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

[Steven D'Aprano]

Following up on my own post.

On Wed, 05 Mar 2014 07:52:01 +0000, Steven D'Aprano wrote:

> 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.
[...]
> 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.

Here's another mathematician who is even more explicit about what she's 
complaining about:

http://blogs.scientificamerican.com/roots-of-unity/2014/01/20/is-the-sum-of-positive-integers-negative/

    [quote]
    There is a meaningful way to associate the number -1/12 to the 
    series 1+2+3+4…, but in my opinion, it is misleading to call 
    it the sum of the series.
    [end quote]

Evelyn Lamb's objection isn't about the mathematics that leads to the 
conclusion that the sum of natural numbers is equivalent to -1/12. That's 
conclusion is pretty much bulletproof. Her objection is over the use of 
the word "equals" to describe that association. Or possibly the use of 
the word "sum" to describe what we're doing when we replace the infinite 
series with -1/12.

Whatever it is that we're doing, it doesn't seem to have the same 
behavioural properties as summing finitely many finite numbers. So 
perhaps she is right, and we shouldn't call the sum of a divergent series 
a sum?


-- 
Steven