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> The “math looks nicer” argument [...] is, for example, the `reason' [...] there are as many integers as rationals. I'm curious what you mean here: The integ

by nimih 8y ago

> The “math looks nicer” argument [...] is, for example, the `reason' [...] there are as many integers as rationals. I'm curious what you mean here: The integers and rationals have quite different algebraic and geometric properties, and the fact that they have the same cardinality has always struck me as a deep result, rather than just a matter of definitions.

pvg 8y ago
Just occurred to me they were perhaps thinking of https://en.wikipedia.org/wiki/Continuum_hypothesis https://en.wikipedia.org/wiki/Continuum_hypothesis and mis-remembered/typed.
Someone 8y ago
I meant that mathematicians struggled with infinities for quite a while. Chances are they initially modified “two sets have the same size iff there’s a 1:1 mapping between items in the set” to “two *finite* sets have the same size iff there’s a 1:1 mapping between items in the set” before ‘bending’ the meaning of ‘same size’ a bit to allow a strict superset of another set to be of equal size as that set.
- deleted 8y ago
  [deleted]
- nimih 8y ago
  As far as I can tell, the idea of “same size means establishing a bijection” appears (in a formal sense) at essentially the same time as it was discovered that there are pairs of infinite collections where you can’t do that (i.e. with Dedekind and Cantor in the late 1800s), so I don’t think that second definition ever actually existed in a meaningful sense, although I think your intuition seems reasonable (albeit “bending” holds the wrong connotation, imo, as this would seem to be a rather principled generalization of an existing concept rather than an arbitrary decision among a number of somewhat reasonable options). Regardless, I’d be very interested to read a survey of the history of ideas about cardinality/ordinality if you have a good one, since modern treatments of these ideas are radically different than how humans understood them even 150 years ago.

by nimih 8y ago

pvg 8y ago

Just occurred to me they were perhaps thinking of https://en.wikipedia.org/wiki/Continuum_hypothesis https://en.wikipedia.org/wiki/Continuum_hypothesis and mis-remembered/typed.

Someone 8y ago

I meant that mathematicians struggled with infinities for quite a while. Chances are they initially modified “two sets have the same size iff there’s a 1:1 mapping between items in the set” to “two *finite* sets have the same size iff there’s a 1:1 mapping between items in the set” before ‘bending’ the meaning of ‘same size’ a bit to allow a strict superset of another set to be of equal size as that set.

deleted 8y ago
[deleted]
nimih 8y ago
As far as I can tell, the idea of “same size means establishing a bijection” appears (in a formal sense) at essentially the same time as it was discovered that there are pairs of infinite collections where you can’t do that (i.e. with Dedekind and Cantor in the late 1800s), so I don’t think that second definition ever actually existed in a meaningful sense, although I think your intuition seems reasonable (albeit “bending” holds the wrong connotation, imo, as this would seem to be a rather principled generalization of an existing concept rather than an arbitrary decision among a number of somewhat reasonable options). Regardless, I’d be very interested to read a survey of the history of ideas about cardinality/ordinality if you have a good one, since modern treatments of these ideas are radically different than how humans understood them even 150 years ago.