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

Steven D'Aprano steve+comp.lang.python at pearwood.info
Wed Mar 5 18:43:02 CET 2014


On Wed, 05 Mar 2014 12:21:37 +0000, Oscar Benjamin wrote:

> On 5 March 2014 07:52, Steven D'Aprano <steve at pearwood.info> 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.
>>
>> 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.
> 
> Why the dig at physicists? 

There is considerable professional rivalry between the branches of 
science. Physicists tend to look at themselves as the paragon of 
scientific "hardness", and look down at mere chemists, who look down at 
biologists. (Which is ironic really, since the actual difficulty in doing 
good science is in the opposite order. Hundreds of years ago, using quite 
primitive techniques, people were able to predict the path of comets 
accurately. I'd like to see them predict the path of a house fly.) 
According to this "greedy reductionist" viewpoint, since all living 
creatures are made up of chemicals, biology is just a subset of 
chemistry, and since chemicals are made up of atoms, chemistry is 
likewise just a subset of physics.

Physics is the fundamental science, at least according to the physicists, 
and Real Soon Now they'll have a Theory Of Everything, something small 
enough to print on a tee-shirt, which will explain everything. At least 
in principle.

Theoretical physicists who work on the deep, fundamental questions of 
Space and Time tend to be the worst for this reductionist streak. They 
have a tendency to think of themselves as elites in an elite field of 
science. Mathematicians, possibly out of professional jealousy, like to 
look down at physics as mere applied maths.

They also get annoyed that physicists often aren't as vigorous with their 
maths as they should be. The controversy over renormalisation in Quantum 
Electrodynamics (QED) is a good example. When you use QED to try to 
calculate the strength of the electron's electric field, you end up 
trying to sum a lot of infinities. Basically, the interaction of the 
electron's charge with it's own electric field gets larger the more 
closely you look. The sum of all those interactions is a divergent 
series. So the physicists basically cancelled out all the infinities, and 
lo and behold just like magic what's left over gives you the right 
answer. Richard Feynman even described it as "hocus-pocus".

The mathematicians *hated* this, and possibly still do, because it looks 
like cheating. It's certainly not vigorous, at least it wasn't back in 
the 1940s. The mathematicians were appalled, and loudly said "You can't 
do that!" and the physicists basically said "Oh yeah, watch us!" and 
ignored them, and then the Universe had the terribly bad manners to side 
with the physicists. QED has turned out to be *astonishingly* accurate, 
the most accurate physical theory of all time. The hocus-pocus worked.


> I think most physicists would be able to tell
> you that the sum of all natural numbers is not -1/12. In fact most
> people with very little background in mathematics can tell you that.

Ah, but there's the rub. People with *very little* background in 
mathematics will tell you that. People with *a very deep and solid* 
background in mathematics will tell you different, particularly if their 
background is complex analysis. (That's *complex numbers*, not 
"complicated" -- although it is complicated too.)


> 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?


>> Well, that is, apart from mathematicians like Euler and Ramanujan. When
>> people like them tell you something, you better pay attention.
> 
> Really? Euler didn't even know about absolutely convergent series (the
> point in question) and would quite happily combine infinite series to
> obtain a formula.

(I note that you avoided criticising Ramanujan's work. Very wise.)

Euler was working on infinite series in the 1700s. There's no doubt that 
his work doesn't meet modern standards of mathematical rigour, but those 
modern standards didn't exist back then. Morris Kline writes of Euler:

    Euler's work lacks rigor, is often ad hoc, and contains blunders, 
    but despite this, his calculations reveal an uncanny ability to 
    judge when his methods might lead to correct results.

http://dept.math.lsa.umich.edu/~krasny/math156_Euler-Kline.pdf

Euler certainly deserves to be in the pantheon of maths demigods, 
possibly the greatest mathematician who ever lived. There is a quip made 
that discoveries in mathematics are usually named after Euler, or the 
first person to discover them after Euler.

Euler also wrote that one should not use the term "sum" to describe the 
total of a divergent series, since that implies regular addition, but 
that one can say that when a divergent series comes from an algebraic 
expression, then the value of the series is the value of the expression 
from which is came. Notice that he carefully avoids using the word 
"equals". (See above URL.)

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.

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



[...]
> 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*.



-- 
Steven D'Aprano
http://import-that.dreamwidth.org/



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