"Believing" is a tall order, and Maddy's paper suggests nothing of that sort, as far as I am aware (though, of course, it is a nice way to connote it). On the contrary: if I were to name one idea from Maddy's paper that helped me shape my view of contemporary set theory, and indeed of the message of her paper, it would be in the vicinity of the following quotation (from the intoductory passages to $\S$1):

Even the most cursory look at the particular axioms of ZFC will reveal that the line between intrinsic and extrinsic justification, vague as it might be, does not fall neatly between ZFC and the rest. The fact that these few axioms are commonly enshrined in the opening pages of mathematics texts should be viewed as an historical accident, not a sign of their privileged epistemological or metaphysical status.

The notion of rank, discussed by Maddy in the paragraph about foundation, must be strongly linked (historically at least) to Russell's theory of types, as the Wikipedia article on Regularity also confirms, quoting Enderton:

The idea of rank is a descendant of Russell's concept of type.

It was probably seen as an enhancement of Russell's way of addressing the paradox.

Treating collections of objects as objects, uniformly across the universe, is what calls for stipulating regularity and what enables its violations. But the scale of uniformity provided by ZFC is perhaps rarely needed. It may be just my illusion, but I think many branches of mathematics do keep their "internal stratification of notions" that makes it pointless to even appeal to regularity.

"Believing" is a tall order, and Maddy's paper suggests nothing of that sort, as far as I am aware (though, of course, it is a nice way to connote it). On the contrary: if I were to name one idea from Maddy's paper that helped me shape my view of contemporary set theory, and indeed of the message of her paper, it would be in the vicinity of the following quotation (from the introductory passages to $\S$1):

Even the most cursory look at the particular axioms of ZFC will reveal that the line between intrinsic and extrinsic justification, vague as it might be, does not fall neatly between ZFC and the rest. The fact that these few axioms are commonly enshrined in the opening pages of mathematics texts should be viewed as an historical accident, not a sign of their privileged epistemological or metaphysical status.

The notion of rank, discussed by Maddy in the paragraph about foundation, must be strongly linked (historically at least) to Russell's theory of types, as the Wikipedia article on Regularity also confirms, quoting Enderton:

The idea of rank is a descendant of Russell's concept of type.

It was probably seen as an enhancement of Russell's way of addressing the paradox.

Treating collections of objects as objects, uniformly across the universe, is what calls for stipulating regularity and what enables its violations. But the scale of uniformity provided by ZFC is perhaps rarely needed. It may be just my illusion, but I think many branches of mathematics do keep their "internal stratification of notions" that makes it pointless to even appeal to regularity.

"Believing" is indeed a tall order, and Maddy's paper suggests nothing of that sort, as far as I am aware (though, of course, it is a nice way to connote it). On the contrary: if I were to name one idea from Maddy's paper that helped me shape my view of contemporary set theory, and indeed of the message of her paper, it would be in the vicinity of the following quotation (from the intoductory passages to $\S$1):

Even the most cursory look at the particular axioms of ZFC will reveal that the line between intrinsic and extrinsic justification, vague as it might be, does not fall neatly between ZFC and the rest. The fact that these few axioms are commonly enshrined in the opening pages of mathematics texts should be viewed as an historical accident, not a sign of their privileged epistemological or metaphysical status.

The notion of rank, discussed by Maddy in the paragraph about foundation, must be strongly linked (historically at least) to Russell's theory of types, as the Wikipedia article on Regularity also confirms, quoting Enderton:

The idea of rank is a descendant of Russell's concept of type.

Indeed it was probably seen as an enhancement of Russell's way of addressing the paradox.

Treating collections of objects as objects, uniformly across the universe, is what calls for stipulating regularity and what enables its violations. But the scale of uniformity provided by ZFC is perhaps rarely needed. It may be just my illusion, but I think many branches of mathematics do keep their "internal stratification of notions" that makes it pointless to even apply to regularity.

"Believing" is a tall order, and Maddy's paper suggests nothing of that sort, as far as I am aware (though, of course, it is a nice way to connote it). On the contrary: if I were to name one idea from Maddy's paper that helped me shape my view of contemporary set theory, and indeed of the message of her paper, it would be in the vicinity of the following quotation (from the intoductory passages to $\S$1):

Even the most cursory look at the particular axioms of ZFC will reveal that the line between intrinsic and extrinsic justification, vague as it might be, does not fall neatly between ZFC and the rest. The fact that these few axioms are commonly enshrined in the opening pages of mathematics texts should be viewed as an historical accident, not a sign of their privileged epistemological or metaphysical status.

The notion of rank, discussed by Maddy in the paragraph about foundation, must be strongly linked (historically at least) to Russell's theory of types, as the Wikipedia article on Regularity also confirms, quoting Enderton:

The idea of rank is a descendant of Russell's concept of type.

It was probably seen as an enhancement of Russell's way of addressing the paradox.

Treating collections of objects as objects, uniformly across the universe, is what calls for stipulating regularity and what enables its violations. But the scale of uniformity provided by ZFC is perhaps rarely needed. It may be just my illusion, but I think many branches of mathematics do keep their "internal stratification of notions" that makes it pointless to even appeal to regularity.