The current dominant mathematical paradigm is based on axiomatic set theory and formal logic. Virtually all mathematical theorems today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is then nothing but the claim that the statement can be derived from the axioms of set theory using the rules of formal logic.
This formalistic approach is unsatisfactory for several reasons. It does not explain why we should use the axioms we do and not some others, and why we should employ the logical rules we do and not some other. It also does not explain why "true" mathematical statements, for instance the laws of arithmetic, appear to be true in the physical world. This was called "The unreasonable effectiveness of mathematics in the physical sciences" by Eugene Wigner in 1960.
The above mentioned notion of formalistic truth could also turn out to be rather pointless: it is perfectly possible that all statements, even contradictions, can be derived from the axioms of set theory. Worse, as a consequence of Gödel's second incompleteness theorem, we can never be sure that this is not the case.
In mathematical realism, sometimes called Platonism, the existence of a world of mathematical objects independent of humans is postulated; the truths about these objects are discovered by humans. In this view, the laws of nature and the laws of mathematics have a similar status, and the "effectiveness" ceases to be "unreasonable". Not our axioms, but the very real world of mathematical objects forms the foundation. The obvious question, then, is: how do we access this world?
Some modern theories in the philosophy of mathematics deny the existence of foundations in the original sense. Some theories tend to focus on mathematical practice, and aim to describe and analyze the actual working of mathematicians as a social group. Others try to create a cognitive science of mathematics, focusing on human cognition as the origin of the reliability of mathematics when applied to the 'real world'. These theories would propose to find foundations only in human thought, not in any 'objective' outside construct. The matter remains controversial.
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