Why are mathematical models so insanely good at making predictions about the physical universe?
Is this a testament to the general modeling power of mathematics to describe any internally consistent system, or is the universe in some sense fundamentally mathematical?
These are the sorts of questions that used to come up in my undergraduate days. I don't recall a good answer, though I was certainly on the dim side of that student body. It may be that the answers were simply over my head.
I wonder about those questions again as I, very slowly, read Brian Greene's The Fabric of the Cosmos (See also: BBC IOT - Theories of Everything With Brian Greene). It's a good book; I'll have more to say on it when I'm done - sometime in the spring of 2008. For now I will say I like the substantial non-string chapters better than the string theory portion.
There are a few reasons for this preference. Greene is a string theorist, and I think most specialists do best describing things outside of their core passion. It's easier to be neutral about things that you haven't poured your heart into. More interestingly, the old question, "Is the universe fundamentally mathematical?", plays a role as well.
Most new physics seems, to a hobbyist like me, to make a reasonable bet that the implausible success of mathematical models will hold in new domains. It's a bit like tossing a plank off the end of a pier, assuming that when one walks to the end of the plank a supporting pillar will be found.
String theory tosses a breathtakingly long plank. It's a daring bet indeed. If we're ever to find out that it holds (proof of those necessary 10 space dimensions?) then we do have to take seriously the old whimsy that the universe, at its heart, is purely mathematical.
Until that day, it sure does feel to the physics hobbyist more like an exercise in mathematical brilliance than even traditional theoretical physics ...
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