This is well written, but I honestly got the feeling that there is nothing worth talking about here. What is the number 4? Easy, 4 is {{}{{}}{{}{{}}}{{}{{}}{{}{{}}}}}. The only thing left to decide is “what is the empty set,” to which the answer is “the unique a such that ∀b: ¬(b ϵ a)”. And I understand how the system which defines what those things mean works… which is just based on defintions and axioms. Maybe this is stupid, but I don’t feel any need to go deeper, and I don’t feel confused about numbers. Set theory has already provided me with all the answers I want.
I think there is something left to be said about how “there exists a city larger than paris” could also be modeled in set theory and ultimately corresponds to a logical formula ranging over quanutm fields – or rather, a set of formulas which may have different truth values because we don’t have a perfect mapping from natural language to formal language. But that’s more of a different topic.
Rather than trying to immediately jump to what things are, I strongly recommend trying to understand why people believe what they do about it. I can claim that 2 is {{}{{}}} or {{}{}} or SS0, but this doesn’t help much more than just calling it 2 - it’s just substituting a symbol for another symbol. What’s needed is to understand the function of 2 in peoples’ thoughts.
Yeah, the statement about cities could be formulated much more mathematically. But ultimately, it’s going to have to refer to empirical facts about the universe in order to be evaluated.
This is well written, but I honestly got the feeling that there is nothing worth talking about here. What is the number 4? Easy, 4 is {{}{{}}{{}{{}}}{{}{{}}{{}{{}}}}}. The only thing left to decide is “what is the empty set,” to which the answer is “the unique a such that ∀b: ¬(b ϵ a)”. And I understand how the system which defines what those things mean works… which is just based on defintions and axioms. Maybe this is stupid, but I don’t feel any need to go deeper, and I don’t feel confused about numbers. Set theory has already provided me with all the answers I want.
I think there is something left to be said about how “there exists a city larger than paris” could also be modeled in set theory and ultimately corresponds to a logical formula ranging over quanutm fields – or rather, a set of formulas which may have different truth values because we don’t have a perfect mapping from natural language to formal language. But that’s more of a different topic.
The formal set definition is only one of many ways of defining the number 4.
Rather than trying to immediately jump to what things are, I strongly recommend trying to understand why people believe what they do about it. I can claim that 2 is {{}{{}}} or {{}{}} or SS0, but this doesn’t help much more than just calling it 2 - it’s just substituting a symbol for another symbol. What’s needed is to understand the function of 2 in peoples’ thoughts.
Yeah, the statement about cities could be formulated much more mathematically. But ultimately, it’s going to have to refer to empirical facts about the universe in order to be evaluated.