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 <ros...@gmail.com> 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-numbers 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 _______________________________________________ Python-ideas mailing list -- python-ideas@python.org To unsubscribe send an email to python-ideas-le...@python.org https://mail.python.org/mailman3/lists/python-ideas.python.org/ Message archived at https://mail.python.org/archives/list/python-ideas@python.org/message/6EEMI4UJBRG427A6Y6UQDSIBYG5OHARQ/ Code of Conduct: http://python.org/psf/codeofconduct/
