First, I want to start with the way in which a blogger hosted by the Guardian, Dana Nuccitelli, informed about the U.S. Senate testimonies by two climate scientists, Judith Curry and Andrew Dessler:

Andrew Dessler presented the usual experimentally falsified crackpottery about the alleged "climate threat"; Judith Curry pointed out that in contrast with the untrue slogan and fabricated \(p\)-values about the "higher confidence", the newest IPCC report reduced the confidence in most of the claims about the climate alarm – because of the reduced estimates of the climate sensitivity, the warming hiatus since the late 1990s, and the mostly growing Antarctic sea ice, among other reasons.Climate scientist to US Senate: 'Climate change is a clear and present danger'

Belief in an instant planet-wide quick-fix, such as blocking sunlight with sulphur, is delusional, US activist declares

Nevertheless, Nuccitelli only mentions the name of Judith Curry thrice while Andrew Dessler is quoted 14 times (and he owns both the title and the subtitle). This is particular weird because relatively to Judith Curry who is the chair of Earth Sciences at the Georgia Tech and whose name appears in 650 papers with over 4,000 citations from the top ten, Andrew Dessler is – using refined diplomatic jargon – an unlikable screwed unhinged bald bespectacled weird prick and a relative scientific nobody with just 280 papers mentioning his name and less than 1,000 citations from the top ten.

Nuccitelli's soulmates have full mouths of women's rights but when it comes to the real world with an ideologically inconvenient woman, they dedicate the space to her and an irrelevant prick in the ratio 1:6 even though the right ratio would be 5:1. This is how all the lies about the "climate threat" and "consensus" are being fabricated. The climate alarmists have no integrity whatsoever. They're ethical pollution that should be removed from the human society; by constantly harming decent people, they have already depleted their moral right to live.

But the title promised you to talk about a conflict between an insane alarmist organization and a mad individual climate alarmist who have actually shared the 2007 Nobel prize in peace.

According to a leaked draft of the summary for policymakers, the third working group (WG3) of the IPCC has advocated the carbon dioxide removal (CDR) fantasy, something that I have already analyzed and determined to be an utterly unrealistic plan to murder 1 billion people (and the sixth of the world's animals, too).

But most of the alarmist media have interpreted this IPCC CDR plan as geoengineering so various outlets tell us that the IPCC thinks that "geoengineering will be essential" or "necessary". It's the word "geoengineering" that the IPCC's Nobel-prize-winning colleague Al Gore has reacted to. Gore thinks that geoengineering is insane.

Because CDR isn't necessarily "geoengineering", at least not the geoengineering based on the spraying of another chemical that Gore talks about, you might suggest that Gore – with his tiny attention span – is just responding to one word because he's not able to get any deeper. However, you may also say that Gore's identification of CDR with geoengineering is legitimate because the CDR fantasy would probably need to fertilize the Earth by lots of other chemicals to actively absorb some existing CO2. The Guardian was talking about sulfur to block the sunlight (the IPCC WG3 isn't necessarily talking about this option at all) but it may also be iron used to fertilize the oceans or something else.

A former Air Force spoiled brat and alarmist Doug Craig has summarized some quotes by Al Gore – who repeated the claims in a phone conference yesterday:

I guess that many readers "partly" sympathize with these "conservative" comments by Gore, especially with the comment that it is not usually wise to beat a smaller disease by a greater one. I don't think that you agree with the quotes that Al Gore has shamelessly plagiarized from a speech of Adolf Hitler about the Weimar Republic, namely that democracy represents a "paralysis in the global political system".Gore:It would be insane, utterly mad and delusional in the extreme to turn to geo-engineering projects to avoid a climate catastrophe.

The idea that we can put a different form of pollution into the atmosphere to cancel out the effects of global warming pollution is utterly insane. The fact that some scientists who should know better are actually engaged in serious discussion of those alternatives is a mark of how desperate some of them are feeling due to the paralysis in the global political system.

The most discussed so-called geo-engineering proposals - like putting sulphur dioxide in the atmosphere to reflect incoming sunlight - that's just insane. Let's just describe that clearly - it is utterly mad.

Such large and untested experiments carried enormous risks while 'doing nothing to address other consequences of climate change such as ocean acidification.

We are already engaged in a planet-wide experiment with consequences we can already tell are unpleasant for the future of humanity. So the hubris involved in thinking we can come up with a second planet-wide experiment that would exactly counteract the first experiment is delusional in the extreme.

At any rate, there is some chance that similar important questions will divide the climate alarmists to two or many camps that will destroy each other. Climate alarmism is indefensible – at the rational or scientific level, it makes no sense whatsoever. And one of the features of ideas that make no sense is that they are highly non-unique and, in fact, diverse. If someone is such an incredible moron to think that we face a "climate threat", he may invent many arrangements of wrong answers to many detailed questions.

Because the climate alarmists don't have any unquestionable leader that would play the analogous role as Adolf Hitler or Joseph Stalin – in fact, the crackpots in the IPCC WG3 dared to openly disagree with their potential Führer Al Gore – there is a chance that unlike Nazism and Stalinism, this particular totalitarian movement (climate alarmism) will become fragmented and ultimately impotent. It could have occurred earlier but for many years, the climate alarmists have been satisfied with superficial religious slogans about climate change and have never discussed any scientific or engineering questions – so the fragmentation hasn't previously occurred.

Because the superficial alarmist prayers are no longer enough and the alarmists are sometimes forced to respond to more detailed questions, the climate alarmist movement is hopefully facing a bloody civil war that will neutralize most of these nasty and potentially dangerous cranks. I hope – and I actually believe – that the self-evident non-existence of the consensus concerning geoengineering and other important questions will terminate one of the driving forces about the ever stronger belief by pathological brainwashed sheep such as Alexander Ač.

The only chemical compound that I am aware of that would be a clear net positive if its concentration were artificially increased is carbon dioxide. But it's still pretty expensive to elevate its concentrations – it's as expensive as the fossil fuels we have to burn. And the benefits are almost certainly not high enough to justify the burning of extra fossil fuels just for the sake of it. So it's silly to spray any chemical all over the globe. If you know about a counterexample, I will be happy to hear about your observations.

## snail feedback (47) :

There isn't such an example, infinitely is a big word. My point was that if you make and use derivations based on weird rules you'll get weird results, like that -1/12 thing, which is not compatible with reality.

On the contrary, the value -1/12 is an experimentally proven fact. It's the claim that the quantity is "infinite" or "the laws of physics break down as soon as we encounter the sum of integers" etc. that are safely experimentally excluded.

Right... I hope that you understand humor when I say that I can almost feel the air flow coming from your waving hands ;-)

Sorry, I didn't mean "measured,", what I meant was it seems that you have assumed (not derived) a value for the sum that gives you a self-consistent physical theory.

This, of course, is merely the impression gotten by a dilettante non-mathematician.

I would dearly like to have a real modern mathematician endorse this.

You will never get "weird results" provided you use only the allowed rules. This is as true in the case of such series as it is in any other mathematics.

The empirical aspects of this issue I leave to Lubos...

That's right, no "weird results" (other than been not in touch with reality but that's not important).

Sorry, hard sciences and maths don't work by counting heads in a committee.

The word "empirical" does mean "related to measurements".

If you agree it is about the internal consistency of a theory using the sum, it is a mathematical condition.

Ask Mickey Mann. I'm sure he can knock one up for you. It's all part of his stage act. "Reality" is whatever he miscalculates it to be.

You'll have to ask nicely though, otherwise you'll get his regular response to disinterested enquirers:

"

YOU WANT THE DATA? YOU CAN'T HANDLE THE DATA — COZ I'M NOT GIVING IT TO YOU!" — Jack Nicholson as Mann in the forthcoming blockbuster on the greatest con in history,A Few REALLY Bad Men.Kimmo,

You might have missed it but Lucretius made a reference to G H Hardy's '

Divergent Series' further down this page. Beneath it I gave a link to Amazon where you can read about six pages of Hardy's introduction. If you haven't done so yet, please take a quick look. You'll have to scroll down the "Look Inside" feature for about ten pages to reach it. I think it will help you out.It's not bollocks! And that's the truth. :)

@Lubos: What a wonderful post! You constantly amaze me. Terry Tao had a nice, relevant post on his blog---

http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

I also do understand your occasional frustrated "...

pearls before swine" comments since on occasion, I am the Sus scrofa domesticus. When you do so much to generate the blog (unless you are, as I suspect, like Mozart, and the complex math/physics just flows effortlessly out of you :)), we can make an effort to read and at least try to understand what we can of the technical posts (I am mostly referring to the string theory posts and my ignorance and difficulty following them). I try to use Penrose's advice to readers of his "Road to Reality" to skim anyway bits you think you don't follow and use the osmosis method, similar to reading a foreign language (I am paraphrasing).

In short, this was an interesting blog post.

Thanks, Gordon - also for the nice Tao essay! He also clarifies what to do with various "inconsistencies" between the formulae - how these ambiguities are encoded in cutoffs that are not visible in the "naive" form of the infinite sum.

I read those free pages and the bottom line was that there is one "reasonable" value for Grandi's series and it should be 1/2. I totally agree with that conclusion.

But because the series is divergent that value is only "reasonable".

Nice article Lubos. Of course, the big surprise ,I cannot get over with, is that after all this mental

gymnastics, some of these calculations end up agreeing with the experiments which people do with things they hold in their hands and which are as real as one can get!!!

Come on — you want chocolate on it too? Be

reasonable! Anyway, chocolate's extra, but you still can't have any because we've run out. :)Fair enough :-)

John, I see, thanks. I didn't realize that what Kimmo needed to swallow these zeta-function values was a chocolate on it.

OK, I attach zeta on chocolate. It should settle all doubts.

http://www.customsweets.com/gallery_images/zeta_chocolate.jpg

Very nice post, your paper from 1995 was simple and impressive

LOL, poor Lenka whom it was dedicated to has only collected 1 citation from someone else. ;-)

Superb! :)

NOW you're being reasonable! See? You don't need chocolate.

And the jam's off too so don't ask. :)

Maybe it was my sweet tooth acting or something like that ;-)

OK, I apologize. I was lazy. I looked up Cais' pdf on Divergent Sums and read it. He has a very clear and convincing argument that the sum of the positive integers is acually -1/12.

So, are the 26 dimensions all spatial or is there a time- dimension in the mix?

The physical signature is 25 spatial plus 1 time but the separation to space and time doesn't follow "just" from the sum of integers.

Wow, Lubos. Great stuff. A finite value for 1+1+1+...seems to be even more involved than a finite value for 1+2+3+4.... since you even have to know the summation index while at school you learn that this is just a pure dummy index. Another indication is that demanding linearity and additivity in the first term from infinite sums (while giving up inifinite associativity) allows you to prove that 1+1+1... is infinite while 1+2+3+4... comes out as -1/12. (see Carl Benders lecture which was linked to your earlier post).

Baby math question: I understand that if a complex function is holomorphic (i.e., differentiable in some open disk around every point in its domain), then this implies that the function is analytic (i.e., able to be expanded as a convergent power series in some open disk around every point in its domain).

However, is the converse true, or, equivalently, do there exist any analytic complex functions that are *not* holomorphic?

Dear Dv, an analytic function is one that is equal to its Taylor expansion. But if you write a power law expansion and it converges, and we just assumed it did, you may substitute a complex variable and it converges in a disk of some radius convergence.

In that disk, the function is inevitably holomorphic. On the boundary of the disk, a circle, there may be (and are, if the radius is finite) singularities. They may be poles or e.g. branch cuts or essential singularities, it depends. But locally, the function is holomorphic.

Dear Mikael, thanks and right, 1+1+1+.... has an "extra subtlety". But in the zeta-regularization perspective, it is exactly as important and settled as 1+2+3+.... This sum of "ones" also appears in string theory, when one counts some fermion charges of the ground state for GSO projections and similar stuff.

david berenstein's article is great! thanks.

Lubos, can it be that deriving the sum 1+2+3+... from the sum 1-1+1-1+... (which has in the real world a mean value, 1/2) is the hearth of the renormalization technique?

Thanks Lumo, yes I am still there :-)

Today I am going to a rehabilitation clinic in Gral Müritz for 3 weeks to further recover. The doctors said the worst trouble I had (I am not at the age for having such trouble at all ...) should now be over for good; I will have to be carful about certain things in the future and then it should be mostly ok.

I hope I will have WLAN access from my room, such that I can unhindered enjoy reading TRF. Reading awsome stuff here and being with the nice community we have here has and will always be very healthy for me :-)

Cheers

Good luck, Dilaton. Most of the assumptions that some health disorders depend on high age are just stereotypes - there's just a correlation but not a waterproof identity.

I didn't realize how close you were e.g. to Insel Rügen where we were some decades ago, fun. ;-)

Ok, cool! That is a clever explanation of why analytic functions are holomorphic. So in fact the set of complex analytic functions is identical to the set of complex holomorphic functions, and all meromorphic functions are (I guess by definition) analytic in the neighborhood around any non-singularity point.

The one thing I don't understand in your explanation is the bit about "branch cuts." I'm reading the Wikipedia article entitled "branch point," right now, but it's very hazy to me. I didn't even realize there was a way to generalize rules and observations about multi-valued functions...

Hi dv, branch cuts are just points like z=0 for the function

sqrt(z)

or z^{1/n} for another value of n or the function

ln(z).

If you make a round trip around z=0 and you evolve the function value continuously, the value of the function at the end is different than at the beginning.

For sqrt(z), this function goes to minus itself if you make the trip around z=0. Similarly, ln(z) additively jumps by 2*pi*i if you go around z=0.

It is as impossible to write the function as a Taylor expansion around the branch cut as it is impossible around any singularity - even though the value at the branch cut is usually finite.

Thanks again Lumo for help trying to save the 1+1+1+...=-1/2 question.

Such and similar attacks on (even high-level advanced topics) technical questions as homework, fundamental physics questions as philosophy, on other excellent questions just for the heck of it abusing any pretext, is what defines the classical action of the now dominating thereview queues trolling dilettantes and aggressive dimwits. And of course those rascals prevent any question they (or a mod) wrongly closed from getting reopend too.

When you look again at the question, you can see that the bastards have migrated it away, disrespecting even the judgments of experts like you, which makes Physics SE look really not good.

I preferred to tell you this here, even though I am no longer capable of asking there anything since long time ago, as the trolling dimwits with too much power attack even good up to excellent questions regularly. This is some kind of sad as I liked the nice answers you and other good people gave me a lot.

I will have to ask Polarkernel soon about good news concerning the upcoming technical private beta of or PhysicsOverflow, which is more healthy for me than tangling with aggressive/destructive know-nothings anyway ... ;-)

Here is a simple reason why this is wrong:

1+1+1+1+1+...=-1/12 <=> 1+1+1+1+... -1/12=0 <=> 11/12 +1+1+1+1+...=0 <=> 1+1+1+1+1+1...=-11/12 (1)

We repeat the same process as in (1) and we will always have different values for 1+1+1+...

Obviously, a sum either diverge or converge to only one number, and here it seems that it converges to more than one number, thus it must be divergent.

You seem to violate this law of mathematics:

A sum of positive integers cannot equal a negative number.

In this natural summation scheme, your law only applies to sums of finitely many numbers and sums that are convergent in the usual sense.

The recent posts above and the pointless migration from physics to math of the negative sum question are great examples of not accepting that nature tells us how it can be described. The "classic" math crowd, like the QM wrong crowd and the SE control freaks all want to impose what they believe is right on the universe . The contexts are different but they all depend on having closed their minds too soon.

I am hoping that you are finding your clinic stay good and your health being restored.

Don't let the trolls bite. They are missing the whole point of this. Those who are never wrong simply don't have capacity to figure out that they were actually never right.

Sorry, but the people you are calling "the classic math crowd" may be a crowd but they are not "classic math crown" but "ignorant of math crowd". The theory of summation of divergent series was alredy fully formalised at the beginning of the 20th century: it has long been a perfectly rigorous and not in any sense mysterious part of modern analysis. You will find few professional mathematicians who will express any doubt or surprise at hearing of these results, and any expert in analysis will no doubt confirm the words of Littlewood from his Introduction to Hardy’s “DIvergent Series”:

“In the early years of the century the subject [Divergent Series], while in no way mystical or unrigorous, was regarded as sensational, and about the present title, now colourless, there hung an aroma of paradox and audacity."

Please ignore all further posts by these ignorant fools.

My use of classic was in quotation marks for that reason.

The math SE Q&A at http://math.stackexchange.com/questions/633887/divergent-series-intuition

Is interesting in this same vein. Ran into it when I was following up on Dilaton's migrated Q. The insistence of the poster that this is all nonsense is well addressed but as so often I find better addressed here.

Anyway I don't usually ignore the ignorant until they confirm that heir ignorance is stupidity. But the outcome usually converges.

Dear Lubos, all nonrenormalizable theories are really renormalizable. Since you have to renormalize an infinite number of parameters, the ACTUAL number of renormalizations you have to do is 1 + 1 + 1 + 1 + 1 +... = 3/4 renormalizations. :)

Dear Lubos,

You said "The method used to "derive" that it is zero or one - by clumping the two neighbors - is just illegitimate in maths because it breaks the algebraic structure of the expression."

The algebraic structure? What algebraic structure do you speak of?

What's wrong with writing 1 - 1 + 1 - 1 ... as (1 - 1) + (1 - 1) +... = 0 + 0 + 0 +... = 0?

I don't see how this "trick" is ANY different from those two guys in the numberphile video computing 1 + 2 + 3 +...

So tell me why I can't do that?

Also, when we deal with infinite sums, it's true we cannot add an infinite number of terms. But that's what taking limits are for.

1 + x + x^2 +... never equals 1/(1-x) because you cannot add an infinite number of terms. Instead, the series above APPROACHES 1/(1-x) as the number of terms approaches infinity for |x| < 1.

However, 1 + 2 + 3 +... approaches infinity as make the series larger and larger.

Lucretius,

You said, "You are all the time thinking of these "sums" as if they were just sums of numbers but first of all you cannot sum infinitely many numbers anyway (even in the convergent case the sum of an infinite series is not a sum in the usual sense)..."

When we deal with infinite sums, it's true we cannot add an infinite number of terms. But that's what taking limits are for.

1 + x + x^2 +... never equals 1/(1-x) because you cannot add an infinite number of terms. Instead, the series above APPROACHES 1/(1-x) as the number of terms approaches infinity for |x| < 1.

However, 1 + 2 + 3 +... approaches infinity as make the series larger and larger.

Lucretius, you said, "The concept of 'sum' of a divergent series requires a definition..."

Why does it require a change in the definition of a sum? What's wrong with saying that the series diverges? The series approaches infinity as the number of terms is increased. Just like the function 1/x approaches infinity as x approaches 0. We accept that. We don't definite a new concept of division so that we get a finite answer.

I didn't think it was known that zeta(3) is transcendental (although I'd place good money that it is). Definitely don't think I'd call it notoriously so.

Just a nitpick. Not every real analytic function on $\mathbb{R}^2 \simeq \mathbb{C}$ is actually (complex) analytic. Consider $z \mapsto \overline{z}$ but IIRC there are examples of real analytic functions on $\mathbb{R}^2$ which are not (complex) analytic with respect to any complex structure on $\mathbb{R}^2$.

For what its worth, lets look at the "series" 1-1+1-1+1-1....

What every one seems to miss, is that addition is communitave.

a+b=b+a, and this is only one way this series could be written.

In fact there are many ways, to write this series and that is the key. One way is 1+1+1+1+....-1-1-1-1...

All must be the same because the sum of a series must be independent of order. This is a weakness of what is being shown. The proofs need order where there is none. One cannot shift the series over by one, because the second series might be in a different order.

lets take the series, 0+0+0+0+0....=0

this could be rewritten 1-1+1-1+1-1...=0

Someone please point out the error in my logic.

Thanks

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