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The philosophy of mathematics

The philosophy of mathematics is a prolific field of inquiry - with questions which go back at least as far as Ancient Greece, when Aristotle and Plato were trying to understand its implications.

Although many of the unanswered questions are extremely esoteric and paradoxical (see examples in the links below) there are also very basic concepts such as : Does mathematics exist in the form of 'Universal Laws' (which would have existed long before humans evolved to contemplate them) or, is mathematics a purely human construct?

For example, hexagons exist in nature (crystal formation, honeycombs etc etc ) and always have six sides. So is the number 6 a 'universal' entity?

Another paradox is the question of whether highly complex and robustly provable mathematical structures have always existed. In other words, do mathematicians simply 'discover' them? If, as seems intuitive, they have always 'been there' then the question arises 'Why?'

For starting points to explore this highly extensive and largely unresolved field of inquiry, see Philosophy of mathematics and Foundations of mathematics - both at Wikipedia

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