Perhaps we should be defining numbers ostensively. In an ostensive way. We don’t need to prove natural numbers have foundations in anything other than themselves. They are not sets, they are not anything except what they are. We learn what the number 2 is in an ostensive way, as a baby, when an adult points towards the difference between 2 and 3 things.
you can do that if you want; pure mathematics, where foundations often lie, is more about what can be done than what should be done. however as always, there remain certain cases where the particular formalism of choice that you use ends up mattering for certain reasons. and in particular, for more and more abstract math, due to the growing nested complexity of proofs it may end up being useful to be *capable* of taking a foundational perspective (e.g. in modern algebraic geometry certain abstractions with underlying "sizes" may grow so large that it becomes useful to briefly switch to a discussion of foundations to clarify the precise meaning of your structures)
Super teacher 👍
Perhaps we should be defining numbers ostensively. In an ostensive way.
We don’t need to prove natural numbers have foundations in anything other than themselves. They are not sets, they are not anything except what they are.
We learn what the number 2 is in an ostensive way, as a baby, when an adult points towards the difference between 2 and 3 things.
you can do that if you want; pure mathematics, where foundations often lie, is more about what can be done than what should be done. however as always, there remain certain cases where the particular formalism of choice that you use ends up mattering for certain reasons. and in particular, for more and more abstract math, due to the growing nested complexity of proofs it may end up being useful to be *capable* of taking a foundational perspective (e.g. in modern algebraic geometry certain abstractions with underlying "sizes" may grow so large that it becomes useful to briefly switch to a discussion of foundations to clarify the precise meaning of your structures)
@@whatno5090 I'm the magic fairy controlling the universe. Plot twist, right?
@@noahkeys-asgill5664 i suppose so