The sum of all natural numbers is not \(-1/12\)It's not hard to see that the title is in some "tension" with at least one of the titles of older TRF blog posts about this issue,

Zeta-function regularization (2007)These texts of mine were actually linked to by some Pharyngula's readers who are my semi-fans – folks who would say how brilliant I am but who would never avoid mentioning that "thee shall not read any Lumo's articles on the global warming" and "he is a [right-wing] as*hole" of a sort. These untrue libels are apparently mandatory in the current Academia.

Why is the sum of integers equal to \(-1/12\) (2011)

At any rate, Pharyngula's "contribution" doesn't say much more than the title. He also copies a segment of a text by Mark Chu-Carroll,

Bad Math from the Bad Astronomer,who didn't like the regulated value of the sum, either. The only person among these notorious over-the-edge left-wing activists who liked the regulated sum was the Bad Astronomer Phil Plait, according to his enthusiastic article

When Infinity Is Actually a Small, Negative Fraction (Slate),which was supplemented with an extra disclaimer after an avalanche of dissatisfied reactions from some other randomly ejaculating liberals.

Plait's text has also embedded an enthusiastic video about the regulated sum:

It was posted on Brady Haran's Numberphile YouTube channel a week ago. It describes/copies the Euler method that I presented both in the 2007 article and the 2011 article.

I have already discussed about four different ways to calculate the natural value of the sum, \[

1+2+3+4+5+\dots = - \frac{1}{12},

\] in the blog posts from 2007 and 2011. In this entry, I would like to focus on some sociological aspects of this issue – and the different scientific disciplines' viewpoint on the sum.

First, let me begin with the high-school or freshman undergraduate perspective on the sum – and folks like P.Z. Myers haven't managed to go beyond this level. What do we mean by the sum of the infinite number of terms? Well, we mean the limit of the partial sums:\[

E_0 = \sum_{n=1}^\infty n = \lim_{N_{\rm max}\to\infty} \sum_{n=1}^{N_{\rm max}} n

\] I will use the symbol \(E_0\) for the sum to remind you that in some units, it is the energy of the ground state of some physical system. To fully appreciate this simple straightforward definition of the infinite sum, you must know what the "limit" means. In this case, the limit exists and is equal to \(L\) if and only if\[

\forall \epsilon\gt 0, \,\, \exists N_\epsilon:\,\, \forall N \gt N_\epsilon:\,\, \abs{ L - \sum_{n=1}^N n } \lt \epsilon.

\] In plain mathematical English, the partial sums have to converge to \(L\) which means that for an arbitrarily narrow neighborhood of \(L\), i.e. for an arbitrarily short interval \((L-\epsilon,L+\epsilon)\), one can find some minimum length of the partial sums \(N_\epsilon\) such that all the partial sums that are this long or longer produce results that belong to this interval.

This is pretty much the only meaning of the infinite sum – the only method to deal with the bizarre symbol \(\infty\) used as the upper limit of the sum – that an undergraduate freshman student who has no extra interests in maths and physics knows. This student may easily prove that the sum of positive integers cannot be negative because all the partial sums are positive and the limit of a sequence of positive numbers cannot be negative. By the same argument, he may prove that the sum cannot be fractional because all the partial sums are integers.

The real problem is that the definition of the sum involving the limit of partial sums – limits that way too often "diverge" or "refuse to exist" – isn't the only definition or the best definition or the most natural definition that may be connected to the sum. There exist better definitions of the infinite sum – numerous definitions that turn out to be more natural in physics applications – and they generally produce the result \(-1/12\). It is no trick or sleight-of-hand. The value \(-1/12\) is really the right one and the rightness may be experimentally verified (using the Casimir effect).

These physics applications of the sum of positive integers are pretty much derived from the master example, the ground state energy of a string. The energy of a relativistic string is given by the Hamiltonian which looks like\[

H = \int\dd \sigma \,\zav{ \frac 12 p(\sigma)^2 + \frac 12 x'(\sigma)^2 }

\] Let's not be picky about the limits of the integral and coefficients, among other things (such as the extra spatial indices that \(x\) and \(p\) usually carry). The point is that once the theory is quantized, the operator \(H\) above – the Hamiltonian – is equivalent to the sum of infinitely many quantum harmonic oscillators' Hamiltonians. Each of these Hamiltonians is linked to one Fourier mode for which the \(x\) and \(p\) are multiplied by functions like \(\cos n\sigma\) (or sine or the complex exponential) whose frequency scales with \(n\). And because the zero-point (ground state) energy of the harmonic oscillator is \(\hbar\omega/2\) and \(\omega\sim n\), we get the contribution scaling like \(n\) from each of these harmonic oscillator which is why the overall zero-point energy of \(H\) will be proportional to \[

E_0 = \sum_{n=1}^\infty n.

\] But an important detail we must realize is that when we play with these Fourier expansions and harmonic oscillator, mathematics and Nature don't tell us\[

E_0 = \sum_{n=1}^\infty n,\\

\text{...please compute the sum...}\\

\text{...using the partial sums, ...}\\

\text{...your loving Nature...}

\] Nature just doesn't give us any similar hints how the sum should be evaluated. The whole concept of "partial sums" is just one possible man-made concept. It implicitly includes the assumption that the low-frequency Fourier modes have a higher priority and the effect of modes with excessively high frequencies may be quantified by simply forgetting about them.

But none of these assumptions – assumptions which are really just man-made social conventions – really follows from the mathematics that encodes the laws of Nature. When we switch between different forms of the formula for the Hamiltonian, some of the manipulations are formal which means that some forms may be more well-defined than others. But again, Nature really doesn't tell us which of the forms is superior. We must find out.

Experiments always measure finite values of quantities such as the energy so from a physics viewpoint, all prescriptions and methods of calculation that produce answers such as \(\infty\) or "the limit doesn't exist" are just immediately excluded. A physics theory isn't allowed to work like that. Getting rid of the "spurious infinities" that just mask the important finite answers isn't just an option for a physicist; it is his unquestionable duty.

So physicists have learned to deal with similar sums in smarter ways, ways that remove the "spurious infinities" that just mask the important finite results. The symbol \(\infty\) is always the same thing and pretty much carries no information. Instead, the physical predictions must carry lots of information. Only finite numbers may convey such information, the dependence on various physical variables, and so on. The infinities in the results are anonymous scum that has to be uprooted.

When the procedures are done properly, we sometimes find out that the correct finite answer is somewhat undetermined and an experimentally measured parameter (e.g. the value of the elementary electric charge) has to be substituted. (In renormalizable theories, the number of infinite sums that are replaced by a finite number is finite – we learn about the values of a finite number of "patterns of diverging integrals" and all others may be expressed as their functions which is what keeps the theory predictive: infinitely many predictions may be made once you measure a finite number of parameters.)

In the case of the sum of the positive integers, the correct finite result that should be assigned to the result is unique. You may reconcile this finite result with your low-brow, straightforward, common-sense freshman undergraduate intuition if you redefine the Hamiltonian and include some terms that subtract the infinite part. The formula for the Hamiltionian without these extra terms is just a "heuristic template" or an "inspiration" and you may say that the "right Hamiltonian" is more complicated – less directly expressible in different forms. Alternatively, you may keep the simple-minded form of the Hamiltonian and redefine the meaning of the infinite sums etc.

Quantum physicists really need similar procedures in their everyday work but they don't have a monopoly over these methods and the finite results acquired with their help. In fact, some of the best mathematicians have determined the right values of these superficially divergent or ill-defined sums long before these sums became important in physics. Leonhard Euler (1707-1783) was able to calculate the value \(-1/12\) for the sum of integers using one of the "old-fashioned" methods that are at risk of being similar to some other methods producing wrong results if the process is performed by a less ingenious mathematician than Euler. The right methods generally preserve the "analytic continuation" of some holomorphic functions of some complex variables but I don't want to turn this text into another technical essay explaining all the possible "wrong regularizations" and how they differ from the right ones.

Many other top mathematicians have spent quite some time by calculating similar sums – sometimes sums that are still in the "ballpark of the physically relevant ones" but that are more complex than anything that physicists have actually needed.

Other mathematicians have developed alternative definitions of the summation that agree for convergent sums but are more likely to produce finite answers if the sum diverges using the trivial limit-of-partial-sums definition. These definitions of the sums that improve the convergence are known under many names borrowed from their inventors. For almost every letter in the alphabet, you find a summation, for example:

Abel summation, Borel summation, Cesàro summation, ..., Lambert summation, Lindelöf summation, ..., Ramanujan summation...and so on. I couldn't forget about Srinivasa Ramanujan who has really gained some profound (according to a modern physicist's opinion) insights on the superficially divergent and ill-defined sums. His prescription is the most far-reaching one in the list – it produces finite answers in many cases (including the result \(-1/12\) for the sum of positive integers) and it is so general that according to some straightforward definitions of the "sum", it can't even be called "a sum". Nevertheless, Ramanujan's summation has the key algebraic properties expected from a sum and this is what really matters for Nature's decision whether some quantity deserves to be called a sum.

Borel summation is helpful for a clarification of the character of the divergence of the perturbative series in most quantum field theories (and perturbative string theory) – and why this divergence doesn't imply an inconsistency.

Especially some of these summation methods are much smarter than the naive man-made prescription using the partial sums. But we must understand that just like the partial-sum definition, they're still man-made inventions. A mathematician may simply define concepts and symbols, including \(\sum_{n=1}^\infty a_n\), in many ways. But what's important to realize is that none of the man-made inventions and none of the man-recommended procedures is automatically relevant for calculations in physics – or natural sciences.

So it's not true that physics is forever constrained to use a particular "Mann summation" named after a particular man (I chose "Mann" for the sake of wittiness – because everyone knows that almost no person boasting this name is able to sum anything). Instead, the truth in physics is determined by Nature's wisdom that is only accessible to us through the experiments and observations (and whatever deeper we may induce from them).

In particular quantum theories, like in the two-dimensional conformal field theory that is used as a world sheet description of string theory, there exist "more natural, less man-dependent" rules that eliminate many wrong ways to sum expressions. Conformal field theories underlying the string world sheet have to obey the axioms – mutual locality of operators, some world sheet duality, and especially modular invariance (two/many ways of visualizing the torus). These conditions restrict the possible value of the true observables (which may be more or less directly extracted from experiments) – such as scattering amplitudes, correlation functions of operators, and so on. In particular, modular invariance is the main physical principle that implies that the result \(-1/12\) for the sum of positive integers is the right one; the constant \(-1/12\) itself is inherited from some identities obeyed by the \(\theta\) and especially \(\eta\) functions.

All the other concepts and quantities that appear in the middle of the calculations as intermediate results are "questionable". In physics, your goal is really to predict particular measurable quantities such as cross sections – and to design algorithms that allow one to determine many such predictions at the same moment. Pretty much any algorithm to produce lots of predictions in physics includes lots of "intermediate calculations" but all conceivable intermediate calculations, not just some randomly selected ones "sponsored" by particular people, are allowed to compete.

That is the reason why it's always naive and ultimately wrong to assume that some particular ways to calculate, like the low-brow limit-of-partial-sums definition of the infinite sums, is the right component of the calculations. It almost never is. It doesn't matter that a stringent instructor was beating you for you to memorize some particular rules or definitions (e.g. the limits and the partial sums); your pain isn't enough to make these algorithms and definitions relevant and right in a quantum field theory, for example. All these seemingly divergent or ill-defined sums and integrals have to interpreted and calculated using the best knowledge and intuition that is rooted in previously successful theories and ways of thinking (those that may produce self-consistent predictions that agree with the empirical data).

So physics is never a permanent slave of some limited axiomatic system in mathematics. In mathematics, people invent concepts which is why these concepts may be rigorous. As Einstein and others liked to emphasize, what is rigorous can't be automatically applied to the real world; what is applicable to the real world is necessarily at least partially non-rigorous. In spite of this independence of mathematics and physics, it is clear that physics is revealing a particularly natural and important way to look at many mathematical structures (not only the infinite sums), a way that good mathematicians who don't want to be stuck at the freshman undergraduate level should get familiar with.

I want to end this essay with a few words on self-confidence.

It's often right to be stubborn about some insights and principles that one is sure about, that have worked many times. But P.Z. Myers' fight against the finite value of the sum of positive integers is a random ejaculation outside his domain of expertise. How can one determine that his effective rules – that have apparently worked pretty well in the past – are no longer good in a given situation and that he should lower his self-confidence?

Well, it is a difficult question, especially because the answer depends on what kind of questions we discuss. However, I still want to give you something like two answers to this general question – or two recipes that should replace the answer.

One point I want to make is that in many cases, you should be able to see that the "simple" answer you are tempted to defend (like in Myers' case) is just too simple and the people who disagree with you can't possibly be stupid enough to misunderstand things like the partial sums. I am convinced that Myers must know that the physicists and mathematicians who still regulate the seemingly divergent sums must be aware of the partial sums and the fact that the partial sums of positive integers are neither fractional nor negative. He must know that they're smarter than himself! He must know that they are calculating many things that he doesn't know and these things have been successfully tested in numerous experiments. That should have been a reason for him to avoid writing his low-brow naive eight-sentence attack claiming that the "sum cannot be \(-1/12\)".

The other point I want to make is that the "limits of our knowledge" (and "limits of validity of our theories" as well!) is something we should be interested in, too; these limits should be a part of our knowledge and our theories, too! This thesis is closely analogous to the recommendation to young experimental physicists to never forget about the error margins of the quantities that they measure. The error margins are just parts of the information. If you don't have a clue how large the error is or might be, the mean value becomes sort of meaningless, especially if you only talk about one such mean value (or a small number of them). Your mean value will almost always disagree (at least a little bit) with someone else's mean value and without any idea about the error margins, you won't know whether you should be worried about the disagreement!

It's similar with the limits of your knowledge and limits of validity of your favorite theories. Unless you are a much better string theory (the theory of everything) expert than Edward Witten, all your theories you use and believe ultimately break down at some point or in some "fuzzy region" and it is your duty to have a clue where these boundaries are located. If you "know" a theory but you have no clue in what class of phenomena it may be trusted, you really know just an "uncertain part" of a theory. P.Z. Myers' knowledge about the infinite sums could be OK to get a passing grade in a freshman undergraduate mathematics exam; but he should know that his body of knowledge is not good enough to evaluate sums and integrals that appear in quantum field theories and string theory because he has never calculated a damn thing in these fields!

To summarize, there are numerous reasons that should have convinced him not to write a naive negative text about a topic he has no clue about – he could have avoided a controllable example showing that he has no clue pretty much about anything that he has assertive opinions about. P.Z. Myers included his scream "the sum cannot be what it is" into his Skepticism category which is bizarre – the incorporation implicitly says that the regulated sum is a supernatural or religious phenomenon. This fact unfortunately shows that the scientific skepticism of these atheists is nothing else than the opposition to everything that these average people don't understand with the help of their common sense combined with their mediocre high-school and freshman undergraduate knowledge. These opinions may sometimes be right but they may sometimes be wrong, too, and they're more likely to be wrong if they talk about topics in which the "skeptics'" opinions have been never tested or verified (that's the case of P.Z. Myers and all of physics or mathematics – and beyond).

A small correction Lubos. It is Ramanujam with an 'm' not 'n'. Otherwise the article is very interesting. I have to understand these infinities better!

ReplyDeleteI'm pretty sure it's spellt both ways.

ReplyDeletehttp://en.wikipedia.org/wiki/Ramanujam

It is indeed weird how people like PJ meyers rely so much on common sense. I wish them and their team would all read this post!

ReplyDeleteInteresting, I have never seen his name with an M at the end, and I copied and pasted the name to be sure that no error relatively to the dominant Western spelling is introduced.

ReplyDeletehttps://en.wikipedia.org/wiki/Ramanujan

But it's fun to see that in the Indian subcontinent, people may sometimes turn M into N or vice versa. It seems like a more modest transformation than the Chinese subcontinent's inability to distinguish L and R. ;-)

I have to appologize Lubos. In Sanskrit "m" as an ending makes more sense. But I looked up on web. Everywhere it is "n". There is a mathematician called Ramanujam but he is not the same as the genius Srinivas Ramanujan. you are much more careful and precise than me! Please do not change. This calls for some more research on my part!!!

ReplyDeleteCool, thanks, so the guy with M exists, too.

ReplyDeleteIt was pretty hard for me to memorize the name of Vijay Balasubramaniam, for example. Some Indian names are so long, others - like Ashoke Sen or Das are so short.

Well, it's similar in Czech, too. Like Mr Jaroslav Červenokostelecký and lots of names with 3 letters, too. ;-)

Dear Lubos,

ReplyDeletesome people, such as Bob Jaffe, like to say that the Casimir effect has nothing to do with the zero-point energy or zero-point fluctuations. Instead it is connected to a one-loop effect with external legs (like any other measurable quantum effect), as opposed to the "true" zero-point "bubble" diagrams. I think he is right about this. Furthermore, the Casimir force can be computed without ever having to encounter this infinite sum. What are your thoughts on this?

Dear Fred, I know that paper.

ReplyDeleteThe zero-point energies and the one-loop graphs promoted by R. Jaffe carry the very same information, they're related. Zero-point energies would of course be immaterial unless they were coupled to something or differentiated with respect to something.

The differentiation of the zero-point energy with respect to the distance between the two plates is equivalent to an insertion, i.e. to adding external legs. These relationships are well-known and universal. It's silly for R. Jaffe to explicitly point out his misunderstanding of these links.

Even when it comes to the question of the cosmological constant, the vacuum energy, it would be unmeasurable unless the curved spacetime it causes could have been seen by other particles. So the zero-point energies really manifest themselves once the vacuum graphs are decorated by external legs only.

"The Casimir force may be calculated while avoiding the infinite sum". Yes, no, what of it? This tautological claim - that only says that one may always obfuscate fact, any fact - has no implications for the right value of the sum.

You may also say that exp(x) may be calculated without encountering the sum of x^k / k! over non-negative integers k. But that doesn't mean that the Taylor expansion isn't equal to exp(x). The Casimir force *may* be expanded into the Fourier modes with different wave numbers or momenta transverse to the plates, and when this is done, we get the identity that the sum of integers (perhaps plus some dull regulating infinite counterterm) is equal to -1/12. If we determine the value -1/12, i.e. by a measurement, we will avoid the sum of positive integers, sure. But it means absolutely nothing. So using this claim as an argument against the fact that the finite value of the sum of integers has to be -1/12 is a basic logical fallacy.

1+2+3+... = -1/12 may look like voodoo math but it is not and is easily explained. In fact those summation machines (Euler, Borel, etc) obey 2 rules

ReplyDeleteRule 1: summation property

S(a0 + a1 + a2+ …) = a0 + S(a1 + a2+ …)

Rule 2: linearity

S( Sum(alpha a_n + beta b_n)) = alpha S( Sum (alpha a_n)) + beta S( Sum(b_n))

It is those rules which guarantee the uniqueness of the answer regardless of the method.

Another misconception is on the domain of convergence. If you stick with Taylor expansion you are dead because in complex numbers the convergence domain is a circle up to the first pole, and if the pole is too close, tough luck. With different methods, the convergence domain is not as dumb and the sum may actually be rigorously well defined.

If the sum is well defined, one can then use any summation technique to get the actual answer because of the 2 rules above guarantees uniqueness. Those are nice tricks of the trade for the perturbation theory practitioner.

Here is the same thing explained by Carl Bender.

https://www.youtube.com/watch?v=VvqeJkT3uyo

This is lecture 4 of 15 from a nice series of lectures on mathematical physics.

All the fun is to be had in the footnotes. Perhaps John Baez should weigh in on this. He is monumental (for things not involving empirical chemistry - climate).

ReplyDeleteThe more or less definitive text on divergent series (from the mathematical point of view), Hardy’s “Divergent Series” was published in 1949 ((I have a copy in Russian translation). In other words, this is a well established and completely understood area, no longer subject to any controversies. Things were, of course, somewhat different in the 19th century when Abel wrote that “divergent series are an invention of the devil”, but mathematics has progressed since then and so has physics.

ReplyDeleteBeing ignorant about these somewhat obscure things is of course no shame unless one is a mathematician but expressing confident opinions on things one is totally ignorant about, without even looking them up on the Wikipedia (e.g. http://en.wikipedia.org/wiki/Divergent_series or http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_·_·_· ) suggests that the Iranians maybe partly right: extraterrestrials are really here but they are stupider than us.

Also @ centralcharge15.

ReplyDeleteThanks.I have completed my research!!! I learnt something new about the genius today. We have to accept "n". There is a Ramanujan Mathematical institute in Madras university (the city is now called Chennai) . Since Srinivas was a Tamilian we have to accept the Tamil spelling. Although the correct Sanskrit word is Ramanujam, meaning Rama's younger brother. The way people react to different names is of course base on what they are used to in their language. I would have thought vasavada was easier than kashyap, because of compound letters, but americans find kashyap easier than vasavada (too many 'A' s!!!

One can read the first few pages of Lucretius's reference for free on Amazon, where Hardy gives a nice little bit of motivation and background on how mathematicians over time have viewed and played with divergent series before the modern approach. It's in English too, so that might make it easier for some. :)

ReplyDeletehttp://www.amazon.com/Divergent-Series-AMS-Chelsea-Publishing/dp/0821826492

http://googlefight.com/index.php?lang=en_GB&word1=ramanujan&word2=ramanujam

ReplyDeleteBoth spellings seem to be common. Adding "Srinivas" actually gives Ramanujam the edge:

http://googlefight.com/index.php?lang=en_GB&word1=Srinivas+Ramanujan&word2=Srinivas+Ramanujam

LOL, but if you write the first name as Srinivasa, "N" safely wins:

ReplyDeletehttp://googlefight.com/index.php?lang=en_GB&word1=Srinivasa+Ramanujan&word2=Srinivasa+Ramanujam

I've only ever seen his name written as Srinivas

ReplyDeletea, with that final 'a'. I've no idea what it's status is, grammatically or otherwise. Are your and Kasyap's 'Srinivas' typos, or is there more than one way of writing his name?P.S. I see Luboš just pipped me to it, sort of.

Interesting point Lucretius. Although I agree completely with Lubos' argument that in physics you should get finite answers and then compare with experiments, I would still like to understand more what prominent PURE mathematicians like Hardy think of divergent series. Can you say something more?

ReplyDeleteThis is becoming more and more entangled!!! You are right. Taking Madras university as authority, since he was from Madras, it is really Srinivasa,with last a, although I have seen his name both with and without last a. As you know the ambiguity in writing Indian names in English is that most Indian languages have 48 to 52 letters. English has only 26. Well, with whatever spelling he was an all time great mathematician. Too bad he died so early. Otherwise we would have seen his name in math textbooks.

ReplyDeleteAh! Thanks for the info. I did wonder if there was more to it.

ReplyDeleteYes, it was sad. He merited a fairy-take ending but some people just have really bad luck. Not in the same league of course, but I read recently that Oliver Heaviside died in poverty. I find the older one gets the more these sadnesses impinge on one.

There is a legend, once idely believed (which you can still find repeated in some places) that Niels Henrik Abel (who really was in Ramanujan's class or maybe even Ramanujan was in his) died practically of malnutrition in abject poverty. However, just like in the case of Mozart, there is only a little bit of truth and much exaggeration in the story. Actually, the enormously talented and productive Abel suffered from being ignored during his visit to France by all the famous French mathematicians of the time. This is how Vladimir Arnold describes what happened:

ReplyDelete“The works of Abel, connecting so many different areas of mathematics were submitted by him for publication to the Paris Academy of Sciences, which asked Cauchy to assess them. As the result, Abel’s texts were lost for many years, since the Academy found itself unable to assess them due to their extreme novelty (in exactly the same way the Academy and Cauchy dealt a few years later with the works of Galois). ”

Arnold adds also : “Abel complained that French mathematicians only wanted to teach but not to learn anything themselves: "one of them is only interested in his celestical mechanics (Laplace), the second only in elasticity theory (Poisson), the third only in the theory of heat (Fourier) and Cauchy only thinks about achieving priority in solving all of the outstanding mathematical problems (including Fermat’s problem). ”

Abel returned to Norway and then contracted tuberculosis and died before taking up professorship in Berlin that was offered to him (the legend says that his work was totally unappreciated). But the real creator of of the "abject poverty" legend was a French newspaper which (I am again quoting Arnold): "published a story according to which when Abel finally left for home, he was so poor that had to return to his part of Siberia known as Norway, on foot, walking across the Atlantic Ocean on ice. Actually Abel returned on one of the first steam ships and had enough money to pay for his ticket.”

Oh! I forgot to mention one thing. Any Indian would pronounce his name as srinivas without a. In Srinivasa , the last two letters may sound like long s e.g. ba in ba ba black sheep!

ReplyDeleteYep, many trolls trolling below too fundamental physics news, blog posts, etc vigourously demand that any valid science has to agree with their specific screwed understanding of common sense etc ... ;-)

ReplyDeletePutting the outragously ridiculous post into the Skepticism category to put well established mathematical knowledge at the same footing as supersticious believes, religion, etc as Lumo says in this nice article, is demeaning ant trolling indeed ...

Is this why Einstein complained about the difficulty of the math involved in working out the general theory of relativity? I'd complain too if I have to work out that the sum of the positive integers converges -1/12! I have enough feel for it to know that what ever math involved, it would be dam hard!

ReplyDeleteWell, I'll admit I find it pretty surprising, but yeah, it's true.

ReplyDeleteThis actually has some interesting application to systems with significant "feedback" where the feedback process can be thought of as representing a sum of geometric series. f=1 implies instability, f>1 implies inverse response (that is, positive perturbation leads to *negative* reaction).

I read E T Bell's

ReplyDeleteMen Of Mathematicsgetting on for five decades ago and barely remember any of it now except that it was kind of romantic. I remember reading a somewhat jaundiced review of it but I enjoyed it at the time. I barely remember anything now about Abel but I do know he was in there. As a check I googled the books (two or maybe three volumes — can't remember although they're somewhere on my book shelves) and came across this: http://en.wikipedia.org/wiki/Men_of_Mathematics. Haha! Not entirely flattering. :)As you suggest, I see one needs to take these stories with a pinch of salt.

I read E T Bell's

ReplyDeleteMen Of Mathematicsgetting on for five decades ago and barely remember any of it now except that it was kind of romantic. I remember reading a somewhat jaundiced review of it but I enjoyed it at the time. I barely remember anything now about Abel but I do know he was in there. As a check I googled the books (two or maybe three volumes — can't remember although they're somewhere on my book shelves) and came across this: http://en.wikipedia.org/wiki/Men_of_Mathematics. Haha! Not entirely flattering. :)As you suggest, I see one needs to take these stories with a pinch of salt.

For completeness, here are the links for all Carl Bender's lectures. I highly recommend to take the time and watch them all. Enjoy.

ReplyDeleteLecture 1:

http://www.youtube.com/watch?v=LYNOGk3ZjFM

Lecture 2:

http://www.youtube.com/watch?v=tV-xIhP7VU8

Lecture 3:

http://www.youtube.com/watch?v=_Sm7SNlNUOI

Lecture 4:

http://www.youtube.com/watch?v=VvqeJkT3uyo

Lecture 5:

http://www.youtube.com/watch?v=LMw0NZDM5B4

Lecture 6:

http://www.youtube.com/watch?v=nWKYrpKn3-0

Lecture 7:

http://www.youtube.com/watch?v=pyEQXMKfEnA

Lecture 8:

http://www.youtube.com/watch?v=YGDTW7mNUKM

Lecture 9:

http://www.youtube.com/watch?v=rs2iJugn808

Lecture 10:

http://www.youtube.com/watch?v=vem9Ltth_Hg

Lecture 11:

http://www.youtube.com/watch?v=9OO-gqj3N0c

Lecture 12:

http://www.youtube.com/watch?v=clpHq5Rs8a4

Lecture 13:

http://www.youtube.com/watch?v=ibR4k-HvRm4

Lecture 14:

http://www.youtube.com/watch?v=31TegmZpIyc

Lecture 15:

http://www.youtube.com/watch?v=RqZ0VsEc4F0

Without the "a", it is the North Indian spelling, Ramanujan is South Indian.

ReplyDeleteNo, the "a" is definitely there in the Southern pronouniciation, and it is the short one, like "u" in "but"

ReplyDeleteWell even though I'm an Indian, I can understand why it could be complicated to pronounce "Vasavada", since it is hard to find out whether it is "Vaasavada" or "Vasavaada", or "Vaasavaada" when the name is written in English...

ReplyDeleteYou see, I agree with almost all of it. But unless you are in favour of completely abolishing laws against defamation and slander (and then we disagree) than you cannot in principle oppose court suits brought up by private individuals or institutions acting on their behave for the slander and defamation (of the worst kind) which is always involved in Holocaust denial.

ReplyDeleteActually, it’s not only Germany and Austria that criminalize Holocaust denial. It is also a crime in Poland, probably in the Czech Republic (I guess Lubos knows) and some other former Eastern block country. By the way, I think in the Czech Republic it is also illegal to deny the crimes of communism and Prince Schwarzenberg has attempted to make it an EU-wide crime. I don’t think he succeeded, though. Again I have mixed feelings, but I think this is as justified as the laws against holocaust denial.

It’s nothing at all do do with trying to prevent such things happening again. If this sort of thing is going to happen again to laws will ever stop it, for reasons that should be pretty obvious.

Personally I am in favour of at least Germany retaining the ban on Holocaust denial, not for the sake of the Jews but for the sake Germany’s own honour.

Henning von Trescow, before committing suicide after the failure of the July plot , said: “ Hitler is the archenemy not only of Germany but of the world. When, in few hours' time, I go before God to account for what I have done and left undone, I know I will be able to justify what I did in the struggle against Hitler. God promised Abraham that He would not destroy Sodom if only ten righteous men could be found in the city, and so I hope for our sake God will not destroy Germany.”

Well, (of course I am speaking metaphorically) but since God decided, after all, not to destroy Germany but make it the leading state in Europe again, I think in return Germany has a special and timeless obligation in this matter.

Poland has a law that makes it illegal to “slander the Polish nation”. I have mixed feelings about such laws but if they are going to exist, then I think laws forbidding Holocaust denial are at least equally justified.

First, a point of terminology. "Slander" is defamatory speech, "libel" is defamatory writings. I am uncomfortable with the concept that (1) holocaust denial defames holocaust survivors collectively; therefore (2) any holocaust survivor can charge criminal defamation or bring an action in tort, even if s/he had not been personally named by the denier.

ReplyDeleteI feel there is something missing in between (1) and (2); it upsets my intuition of how the law works.

Eugene, FYI the term "extreme right" to qualify MLP's political party is in debate these days. Journalists still call this party Extreme Droite and she hasn't sued anyone so far. Yesterday there was a TV programme with a short movie debating about this precise point but I didn't get to see it. What is funny is that it makes it clear now which side the journalist is when talking about the FN. Maybe that is what she wanted to achieve.

ReplyDeleteI was only assuming. I thought "Werdna" must surely be feminine.

ReplyDeleteI like the fact that she is offended by that. Personally I am quite happy to be called exactly that (in my own meaning of "right" of course) and and form time to time I do get called that (even on this blog ;-) ). But since on almost every issue Ms. Le Pen and I are on the opposite sides, I am sure the description can't apply to both of us. Also, it is telling that a large proportion of her supporters, whom she inherited from her father, are known to be former Communist Party voters.

ReplyDeleteMay the philosophical quandary here be that our own mental biocomputers are poorly understood, but each type of summation method may be rightly seen as a literal mechanical machine in the form of various types of existing physical computer circuits that may spit out the same numerical answer as a mathematician does? What is there to be skeptical of about a physically deterministic machine producing a reproducible result? The social argument seems to amount to mere semantics. Now where can I find my -1/12 computer program to translate into electrical charges? Does it exist? A Mathematica notebook, perhaps?

ReplyDeleteOh yes, I will definitively have to watch these :-)

ReplyDeleteIt would be a surprise for me. Yes, I tended to assume you were male, because of the focus of interests in some way. ;-)

ReplyDeleteGlad you like it. The lectures are a gem. In those lectures I would have changed only the ending after the WKB is presented and shown the link with soliton theory using reflection-less potentials for inverse scattering, but Mr. Bender ran out of time and many other things were left unsaid.

ReplyDeleteThat is right Lucretius. More and more communists are voting for her now. A lot of her economic programme is socialist and nationalist (yes, national-socialist, but she doesn't like this terminology either for the reasons everybody knows -Nazi-). She rejects Europe and the Euro. She is similar to the UKIP of Nigel Farage even though Farage denies it. I like to listen to what she has to say because her party is the only true opposition in the French political scene. Her father was more liberal than her on the economic side. However she tends to defend small firms against the big CAC40 ones for example.

ReplyDeleteIn France the words "extreme right" is equal to fascists, Nazis, racists, antisemists, xenophobes etc....

Marine Le Pen uses any legal tools she can to fight against this image.

Lucretius, there is a big difference to be called extreme right in the US compared to France, and even Europe.

both right and left wing extremists share the hatred of free speech ...and freedom as well..

ReplyDeleteMy problem with this result is that they guys in the video are smugly selling it as an unqualified statement. It is often the case that in mathematics you want to extend some concept to a larger family of objects and you show that the extended version has some nice properties and it is consistent with the previous version; but you should state very clearly that it is the extended version of that concept. For example, there are some functions that are not Riemann integrable, but are Lebesgue integrable. Because of that, it is common to mention explicitly write "f(x) is Lebesgue integrable".

ReplyDeleteAlso, when dealing with convergence of functions,

you don't say f_n converges to f. You say f_n converges to f *pointwise*, or, f_n converges to f *uniformly*, or, f_n converges to f in *mean*.

As I am not a physicist, I couldn't care less about the physical implication of the results. There are many areas in mathematics where people work with the extended real line and it is natural to work with infinity instead of assigning some finite value. Open any book on convex analysis and you see people playing rather comfortably with infinity.

I have no doubt that the result is correct under the appropriate definitions and notions; but they should cautioned the viewer about that.

Don't say "the sum is -1/2". Say

"the XYZ sum of the series is -1/2" or "the series converges to ABC, in the XYZ mode of convergence". Not being precise in this is misleading.

Hmmm, I was with you (Lubos) right up to the middle of the last paragraph when you gratuitously threw in the word "atheists". Do you think that "believers" would be any better?

ReplyDeleteIf the term just refers to PZ and, say Sam Harris, then O.K...even I think they are overly rude and self-impressed. I could add Krauss.

I think that the problem is with their reliance on "common sense", and is similar to those who hang on to hidden variables in QM. But it also has something to do with intelligence...who would not

stop and think before pitting oneself against Euler

and looking into what the result could mean?

Appeal to common sense is often like appealing to the mob.

We are actually on the same track ;-)

ReplyDeletehttp://www.giantbomb.com/werdna/3005-8642/

Gordon, my sentence wasn't meant to say anything you would disagree with. It was really a suggestion that sometimes the professional atheists are acting and deciding following the same kind of mindless group think that is often associated with religions - just with different labels.

ReplyDeleteYour comment about the war on Euler is a good one - it's another sociological reason to be careful before one jumps to too far-reaching conclusions.

People may be used to employ common sense for many conclusions that are widespread. For example, our history teacher at the basic school would be making fun of philosophers who were trying to decide whether objective reality existed. At that time, I would also think about the world in terms of classical physics, so I was completely on her side: it's so ludicrous to question objective reality.

Well, with quantum mechanics that she couldn't understand, things are different. It's likely that all founders of quantum mechanics - and all people who ever learn it - came through the stage in which they believed that objective reality was inevitably true, a part of common sense. But one must simply try and be able to feel that Euler and Heisenberg may be understanding something that goes "beyond" common sense.

Hi Ctulhu, the video is oriented to lay audience, overly enthusiastic, somewhat too heuristic, and I noticed that the method is slightly different than what I was calling Euler's method.

ReplyDeleteBut this method may actually be supplemented with all the hair so that it becomes a legitimate calculation rooted in some analytic continuation. It is not a coincidence that they got the right result by this method.

So I think it's at least morally right to say that the sum is -1/12 without extra "qualifications". On the contrary I have problems with your description of the situation that involves the words "extending some concept".

This suggests - at least I think that it does - that the "unextended" definitions of the sum (you probably mean the limit of the partial sums) are more true, more acceptable, more agreeable, more profound, more standard. I just disagree with that. The sum's being equal to -1/12 is the more standard, more natural result etc., and the high-school calculations of the sum that would end up with "infinity" or "divergence" are a castrated, modified, reduced, dumbed down version of the deeper maths.

I am familiar with all the types of convergences, pointwise, integrability of integrals, and so on - we had to learn these things in the college in quite some detail and I've never forgotten it - but none of these special words can really tell you which of the types of convergence or integrability etc. is more natural, more relevant, more physical, and so on. You are clearly assuming some answer about the "default" answer to this question and I think that your assumption is just wrong.

I know that mathematicians are satisfied with having "infinity" everywhere. A caculator shows "E" (for error) and you keep on pressing the keys hundreds of times without noticing that what you're doing is really stupid. But the need to get meaningful structures and functions out of these functions - via analytic continuation etc. - isn't needed just for physical reasons because physical quantities are always measured to be finite. It's needed for this playing to be really about the structure of functions and sums and identities. Getting "infinity, nonsense, divergence" most of the time shouldn't really be called "calculus" or "analysis". It's stupid even at the mathematical level.

The conclusion in the previous sentence has a sort of moral character so I can't prove it. But this conclusion of mine is based on a comparison of the amount of true and important results and "hidden treasures of insights" that can be uncovered by each approach to the sum. The games that are satisfied with the verdict "it's divergent, infinity, shut up", don't really uncover any important insights in physics - and in meaningful mathematics. The verdict "it's just infinite, shut up" is fully analogous to "God is great, infinite, omnipresent, and shut up". It's just not a good starting point for learning how the world works even though it may be formally an implications of the axioms. But the problem is that the detailed form of the axioms that these people are insisting upon are "wrong" - wrong as starting points to doing deep and interesting things.

Don't say "the sum is -1/2". Say

"the XYZ sum of the series is -1/2" or "the series converges to ABC, in the XYZ mode of convergence". Not being precise in this is misleading.

Phil Plait is a fool masquerading as a rational being. I have read some of the stuff on his blog. He regularly debunks UFO stories and seems to think that by doing so he convinces everyone he's rational. Great. Debunking an idiot like Stanton Friedman doesn't qualify you to talk rationally about climate science. He's just another left wing toe-the-line alarmist with no scientific credibility. Furthermore using labels like "denier" automatically categorizes you as a cretin. It's time people started replying in kind to this nonsense. Children cannot be reasoned with.

ReplyDeleteNice article, thanks.

ReplyDeleteMy favorite point of view is the following, because it combines a formal math result with clear intuition. It is a theorem* that one may obtain any desired finite value from a divergent series by reordering its terms. Hence, a priori, any finite value is as good as any other or infinity. Additional input is needed to pick the right value. In particular the usual algebra to obtain the series (e.g. the sum over infinitely many oscillators on the string world sheet) tells us *what terms* to sum but not in *which order*. Additional input, for instance symmetry considerations, help us pick out the right order or, at least, the correct finite value of the series.

Best, Michael

* See e.g. Section 12-13 of Apostol's "Mathematical Analysis" or the rearrangements section of Chapter 3 of Rudin's "Principles of Mathematical Analysis." They quote a result due to Weierstrass, that if one has a series that is only conditionally convergent, and one choose two real numbers x<y then there is a rearrangement of the series such that the sequence of partial sums of the rearrangement has lim inf = x and lim sup = y. Choosing x=y we can rearrange so that the new series converges to any real number.

OFF TOPIC

ReplyDeleteMichio Kaku's forthcoming book ''The Future of the Mind'' Look like string theorist are becoming spiritual guru one by one,

http://www.digitaljournal.com/tech/science/is-the-perception-of-reality-a-homocentric-experience/article/366464

http://mkaku.org/

Let's try the mangled part again. :-/

ReplyDelete[...] Each quotes a result due to Weierstrass, that if one has a series that is only conditionally convergent, and one choose two real numbers x<y then there is a rearrangement of the series such that the sequence of partial sums of the rearrangement has lim inf = x and lim sup = y. Choosing x=y we can rearrange so that the new series converges to any real number. Behavior like this is proof that infinite sums need not behave the way we are used to finite sums behaving.

OK, I give up. :-(

ReplyDeleteOne thing I might add is that the Casimir force is not sensitive to the high frequency terms in the sum, i.e., the ones that gives rise to an infinite number of terms. The Casimir force is primarily due to the low frequency modes (the ones with wavelength comparable to the plate separation) and doesn't carry much about the UV modes. In fact the UV modes are not conducted by any real metal. So the experimental confirmation of the Casimir force is pretty irrelevant to all this discussion about how to make sense of this infinite series.

ReplyDeleteThanks for pointing out this lecture . Finally I heard it. It is a great lecture.I wish I had a teacher like Bender. In the last few sentences he answers all the basic questions: why is this correct and why is this useful. He completely agrees with Lubos.

ReplyDeleteAs a mere mortal (a benchtop chemist by training), even with hierarchical entrainment of atomic level order by conscious brain function, this debate is still just glowing photochemical patterns on a smart phone screen, created by the finger muscles of a blogger, controlled by biochemistry in a living brain, the result presented being a conceptual rule leading to a result akin to following a recipe to make dinner. In that, I fail to see the conflict except in wordplay that has no real bearing on the recipe for -1/12.

ReplyDeleteYou are probably referring to a theorem of Riemann (http://en.wikipedia.org/wiki/Riemann_series_theorem) which refers only to conditionally convergent series and says that by permuting terms one can obtain a conditionally convergent series whose sum (in the sense of limit of the sequence of partial sums) is any desired number and also infinity or minus infinity. This applies only to convergence in the classical sense of convergence of the sequence of partial sums. In the case of methods of summing divergent series the situation is more complicated because they also work for sums of series with positive terms (such as the one that is the subject of this post), to which the Riemann theorem does not apply,

ReplyDeleteThanks Lubos.

ReplyDeleteOne thing I find very interesting here is that this is, among many others, a very striking and accesible example where physics enlightens us about math, rather than the opposite.

Don't let the detractors about the social aspects of climate change and macroeconomy get you down. I do agree with them sometimes, but so what!

Hi Lubos,

ReplyDeleteany thoughts on Max Tegmark's book "The Mathematical Universe", where he claims that "mathematics=reality"? I thought this was a claim going back to the ancient greeks, but Tegmark claims he is the first to claim this and wrote a 462 page book about it (apparently stemming from his 2007 paper of the same title; that has obtained 30 citations in 7 years). Thanks.

Thanks for your reply!

ReplyDeleteAs 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.

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.

Can't we all just be friends?

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.

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.

Perhaps, in the

Here's some common sense :-) http://www.toebi.com/blog/mathematics/grandis-series/

ReplyDeleteDear Kimmo, it's really the same low-brow common sense as everywhere.

ReplyDeleteWhen 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.

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.

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.

> and sometimes basic human rights including the free speech

ReplyDeleteAn excellent point. Though you seem to have missed the fact that the papers are still available.

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!

http://scienceblogs.com/stoat/2014/01/17/ship-of-fools/

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.

ReplyDeleteThe 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.

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.

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.

ReplyDelete> Nicola Scaffeta

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?

> Will the climate Nazis destroy...

Godwin! Though you lost on that loooong ago.

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 Wikipedia block log. 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.

ReplyDeleteIt 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.

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 Big City Lib write-up the most informative one.

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.

Nonetheless,

> You have also had other sanctions imposed on you there

ReplyDeleteSure have. Though the relevance of that here is unclear, especially since none related to scientific content.

> our host answer for the actions of the hosts of other weblogs

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.

> The science in the journal in question may also not be up to high standards...

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.

But perhaps you care. Why don't *you* try reading one?

> publishers initially justified cancellation of the journal

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."

This is a little worrying.

ReplyDeleteI 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.

Just a regular male, although that is still an interesting sociological question, I guess.

ReplyDeleteYes, I am most definitely a warlock of the most evil caliber.

ReplyDeleteYeah, makes sense to me I guess.

ReplyDeleteIt 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.

ReplyDeleteWhat I don't get about the "God" believers, Gordon, is that here we are, most of

ReplyDeleteour atoms were made inside exploding stars. They then coalesced, formed a

solar system with stable orbits around a very stable star, (of course

this has almost certainly happened many, many times in many different

places), which eventually led to complex life that in time became

sentient. In a very real sense, we are the universe, maturing,

becoming self aware and beginning to ask questions about where it came

from, where it's going, what it's purpose is. This is all backed by

sound science and with a reality like that, who needs the "God" myth?

What could be more heavenly than that?

This has nothing to do with convergence!

ReplyDeleteThere 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. )

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.

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”.

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.

Well, I have no problem with that.

ReplyDeleteI 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.

"It created itself out of nothing" (the ironically named R. Gott, and others) is interesting but depends on Planck time CTCs.

Well, actually, not anymore it seems.

ReplyDeleteRacist cetaceans, whodathunkit?

ReplyDeleteAlthough 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.

ReplyDeleteTo 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, 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.

ReplyDeleteThe 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.

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.

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.

ReplyDeleteProbably Lucretius, however in those days the people was less educated and the Church was doing the censorship...

ReplyDeleteI 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.

ReplyDeleteWhat 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"?

ReplyDeleteUsing that "sum" in other derivations guarantees that those results won't be compatible with reality.

ReplyDelete