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Wednesday, May 03, 2023

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>First of all a pair of shoes is not a term, collective or singular. <

Unreasonably picky. Brightly clearly meant "a pair of shoes".

He probably did. But in a discussion which turns on subtleties, one must strive for precision of expression. The use-mention distinction must be observed.

If I am being unreasonably picky -- and perhaps I am -- you are unreasonably ignoring the substantive points I made.

Thanks, Bill, for the extended response. I am trying to nudge people away from thinking of sets as objects towards thinking of the notion of 'set' as an extension of the referential capacity of natural language. On my view '{1, 2, 3}' is a referring term that plurally refers to 1, 2, and 3. If I then say, 'Let S = {1,2,3}', I am introducing a new referring term, 'S', that also refers plurally to 1, 2, and 3. We would normally express this by saying 'S is the set containing exactly 1, 2, and 3', reinforcing the 'sets as objects' mode of thought.

When I say 'mathematical abstract objects have relations between their parts'I don't intend 'part' in the mereological sense. Take the requirements for a (number) field. Among other requirements, there has to be an addition operation and a multiplication operation and multiplication must distribute through addition. I think of these operations as 'parts' or 'aspects' of the field. There are no such 'parts' or 'aspects' required for a set.

One difficulty that you raise for this view is that math sets do seem to have a bit of structure. A set can have a set as an element in addition to ur-elements. But if 'the Hatfields and the McCoys' makes sense as a referring term in natural language, it seems we can make sense of '{{1, 2, 3}, {4, 5, 6}}' as a referring term in an artificially extended language. Set notation avoids the ambiguities of natural language plural referring terms that I drew attention to here.

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