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> Well, I think the idea that one can buy a set from a fruitier is somewhat absurd. Suppose a fruitier has only three apples left in stock – let us name them Ap

> Well, I think the idea that one can buy a set from a fruitier is somewhat absurd. Suppose a fruitier has only three apples left in stock – let us name them Apple1, Apple2, Apple3. Could I buy the set {Apple1, Apple2, Apple3} from the fruitier? Could I instead buy the set {{Apple1, Apple2}, {Apple1, Apple3}}? How would these two purchases differ? Is it possible to buy two apples or you can only buy “this” apple and “that” apple? How would these two purchases differ? I would say depends on how you define the structure and operationalize the problem. In your case the straightforward way would be to allow only for the first variant to make sense. > Many set theory texts try to justify set theory based on those simple examples while ignoring the more complex ones, and ignoring the alternative of mereology which claims to handle those more complex cases in a way which better respects common sense. I am not sure what you mean by “the more complex ones”, but the advancements in the set theory allowed to develop the measure theory, which is the basis of rigorous probability theory and statistics.

skissane 6y ago
> I am not sure what you mean by “the more complex ones” What I mean is that when you introduce nested sets (sets of sets, sets of sets of sets), and then allow repetition of elements in nested sets, it isn't clear what that means (if anything) if you think of sets as groups of physical objects. Mereology avoids this particular issue. > the advancements in the set theory allowed to develop the measure theory, which is the basis of rigorous probability theory and statistics. Well, there are two different questions about set theory (1) is it an accurate model of everyday human thinking about grouping objects? (2) is it useful as a foundation for developing various useful mathematical theories? A lot of defenders of set theory assume the answer to both questions is "Yes", or even fail to clearly keep the two questions clearly distinguished. The correct answers could well be "No" and "Yes".
- randallsquared 6y ago
  > it isn't clear what that means (if anything) if you think of sets as groups of physical objects. Isn't this similar to the idea that you can count apples or birds: one, two, three, four, etc, but there's nothing countably physical about infinity, or even arbitrarily large finite numbers. That doesn't impede the usefulness of infinity as an abstraction of counting, in a similar way to the fact that sets are useful as an abstraction of treating groups of things as a thing (and, in fact, we have to treat groups of things as a thing -- most of the physical things we are familiar with have fuzzy boundaries and definitions, because they're all groupings of smaller things).
- LudwigNagasena 6y ago
  >What I mean is that when you introduce nested sets (sets of sets, sets of sets of sets), and then allow repetition of elements in nested sets, it isn't clear what that means (if anything) if you think of sets as groups of physical objects. Mereology avoids this particular issue. It avoids it by cost of being useless for any serious mathematical endeavor (mereology is essentially a complete Boolean algebra without a zero element). Yes, probably compactness and differentiability is not something clear and easy to grasp, especially when you have groups of physical objects in mind, but they are very useful constructs grounded in modern set theory. I can't comment on how useful set theory or mereology is in reagrds to classical Western ontology, but I have no reason to doubt that many people in humanities may abuse and misuse mathematical and scientific apparatus. >(1) is it an accurate model of everyday human thinking about grouping objects? (2) is it useful as a foundation for developing various useful mathematical theories? A lot of defenders of set theory assume the answer to both questions is "Yes", or even fail to clearly keep the two questions clearly distinguished. The correct answers could well be "No" and "Yes". I think what you perceive as "failing to clearly keep the two questions clearly distinguished" may simply be a misinterpration of deep disinterest in the question, I think most mathematicians wouldn't perceive the question of "how everyday human thinking works" to be in the realm of math. Trying to tie in math with metaphysics fell out of favor since times of Gödel.
  - skissane 6y ago
    > I think what you perceive as "failing to clearly keep the two questions clearly distinguished" may simply be a misinterpration of deep disinterest in the question, I think most mathematicians wouldn't perceive the question of "how everyday human thinking works" to be in the realm of math. Well, how "everyday thinking works" is very much related to the realm of math in my mind. Most commonly classical logic is selected as a foundation for mathematics, although there has been some work done on alternative foundations (most significantly constructivism/intuitionism, although there are less notable projects trying to build out mathematics on yet other foundations.) But, classical logic is commonly criticised by philosophers as being a poor model of everyday human thought, as represented by issues such as the paradoxes of material implication. So, alternative logics get proposed which attempt to answer those criticisms – for example, relevant/relevance logic. The study of these alternative logical formalisms is itself part of mathematics (mathematical logic, proof theory, etc), and the question of whether any of those alternative logical formalisms can be used as a foundation to develop other parts of mathematics (such as analysis) is an interesting mathematical question. And, added to all that, studying alternative logics is also interesting from the viewpoint of their possible practical applications in computer science (in fields such as automated reasoning.) On the topic of set theory, while most mathematics assumes ZFC, there has been a lot of work on alternative set theories [1]. For each of these alternatives, we can ask (a) the philosophical question of whether it does a better job of modelling naïve human thought than ZFC does; (b) the metamathematical question of how easy it is to build out the rest of mathematics on that foundation; (c) the question of whether the theory has any useful applications in other fields such as computer science. All three questions are interesting, and they are all interconnected. [1] https://en.wikipedia.org/wiki/Alternative_set_theory https://en.wikipedia.org/wiki/Alternative_set_theory

skissane 6y ago

> I am not sure what you mean by “the more complex ones” What I mean is that when you introduce nested sets (sets of sets, sets of sets of sets), and then allow repetition of elements in nested sets, it isn't clear what that means (if anything) if you think of sets as groups of physical objects. Mereology avoids this particular issue. > the advancements in the set theory allowed to develop the measure theory, which is the basis of rigorous probability theory and statistics. Well, there are two different questions about set theory (1) is it an accurate model of everyday human thinking about grouping objects? (2) is it useful as a foundation for developing various useful mathematical theories? A lot of defenders of set theory assume the answer to both questions is "Yes", or even fail to clearly keep the two questions clearly distinguished. The correct answers could well be "No" and "Yes".

randallsquared 6y ago
> it isn't clear what that means (if anything) if you think of sets as groups of physical objects. Isn't this similar to the idea that you can count apples or birds: one, two, three, four, etc, but there's nothing countably physical about infinity, or even arbitrarily large finite numbers. That doesn't impede the usefulness of infinity as an abstraction of counting, in a similar way to the fact that sets are useful as an abstraction of treating groups of things as a thing (and, in fact, we have to treat groups of things as a thing -- most of the physical things we are familiar with have fuzzy boundaries and definitions, because they're all groupings of smaller things).
LudwigNagasena 6y ago
>What I mean is that when you introduce nested sets (sets of sets, sets of sets of sets), and then allow repetition of elements in nested sets, it isn't clear what that means (if anything) if you think of sets as groups of physical objects. Mereology avoids this particular issue. It avoids it by cost of being useless for any serious mathematical endeavor (mereology is essentially a complete Boolean algebra without a zero element). Yes, probably compactness and differentiability is not something clear and easy to grasp, especially when you have groups of physical objects in mind, but they are very useful constructs grounded in modern set theory. I can't comment on how useful set theory or mereology is in reagrds to classical Western ontology, but I have no reason to doubt that many people in humanities may abuse and misuse mathematical and scientific apparatus. >(1) is it an accurate model of everyday human thinking about grouping objects? (2) is it useful as a foundation for developing various useful mathematical theories? A lot of defenders of set theory assume the answer to both questions is "Yes", or even fail to clearly keep the two questions clearly distinguished. The correct answers could well be "No" and "Yes". I think what you perceive as "failing to clearly keep the two questions clearly distinguished" may simply be a misinterpration of deep disinterest in the question, I think most mathematicians wouldn't perceive the question of "how everyday human thinking works" to be in the realm of math. Trying to tie in math with metaphysics fell out of favor since times of Gödel.
- skissane 6y ago
  > I think what you perceive as "failing to clearly keep the two questions clearly distinguished" may simply be a misinterpration of deep disinterest in the question, I think most mathematicians wouldn't perceive the question of "how everyday human thinking works" to be in the realm of math. Well, how "everyday thinking works" is very much related to the realm of math in my mind. Most commonly classical logic is selected as a foundation for mathematics, although there has been some work done on alternative foundations (most significantly constructivism/intuitionism, although there are less notable projects trying to build out mathematics on yet other foundations.) But, classical logic is commonly criticised by philosophers as being a poor model of everyday human thought, as represented by issues such as the paradoxes of material implication. So, alternative logics get proposed which attempt to answer those criticisms – for example, relevant/relevance logic. The study of these alternative logical formalisms is itself part of mathematics (mathematical logic, proof theory, etc), and the question of whether any of those alternative logical formalisms can be used as a foundation to develop other parts of mathematics (such as analysis) is an interesting mathematical question. And, added to all that, studying alternative logics is also interesting from the viewpoint of their possible practical applications in computer science (in fields such as automated reasoning.) On the topic of set theory, while most mathematics assumes ZFC, there has been a lot of work on alternative set theories [1]. For each of these alternatives, we can ask (a) the philosophical question of whether it does a better job of modelling naïve human thought than ZFC does; (b) the metamathematical question of how easy it is to build out the rest of mathematics on that foundation; (c) the question of whether the theory has any useful applications in other fields such as computer science. All three questions are interesting, and they are all interconnected. [1] https://en.wikipedia.org/wiki/Alternative_set_theory https://en.wikipedia.org/wiki/Alternative_set_theory