On Mon, Dec 30, 2019 at 3:52 PM Andrew Barnert <abarn...@yahoo.com> wrote:
>
> 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.
>
Gotcha. And yeah, it's hard to come up with any sort of intuition
about numbers that doesn't break down *somewhere*. I mean, what does
"equal" mean? Is it true that a 100mm stick is "equal" in length to
another 100mm stick? And if you cut a stick 100mm long, then cut
another stick the same length as it, and keep taking the most recent
stick and cutting another stick the same length, are they all equal?
(If you say "yes", then I encourage you to try it.) Similarly, is it
valid to say that the infinite sum 1+2+4+8+16.... is "equal" to -1?
> > 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?
>
3blue1brown discussed this as regards the term "vector". What IS a
vector? Is it an ordered tuple of numbers representing N-dimensional
coordinates? A point in N-dimensional space? An arrow pointing from
somewhere to somewhere else? Or is it, instead, "a thing that you can
scale by a unitless value, and which you can add to another vector to
produce a vector"?
But the question is: What should the statistics module do with a nan?
This can only be answered by understanding what "nan" means in a
statistical context, which is only tangentially related to the
question of whether nan is a number. And I think this thread has
proven that there is no single obvious meaning for a nan. It could
mean missing data. It could mean out-of-bounds or illogical data. It
could mean something different entirely.
ChrisA
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