> One can do the thinking without specifying what. After one has figured out a rigorous approach, one can generally ignore it in day-to-day work. But you need

> One can do the thinking without specifying what. After one has figured out a rigorous approach, one can generally ignore it in day-to-day work. But you need to have that foundation there or you can go horribly wrong. (Analogy: we don't generally think about limits when doing day-to-day calculus. But when people tried to do calculus pre-Weierstrass, unless they were geniuses they "proved" all sorts of nonsense and got crazy results). > I believe this is the notion of a 'universal property' in category theory. That's one of the good answers to this problem. But you still need the constructions (if only to prove that an object with the required properties exists at all) and you need a lot of the kind of work the article is talking about, where you establish that these different constructions are equivalent and you can lift all your theorems along those equivalences. > I'm saying that zero is the concept of taking two things (one an initial thing and the other a method to take one thing and get the next thing) and then returning the initial thing. While three is the concept of taking the same two things and then doing the method three times on the first thing. I used lambda notation because that is an easy way to convey the idea, not because that is what three is. OK but then what kind of things are these "concepts"? What kind of questions can and can't we ask about them? If you care about "what the natural numbers are" then these kind of questions are worth asking.