On 12/30/19 2:10 AM, Chris Angelico wrote:
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?
Perhaps the intuitive understanding of numbers includes a finite but
perhaps arbitrary precision, but if anything that makes it a better test
for what is a number in the case of the float type, since it is a finite
approximation. In the case of comparing two 100mm sticks, conceptually,
it is simple, align one end of them together, then look at the other end
and see which is longer. Intuitively simple. Yes, if we actually try to
do this, we are constrained by the limits of precision of out senses,
perhaps augmented by tools, and by the quantization of reality. Perhaps
it is better to say that the mathematical concept of 'Real Numbers' is
what is off, and no such thing can actually exist
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
My first answer is that to statistics in general, NaN is meaningless.
statistics as a Science, deals in the domain of Real Numbers (as in
Numbers that really exist), it is a Science to help us measure and
understand reality from those measurements. As such, a value like 'NaN'
has no meaning.
Historically, based on the history of using languages like Fortran, some
statistical packages have used it as a convenient indicator for no data.
Originally, to handle this case, an application would use some absurdly
out of bounds value for this, to allow the package to filter them out
later. Due to language constraints, you had to use some floating point
value as this marker. When the IEEE format came out, and had this
strange value, it became a somewhat natural choice, as not being a
number, it couldn't really be part of the statistics. This did break the
ability to use the generation of NaNs as a way to detect calculation
errors, but statisticians were used to having to do analysis ahead of
time to avoid those avoidable loss of precision that NaN caught when the
hit the extremums, so losing its diagnostic value wasn't that bad, and
you could still test for them being generated before adding them to the
list to confirm things.
Personally, I don't think the simple statistics package needs to support
this usage, as if you really want to use it, you really should
understand the meaning of the NaN, and how to program with it, and can
the either handle it yourself before passing to the statistics package,
or use a higher grade package that is built with that interpretation
built in. It also likely gives better control over how the values are
computed.
--
Richard Damon
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