However if you take the set of atoms in H-O-H: it is {H, O}! not {H, O, H}, and it has a cardinality of 2, which doesn’t map to the 3 atoms of our water molecule, because elements of sets must be distinct.
no… it contains { this H, O, that H }.
my lay understanding is that some particles cannot be distinguished. but it is unlikely that children are commonly coming across these while developing intuition for counting.
What matters in the water example is whether we care about distinguishing hydrogen atoms, not whether we can.
Let’s picture a chemist experimenting with water. Will they get a different experimental result if they experiment with water molecule H_left—O -- H_right as opposed to the same molecule but with swapped hydrogen atoms H_right—O -- H_left?
Probably not. This means that H_left is indistinguishable from H_right for their purpose (I am not making a claim about atoms at the quantum level). So it makes perfect sense that they end up with a mathematical model where H is not distinct from H and where the set of water atoms is {H, O}.
Situations like that seem pretty common in daily life, where we care about collections of objects and their cardinality, but define equality of the members in terms of a high-level property we care about, rather than in terms of deep identity.
For example, if you set the table for several people, you don’t distinguish between the spoons: you don’t try to give a particular spoon to a particular person, even though you could. The spoons are interchangeable. But you do care about how many spoons you bring to the table. In other words that’s an instance of treating spoons as a multiset, where you have multiple occurrences of one identical member.
Same for dollar bills, if you have multiple bills of the same denomination in your wallet, it matters how many bills you have, but you don’t care about their serial number when buying groceries. Same for shares of a given stock, etc. Probably all fungible assets are like that.
So, if you’re in the spoon situation, and want to count the number of spoons using the cardinality of a set, you need to change your definition of equality to consider each spoon as distinct. But then the spoons are not interchangeable anymore, it’s a bit as though the spoons were now bent and you have to decide which person gets which bent spoon, it’s much less convenient than the multiset.
thanks, i appreciate your write up! this is a good discussion of fungibility, and the desirable properties of numbers as a generalization of cardinality.
for you, then, you would say that multisets feel “intuitively prior” to sets/cardinality? if so, could you help me understand that intuition?
to me, ‘bijective correspondence’ feels like an earlier notion than counting, and counting feels like an earlier notion than multisets.
no… it contains { this H, O, that H }.
my lay understanding is that some particles cannot be distinguished. but it is unlikely that children are commonly coming across these while developing intuition for counting.
What matters in the water example is whether we care about distinguishing hydrogen atoms, not whether we can.
Let’s picture a chemist experimenting with water. Will they get a different experimental result if they experiment with water molecule H_left—O -- H_right as opposed to the same molecule but with swapped hydrogen atoms H_right—O -- H_left?
Probably not. This means that H_left is indistinguishable from H_right for their purpose (I am not making a claim about atoms at the quantum level). So it makes perfect sense that they end up with a mathematical model where H is not distinct from H and where the set of water atoms is {H, O}.
Situations like that seem pretty common in daily life, where we care about collections of objects and their cardinality, but define equality of the members in terms of a high-level property we care about, rather than in terms of deep identity.
For example, if you set the table for several people, you don’t distinguish between the spoons: you don’t try to give a particular spoon to a particular person, even though you could. The spoons are interchangeable. But you do care about how many spoons you bring to the table. In other words that’s an instance of treating spoons as a multiset, where you have multiple occurrences of one identical member.
Same for dollar bills, if you have multiple bills of the same denomination in your wallet, it matters how many bills you have, but you don’t care about their serial number when buying groceries. Same for shares of a given stock, etc. Probably all fungible assets are like that.
So, if you’re in the spoon situation, and want to count the number of spoons using the cardinality of a set, you need to change your definition of equality to consider each spoon as distinct. But then the spoons are not interchangeable anymore, it’s a bit as though the spoons were now bent and you have to decide which person gets which bent spoon, it’s much less convenient than the multiset.
thanks, i appreciate your write up! this is a good discussion of fungibility, and the desirable properties of numbers as a generalization of cardinality.
for you, then, you would say that multisets feel “intuitively prior” to sets/cardinality? if so, could you help me understand that intuition?
to me, ‘bijective correspondence’ feels like an earlier notion than counting, and counting feels like an earlier notion than multisets.