Thank you for answer Phil. If you don’t mind, I would like to defend my evolutionary explanation a bit (neglecting the other parts of your answer for the moment).
In a way, there is lot of applied mathematics in biology – a spectacular example would be echolocation in bats or dolphins but there are many others. Admittedly, those are mostly specialized systems and not all-purpose processors like the human mind, but I still think they illustrate that advanced mathematical abilities can be acquired by evolution.
The human ability to apply mathematical rules and – probably more important for real mathematics - to combine them freely might be a chance product of their coexistence in the same mental apparatus. The chance element is of course a weak point here, but then again, chance is an essential part of evolution and there are quite a few examples where unsuspecting inventions have been coopted by evolution for some unforeseen purpose (e.g. feathers for flight).
I would also like to outline a possible explanation for the ability of mathematics to work in counter-intuitive realms which is in line with an evolutionary approach: The rules used to interpret the physical world (or to manipulate it) would be more useful the more general (and accurate) they were. I would further posit that intuition is based on what we can directly experience through our senses or actions but those experiences are constrained by the implementation of our physical body. However, mental rules that are able to interpret data accurately in the working range of our body do not necessarily fail where the body fails, in the same way that the validity of a temperature scale does not stop where a particular thermometer reaches the end of its dynamic range.
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