By analogy with stacks, my intuition is that Spec(N) still has a one-dimensional geometry, but with a (-1)-dimensional tier from quotienting out the prime points by a semiring action.
Upvoted. USAID has been misused plenty: political meddling in Bangladesh, Cuba's ZunZuneo op, boomerang aid in Haiti, Afghan contractor enrichment, etc. I could go on.
But understand: the soft power isn't going away, just the doctors, food shipments, and vaccines.
It looks like we've hit a reply depth limit, which is maybe for the best, because I don't think we're making any progress here.
> You did use tautologies...
You seem to think calling something a tautology is a way to dismiss it. Almost everything in mathematics is a tautology-- most of what I say is a tautology. Any rigorous argument is tautological; it's the aspiration of literally all formal reasoning.
> Lots of uninteresting and bad music is also entropic.
And here, you seem to think someone is claiming that entropy is equivalent to music quality, not just a useful correlate or eg. indicator of something that might be more likely to show up in good music than bad music. I don't know of anyone making that claim; all the examples I gave require mild correlation.
You said I was using tautologies as straw men, which is incoherent and suggests you’re not arguing in good faith.
Anyhow, of course entropy correlates to music quality; maximum entropy music is white noise! I’ve even had luck finding interesting jazz musicians from the distribution of key signatures they use—- anything more entropic than the Real Book is a great indicator. Similarly, network entropy makes it easier to identify musicians with a flexible arsenal of riffs. You could adapt it to chord progressions to find unusual reharmonizations in live jazz to study and practice. It could be a helpful regularizer for neural network music generation. Entropic methods are among the most powerful in statistics.
Sorry for the cliché, but "the most exciting phrase to hear in science, the one that heralds the most discoveries, is not "Eureka!" (I found it!) but 'That's funny...'" --Asimov
My point is just that proving known facts can be useful and interesting.
As for the paper, network entropy and node heterogeneity seem to be perfectly sensible statistical concepts, and encode useful information. They also dovetail conveniently with powerful tools in machine learning. Criticizing this paper for lack of potential applications feels unreasonable.
First, the methods of the paper don’t have to be a Mendelssohn replacement to be useful. Second, if you don’t like that potential application, consider all the other predictive models that could benefit from these features.
You might be missing the point somewhat. With these methods, you could eg. test the vast swaths of potentially overlooked composers to see if any of them merit closer listening.