Joel David Hamkins: Philosophy of Mathematics and Truth
Paradigm
Mathematical Statements, Extrinsic Support, and Independence
In mathematics, there are statements that we can't determine if they are true or not. However, we can rely on extrinsic support to guide our decision. This is similar to relying on intuition for intuitive concepts. We often encounter this dilemma in strong theories, where mathematical principles express clear ideas but their truth is unknown. In such cases, we turn to extrinsic support to help us make a judgment. This regrettable fact leads us to search for alternative methods to decide on these statements. Extrinsic support involves proving a statement is independent by demonstrating models in which the statement is true or false, while other statements remain true. This is seen in examples like Euclidean and non-Euclidean geometries, where the parallel postulate is either true or false. The same applies to set theory, where the continuum hypothesis can be true or false in different models. These independence results are common, and for many of these statements, we are left unsure of their truth. Therefore, we must grapple with questions of whether to accept them or their negation, using philosophical justifications rather than proofs.
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