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I understand and to an extent agree with the gist of your argument, but I can't agree with this: > Mathematics would be a failure if it wasn't usable to model

I understand and to an extent agree with the gist of your argument, but I can't agree with this: > Mathematics would be a failure if it wasn't usable to model nature. In fact I don't understand why you would make such a statement, unless you have a very different description of nature (or failure, or both) to what I have. Mathematics has many applications beyond modelling physical reality. This topic has the potential to raise many philosophical questions, but even at the most superficial level there is a difference between "describing" and "solving". This doesn't sit quite well with me either: > it seems hardly amazing that something created to describe nature produces descriptions that seem to describe nature very well. Pure mathematics isn't created to describe nature. Physical applications of pure mathematical concepts are usually only found many years after they are devised.

Confusion 17y ago
My belief (I call it a belief, because I haven't done rigorous research to confirm it; it is based on what I learned during studying physics) is that all early, 'basic', mathematics was invented (or discovered, I don't want to get into that) to describe nature. Initially, the natural numbers were nothing but a convention to describe and differentiate between sets of multiple objects that were to be considered equal for the purposes of discussion. IV apples vs. V apples. II rocks vs. VII garments (of course, the Romans didn't invent this, but I find it a distraction to use 'our' Arabic numerals). Another example: Newton invented differential calculus specifically to describe nature. Shortly after that field was invented, mathematics in it would be done for it's own sake, but that doesn't detract from the fact that originally it was meant to describe nature. More recently, mathematics has been invented before it was shown to be applicable to physics. What I earlier meant is that this may be a result of the origins of mathematics and doesn't prove what it seems to imply.

Confusion 17y ago

My belief (I call it a belief, because I haven't done rigorous research to confirm it; it is based on what I learned during studying physics) is that all early, 'basic', mathematics was invented (or discovered, I don't want to get into that) to describe nature. Initially, the natural numbers were nothing but a convention to describe and differentiate between sets of multiple objects that were to be considered equal for the purposes of discussion. IV apples vs. V apples. II rocks vs. VII garments (of course, the Romans didn't invent this, but I find it a distraction to use 'our' Arabic numerals). Another example: Newton invented differential calculus specifically to describe nature. Shortly after that field was invented, mathematics in it would be done for it's own sake, but that doesn't detract from the fact that originally it was meant to describe nature. More recently, mathematics has been invented before it was shown to be applicable to physics. What I earlier meant is that this may be a result of the origins of mathematics and doesn't prove what it seems to imply.