Try Ridley’s The Rational Optimist, as I mentioned above.

You might also like Matt Ridley’s The Rational Optimist (2009). I read that first and was struck by his many insights into how human society has prospered by collaboration and exchange. Reading Sapiens after that book gave me the feeling of reading a somewhat watered-down and much more pessimistic take on the same millenia-long story. On the topic of wheat for eaxmple Ridley, who is originally a biologist, tells it as how humans created wheat for our own benefit - modern wheat simply does not exist as a wild plant. (Have you ever seen a wheat weed?)

I just read that story based on your recommendation - thank you! I might be just a tiny bit better as a person now than I was before reading it.

For fellow fans of le Guin, Sturgeon and Smith looking for more authors, I would recommend Lois McMaster Bujold. Space opera adventures on the surface, but with subtle depths and insights into human nature. Or for something more recent, Ada Palmer’s Terra Ignota series; we have solved world hunger, achieved gender equality and invented flying cars, but what are the implications for world politics and human interactions?

Neal Stephenson’s Anathem, Norman Spinrod’s Child of Fortune.

I’ve thought about that as well. The only addition I can think of to your short list is Ada Palmer, I highly recommend her Terra Ignota series for social commentary and some refreshing optimism - and flying cars.

I did some work in Ramsey theory 20 years ago (improved the upper boundaries for R(3,12) and R(3,15) by one digit each [1]) and I agree with Erdös: we could do it, if there was a compelling reason to devote a lot of effort to it.

As others have said, brute force is just plain impossible. But we can limit the search space significantly by proving sub-results about the structures of the possible graphs. Somewhat trivial example to give the flavor: let's state the problem as "find the smallest number of vertices needed such that any graph must have either a 5-connected component (a "pentagram") or 5 independent vertices". We know that the number is at least 43. So if we are looking for a counterexample, a graph on 43 vertices that does not have either of these two subgraphs, what can we say about the possible number of edges that each vertex must have? (the degree, for graph theorists). We can immediately say that if we have 5 or more vertices with no edges (degree 0) then we have our independent set already, so any counterexample can have at most 4 vertices of degree 0.

By the way, my favorite Explain-like-I'm-5 version of Ramsey's theorem is "Total disorder is impossible": If a structure is large enough, there must be some substructure that is ordered.

[1] https://www2.math.su.se/ramseytheory/sciramseyams.pdf

As a mathematician, I would claim that math at its best is exactly about riffing and jamming on shared fundamentals and adding your own insights. But it is often misunderstood what these fundamentals are: not the proofs or the formulas, but the elements of logical reasoning that make it possible to say with absolute certainty that given some conditions, some statement must be absolutely true.

A riff on a proof in geometry isn't changing a few of the letters around to see if still works - it might instead be about changing the assumptions. In plane geometry, the angles of a triangle sum to 180 degrees. But what if we are not in the plane, but on the surface of a sphere, such as the earth? Does a triangle connecting the north pole to two points on the equator still have the sum 180 degrees? If not, can we prove something else about it? In the context of the original article, it might be something like "Can we use any parts of our proof about pentagons (5-cycles) for some other shapes? What about hexagons (6-cycles)? Or is there even any insight we can generalize so that it becomes a statement about cycles of any length?"

Sadly, this is way too seldom the way math is taught. I didn't enjoy the endless calculations of long division in grade school, or the memorization of different tricks for solving trigonometric integrals in college, either.

For a famous mathematician quote, how about Paul Erdös concept of "a proof from the Book" - said about math proofs that are so perfect and clear that they must be in God's celestial collection. Often, there is more than one way to prove something true - checking every example by brute force would be the most extreme - but sometimes you can discover an elegant argument that just convinces everyone who reads it that it simply must be true. That could also be thought of as riffing on a proof: Ok, you convinced me that this is true, after many boring pages of calculations - can I find a simpler way to convince myself of the same thing, and improve both our understandings?" Quanta magazine had a nice article about this as well: https://www.quantamagazine.org/gunter-ziegler-and-martin-aig...

Yes, this is possible and legal in Sweden. The other way to obtain cheap alcohol is to take a cruise via the island of Åland, which is not in the EU. If you plan to visit Åland, make sure to pick a ferry that actually stops there as opposed to just symbolically tossing out a mooring rope for a few minutes to satisfy the legal requirements.

Pre-covid, this was a thriving and mostly legal industry. There are quotas on how much may be imported for personal consumption.

Meanwhile, the Swedes go to Denmark for cheap alcohol, the Danes go to Germany, the Germans to Poland and the Poles to Ukraine. Thus Ukraine is the fixed point of the alcohol-flow function. This theorem was proven when I was a PhD student in a research group which contained a Norwegian, a Swede, a German, a Pole and a Ukrainian - we extrapolated the Danish connection point.

Yes, it’s ”The time traveler”, first published in Analog, April 1974. Reprinted in the collection “Callahan’s Crosstime Saloon”, and in the meta-collection “The Callahan Chronicals”. Is shows how much cultural shift can take place in a decade: the protagonist is taken prisoner in 1963 and returns to a changed society in 1973. Highly recommended!