I have always defined "mathematics" as the abstract study of internally self consistent structures. Against this one has reality -- a totally unknown thing. Somehow our subconscious tries to make sense of it. What do we mean by "sense of it"? That we have an internally self consistent model of what is going on.
There are those who will hold that self-consistency is an unnecessary constraint; however, I hold that the definition of self-consistency is that the same result is obtained from any analysis of the model. If the result of the analysis depends on how you attack the problem then the model will give different answers for the same circumstances which defeats the purpose of the model: i.e., to provide an easy way to keep track of results.
If your model of reality is to be self-consistent than is it any surprise that mathematics should be applicable?
>>>But, wait a minute. One apple plus one apple does not always equals two apples. Leave the apples unattended for a few years and see what happens: one apple plus one apple = zero apples! OK, so it's only a matter of applying math "the right way". The apples are gone but all their constituent atoms are still out there in the world.>The validity of logic is preserved. Or is it? Aren't we simply shifting the logical inconsistency to a time scale where it can't be invalidated, in other words, restricting the amount of data so that it conforms to our theory? I'm not sure what modern physics says about it but my understanding is that particles are not logical at all, that is, 1+1=2 does not apply.>space and time in which we live