Thanks for the link. I'll also mention a year old book that may be relevant for interested people: "Godel's Way" by Chaitin, De Copsta, Doria; CRC Press.
Personally, I would feel closest to Steve Simpson of those mentioned in the article (not relevant, but I am PSU alum as well.)
But, if it just happens that there are two (or more) different logically consistent universes, and there is absolutely no empirical deciding between the lot, well, that's just the way it is. Too bad. They talk about "proof" in the article as if the past 150 years of mathematics hadn't occurred, wherein it was discovered more than once that honest people can disagree at on which axioms to pursue, and neither party turns out to be facing any inconsistencies on account of those preferences. It thus (seems to me) turns into a war of fashion: short skirts or calf length?
>From article: "The back-and-forth will likely continue, they said, until one or the other candidate falls by the wayside."
Really? Perhaps they mean until one set of bullies takes over the playground? Maybe, but I'd prefer that the issue not be decided by bullies of any stripe. Not that I'm unaware that the history of mathematics has already had its share of bullying one way or another.
>From article: "Mathematics has a reputation for objectivity. But without real-world infinite objects upon which to base abstractions, mathematical truth becomes, to some extent, a matter of opinion?which is Simpson?s argument for keeping actual infinity out of mathematics altogether."
Well, I'd say, let's all live and let live, but the view that actual infinities are some flavor of game we don't really care much for (which Gauss more or less agreed with!) has been the poor bullied kid cowering in the corner for some time now. I'm glad there are still champions for that view, including say, Ed Nelson at Princeton.
>From article: "According to Koellner, "one side is going to have to be wrong."