On Mon, Oct 14, 2019 at 12:54:13AM -0700, Andrew Barnert via Python-ideas wrote:
On Oct 13, 2019, at 22:54, Chris Angelico
wrote: Mathematically, what's the difference between '1' and '1+0j' (or '1+0i')?
The details depend on what foundations you use, but let’s go with the most common construction.
More importantly, the details depend on where you sit on the Practicality vs Purity axis. I think it is important to state that most working mathematicians would have fallen asleep as soon as you mention "foundations".^1 I think the best answer (note plural) to the question of whether the Integers are a subset of the Reals can be paraphrased from this thread: https://math.stackexchange.com/questions/14828/set-theoretic-definition-of-n... Yes. But no. When we say that the Integers form a subset of the Reals, what we really mean is that there exists a subset of the reals that is isomorphic to ("exactly the same in every way that matters") to the integers. Distinguishing between that "image under the canonical embedding" and a true subset becomes tiresome very quickly, so anyone who isn't mad about foundational issues quickly drops the distinction. (I've paraphrased a number of different people in that discussion.) Mathematicians don't typically talk about "duck-typing", but if they did, they'd be saying that the Integers quack like a subset of the Reals no matter what those weird guys who care about foundational issues say :-) ^1 My cousin is a mathematics professor, ex of Harvard and Yale, currently at Rutgers. I once asked her opinion on Gödel's incompleteness theorems, and her response was more or less "Who?". At the time she was working in a field where the foundational issues raised by Gödel simply had no impact. Mathematics is a HUGE area of knowledge, and it is possible to have a long and successful career without once being the least bit concerned about the difference between Integers and Reals, beyond the obvious stuff they teach in secondary schools. -- Steven