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

[Steven D'Aprano]

On Tue, 04 Mar 2014 23:25:37 -0500, Roy Smith wrote:

> I stopped paying attention to mathematicians when they tried to convince
> me that the sum of all natural numbers is -1/12.  

I'm pretty sure they did not. Possibly a physicist may have tried to tell 
you that, but most mathematicians consider physicists to be lousy 
mathematicians, and the mere fact that they're results seem to actually 
work in practice is an embarrassment for the entire universe. A 
mathematician would probably have said that the sum of all natural 
numbers is divergent and therefore there is no finite answer.

Well, that is, apart from mathematicians like Euler and Ramanujan. When 
people like them tell you something, you better pay attention.

We have an intuitive understanding of the properties of addition. You 
can't add 1000 positive whole numbers and get a negative fraction, that's 
obvious. But that intuition only applies to *finite* sums. They don't 
even apply to infinite *convergent* series, and they're *easy*. Remember 
Zeno's Paradoxes? People doubted that the convergent series:

1/2 + 1/4 + 1/8 + 1/16 + ... 

added up to 1 for the longest time, even though they could see with their 
own eyes that it had to. Until they worked out what *infinite* sums 
actually meant, their intuitions were completely wrong. This is a good 
lesson for us all.

The sum of all the natural numbers is a divergent infinite series, so we 
shouldn't expect that our intuitions hold. We can't add it up as if it 
were a convergent series, because it's not convergent. Nobody disputes 
that. But perhaps there's another way?

Normally mathematicians will tell you that divergent series don't have a 
total. That's because often the total you get can vary depending on how 
you add them up. The classic example is summing the infinite series:

1 - 1 + 1 - 1 + 1 - ... 

Depending on how you group them, you can get:

(1 - 1) + (1 - 1) + (1 - 1) ...  
= 0 + 0 + 0 + ... = 0

or you can get:

1 - (1 - 1 + 1 - 1 + ... ) 
= 1 - (1 - 1) - (1 - 1) - ... )
= 1 - 0 - 0 - 0 ... 
= 1

Or you can do a neat little trick where we define the sum as "x":

x = 1 - 1 + 1 - 1 + 1 - ... 
x = 1 - (1 - 1 + 1 - 1 + ... )
x = 1 - x
2x = 1
x = 1/2


So at first glance, summing a divergent series is like dividing by zero. 
You get contradictory results, at least in this case.

But that's not necessarily always the case. You do have to be careful 
when summing divergent series, but that doesn't always mean you can't do 
it and get a meaningful answer. Sometimes you can, sometimes you can't, 
it depends on the specific series. With the sum of the natural numbers, 
rather than getting three different results from three different methods, 
mathematicians keep getting the same -1/12 result using various methods. 
That's a good hint that there is something logically sound going on here, 
even if it seems unintuitive.

Remember Zeno's Paradoxes? Our intuitions about equality and plus and 
sums of numbers don't apply to infinite series. We should be at least 
open to the possibility that while all the *finite* sums:

1 + 2
1 + 2 + 3
1 + 2 + 3 + 4
...

and so on sum to positive whole numbers, that doesn't mean that the 
*infinite* sum has to total to a positive whole number. Maybe that's not 
how addition works. I don't know about you, but I've never personally 
added up an infinite number of every-increasing quantities to see what 
the result is. Maybe it is a negative fraction. (I'd say "try it and 
see", but I don't have an infinite amount of time to spend on it.)

And in fact that's exactly what seems to be case here. Mathematicians can 
demonstrate an identity (that is, equality) between the divergent sum of 
the natural numbers with the zeta function ζ(-1), and *that* can be 
worked out independently, and equals -1/12.

So there are a bunch of different ways to show that the divergent sum 
adds up to -1/12, some of them are more vigorous than others. The zeta 
function method is about as vigorous as they come. The addition of an 
infinite number of things behaves differently than the addition of finite 
numbers of things.

More here:

http://scitation.aip.org/content/aip/magazine/physicstoday/news/10.1063/PT.5.8029

http://math.ucr.edu/home/baez/week126.html

http://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_%E2%8B%AF

and even here:

http://scientopia.org/blogs/goodmath/2014/01/20/oy-veh-power-series-analytic-continuations-and-riemann-zeta/

where a mathematician tries *really hard* to discredit the idea that the 
sum equals -1/12, but ends up proving that it does. So he simply plays a 
linguistic slight of hand and claims that despite the series and the zeta 
function being equal, they're not *actually* equal.

In effect, the author Mark Carrol-Chu in the "GoodMath" blog above wants 
to make the claim that the divergent sum is not equal to ζ(-1), but 
everywhere you find that divergent sum in your calculations you can rub 
it out and replace it with ζ(-1), which is -1/12. In other words, he's 
accepting that the divergent sum behaves *as if* it were equal to -1/12, 
he just doesn't want to say that it *is* equal to -1/12.

Is this a mere semantic trick, or a difference of deep and fundamental 
importance? Mark C-C thinks it's an important difference. Mathematicians 
who actually work on this stuff all the time think he's making a semantic 
trick to avoid facing up to the fact that sums of infinite sequences 
don't always behave like sums of finite sequences.



-- 
Steven