[inscrutablemike]

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. The natural numbers don't merely correspond to countable things - countable things are the original source of the much more abstract idea of "natural numbers". Whether or not any further baste actions are a useful tool for measuring and describing the physical world is an entirely separate question from the nature and source of mathematics as a field of study.

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.

[webnrrd2k]

> 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. This sounds like a contradiction -- epistemology is, more-or-less concerned with what's true. If mathematics is simply a field of epistemology (with is own methods), then it's not a stretch to say that there is a mathematical truth -- a truth that satisfies the methods of the filed of mathematics.

> 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.

[AnimalMuppet]

> Mathematics has no separate truth from the physical world - it's just a field of epistemological methods.

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.

[cristianpascu]

> mathematics is derived from nature

And where does 'derivation' come from?

[eli_gottlieb]

It's taking place in the brain of the mathematician.