Finding size of Variable

[Roy Smith]

I cannot find the exact quote so from memory Weyl says something to this
effect:

Cantor's diagonalization PROOF is not in question.
Its CONCLUSION very much is.
The classical/platonic mathematician (subject to wooly thinking) concludes
that
the real numbers are a superset of the integers

The constructvist mathematician (who supposedly thinks clearly) only
concludes
the obvious, viz that real numbers cannot be enumerated

To go from 'cannot be enumerated' to 'is a proper superset of' requires the
assumption of 'completed infinities' and that is not math but theology

I stopped paying attention to mathematicians when they tried to convince
me that the sum of all natural numbers is -1/12. Sure, you can
manipulate the symbols in a way which is consistent with some set of
rules that we believe govern the legal manipulation of symbols, but it
just plain doesn't make sense.

 

[Rustom Mody]

Roy Smith writes:

I stopped paying attention to a particular person when they said "I
stopped paying attention to an entire field of study because one
position expressed by some practicioners was disagreeable to me".

In general this is a correct response
In this particular case (apart from Roy speaking tongue-in-cheek)
it (Roy's viewpoint) is more appropriate and central to our field than you
perhaps realize:

Nonsensical results believed in by a small minority (Cantor's time)
became full scale war between platonists (Hilbert) and constructivists
(Brouwer) a generation later.

Gödel staunchly in Hilbert camp made his incompleteness theorem to rebut the
constructivists

Turing unable to disagree with Gödel's result but disagreeing with platonic
philosophy made his 'machine'. The negative result that he did not like but
had to admit was uncomputability/undecidability. However he trumped Gödelin
making a 'universal' machine

And so we are here

 

[Roy Smith]

I stopped paying attention to a particular person when they said “I
stopped paying attention to an entire field of study because one
position expressed by some practicioners was disagreeable to meâ€.

Would you think “I stopped listening to logicians when some of them
expressed Zeno's paradox of the impossibility of motion†to be a good
justification for ignoring the entire field of logic?

Rather, a more honest response is to say why that position is incorrect,
and not dismiss the entire field of study merely for a disagreement with
that position.

I *was* partly joking (but only partly).

Still, there's lots of stuff mathematicians do which I don't understand.
I cannot understand, for example, Andrew's Wiles's proof of Fermat's
Last Theorm. I can't even get past the first few paragraphs of the
Wikipedia article. But, that doesn't sour me on the proof. I can
accept that there are things I don't understand. I don't know how to
speak Chinese. I don't know how to paint a flower. I don't know how to
run a mile in 4 minutes. But I accept that there are people who do know
how to do those things.

I can watch a friend pick up a piece of paper, a brush, and some
watercolors and 5 minutes later, she's got a painting of a flower. I
watched her hands hold the brush and move it over the paper. There's
nothing mystical about what she did. Her hands made no motions which
are fundamentally impossible for my hands to make, yet I know that my
attempt at reproducing her work would not result in a painting of a
flower.

But, as I watch the -1/12 proof unfold, I don't get the same feeling. I
understand every step. I wouldn't have thought to manipulate the
symbols that way, but once I've seen it done, I can reproduce the steps
myself. It's all completely understandable. The only problem is, it
results in a conclusion which makes no sense. I can *prove* that it
makes no sense, by manipulating the symbols in different ways. The sum
of any two positive numbers must be positive. I can group them and add
them up any way I want and that's still true.

But, here I've got some guy telling me it's not true. If you just slide
this over that way, and add these parts up this way, it's -1/12. That
does not compute. But it doesn't not compute in the sense of, "that's
so complicated, I have no idea what you did", but in the sense of "thats
so simple, I know exactly what you did, and it's bullshit"

 

[Steven D'Aprano]

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_+_⋯

and even here:

http://scientopia.org/blogs/goodmat...ries-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 D'Aprano]

Following up on my own post.

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.../20/is-the-sum-of-positive-integers-negative/

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?

 

[wxjmfauth]

Mathematics?
The Flexible String Representation is a very nice example
of a mathematical absurdity.

jmf

PS Do not even think to expect to contradict me. Hint:
sheet of paper and pencil.

 

[Ned Batchelder]

Mathematics?
The Flexible String Representation is a very nice example
of a mathematical absurdity.

jmf

PS Do not even think to expect to contradict me. Hint:
sheet of paper and pencil.

Reminder to everyone: JMF makes no sense when he talks about the FSR,
and absurdly seems to think hinting at paper and pencil will convince us
he is right. Don't engage with him on this topic.

 

[Oscar Benjamin]

One problem with complexity claims is that it's easy to miss some
contributing time eaters. I haven't done any measuring on modern
machines nor in python, but I'd assume that multiplies take
*much* longer for large integers, and that divides are much
worse. So counting iterations isn't the whole story.

Agreed but there's a big difference between log(N) iterations and N iterations!

On the assumption that division by 2 is very fast, and that a
general multiply isn't too bad, you could improve on Newton by
observing that the slope is 2.

err = n - guess * guess
guess += err/2

I gues you mean like this:

def sqrt(n):
err = guess = 1
while err > 1e-10:
err = n - guess * guess
guess += err/2
return guess

This requires log2(10)*N iterations to get N digits. So the penalty
for using division would have to be extreme in order for this to
better. Using Decimal to get many digits we can write that as:

def sqrt2(n, prec=1000):
'''Solve x**2 = y'''
eps = D(10) ** -(prec + 5)
err = guess = D(1)
with localcontext() as ctx:
ctx.prec = prec + 10
while abs(err) > eps:
err = n - guess*guess
guess += err/2
return guess

This method works out much slower than Newton with division at 10000
digits: 40s (based on a single trial) vs 80ms (timeit result).

Some 37 years ago I microcoded a math package which included
square root. All the math was decimal, and there was no hardware
multiply or divide. The algorithm I came up with generated the
answer one digit at a time, with no subsequent rounding needed.
And it took just a little less time than two divides. For that
architecture, Newton's method would've been too
slow.

If you're working with a fixed small precision then it might be.

Incidentally, the algorithm did no divides, not even by 2. No
multiplies either. Just repeated subtraction, sorta like divide
was done.

If anyone is curious, I'll be glad to describe the algorithm;
I've never seen it published, before or since. I got my
inspiration from a method used in mechanical, non-motorized,
adding machines. My father had shown me that approach in the
50's.

I'm curious.


Oscar

 

[Mark Lawrence]

Why the dig at physicists? 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.

I'll put that one to the test tomorrow morning when the bin men come
round. I fully expect them to dial 999 and ask that the paramedics are
armed with plenty of sedatives.

 

[Dave Angel]

Agreed but there's a big difference between log(N) iterations and N iterations!


I gues you mean like this:

def sqrt(n):
err = guess = 1
while err > 1e-10:
err = n - guess * guess
guess += err/2
return guess

This requires log2(10)*N iterations to get N digits.

No idea how you came up with that, but I see an error in my
stated algorithm, which does surely penalize it. The slope isn't
2, but 2x. So the line should have been
guess += err/(2*guess)

Now if you stop the loop after 3 iterations (or use some other
approach to get a low-precision estimate, then you can calculate
scale = 1/(2*estimate)

and then for remaining iterations,
guess += err *scale

So the penalty
for using division would have to be extreme in order for this to
better. Using Decimal to get many digits we can write that as:

def sqrt2(n, prec=1000):
'''Solve x**2 = y'''
eps = D(10) ** -(prec + 5)
err = guess = D(1)
with localcontext() as ctx:
ctx.prec = prec + 10
while abs(err) > eps:
err = n - guess*guess
guess += err/2
return guess

This method works out much slower than Newton with division at 10000
digits: 40s (based on a single trial) vs 80ms (timeit result).

Well considering you did not special-case the divide by 2, I'm not
surprised it's slower.

If you're working with a fixed small precision then it might be.


I'm curious.

A later message, I guess. I can't write that much on the tablet.

 

[Dave Angel]

A later message, I guess. I can't write that much on the tablet.

Given a microcodable architecture with no hardware support for multiply or divide, clearly multiply will be several times as fast as divide (at least).  There was a BCD ALU, so add and subtract of decimal values was quite reasonable.  All floating point logic, however, is just microcode.

Divide is implemented via repeated subtraction of the divisor from
the dividend.  The count of how many subtracts is done is the
quotient. Naturally, this is combined with digit shifts, so you
find one quotient digit at a time.  For a 13 digit result, the
maximum subtracts are 13*10.

Multiply is much faster, as you know ahead of time for each column
how many adds you're supposed to do.  So you can have
precalculated multiples of the divisor on hand, and you can
subtract instead of add when appropriate.

Square root is implemented as a kind of variable division, where
the "divisor" is changing constantly.  Everyone knows that the
sum of the first n odd numbers is n squared.  So if you started
with a square, you could repeatedly subtract odd numbers from it
till you reached zero, and the square root will be roughly half
the last odd number subtracted.

So to make this work across multiple columns it turns out you can
accumulate these odd numbers, doing the appropriate shifts after
each column, and if you take the last number shifted, you can
just add 1 and divide by 2.

In many architectures, that would be as far as you can go, but in
the particular one I was using, generating those pesky odd
numbers was more expensive than you'd expect.  So it turned out
to be quicker to just do twice as many subtracts.

Instead of subtracting 1,3, 5, etc., till the value went negative,
we subtract 0 and 1, 1 and 2, 2 and 3, etc.  You have twice as
many subtracts, but no divide by 2 at the end.  And for each
column, you need not go beyond 8 + 9, since if it were more than
that, we would have picked it up in the previous column.  So you
do not have to propagate the carry across the trial
divisor.

Supposing the correct result will be 7.1234567, you will at one
stage of operations, be subtracting
       71230
       71231
       71231
       71232
       71232
       71233
       71233
       71234
The next subtract will make the result go negative, so you either
do it, detect negative and undo it, or you do some compare
operation.


I am here glossing over all the details of normalizing the
dividend so the exponent is even, and calculating the final
exponent, which at first approximation is half the original
one.

 

[Steven D'Aprano]

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?

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]

I'll put that one to the test tomorrow morning when the bin men come
round.

Do you seriously think that garbos (bin men) know more about mathematics
than mathematicians?

I fully expect them to dial 999 and ask that the paramedics are
armed with plenty of sedatives.

You know that rather large piece of machinery in Europe called the Large
Hadron Collider? The one which is generating some rather extraordinary
proofs of fundamental physics, such as the Higgs Boson? A lot of that
physics is based on theory which uses the same logic and mathematics that
you are mocking.

Laugh away, but the universe behaves as if the sum of the natural numbers
is -1/12.

 

[Chris Angelico]

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.

Everything is, except what isn't.

That's my theory, and I'm sticking to it!

ChrisA

 

[Chris Kaynor]

On Wed, Mar 5, 2014 at 9:43 AM, Steven D'Aprano <

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

This makes me think that maybe the universe is using ones or two complement
math (is there a negative zero?)...

Chris

 

[Grant Edwards]

On Wed, Mar 5, 2014 at 9:43 AM, Steven D'Aprano <


This makes me think that maybe the universe is using ones or two complement
math (is there a negative zero?)...

If the Universe (like most all Python implementations) is using
IEEE-754 floating point, there is.

 

[Oscar Benjamin]

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?

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".

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

 

[Roy Smith]

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.

A mathematician, a chemist, and a physicist are arguing the nature of
prime numbers. The chemist says, "All odd numbers are prime. Look, I
can prove it. Three is prime. Five is prime. Seven is prime". The
mathematician says, "That's nonsense. Nine is not prime". The
physicist looks at him and says, "Hmmmm, you may be right, but eleven
is prime, and thirteen is prime. It appears that within the limits of
experimental error, all odd number are indeed prime!"

 

[Steven D'Aprano]

A mathematician, a chemist, and a physicist are arguing the nature of
prime numbers. The chemist says, "All odd numbers are prime. Look, I
can prove it. Three is prime. Five is prime. Seven is prime". The
mathematician says, "That's nonsense. Nine is not prime". The
physicist looks at him and says, "Hmmmm, you may be right, but eleven is
prime, and thirteen is prime. It appears that within the limits of
experimental error, all odd number are indeed prime!"

They ask a computer programmer to adjudicate who is right, so he writes a
program to print out all the primes:

1 is prime
1 is prime
1 is prime
1 is prime
1 is prime
....

 

[Chris Angelico]

They ask a computer programmer to adjudicate who is right, so he writes a
program to print out all the primes:

1 is prime
1 is prime
1 is prime
1 is prime
1 is prime

And he claimed that he was correct, because he had - as is known to be
true in reality - a countably infinite number of primes.

ChrisA