3 ms·

> what distinguishes a from {a}? Heavens. Stopped there.

zozbot234 6y ago
It's not a bad question. Set theories answer that question somewhat differently, depending on whether they are 'structural' (like ETCS, SEAR etc.) or membership-based (like ZF(C) or NF).
- danielam 6y ago
  Here's a nice early 20th century example[0]. Lesniewski's hatred of the singleton (and set theory in general) is legendary. In his case, the motivation was his own nominalism. [0] https://plato.stanford.edu/entries/lesniewski/ https://plato.stanford.edu/entries/lesniewski/
uryga 6y ago
the sentence you quoted is prefixed by "set theory has no natural interpretation.". that's the problem. if you're trying to talk about herds of sheep, you might decide to represent them with sets – that sounds like what sets are for! but then is a single-sheep-set meaningfully different from an "unwrapped" sheep? how? in general the article seems to be talking about the issues with using set theory to talk about real-world stuff; it's not questioning the math.
- smadge 6y ago
  There is an obvious isomorphism between the set of all sheep and the singleton sets of sheep, so practically speaking it shouldn’t be much of a barrier.
- HelloNurse 6y ago
  Mathematics is about theories, philosophy is about describing reality. Theories that have little to do with reality are philosophically irrelevant, confusing and to be avoided despite being mathematically interesting.
sinkasapa 6y ago
It is at least a good reason to reject set theory for certain theoretical applications. Imagine you're a linguist trying to develop a theory of compositional natural language semantics. You want to have a good theory of what is entailed by a sentence of "Bill ate cheerios." You decide that pluralities are sets because that seems intuitive. Does that mean that when you say that Bill ate cheerios, he might have only eaten a singleton cheerio? That seems like the wrong prediction. It is clear that one cannot simply say that pluralities are sets, they are sets plus certain conditions, at least. This isn't only a philosophical exercise. This is about building a predictive model of natural language meanings where one doesn't want to make spurious predictions. The complexity of using sets to represent plurals, collections and groups quickly ramps up in this domain. This is a practical issue.

It's not a bad question. Set theories answer that question somewhat differently, depending on whether they are 'structural' (like ETCS, SEAR etc.) or membership-based (like ZF(C) or NF).

danielam 6y ago
Here's a nice early 20th century example[0]. Lesniewski's hatred of the singleton (and set theory in general) is legendary. In his case, the motivation was his own nominalism. [0] https://plato.stanford.edu/entries/lesniewski/ https://plato.stanford.edu/entries/lesniewski/

uryga 6y ago

the sentence you quoted is prefixed by "set theory has no natural interpretation.". that's the problem. if you're trying to talk about herds of sheep, you might decide to represent them with sets – that sounds like what sets are for! but then is a single-sheep-set meaningfully different from an "unwrapped" sheep? how? in general the article seems to be talking about the issues with using set theory to talk about real-world stuff; it's not questioning the math.

smadge 6y ago
There is an obvious isomorphism between the set of all sheep and the singleton sets of sheep, so practically speaking it shouldn’t be much of a barrier.
HelloNurse 6y ago
Mathematics is about theories, philosophy is about describing reality. Theories that have little to do with reality are philosophically irrelevant, confusing and to be avoided despite being mathematically interesting.

sinkasapa 6y ago

It is at least a good reason to reject set theory for certain theoretical applications. Imagine you're a linguist trying to develop a theory of compositional natural language semantics. You want to have a good theory of what is entailed by a sentence of "Bill ate cheerios." You decide that pluralities are sets because that seems intuitive. Does that mean that when you say that Bill ate cheerios, he might have only eaten a singleton cheerio? That seems like the wrong prediction. It is clear that one cannot simply say that pluralities are sets, they are sets plus certain conditions, at least. This isn't only a philosophical exercise. This is about building a predictive model of natural language meanings where one doesn't want to make spurious predictions. The complexity of using sets to represent plurals, collections and groups quickly ramps up in this domain. This is a practical issue.