  So far we have considered three uniform types of methodological naturalism relating to mathematics: scientific, mathematical, and mathematical-cum-scientific. Consider now a heterogeneous methodological naturalism that accepts mathematical standards when it comes to mathematics proper but adopts scientific standards for the philosophy of mathematics and for philosophy more generally. Heterogeneous naturalism has been advanced by Penelope Maddy (1997), whose rich contributions have animated and incomparably influenced the debate on naturalism in the philosophy of mathematics over the past twenty years. (Maddy now prefers to call her naturalism ‘Second Philosophy’, as in the title of her 2007 book, but here we maintain the label ‘naturalism’ in keeping with the rest of the entry.)