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Let me share some illustrations I've recently created to discuss the topic:
What is a number?
Two sets are equinumerous—they have the
same cardinal size—when they can be placed into a one-to-one correspondence, like the shepherd counting his sheep off on his fingers.
What is a number?
Two sets are equinumerous—they have the
same cardinal size—when they can be placed into a one-to-one correspondence, like the shepherd counting his sheep off on his fingers.
Hume's principle asserts that the numbers of F's is the same as the number of G's just in case these classes are equinumerous.
{ x | Fx } ~ { x | Gx }
The equinumerosity relation thus partitions the collection of all sets into equinumerosity classes of same-size sets.
{ x | Fx } ~ { x | Gx }
The equinumerosity relation thus partitions the collection of all sets into equinumerosity classes of same-size sets.
In his attempts to reduce all mathematics to logic—the logicist program—Frege defined the cardinal numbers simply to be these equivalence classes. For Frege, the number 2 is the class of all two-element sets; the number 3 is the class of all three-element sets, and so on.
Frege proceeded to interpret the arithmetic structure and implement Dedekind's arithmetic with zero, successor and induction.
Meanwhile, Hume's principle can be understood in a structuralist light, asserting merely that numbers are an invariant of the equinumerosity relation. Whatever numbers are, equinumerous sets get assigned the same number, and non-equinumerous sets get assigned different numbers.
On this structuralist perspective, Hume's principle is all we need to say about what number are. They are an equinumersoity invariant, and that is all that matters.
