion/iteration) going upstairs... it is like we are generalizing this graded structure of polinomials (well behaved, commutative, familiar objects) to more exotic, non-commuative frameworks (like group of functions, monoids and so on..).

Amazingly this can be made very precise.

Quote:But aside form these speculations there is a subtle connection going on here between ranks, computation complexity and dimension.(*)

I didn't mean that. The paper does not talk about ranks but only about complexity, dimension and entropy.

About the analogies... I guess that it's easy to see! There is a subtle and fascinating network of analogies between all these concepts.

And if it is not, start by what is probably the most easy-to-spot analogy. (remember that I'm not making the domains precise, I'm just trying to be suggestive)

The ackermann function and all the recursive(but not-primitive) functions that are "beyond" the hierarchy (finite rank) are a bit like the transcendental functions (or smooth, or analytic) that are beyond the polynomial (finite degree): in both cases we have a procedure going back "through dimensions" (differences and anti-recursion) and in the other direction a mutivalued procedure (indefinite sum and not based-recursion/iteration) going upstairs... it is like we are generalizing this graded structure of polinomials (well behaved, commutative, familiar objects) to more exotic, non-commuative frameworks (like group of functions, monoids and so on..).

Amazingly this can be made very precise.

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