On Dec 29, 2019, at 18:50, Chris Angelico <ros...@gmail.com> wrote:
>
> On Mon, Dec 30, 2019 at 1:40 PM Andrew Barnert <abarn...@yahoo.com> wrote:
>>
>>> On Dec 29, 2019, at 18:20, Chris Angelico <ros...@gmail.com> wrote:
>>
>> Counting numbers are intuitively numbers. So are measures. And yet, they’re
>> different. Which one is the “one true numbers”? Who cares? Medieval
>> mathematicians did spend thousands of pages trying to resolve that question,
>> but it’s a lot more productive to just accept that the intuitive notion of
>> “number” is vague and instead come up with systematic ways to define and
>> compare and contrast and relate different algebras (not just those two).
>>
>
> That's what I said. You cannot use intuition to define numbers unless
> you're willing to restrict it to counting numbers.
I was agreeing with you in general, but I think the truth is even stronger than
you’re claiming.
You can also use intuition to define numbers if you’re willing to restrict it
to measures. In fact, Steven is doing so in at least half his arguments. And
anyone who tries to make their intuition rigorous by starting off with “numbers
are elements of fields that…” is using the measures intuition, not the counting
intuition. And that’s the real problem: even counting numbers are not “the
numbers”; we really do have conflicting intuitions, and there really is no way
around that.
> And the same is true of the debate about float("nan"). You cannot use
> intuition to figure out whether this is a number or not, because
> intuition has ALREADY failed us. "Commonsense tests" such as Steven
> put forward are not a valid way to debate the edge cases, because they
> fail on what we would consider clear cases.
Agreed.
There actually is an important place for common sense tests, but that place is
in coming up with new systems, not in deciding which system is “right”.
Obviously complex numbers are numbers. Obviously numbers can be ordered.
Complex numbers don’t have a natural order. Which one of those intuitions was
wrong? Neither; they’re both right, and therefore we just found a new way to
distinguish between two useful classes of “number” structures. We’re farther
from ever from knowing which things are “really numbers”, but who cares?
_______________________________________________
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/VJDFTXDR7Z3GUGVO3IUN4XDAV6IFOFUC/
Code of Conduct: http://python.org/psf/codeofconduct/
