I was bored and felt like writing some lengthy reply here... It's not like I thought through all of this but would be nice to get some replies
What does it mean to "exist"? I guess sciences requires existing objects to be material in some sense but then, what is matter? According to modern physics, all sorts of matter can be decomposed into a set of elementary particles. So all the things around us in everyday life ultimately are made of electrons, protons, neutrons and phonons. However, there is more to it, clearly, because an existing object is more than the sum of its elementary consituents. It is important in what way the consituents are arranged, it's the structure which gives a certain material object its characteristic properties. So in the end, matter consists of some elementary consituents (maybe there is only one kind of consituent in a really fundamental physical theory) arranged in certain patterns, where the pattern is actually the more important part. Maths is the science which deals with patterns and patterns can be seen as mathematical objects. So in fact, existing things are made of mathematical objects. So Maths deals with real things just as any other natural science.
Now you could argue that maths is only some sort of "language" we use to understand the patterns of nature. We therefore have to make a difference between the abstract pattern as an object of mathematics and the pattern realized in nature. However, elementary particles are indistinguishable in a fundamental way (it's a principle of quantum theory). For example, let's take some coins and arrange them in e.g. a triangular pattern. Someone else could arrange a different set of coins in exactly the same pattern. One could always tell the difference between the two patterns because they consist of different coins (they can be marked if necessary). Obviously, what is real are the coins but not the pattern. However if you arrange a number of particles in a certain pattern and exactly the same pattern of particles is created by someone else, then (according to physics) they are identical. If the same number of particles was arranged in a different pattern, then it could be identified as a different object (on the whole). Does it still make sense to distinguish between abstract patterns and real patterns if, in fact, the pattern is all there is?
