A naturalist account of the reasonable effectiveness of mathematics in physics [pdf]
arxiv.org
arxiv.org
> The universe exists apart from being evoked by the human imagination, while mathematical objects do not exist before and apart from being evoked by human imagination.
Smolin says this in his conclusion. But when people talk about the unreasonable effectiveness of mathematics (at least when I've heard it), this is what they're talking about - not that mathematical objects exist in some nonphysical platonic space, but that they exist in our heads as a game we play - a formal axiomatic system. The question is, why does our formal game, which we think is mostly abstract, suddenly and surprisingly turn out to work so well to model the physical universe? (We don't find that chess works as a model, for example.)
Smolin answers that, sort of. He says that since the basics of mathematics are in nature, it's reasonable that as math progresses, it will continue to correspond to nature. But it seems to me quite a stretch to say that, because the natural numbers correspond to the existence of countable things in nature, and natural objects take up space, therefore pseudo-Riemannian manifolds will correspond to general relativity. To say that is reasonable, it seems to me, requires making a mysticism around nature.
It would probably be best to do away with the term "mathematical truth" altogether. It's confusing and sloppy. Mathematics has no separate truth from the physical world - it's just a field of epistemological methods.
And where does 'derivation' come from?
I don't think that can be right. First of all, there are many mathematical things that don't correspond to anything in the physical world. Second, I'm not sure that "epistemological methods" correspond very well to the physical world, either. (I mean, yes, in one sense it's something humans do in their heads, so it's part of biology, so it's part of the physical world, but that seems like a bit more than what we usually mean by "the physical world".)
More that that, though, we don't think of everything as physics. Biology, and even chemistry, we think of as separate disciplines. Thinking of mathematics as also a separate discipline seems perfectly reasonable.
> The simple fact is that mathematics is derived from nature first by observation of countable entities and then a process of abstraction from those observations.
I think it's more complex than that... I suspect that math (and many other fields) starts out this way, going from concrete to abstract, but once it gets to the abstraction phase then it takes on a life of it's own. People find the abstractions interesting in their own right, they seem to develop a sort of historical direction, and they have compelling properties on their own merit, so people felt compelled to develop them. People's intuitions, however, are strongly shaped by physical reality. And those mathematical abstractions are often useful for modeling physical reality, causing a strong interplay between reality and mathematical abstractions.
That would cause problems for super-string theory, the big bang and many other models of the origin of the universe.