This post over at Thought Mesh raises to related philosophical questions that are fun to think about. The first is the nature of mathematics, which I'll discuss in this post, and the second is the nature of science (or the problem with Einstein), which I'll discuss in a post to come.
Alternate universes, the subject of AOG's post, is an example of a set of scientific speculations (time travel is another) that is allowed, but not compelled, by our current models of reality. One question this raises is the reliability of our models, which, in turn, raises the question of the nature of mathematics.
Is mathematics inherently tied to the nature of the universe, or is it just a really good modeling tool that, if done right, has not yet been inaccurate in our experience. Put another way, does putting two objects down next to two other objects have to result in four objects, or does it just so happen to work out that way?
We tend to assume that mathematics is inherently tied to the nature of reality for two reasons. First, as I've already said, because our experience doesn't include an example of where it doesn't. Second, because math has been usefully predictive. (Are these actually two different reasons, or just separate aspects of one reason? It is in any event useful to discuss them separately.)
The first reason seems compelling -- or, rather, feels compelling -- but is obviously problematic. We have, in fact, a rather small sample of different situation in which math has been a useful modeling tool -- those that can be experienced on Earth, with the tools available to us, over the last 10,000 years. How many different types of situations have we really been in, compared to the age and size of the universe. Plus, we're by nature connection seeking, satisficing animals. Once we've experienced an event, the fact that we can go back and show that the math works isn't really all that interesting. We can always find connections after the fact (connection seeking) and once we've found one tend to stop looking (satisficing). So that fact that mathematics conforms to our experience, as we've experienced it, might not be great evidence about the nature of mathematics.
The second reason, because it avoids the problem of looking backwards, seems the stronger. If we can use math to model unknown facts about the universe, and then test to conform them (Popper looks around triumphantly) then it is that much more likely that mathematics is inherently tied to the nature of the universe. But this, too, proves less than it seems. Because we also do make predictions based on mathematics that, when we test them, are disproved -- otherwise, we wouldn't bother to check, would we. In other words, science, in the Popperian sense, requires that we be skeptical that mathematics, or at least our ability to do mathematics, does accurately mirror reality. We don't accept our models until we confirm them.
So it seems that, speaking of Popper, our intuitive sense that mathematics is inherently tied to the nature of reality is unfalsifiable.
Does this matter? Actually, it seems to matter quite a bit. The rocket-sciences who gravitated to Wall Street over the last 15-20 years made billions with models of the financial market used to value options and derivatives without a history of performance or any tangible, real world, analog. The mathematicians and their banks all assumed that their mathematical models were tied to some underlying reality; that the constructs "really" had the value that the models suggested. We know how that turned out.
Now we're about to engage in an even more drastic economic experiment based on the assumption that global climate models can reliably reflect the future, a proposition that, to the extent it can be falsified, has been. Moreover, we not only have to consider whether mathematical models can model future climate, but also whether the people performing the modeling are doing so honestly and competently. Of course, every time you fly, you're also betting your life (albeit with much better evidence) that there is some relationship between mathematics and reality.