I’m posting this here because I find that I don’t get the feedback or discussion that I want in order to improve my ideas on Medium. So I hope that people leave comments here so we can discuss this further.
Personally, I’ve come across two other models of how humans intuitively compare infinities.
One of them is that humans use a notion of “density”. For example, positive multiples of three (3, 6, 9, 12, etc.) seem like a smaller set than all positive numbers (1, 2, 3, etc.). You could use the Bruce Framework here, but I think that what we’re actually doing something closer to evaluating the density of the sets. We notice that 3 and 6 and 9 are in both sets (similar to the Bruce Framework), but then we look to see how many numbers are between those “markers”. In the positive numbers set, there are 3 numbers between each “marker” (3, 4, 5 and then 6, 7, 8), whereas in the set of positive multiples of three there is only 1 number between each “marker” (3 and then we immediately go to 6). Thus, the cardinality of the positive numbers must be 3 times bigger than the cardinality of positive multiples of three.
If you expand this sort of thinking further, you get to a more “meta-model” of how humans intuitively compare sets, which is that we seem to build simple and easy functions to map items in one set to items in the other set. Sometimes the simple function is “are these inherently equal”, as in the Bruce Framework. Other times it’s “obvious” function like converting a negative number to a positive number. Once we have this mapping of “markers”, we then use density to compare the sizes of the two sets.
I’m not 100% sure if density is the only intuitive metric we use, but from the toy examples in my head it is. What are your thoughts? Are there any infinite sets (numbers or objects or anything) where your intuitive calculation doesn’t involve pairing up markers between the sets and then evaluating the density between those markers?
I’m posting this here because I find that I don’t get the feedback or discussion that I want in order to improve my ideas on Medium. So I hope that people leave comments here so we can discuss this further.
Personally, I’ve come across two other models of how humans intuitively compare infinities.
One of them is that humans use a notion of “density”. For example, positive multiples of three (3, 6, 9, 12, etc.) seem like a smaller set than all positive numbers (1, 2, 3, etc.). You could use the Bruce Framework here, but I think that what we’re actually doing something closer to evaluating the density of the sets. We notice that 3 and 6 and 9 are in both sets (similar to the Bruce Framework), but then we look to see how many numbers are between those “markers”. In the positive numbers set, there are 3 numbers between each “marker” (3, 4, 5 and then 6, 7, 8), whereas in the set of positive multiples of three there is only 1 number between each “marker” (3 and then we immediately go to 6). Thus, the cardinality of the positive numbers must be 3 times bigger than the cardinality of positive multiples of three.
If you expand this sort of thinking further, you get to a more “meta-model” of how humans intuitively compare sets, which is that we seem to build simple and easy functions to map items in one set to items in the other set. Sometimes the simple function is “are these inherently equal”, as in the Bruce Framework. Other times it’s “obvious” function like converting a negative number to a positive number. Once we have this mapping of “markers”, we then use density to compare the sizes of the two sets.
I’m not 100% sure if density is the only intuitive metric we use, but from the toy examples in my head it is. What are your thoughts? Are there any infinite sets (numbers or objects or anything) where your intuitive calculation doesn’t involve pairing up markers between the sets and then evaluating the density between those markers?