Plato Was Not a Mathematical Platonist

This Element shows that Plato keeps a clear distinction between mathematical and metaphysical realism and the knife he uses to slice the difference is method. The philosopher''s dialectical method requires that we tether the truth of hypotheses to existing metaphysical objects. The mathematician''s hypothetical method by contrast takes hypotheses as if they were first principles so no metaphysical account of their truth is needed. Thus we come to Plato''s methodological as-if realism: in mathematics we treat our hypotheses as if they were first principles and consequently our objects as if they existed and we do this for the purpose of solving problems. Taking the road suggested by Plato''s Republic this Element shows that methodological commitments to mathematical objects are made in light of mathematical practice; foundational considerations; and mathematical applicability. This title is also available as Open Access on Cambridge Core.