For mathematical insight, according to the structuralist imperative, investigate mathematical structure and consider mathematical concepts only as invariant under isomorphism.
I shall be discussing the indispensibility argument in a later post (it is in chapter 2 of the book, whereas structuralism is from chapter 1). In my view, there is a way of viewing structuralism as undermining the success of the indispensibility argument, because no particular mathematical structure is ever indispensible, since it can be interpreted via alternative structure. For example, we don't need the complex numbers explicitly if we have real numbers, since we can simulate them; we don't need sets explicitly if we have sufficiently rich category theory; and so forth. At heart, Field's nominalization program is an interpretation effort, to interpret mathematical structure via physical phenomenon. And so I view structuralist issues as being at the center of that debate.
Great article! I find the discussion of the different strands of philosophical structuralism illuminative.
What, if any, implications those varieties of structuralism have on "The Unreasonable Effectiveness of Mathematics in the Natural Sciences"?
I shall be discussing the indispensibility argument in a later post (it is in chapter 2 of the book, whereas structuralism is from chapter 1). In my view, there is a way of viewing structuralism as undermining the success of the indispensibility argument, because no particular mathematical structure is ever indispensible, since it can be interpreted via alternative structure. For example, we don't need the complex numbers explicitly if we have real numbers, since we can simulate them; we don't need sets explicitly if we have sufficiently rich category theory; and so forth. At heart, Field's nominalization program is an interpretation effort, to interpret mathematical structure via physical phenomenon. And so I view structuralist issues as being at the center of that debate.