tag:blogger.com,1999:blog-8666091.post1278886268300815159..comments2020-01-17T07:38:07.311+01:00Comments on The Reference Frame: Sum of integers and oversold common senseLuboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.comBlogger82125tag:blogger.com,1999:blog-8666091.post-21037092377600606862014-01-21T13:06:06.749+01:002014-01-21T13:06:06.749+01:00Using that "sum" in other derivations gu...Using that "sum" in other derivations guarantees that those results won't be compatible with reality.Kimmo Rouvarihttp://www.toebi.com/blog/noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-82539706145607360132014-01-21T12:28:49.900+01:002014-01-21T12:28:49.900+01:00What do you mean by "we can actually see"...What do you mean by "we can actually see"? You can "see" that the series diverges. But nobody is denying it. We are talking about the "sum" of a divergent series. So what else can you "see"?lucretiusnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-75126494092673164152014-01-21T11:26:14.518+01:002014-01-21T11:26:14.518+01:00I do understand what you are saying, no confusion ...I do understand what you are saying, no confusion at all. In math, one can do what ever (s)he wants based on ones rules. However, in this case we can actually see where the deviation from reality happens.Kimmo Rouvarihttp://www.toebi.com/blog/noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-11589998476643417832014-01-21T09:55:47.547+01:002014-01-21T09:55:47.547+01:00Probably Lucretius, however in those days the peop...Probably Lucretius, however in those days the people was less educated and the Church was doing the censorship...Shannonnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-78103252789796780592014-01-21T09:09:26.894+01:002014-01-21T09:09:26.894+01:00I think the last time France had a government that...I think the last time France had a government that was really in favour of freedom of speech was ... at the beginning of the 18th century. When Philippe II, Duke of Orleans became Regent after the death of Louis XIV he ordered every book that had been banned previously to be printed. He was also in favour of free markets, low taxes and small government. I can't think of any other French government since the Middle Ages about which this could be said.lucretiusnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-55526736476082898882014-01-21T08:57:13.682+01:002014-01-21T08:57:13.682+01:00Kimmo, you are just completely confused, that'...Kimmo, you are just completely confused, that's all. You are thinking about convergence, but this series is not convergent and nobody, I repeat, nobody, is saying that it is.<br />The sum "is" 1/2 when you define it by a special method. The justification for doing so is that you can derive many true and useful formulas when you do that and all these derivations are perfectly rigorous. <br /><br /><br />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) and secondly, these series are "divergent" and that is indeed the whole point. The concept of "sum" of a divergent series requires a definition and it is idiotic to argue with definitions using arguments that have nothing at all to do with them.lucretiusnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-24430527378501292862014-01-21T05:05:49.508+01:002014-01-21T05:05:49.508+01:00Although we "live" on totally different ...Although we "live" on totally different planets when it comes to theoretical physics I do have a great respect towards you. So, if I'm sometimes disrespectful towards e.g. string theory that is meant directly to the theory itself. But naturally, one can't always constrict his/her tongue in the heat of the moment ;-) You know.<br /><br /><br />To the point... it's absurd to find out the result for Grandi's series because the sum is altering between 1 and 0. You don't agree on this one, do you? To you, the result is +1/2, but it's not.Kimmo Rouvarihttp://www.toebi.com/blog/noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-37590804336299037202014-01-20T23:46:09.874+01:002014-01-20T23:46:09.874+01:00Racist cetaceans, whodathunkit?Racist cetaceans, whodathunkit?MoptopTheLibertariannoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-80858557173932027162014-01-20T22:38:16.857+01:002014-01-20T22:38:16.857+01:00Well, actually, not anymore it seems.Well, actually, not anymore it seems.Shannonnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-87185877456197833352014-01-20T20:45:24.077+01:002014-01-20T20:45:24.077+01:00Well, I have no problem with that.
I am still per...Well, I have no problem with that. <br />I am still perplexed with the question, "Why is there something rather than nothing?", but like (Sidney Coleman's, I think) answer, "Because Nothing is unstable." But the "God" answer doesn't answer an infinite regress, and "It was always here" isn't very satisfying.<br />"It created itself out of nothing" (the ironically named R. Gott, and others) is interesting but depends on Planck time CTCs.Gordonnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-56085545398832549822014-01-20T20:07:58.824+01:002014-01-20T20:07:58.824+01:00This has nothing to do with convergence!
There is...This has nothing to do with convergence!<br /><br />There is nothing mysterious about this fact when you consider it as a mathematical identity.( I admit that I don’t yet understand fully it’s relevance in physics as I have not yet had the time to read Lubos’ latest post but the purely mathematically everything is perfectly in order. )<br /><br />Remember that we are dealing with an infinite series and not a finite sum, so you can’t just add the numbers. Mathematics is all about consistency and there are more then one consistent ways to define the “sum” of such a series. One way is to define it as the limit of partial sums and that is, of course, infinity. That corresponds to the fact that the partial sums grow arbitrarily large. But “sum” does not have to mean the limit of such approximations. <br /><br />A natural way to understand an equality A=B is to say that whenever you encounter a in some expression A and replace it by B, you will get the “right answer”. Well, we can prove using completely rigorous mathematics that if you start with an “ordinary” numerical or symbolic expression (i.e. not involving an infinite series), and then expand it in certain allowable ways that result in the appearance of this infinite series , and then you replace the sum of this series by -1/12, then you will always get the correct answer. In this sense the sum of the series is -1/12: because replacing it by -1/12 always gives the right answer (provided all the transformations were of the allowable kind). As far as mathematics is concerned that is all there is to this “equality”.<br /><br />If you find this hard to accept, think about how people used to think about imaginary numbers (before they found a geometric representation for them). They started with certain equations with real coefficients and they wanted to find the real roots. But they discovered that in order to do so they had first to introduce this weird number i whose square was -1. It seemed as weird as the sum of this infinite series of positive numbers being negative, but they found that when they followed certain rules these weird i’s disappeared and they ended up with real roots. You can still think of complex numbers as such “formal” mathematical objects defined by certain rules that can be applied to them, but because we now have a nice geometrical interpretation we no longer need to do so. In the case of divergent series mathematicians don’t really have anything but a formal set of rules but maybe physicists do.lucretiusnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-18523441526804029342014-01-20T19:21:04.364+01:002014-01-20T19:21:04.364+01:00What I don't get about the "God" bel...What I don't get about the "God" believers, Gordon, is that here we are, most of <br />our atoms were made inside exploding stars. They then coalesced, formed a<br /> solar system with stable orbits around a very stable star, (of course <br />this has almost certainly happened many, many times in many different <br />places), which eventually led to complex life that in time became <br />sentient. In a very real sense, we are the universe, maturing, <br />becoming self aware and beginning to ask questions about where it came <br />from, where it's going, what it's purpose is. This is all backed by <br />sound science and with a reality like that, who needs the "God" myth? <br />What could be more heavenly than that?cynholtnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-80511290401632776532014-01-20T19:06:14.341+01:002014-01-20T19:06:14.341+01:00It is extremely relevant to the fact that you are ...It is extremely relevant to the fact that you are an un-credible person who isn't allowed to post in many places because you are of exactly the sort of ilk decent people don't have to put up with if they don't want to.Werdnanoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-61294386832660702622014-01-20T19:01:00.309+01:002014-01-20T19:01:00.309+01:00Yeah, makes sense to me I guess.Yeah, makes sense to me I guess.Werdnanoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-21859279737376707222014-01-20T18:59:37.349+01:002014-01-20T18:59:37.349+01:00Yes, I am most definitely a warlock of the most ev...Yes, I am most definitely a warlock of the most evil caliber.Werdnanoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-30316825398557978462014-01-20T18:59:08.611+01:002014-01-20T18:59:08.611+01:00Just a regular male, although that is still an int...Just a regular male, although that is still an interesting sociological question, I guess.Werdnanoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-87656056380896220492014-01-20T14:54:32.560+01:002014-01-20T14:54:32.560+01:00This is a little worrying.
I have long thought th...This is a little worrying. <br />I have long thought that in the current situation the best tactics for the falling climate alarmist Zeppelin would be to start dumping ballast and it looks like they may have started doing this and some of it has fallen on TRF.lucretiusnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-86460281608451835222014-01-20T14:50:32.081+01:002014-01-20T14:50:32.081+01:00> You have also had other sanctions imposed on ...> You have also had other sanctions imposed on you there<br /><br />Sure have. Though the relevance of that here is unclear, especially since none related to scientific content.<br /><br />> our host answer for the actions of the hosts of other weblogs<br /><br />I'm not asking him to answer for them. I'm asking him to condemn them. Or to admit that his interest in "free speech" is really rather limited. You understand the distinction, I hope.<br /><br />> The science in the journal in question may also not be up to high standards...<br /><br />I don't think *anyone* think its up to *high* standard. The question is whether its up to even rather low standards. To answer that, you'd need evidence of a robust review process - which is lacking - or independent evidence from someone actually reading one of the wretched things. Up to now people are voting with their feet - happy to talk about the politics, totally uninterested in reading the articles.<br /><br />But perhaps you care. Why don't *you* try reading one?<br /><br />> publishers initially justified cancellation of the journal<br /><br /><br /><br />Yeah, I think they screwed that up a bit. They should just have said "this is a bunch of wacko pals gathered together to publish their junk. They fooled us for a bit but now we've rumbled them."William Connolleynoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-72167904262433735922014-01-20T14:16:02.842+01:002014-01-20T14:16:02.842+01:00Hi, even on Wikipedia where alarmists hold sway ov...Hi, even on Wikipedia where alarmists hold sway over climate articles, you are considered one of the most extreme and rabidly partisan people. For evidence, here is your <a href="https://en.wikipedia.org/w/index.php?title=Special%3ALog&type=block&user=&page=William_M._Connolley&year=&month=-1&tagfilter=&hide_review_log=1&hide_thanks_log=1" rel="nofollow">Wikipedia block log</a>. I wanted to count the times you've been blocked but I ran out of fingers. You have also had other sanctions imposed on you there.<br /><br />It is bizarre for you to demand that our host answer for the actions of the hosts of other weblogs, especially so since you complain about having comments suppressed when you yourself do so on your own blog.<br /><br />Reasonable people may disagree on the weighting of the reasons cited by the management of Copernicus. Specifically, one may conclude that the "nepotism" charged by them has a basis in fact. I have read all four of the external links thoughtfully provided by our host at the end of this article and find the <a href="http://bigcitylib.blogspot.cz/2014/01/on-pattern-recognition-in-physics.htmlThanks%20to" rel="nofollow">Big City Lib</a> write-up the most informative one.<br /><br /><br />The science in the journal in question may also not be up to high standards (see the relevant article by Anthony Watts on WUWT). That is beside the point, however. What should be troubling to every scientist, even you Dr. Connolley, is that the publishers initially justified cancellation of the journal because of drawing a conclusion that runs counter to climate orthodoxy -- in short, for political reasons.<br /><br />Nonetheless,Eugene Snoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-32647524435827627132014-01-20T13:03:34.580+01:002014-01-20T13:03:34.580+01:00Thanks for letting my humble comment thorugh. It s...Thanks for letting my humble comment thorugh. It seems that you're interested in free speech in journals (good!) and here (of course) but utterly uninterested in FS at, say, WUWT.<br /><br />> Nicola Scaffeta<br /><br />I notice that although you (very briefly) discuss NS's paper - enough to indicate that you've skimmed it - you don't offer any opinion as to its quality. Now you might say "I wasn't interested to read it in detail" but even that would be a condemnation. So do you care to offer an opinion? What about some of Tallblokes numerology - can you even be bothered to glance at it?<br /><br />> Will the climate Nazis destroy...<br /><br /><br /><br />Godwin! Though you lost on that loooong ago.William Connolleynoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-68077331124441457612014-01-20T12:45:09.695+01:002014-01-20T12:45:09.695+01:00Dear William, you still have the freedom to expres...Dear William, you still have the freedom to express your dumb opinions on my blog, at least to the extent when you're annoying beneath a certain threshold.<br /><br /><br />The key problem you don't understand is that the termination of the journal has killed *future* papers with certain similarities of the spirit, and by doing so, the termination is exerting pressure on what kind of results "may" be obtained by papers. It is an ideologically motivated pressure which is just not fine.<br /><br /><br />The bloggers you mentioned as well as myself sometimes erase comments but I am personally doing so according to ideology-blind quality rules. That's why you will find comments from all corners of the spectrum of thinking. Banning any papers that could imply that natural drivers are more important for something in the climate is a scientific defective, Stalinist-like distortion of the research.Luboš Motlhttp://motls.blogspot.com/noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-80648620438088614582014-01-20T09:50:22.100+01:002014-01-20T09:50:22.100+01:00> and sometimes basic human rights including th...> and sometimes basic human rights including the free speech<br /><br />An excellent point. Though you seem to have missed the fact that the papers are still available.<br /><br />Doubtless with your fine upstanding concern for free speech you'll thunderously condemn WUWT, Tallbloke, NoTricksZone and all the other blogs that have suppressed my comments? Come on, don't be shy!<br /><br />http://scienceblogs.com/stoat/2014/01/17/ship-of-fools/William Connolleynoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-36550980505548959082014-01-20T07:36:26.381+01:002014-01-20T07:36:26.381+01:00Dear Kimmo, it's really the same low-brow comm...Dear Kimmo, it's really the same low-brow common sense as everywhere.<br /><br /><br />When 1-1+1-1+1-.... appears anywhere where it matters, the value is always +1/2. The method used to "derive" that it is zero or one - by clumping the two neightbors - is just illegitimate in maths because it breaks the algebraic structure of the expression.<br /><br /><br />What it depends upon is the assumption that the adjacent two terms are always equal, up to the sign. But this is clearly just a very special feature of this particular sum with the numbers 1, -1, ..., so it does *not* admit any analytical continuation in any conceivable variable on which the dependence is different than proportionality. That's why you can never get 0 or 1 as the result of any analytical continuation method and/or any subtraction/renormalization scheme that respects any analytic continuation.<br /><br /><br />Grandi's series is +1/2 at the same level of uniqueness and logic at which the sum of positive integers equals -1/12. It's the same kind of maths. After all, in the video embedded in this blog post, Grandi's series was used to calculate the sum of integers.Luboš Motlhttp://motls.blogspot.com/noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-65487892800828606812014-01-20T05:41:20.116+01:002014-01-20T05:41:20.116+01:00Here's some common sense :-) http://www.toebi....Here's some common sense :-) http://www.toebi.com/blog/mathematics/grandis-series/Kimmo Rouvarihttp://www.toebi.com/blog/noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-87667544918439235062014-01-20T04:51:38.544+01:002014-01-20T04:51:38.544+01:00Thanks for your reply!
As I tend towards the math...Thanks for your reply!<br /><br />As I tend towards the mathematical side, if I could have my way I would always avoid using "sum without adjectives" and would mention explicitly what kind of sum it is, even if it just limit of partial sums.<br /><br />However, I do not agree that having infinity all the time is nonsense. As I mentioned, infinity plays a very important and subtle role in Convex Analysis. Just open the classical book by Rockafellar and you see the insight that is gained when you consider the epigraph of functions defined in the extended real line. This is not naive mathematics.<br /><br />Can't we all just be friends? <br />Is -1/12 right under the appropriate notion and definitions? Yeah, I guess so. Is it useful and provides insight? I am guessing from your post that the answer is yes. Is +infinity right under the appropriate notion and definitions? Yeah, I guess so too. Does it also provides insight and is useful? I think so and it is not naive to think like that; but I agree it is a matter of taste.<br /><br />You may think that people are generally biased towards partial sums and I agree with that. But rather than fighting to decide which one gets to be default meaning of "sum without adjectives", I think it is better to convince people to be precise and state explicitly what kind of sum they are talking about. <br /><br />Perhaps, in theCthulhunoreply@blogger.com