Mathematical explanations

Mathematical facts appear in many scientific explanations, since (paraphrasing Galileo) the universe’s laws are written in the language of mathematics. But some scientific explanations are distinctively mathematical in that they do not appeal to laws of nature or derive their explanatory power from describing the world’s causal connections. Rather, distinctively mathematical explanations transcend natural laws and causal connections.

Several varieties of distinctively mathematical explanations can be distinguished. Some of these explanations reveal the natural facts being explained to be mathematically necessary. Other distinctively mathematical explanations show that the reason that different laws of nature (such as laws concerning electrostatics and thermodynamics) are mathematically analogous is that they derive in mathematically analogous ways from mathematically analogous more fundamental laws.

Philosophers have investigated how distinctively scientific explanations operate – for instance, the source of their explanatory asymmetry. Philosophers have also investigated what mathematical facts could be, such that they can explain some physical facts. Distinctively mathematical explanations have been used to argue for mathematical platonism, on the grounds that platonist mathematical objects are indispensible to those scientific explanations. But some philosophers have regarded facts about platonist mathematical objects as incapable of explaining physical facts. These philosophers have regarded distinctively mathematical explanations as supporting non-platonist interpretations of mathematics, such as Aristotelian realism.