My conception of mathematics is that you start with a set of axioms and then explore the implications of them. There are infinitely many possible sets of starting axioms you can use [1] .
This is a popular view but in my opinion it is wrong. My conception of math is that you start with a set of definitions and the axioms only come after that, as an attempt to formalize the definitions. For example:
The natural numbers are defined as the objects that you get by starting with a base object “zero” and iterating a “successor operation” arbitrarily many times. Addition and multiplication on the natural numbers are defined recursively according to certain basic formulas. The axioms of Peano arithmetic can then be viewed as simply a way of formalizing these definitions: most of the axioms are just the recursive definitions of addition and multiplication, and the induction schema is an attempt to formalize the fact that all natural numbers result from repeatedly applying the successor operation to 0.
The universe of sets is defined as the collection you get by starting with nothing, and repeatedly growing the collection by at each stage replacing it with the set of all its subsets (i.e. its powerset). The axioms of Zermelo-Fraenkel set theory are an attempt to state true facts about this universe of sets.
Of course, it’s possible to claim that the definitions in question are not valid—they are not “rigorous” in the sense of modern mathematics, i.e. they do not follow from axioms because they are logically prior to axioms. This is particularly true for the definition of the universe of sets, which in addition to being vague has the issues that it presupposes the notion of a “subset” of a collection while we are currently trying to define the notion of a set, and that it’s not clear when we are supposed to “stop” growing the collection (it’s not at “infinity”, because the axiom of infinity implies that we are supposed to continue on past infinity). But Peano arithmetic doesn’t have those problems, and in my opinion is therefore on an epistemologically sound basis. And to be honest much (most?) of modern mathematics can be translated into Peano arithmetic; people use ZFC for convenience but it’s actually not necessary much of the time.
This is a popular view but in my opinion it is wrong. My conception of math is that you start with a set of definitions and the axioms only come after that, as an attempt to formalize the definitions. For example:
The natural numbers are defined as the objects that you get by starting with a base object “zero” and iterating a “successor operation” arbitrarily many times. Addition and multiplication on the natural numbers are defined recursively according to certain basic formulas. The axioms of Peano arithmetic can then be viewed as simply a way of formalizing these definitions: most of the axioms are just the recursive definitions of addition and multiplication, and the induction schema is an attempt to formalize the fact that all natural numbers result from repeatedly applying the successor operation to 0.
The universe of sets is defined as the collection you get by starting with nothing, and repeatedly growing the collection by at each stage replacing it with the set of all its subsets (i.e. its powerset). The axioms of Zermelo-Fraenkel set theory are an attempt to state true facts about this universe of sets.
Of course, it’s possible to claim that the definitions in question are not valid—they are not “rigorous” in the sense of modern mathematics, i.e. they do not follow from axioms because they are logically prior to axioms. This is particularly true for the definition of the universe of sets, which in addition to being vague has the issues that it presupposes the notion of a “subset” of a collection while we are currently trying to define the notion of a set, and that it’s not clear when we are supposed to “stop” growing the collection (it’s not at “infinity”, because the axiom of infinity implies that we are supposed to continue on past infinity). But Peano arithmetic doesn’t have those problems, and in my opinion is therefore on an epistemologically sound basis. And to be honest much (most?) of modern mathematics can be translated into Peano arithmetic; people use ZFC for convenience but it’s actually not necessary much of the time.