Working with the set of real numbers (was: Finding size of Variable)

[Oscar Benjamin]

On 5 March 2014 17:43, Steven D'Aprano
<steve+comp.lang.python at pearwood.info> wrote:
> On Wed, 05 Mar 2014 12:21:37 +0000, Oscar Benjamin wrote:
>>
>> The argument that the sum of all natural numbers comes to -1/12 is just
>> some kind of hoax. I don't think *anyone* seriously believes it.
>
> You would be wrong. I suggest you read the links I gave earlier. Even the
> mathematicians who complain about describing this using the word "equals"
> don't try to dispute the fact that you can identify the sum of natural
> numbers with ζ(-1), or that ζ(-1) = -1/12. They simply dispute that we
> should describe this association as "equals".
>
> What nobody believes is that the sum of natural numbers is a convergent
> series that sums to -1/12, because it is provably not.
>
> In other words, this is not an argument about the maths. Everyone who
> looks at the maths has to admit that it is sound. It's an argument about
> the words we use to describe this. Is it legitimate to say that the
> infinite sum *equals* -1/12? Or only that the series has the value -1/12?
> Or that we can "associate" (talk about a sloppy, non-vigorous term!) the
> series with -1/12?

This is the point. You can "identify" numbers with many different
things. It does not mean to say that the thing is equal to that
number. I can associate the number 2 with my bike since it has 2
wheels. That doesn't mean that the bike is equal to 2.

So the problem with saying that "the sum of the natural numbers equals
-1/12" is precisely as you say with the word "equals" because they're
not equal!

If you restate the conclusion in more accurate (but technical and less
accessible) way that "the analytic continuation of a related set of
convergent series has the value -1/12 at the value that would
correspond to this divergent series" then it becomes less mysterious.
Do I really have to associate the finite negative value found in the
analytic continuation with the sum of the series that is provably
greater than any finite number?

<snip>
>
> At one time, Euler summed an infinite series and got -1, from which he
> concluded that -1 was (in some sense) larger than infinity. I don't know
> what justification he gave, but the way I think of it is to take the
> number line from -∞ to +∞ and then bend it back upon itself so that there
> is a single infinity, rather like the projective plane only in a single
> dimension. If you start at zero and move towards increasingly large
> numbers, then like Buzz Lightyear you can go to infinity and beyond:
>
> 0 -> 1 -> 10 -> 10000 -> ... ∞ -> ... -10000 -> -10 -> -1 -> 0
>
> In this sense, -1/12 is larger than infinity.

There are many examples that appear to show wrapping round from
+infinity to -infinity e.g. the tan function. The thing is that it is
not really "physical" (or meaningful in any direct sense).

So for example I might consider the forces on a particle, apply
Newton's 2nd law and arrive at a differential equation for the
acceleration of the particle, solve the equation and find that the
position of the particle at time t is given by tan(t). This would seem
to imply that as t increases toward pi/2 the particle heads off
infinity miles West but at the exact time pi/2 it wraps around to
reappear at infinity miles East and starts heading back toward its
starting point. The truth is less interesting: the solution tan(t)
becomes invalid at pi/2 and mathematics can tell us nothing about what
happens after that even if all the physics we used was exactly true.

> Now of course this is an ad hoc sloppy argument, but I'm not a
> professional mathematician. However I can tell you that it's pretty close
> to what the professional mathematicians and physicists do with negative
> absolute temperatures, and that is rigorous.
>
> http://en.wikipedia.org/wiki/Negative_temperature

The key point from that page is the sentence "A definition of
temperature can be based on the relationship...".  It is clear that
temperature is a theoretical abstraction. We have intuitive
understandings of what it means but in order for the current body of
thermodynamic theory to be consistent it is necessary to sometimes
give negative values to the temperature. There's nothing unintuitive
about negative temperatures if you understand the usual thermodynamic
definitions of "temperature".

>> Personally I think it's reasonable to just say that the sum of the
>> natural numbers is infinite rather than messing around with terms like
>> undefined, divergent, or existence. There is a clear difference between
>> a series (or any limit) that fails to converge  asymptotically and
>> another that just goes to +-infinity. The difference is usually also
>> relevant to any practical application of this kind of maths.
>
> And this is where you get it exactly backwards. The *practical
> application* comes from physics, where they do exactly what you argue
> against: they associate ζ(-1) with the sum of the natural numbers (see, I
> too can avoid the word "equals" too), and *it works*.

I don't know all the details of what they do there and whether or not
there's a better way of doing it or perhaps a better way of thinking
about the mathematical procedures they apply. (I'm assuming you're
talking about the Casimir effect here).

Let's use a more down to earth example though. Every day from now I'll
give you N pounds where N is the number of days from today. so
tomorrow I'll give you 1 pound, the next day 2 pounds and so on. If
this continues for an infinitely long time then you will have been
given an infinite amount of money. If you phrase the question like
this then I think the professional mathematicians you're referring to
will agree that the sum is infinite.

(It's possible that the money I said I'd send will not materialise. If
you receive a bill for 8 pence you'll know that I was wrong which
should console you for the missing infinite amounts of money).


Oscar