In reference to the attached presentation material regarding the sum of the following infinite series 1+2+3+4+... the following comments are due
• The obvious answer is a huge positive whole number beyond our imagination
• But what if a theory wants the answer not to be so?
• Are we allowed to challenge the theory?
• Are we allowed to bend the rule or even cheat to get the desired value?
• Is this bending, of the rule, only applies for exceptional cases or can be exercised freely?
My conclusions are
- It is a well-known fact that how feeble tricks are used in mathematics to obtain some haphazard values from divergent series to baptize certain theories in physics.
- Three types of tricks are used to obtain the desired results
- These tricks simply erode confidence in mathematics as a sure scientific tool.
- It is a legitimate question that if these flagrant deceptions are exercised to fool ourselves, who knows what other tricks are used to obtain desired results from complicated mathematical derivations?
The main question is; why for heaven’s sake, mathematics needs cheating in dealing with new challenges in science. Either it is not competent enough to cope or it is just a subjugated slave in the hand of any popular theory
Ziaedin Shafiei Dear Ziaedin,
The problem depends on how one defines and accepts a trick. If we understand by a trick a "skillful act" that can be put in simple sequences of logical steps, then it is a mathematical act. If we understand by a trick an act intended to "deceive or mystify" others or oneself, then this clearly cannot be accepted. The problem is accepting illogical ideas put in deceptive bright capsules.
The short answer is that 'tricks' are absolutely not allowed in mathematics.
The 'tricks' you refer to here, such as the infinite sum of the natural numbers being equal to -1/12, are plain outright mistakes - nothing more.
Interestingly, this particular error, for some unfathomable reason, has been used by some people - such as some string physicists etc.
A glaring and frankly quite baffling example of its being trotted out to make a point is by Andrew H. Thomas in the 6th outing of his 'Hidden in Plain Sight' eBook series (Hidden in Plain Sight 6: Why Three Dimensions'.)
Zbigniew Filek effectively debunks this calculation in his Amazon.com review ("Parlour Trick Mathematics") of Hidden in Plain Sight 6.
I find it absolutely stunning that Andrew Thomas called this glaring mistake Quote One of the greatest discoveries of science Unquote. I personally hope he was taking the mickey, but who knows?
It's just all demonstrable nonsense. I would not lose sleep over it. There are real issues in mathematics (the Millennium problems, etc.) that call for resolution. Let's leave the bunkum to those who think it's cool to waste time.
There is a different type of "tricks" which is indispensible for the advance of Mathematics and the sciences which rely on Mathematics. To illustrate such tricks, late Professor Fritz Bopp had this strory: In a pub mainly visited by students they had tablecloth of paper which where often used for scetching ideas or asking challenging questions.
Once a long calculation covered a table starting from a 23-fold integral and working it down to a quite complicated single integral where the art of the guy came to an end so that he left his result on a new paper with a thick question mark.
Next day one found a shorter computation which transformed the result of the first guy into a 23-fold integral, did a coordinate transformation in R^23 that separated the variables in the integrand and so got the solution as a product of 23 factors.
Expressed in general terms, here the trick is to change the point of view so that 'things' become visible that were hidden before.
One interesting aspect of the matter is that (as a rule) it is easy to verify that the trick does what it was invented for, but, that it is impossible to find it in an algorithmic manner.
Dear Ulrich,
Obviously the Prof. Fritz Bopp was not a mathematician because he didn't distinguish what are mathematics and what is calculation. Nowadays the calculation of long primitive integrals (or other operations) are solved directly by compute and if not using different numerical approaches. In such oparations it is normal to use "tricks" as reducing many sums to one only multiplication, but the question is thought for Mathematics as divergent infinite series convergence of Ramanujan, Euler-Maclaurin, Cesaro, Borel... It is true that in such a case, there are also almost always some degree of choosing the grouping terms that could appear as real "tricks", but in most cases the things works quite well and their resuls were employed in many branches of the Physics mainly related with the group of renormalization or with perturbative methods in QFT.
In other branches of the Mathematics there are not so degrees of freedom and no possibility of "tricks" of calculation are impossibles, even the calculation by itself doesn't play one important role. That happens much more in Physics, Chemistry or Engineering.
@Daniel
did you mean 'respondent'? So far I did not manage to learn anything useful from you. You could change this by giving me the complete phrase. Never heard it; the same seems to be true for the internet.
No Ulrich no. I thought that you know the language that you used in your last post. O my God!
HCR “It's just all demonstrable nonsense. I would not lose sleep over it.”
Dear Chris
As you correctly pointed out these tricks are used in theoretical physics such as string theory. But it is also a common practice in QFT. For example, if one tries to find out the experimental evidence for virtual particles one comes across Casimir effect. There are also several offshoots of the theory such as dynamic Casimir effect. As I have mentioned in the attached material, Casimir effect relies on the sum of the cubes of natural numbers to be 1/120. So, the trick we are talking about has helped a few theories to be establish firmly within physics. When we consider the amount of papers written on this subject and zero-point energies of quantum fields then we realise the vast impact of the trickery.
It seems science and mathematics not only have not distanced themselves from this hidden enemy of science but also have doggedly embraced it. What I want to get out of the question is to make sure I am not dreaming this.
Daniel,
don't escape and write down the phrase. Many Latin speakers are waiting for you.
@Ziaedin
Please take into account that there are also treatments of the Casimir effect that dont make use of what Chris rightfully terms 'demonstrable nonsense'. I just consulted the treatment of the Casimir effect in the book 'Quantum Field Theory' by Itzykson and Zuber and found everything OK.
By the way, the formulas for the Casimir effect today are something that even mechanical engineers in the industry work with. I once was called up by a former collegue of mine (a gifted mechanical engineer) who was desperate about a discrepency between practical tests and theory (based on the Casimir effect). When I told him that the strange ornamented h in his formulas means h/(2*pi) everthing fitted together and the collegue was happy.
RPG “Try to understand and enjoy the logic behind these so called "tricks".”
Dear Romeo
I tried to understand but unfortunately, I do not like to be tricked, especially by science and mathematics; magicians? maybe.
Please do not forget that we are supposed to find the facts of our universe in our scientific endeavour. How can trickery support us with this is beyond me.
Dear Ziaedin,
There are two parts in your question:
1. The infinite series convergence that can give mathematical results very strange and difficult to understand as the sum of the natural namber is -1/12. But this is a mathematical result obtained by Ramanujan and in fact started by Euler. This is true if accept the concepts of infinity sum in this serie.
2, But the application I QFT is more questionable and difficult to say that in fact is necessary to employ these results. The only thing is that they simplify the calculations. If you look at Casimir effect:
Wikipedia(http://en.wikipedia.org/wiki/Casimir_effect, part "Derivation of Casimir effect assuming zeta-regularization", version from 13 January 2014, 20:07)
you can see that clearly shows the most important point for solving the infinite series convergences in QED:the regularization ! The Wikipedia article does not merely sum 1^3 + 2^3 + 3^3 + 4^3 + ... = 1/120. Instead, it sums 1^(3-s) + 2^(3-s) + 3^(3-s) + 4^(3-s) + ..., where s is a (complex) regularization parameter. In the end, s=0 is needed, but already at an early point of the calculation, this parameter s is introduced in order to make the calculational steps mathematically well-defined. The sum which appears only converges for s>4 (or, rather, the real part of s must satisfy Re(s)>4). Under this condition, the sum yields the Riemann zeta function, 1^(3-s) + 2^(3-s) + 3^(3-s) + 4^(3-s) + ... = zeta(s-3), for Re(s)>4. So we get the result for the Casimir energy as a function of the (complex) regulator s. This function is analytic for Re(s)>4, so it can be analytically continued to other values of s, including s=0 which is the desired value for s where the regularization disappears. This function value, zeta(-3) = 1/120, is known, it can be derived by techniques from complex analysis. But zeta(-3) is *not* given by the divergent sum 1^3 + 2^3 + 3^3 + 4^3 + ... which has no well-defined value. Thus it is difficult to accept the results of Wikipedia as it stand.
@Daniel
Fair enough, you forgot your Latin.
In future, please, follow these rules:
Respond to any of my contributions only in order to correct errors.
This would save me the anger that I can't avoid feeling when reading narrowminded epistels such as that on Professor Bopp's understanding of mathematics.
Don't ask questions to me, since painful experience told me that you never would understand my answer the way it was intended.
This rules should avoid the perpetuation of the offensive tone into which all our discussions slipped after a few rallies.
@All
BTW the initiated will recognize Professor Bopp's story as an exaggeration of the genial trick in doing the integral from 0 to infinity of exp(-x^2) dx.
Dear Daniel
Ramanujan and Euler were great mathematicians but it does not mean we should agree with their known feeble ploys. In the attached document, I tried to summarize all three steps used for this tricks, to obtain a finite result.
Please also note any other methods, say, using Riemann zeta function, rely on exactly the same sleight of hand. I have shown this at the end of the file.
@Ziaedin
It is certainly not completely wrong to say that Ramanujan was a great mathematician. But, without additional information, this gives a wrong impression. Ramanujan is said to have not known the rules of a mathematical proof and would have not accepted such rules anyway. That great achievements in mathematics on such a basis are possible is even a larger miracle than that traditional mathematics works.
Rich and insightful information on Ramanujan can be found here:
http://blog.stephenwolfram.com/2016/04/who-was-ramanujan/
Dear Ulrich
I mentioned Ramanujan only because Daniel mentioned him and the three steps derivation of the infinite sum is attributed to him.
What I meant was that we should not argue under banners-of-names or formulas but for the integrity of mathematics and science. If someone uses a trick here and there, just to get a desired result, then what is left to be trusted.
Moreover, if we know this practice is wrong, then we should challenge rather than bend over backwards to support it only because it has come from a great scientist or mathematician.
Unfortunately, it seems all are jolly with the results and we have ended up with several established theories.
Dear Ziaedin,
I fully agree with you that scientists should never accept a mathematical argument that is based on illegal transformations such as reordering of non-(absolutely convergent) series. I never encountered such an mathematically wrong argument in a respectable publication. However, in these days there are much more non-respectable sources than used to exist in past decades. Your statement is certainly not supported by any respectable source. Unfortunately I don't know string and brane theories from respectable sources and therefore I'm not much more concerned about much of the nonsense to be read in secondary (or ternary) sources than I'm concerned about much non-sense in my newspaper. However, what has to be said is that your compilation 'Tricks in Mathematics ...' is fun to read and very instructive. Thank you for it.
Dear Ziaedin,
Numerical calculus is a part of mathematics full of intuitions and difficult to distinguish sometimes if there is some "trick" behind a given result. You have introduced many different interesting mathematical results in your file. but you started with the sum of natural numbers. This is the Riemann zeta (-1) (using regularization procedure and analytical continuation) number associated to an infinite serie converging to -1/12, but curiously it is nowadays better known within the infinite series of Ramanujan (at least for me and for my nearest neigbours in the faculty). It is obviously one spectacular and not easy to believe result, but not difficult to prove its validity (at least following the associative properties of the integer numbers). I remember some years ago to be expending some time with this issue and even here in RG I had participate in one interesting discussion about this result. It is clear than some people, in his full right, can say that this mathematical result is not right as Chris Ransford. Obviously I don't agree, but I understand it, although sometimes pehaps it would be good to present a reference where it is shown or better to show it, if possible.
What is very curious is that although the result of Ramanujan for this infinite serie is the same than the Riemann zeta (-1). In fact he used the method to isolate the constant term of the Euler-Maclaurin formula which uses Bernouille numbers instead of the Dirichlet eta function. This last one was the used in fact in the regularization and renomalization in QFT and sometimes not in a proper form, even in classical texts used for several generations of students.
In summary, I was saying that the mathematical result obtained in different forms in mathematics as the sum of the natural numbers is for me well shown and what is not well done is its application mainly in some aspects as the Casimir effect. For such aim I was chosen the wikipedia where everybody can understand what is my criticism. I'm not religious to accept by just faith of great authorities, as Euler or Riemann, because it is quite stupid to do it in science and more in mathematics, without trying to enjoy to follow their fantastic work done. It is a privilegious to name them after understanding their work and without necessity of trickery for obtaining their results.
Dear all,
there is this other thread related to the topic
https://www.researchgate.net/post/Is_1_2_3-1_12_How_can_sum_of_positive_numbers_make_a_negative
I think it is worth to have a look.
Things are so simple (after the work of generations of gifted mathematcians):
There are two interesting functions:
f(z) := 1+2^(-z)+3^(-z)+4^(-z)+... for Re(z) > 1
g:= analytic continuation of f to a holomorphic function on the complex plane with the point z=1 removed.
We have:
g(-1) = -1/12
and
f(-1) not defined (or infinity if one wants to work with extended real numbers).
Of course, f is the Dirchlet series viewed as a function and g is the Euler/Riemann zeta function.
The idea that f(-1) should equal g(-1) ( and thus 1+2^(-1)+3^(-1)+... = -1/12) is plainly wrong and with the mathematical experience of today really silly.
Why should the analytic continuation of an expression be representable by the same expression????
Dear Ulrich,
let me understand, so according to what you just wrote
the sum of natural numbers =-1/12
is devoid of any meaning..does not make sense
My dear Ulrich,
Your simple mathematics fail. The analytical continuation that you have made is not so simple. For computing the zeroes it is not possible use the normal definition of the Riemann Zeta function. You cannot do it so straightforward, all that you have written is a mistake or error, as you want to call it.
Dear Stefano,
it simply is a wrong statement in the sense of logic (you remember: 'tertium non datur'). That it is a statement with a history and that it is based on observations and considerations of deep thinkers (such as Ramanujan) is a different matter. But we can't hold Mathematics (one of the greater endeavors of mankind) in good shape if we leave mathematical truth to the dicretion of ignorants (fortunately the impression that this is what's just happening comes more from the internet than from the science community).
The analytic continuation of a zeta function, zeta(s), to the manifold where all the non-trivial zeros have been found so far, you need another function. After substracting zeta for the auxiliary function you have
Zeta(s)=(1-2^(s-1))eta(s)
And s is associated to a winding number n
s=1+2 pi i n/log 2
and this means that you always can find removable singularities for finding your analytical continuation. But you need to introduce this condition before claiming that you have got an analytical continuation.
Dear Ziaedin,
Interesting analysis, however we have not to forget next quite simple rule:
- we are not allowed to re-arrange the terms of a non convergent series
So, all presented tricks are definitely wrong.
As for your main argument: I fully agree with you and I add that, instead of trying to re-normalize something, it would be better if we use another theoretical frame, without such a need.
Daniel,
did I say that the analytic continuation is simple to obtain? What counts is that it is uniquely determined as a holomorphic function on C\{1}. That analytic continuations, if existing, are unique is common place. My reference for this is a function theory book by Konrad Knopp who conjectured the globally convergent series of formula (21) in this instructive article:
http://mathworld.wolfram.com/RiemannZetaFunction.html
Further, did I say anything about ways to compute the zeros of the zeta function? Instead of stating you better tell me a single wrong statement that you can identify.
BTW in the world of holomorphic or meromorphic functions there are no removable singuarities. Either singularity or not, tertium non datur.
Since again I slipped into Latin, may I ask you to try to refresh your memory and complete your partial Latin citation?
Ulrich wrote:
We have:
g(-1) = -1/12
and
f(-1) not defined (or infinity if one wants to work with extended real numbers).
Of course, f is the Dirchlet series viewed as a function and g is the Euler/Riemann zeta function.
The idea that f(-1) should equal g(-1) ( and thus 1+2^(-1)+3^(-1)+... = -1/12) is plainly wrong and with the mathematical experience of today really silly.
Why should the analytic continuation of an expression be representable by the same expression????
And I thought that he has (better he believed) introduced here the analytical continuation. I must say that I don't understand that you have made and this is for me a nonsense. If you tried to prove something this is a mistery for my intelligence !!!
Daniel,
OK, you don't understand the normal mathematical idiom, which defines an object (here the zeta function) by application of a uniquely defined operation (here the analytic continuation) to a well defined object ( here the Dirichlet series), without giving a constructive definition of the operation.
I was quite sure that you would be unable to understand this since in the near past I had to experience that you were unable to understand a simple chapter in one of our best and most pedagogic quantum mechanics textbooks. So I will not again make the mistake and start a endless loop of fruitless arguments.
As you suggest it's probably not too wrong that it has something to do with intelligence, at least with the kind of intelligence that one normally finds in theoretical physicists, and which allows them normally to engage in vivid and fruitful discussions.
What I wanted to prove, and have proved, is that 1+2+3+..=-1/12 is not a logical consequence of any true mathematical proposition, but the result of misunderstanding the well-known facts on Dirichlet series, analytic continuation, and zeta function. For the convenience of the reader I summarized these facts in the shortest possible form and added for details the link to the mathworld page. In this page I found the Knopp formula (21) a jewel. So, at least for me, plunging into this matter was beneficial, and the thank for this goes to Ziaedin.
There are no tricks of falsity like the harmonic series that equals -1/12. Such things are acts in which tricksters play to entertain naive audiences. Magicians display sensual presentations which mesmerize audiences but all are tricks.
Chopping a horizontally lying person in two pieces and bringing the person alive is a trickster putting the person in one side of the table alone but seemingly stretching legs on the other end to look like the person is lying.
Mathematics is a discipline of truth, not tricks. There are no tricks out there in the universe that make things function as we see them but truth.
Dear Dejenie,
if a german mathematician tells his friend that he found a nice trick to solve some problem he does not imply to have done something that comes close to fraud. Rather he means to have had a happy idea that would not arise automatically by mere systematic thinking. My first contribution to this thread was about tricks in this sense. My question to you is whether in the US such an interpretation of 'trick' is also common or would be considered strange.
Dear all,
the sum of natural numbers is indeed infinite, otherwise we had found a counterexample according to which the criteria of convergence of series do not work and this would be a very serious issue!!!!
But so far the convergence works very well, also for integrals we use everyday!!!
It is also true that arbitrary re-arrangements of numbers, like associative properties are not suitable/defined for infinite sum of numbers, so in principle they lead to unpredictable results.
Casimir was certainly a genius by applying the Ramanujan/ Reimann, Euler (Zeta function) pseudo-sum computation to a physical problem and find that the outcome of the model fits the measurements.
The re-normalization is an "incredible" practice and has to be applied as it is *as long as it works*. On the other hand it has to be considered *abnormal*, (abnormalization...), handle with care and think twice...
Such practice has infact drawbacks: it is an invitation/exortation to Physicists to give up with an actual understanding of reality, it is like a black box approach with a *magic* number crunching in it.
IF no further investigation is made in order to understand why such methods work, we may continue to miss a wide part of Physical reality, since we cover instead of discovering pheonomena!!!
Dear ulRICH,
Let me start with some of your sentences, which curiously deserve the vote of some "scientists" of RG:
you don't understand the normal mathematical idiom,
Although I'm mathematician and physicists, you can look at my publications for seeing if I understand the language of mathematics (not your language of mathematics).
I was quite sure that you would be unable to understand this since in the near past I had to experience that you were unable to understand a simple chapter in one of our best and most pedagogic quantum mechanics testbooks.
Yes, you were trying to explain that the second law of Newton worked in quantum mechanics thanks to the Ehrenfest. This is, besides ignorance a lack of clear intelligence of what physics is. But you can find tunnel quantum effects everyday writing without the minimum of politeness and from a podium that sombody rent you as ulRICH the great. In any case, at the end I'm sure that this posts are written and there are many strenght intelligent people who can distinguish each one.
I share the understanding of mathematical trick with Ulrich Mutze (Rather he (german mathematician in original post by UM) means to have had a happy idea that would not arise automatically by mere systematic thinking). In my opinion the "trick" in mathematics is rather a lucky association of one or more ideas from seemingly unrelated area(s) making possible to perfectly solve the original problem. This is often how sometimes the theorems are proved (but not necessarily lemmas). The examples shown at the beginning of this thread (S = -1 +1 -1 + 1 -...) only show that the number S is poorly defined at least, those are no "tricks". We should treat them rather as clever counterexamples to the statement "presented infinite sum has a well defined value", not "tricks".
Daniel,
specifically I spoke about one mathematical idiom, that I characterized both in general terms and in terms of the problem at hand. From the grotesque way you misunderstood my reference to analytic continuation: I indeed concluded that you lack experience with the more abstract levels of mathematics.
Dear ulRICH,
What I tried to say is that is not possible to reach a contradiction writing the things you have done and making analytical continuation is not so simple as writing just functions. You must be sure to avoid the singularities on the complex plane introducing an auxiliar functional equation that in your "theorem" doesn't exist or my eyes didn't see it
f(s)=2^s ᴫ^(s-1) sin(ᴫs/2) ᴦ(1-s) f(1-s)
(this ᴦ is capital gamma that I don´t get it higher)
I rewrite for your memory what has you written, without taking into account anything about this necessary things (recognizing your high level of mathematics that a simple mathematician as me cannot reach following your ulRICH criterium)
Things are so simple (after the work of generations of gifted mathematcians):
There are two interesting functions:
f(z) := 1+2^(-z)+3^(-z)+4^(-z)+... for Re(z) > 1
g:= analytic continuation of f to a holomorphic function on the complex plane with the point z=1 removed.
We have:
g(-1) = -1/12
and
f(-1) not defined (or infinity if one wants to work with extended real numbers).
Of course, f is the Dirchlet series viewed as a function and g is the Euler/Riemann zeta function.
The idea that f(-1) should equal g(-1) ( and thus 1+2^(-1)+3^(-1)+... = -1/12) is plainly wrong and with the mathematical experience of today really silly.
Dear ulRICH,
I apologize if my language with you is sometimes too aggressive. This cannot be never justified when the discussion is between two scientists, assuming rational behaviour. But I must say that your sarcasm is for me excesive and precludes a normal argumentation. If you sistematically ask me to write in latin, or to show a needle compass (made with 1T on your gardin) as the achievement in magnetism or to read a sentence of the Sakurai out of the context. This can be interpreted as mocking people trying to expend time and no to really answer a question sincerely.
ulRICH it took me some time to understand you but as you can see if this is your fun, why not? I can do exactly the same than you or even perhaps to try to do it better. You can follow with your humour and I'm grateful for having some relax after different jobs. Writing a post as your last one is to laugh all the rest of the day, even if you put an angel face in the photography of the post
specifically I spoke about one mathematical idiom, that I characterized both in general terms and in terms of the problem at hand. From the grotesque way you misunderstood my reference to analytic continuation: I indeed concluded that you lack experience with the more abstract levels of mathematics.
Dear Stefano
Please note we are not denying the brilliance of Ramanujan and Casimir or any other mathematicians or scientists. The concern is about the methodology and veracity. It is not acceptable that we ignore the conditions we introduced ourselves and this is exactly what is happening with the tricks we are dealing with.
As it pointed out in page 13, if we accept this kind of cleverness, mathematics becomes a junkyard. For example, we agree that sin(x) = x for very small angles and then somebody comes along and say, what if we just ignore the condition and use sin(2)=2 because I need it to be so, and we say, why not, if you can use it in your theory. Good luck.
The second problem is the possibility of existence of countless numbers of wrong results, i.e. please see page 16.
We can now realize another danger of this approach, the critical teeth of mathematics are pulled out by this type of trickery and this is detrimental to science.
Dear Ziaedin,
I think that there is a great confusion about the infinite series applications in Physics. For instance, Casimir never employed the Ramanujan result or Riemann zeta functions. He only employed the Maclaurin series usual in complex analysis. I attach you his original paper, written only some mounts to a previous one using the dipole-dipole interaction and which is nowadays in fashion in condensed matter for nanotechnology.
The other application is also false. The strange (although mathematically well obtained) was only applied in the old bosonic strings of 26 dimensions that is nowadays absolutely out of the reallity, mainly due to its lack of real fundamental state well defined. In superstrings (with only 10 dimensions) it is not necessary to employ these functions at all for their regularization or renormalization within the supersymmetric algebrae, although some publications has had done for simplicity and elegance if the calculation results. But they are not necessary.
Thus, there are not tricks at all in the mathematics involved and well known, and its defense with the application in Physics is not well sustained.
Dear Ziaedin,
I have been scanning your quite interesting list of mathematical rule benders, and I am in complete agreement that mathematical logical rules never should be bent.
There is a quite contemporary case that I know of that in my view should belong to a fourth type:
4. Restricting the domain of a formula.
I have been waiting for an opportunity to submit this case to a mathematician first and foremost to have his opinion. This thread seems to provide such an opportunity.
There was a discussion a while back on RG about the energy-momentum equation that Dirac developped to represent the total energy of the electron in motion.
His original form was E=sqrt(c2p2 + (moc2)2) + V , (V representing potential energy)
Dirac built this equation by combining 2 simpler relativistic equations (See pdf below).
It so happens that for decades, the energy-momentum equation has been deemed to be true without the (+ V) part. Ref: wikipedia and quite a few formal references:
https://en.wikipedia.org/wiki/Energy%E2%80%93momentum_relation
No reason could be found anywhere for the disapearance of the (+ V) part.
Intrigued by the absence of (+ V) that Dirac deemed was part of the equation, I tried to locate Dirac's original derivation, but found it nowhere.
The closest source I chanced upon was a mention in Sears F, Zemansky M, Young H (1984) University Physics, 6th Edition, Addison Wesley (Reference [28] mentioned in the pdf), simply mentioning the two initial equations and stating that when combined will produce the energy-momentum equation but no derivation was provided.
So I decided to proceed myself to the derivation to identify the discrepancy.
I submitted my derivation to the thread where this was being discussed and found that all physicists consider that the gamma/gamma ratio to the right of equation (B4) must to be simplified to 1, which results in non-relativistic equation E=sqrt(c2p2 + (moc2)2) (without the + V component that Dirac rightfully included -- for reasons that I can completely elaborate on if you wish). Relativistic mechanics equations are deemed "relativistic" if they involve the gamma factor, and the energy-momentum equation is supposed to be the ultimate relativistic equation.
I am of the opinion that the gamma/gamma ratio to the right of equation (B4) must not be simplified to 1 (thus eliminating the non-linear relativistic growth function from the equation), which is the logical rule bender that I see here, but should rather be squared to be correctly re-associated with the mo elements that they were initially associated with in the initial equations, that now are squared.
My conclusion was met with general disapproval from all physicists on RG and also in private conversations.
I would like to know your opinion from a strict mathematical viewpoint.
Best Regards
André
Dear Daniel
Totally agree with you, that is why in page 4, I have mentioned “Modern derivation”. But, Casimir could not escape the dilemma anyway and this is clear in the paper. For example, “In order to obtain a finite result it is necessary to multiply the integrands by a function…”.
I cannot comment on why a new derivation was necessary for his theory but it is there and is not proposed by crackpots.
Please also note this practice is not limited to a one or two cases. For example, see sixty symbol video.
https://www.youtube.com/watch?v=bHqkl51KzmM&t=37s
Dear Ziaedin,
in the Ramanujan and zeta problems, the answer to the divergent series is always unique and there are not "tricks". Going to particular L-loop diagrams of QFT, it isn't unique but the infinite part can be subtracted for obtaining something which is a finite part able to be compared with the experiments. This is the main interest for calculating effects as Casimir.
In the bosonic strings there is wonderful result that every body likes if accept the Ramanujan mathematical result
( D-2) (Sum natual numbers) +1=0
implies the dimensionality D must be 26. This is beautiful but with few practical resultas
Dear André,
When you write E=gamma E , this implies that you are transforming a relativistic problem in one classical one, i.e. instead of the Lorentz group of transformations you obliges to use the Galilean one.
There are more things to say but this one is fundamental for your question.
Dear Daniel,
I understand your argument, but final E simply becomes a larger quantity of the same energy "substance" as initial E, the difference being related to the varying energy increment related to the varying velocity of the electron.
If you wish, Efinal = Einitial + ((gamma Einitial) - Einitial)
I thought it obvious that left E of (E=gamma E) was an increased amount of energy by a factor "momentary gamma".
Best Regards
André
Dear André,
Such condition is only in your imagination of the physical process but when you do gamma =1 immediatally v
Dear Daniel,
gamma does not equal 1, so your argument is meaningless.
In context, it is a mathematical function equal to 1/sqrt(1-(v/c)2), c being the asymptotic velocity limit for a moving electron.
Energy is energy, whether increasing linearly or exponentially.
I already have had this discussion with numerous physicists and won't indulge again. I already know your opinion. Noted and filed. Thank you for your contribution.
I am simply looking forward to Ziaedin's opinion, or other true mathematicians' opinions from a pure mathematics perspective.
Best Regards
André
Dear Ziaedin,
infact I wrote:
Such practice has infact drawbacks: it is an invitation/exortation to Physicists to give up with an actual understanding of reality, it is like a black box approach with a *magic* number crunching in it.
IF no further investigation is made in order to understand why such methods work, we may continue to miss a wide part of Physical reality, since we cover instead of discovering pheonomena!!!
Dear Stefano,
I think that your last message is with a lot of degrees of freedom and can suggest that we have in this issue many thinks to discover. I don't think so, for me this is this kind of spectacular things that we obtain when we work with concepts as infinity. This concept defined as the set which is equivalent to one of its subsets has leading to the concept of transfinite numbers, alephs classification and so which were without any kind of application. Here we are involved with sums of infinite terms that is very difficult to find a physical application, only when you can translate problems of renormalization or regurization is this justified, but even in such a case is not more than a mathematical method.
On the other hand, let me to say that everybody here coincides, that there are not tricks in mathematics, at least as I interpret what Ziaedin want to say (Ulrich and Marek explained one accepted meaning in the same logic as when my children say that they came to rober me some money for going to a party). Within the synonimes of this word never is idea or solution but we understand perfectly what we want to say with "trick" in mathematics. Chris started this thread saying this very clearly and fully agree with him in this aspect.
In summary, Stefano I have presented the two parts where usually people says that these concepts ara used for explaining that, in practice, they are not necessary and which only cover an elegant form (or sophisticated and mysterious) to present some results. Mathematically this issue is quite serious and difficult to justify while physically is quite restricted its application from my humble point of view.
Dear Daniel,
>
We found an "effective" way to connect things, but we are giving up the actual understanding of why such method works.
I've been involved in several exams of *system identification* with black box approach (find math models of dynamical system which could fit I/O of acquisition from real systems) at university . I remember when we modeled complex systems with an insufficient number of variables, we had divergent behavoirs which in some cases were modeled as an "external disturbance" in the control chain. This modeling was not eventually satisfactory, and the only solution was to increase degrees of freedom inside the model itself.
The first question should be :
are we really sure that renormalization does not hide some degrees of freedom which the actual models of Physics neglected? Isn't it a way to patch non suitable models which reached a limit?
Are we trying to extend a non suitable model to work where it should not??
Why should phyiscal phenomena apply the associative rule in that way?
Let's set aside math which tell us clearly that the series of the natural numbers diverge. What is done with the Zeta is a manipulation which works eventually. But we have to acknowledge that It is totally arbitrary to replace a positive integer divergent arbitrarily large number with a negative fractionary finite number.
Is this the victory of "shut up and calculate" and "the ends justifies the means"???
Dear Stefano,
I don't agree with your sentence:
we have to acknowledge that It is totally arbitrary to replace a positive integer divergent number with a negative fractionary finite number
although I understand what you want to say, which is quite reasonable although don't follow properly the abstract mathematical rules. Please don't take me as a pedantic person and let me try to explain it trying to show how deep is necessary to make what you call "totally arbitrary to replace".
The zeta Riemann functions were (partially) made by one absolute genius as Riemann. The schematic idea is, you have a real derivable function f(x) that you want to extend its existence to the complex plane (i.e. with one dimension more) where f(z) is holomorphic. Thus you need to go to one new domain with can present branch points: algebraic branch points, transcendental branch points, and logarithmic branch points. That is to say the topology of the Riemann sheets can be a highly non trivial monodromy with an essential singularity (this the falls in the domain of the algebraic topology).
After defining somewhere the domain D where the function is well defined. We might figure out a way to construct another function g(z) that is defined in a larger region such that f(z)=g(z) whenever z is in D. So the new function g(z) agrees with the original function f(z) everywhere f(z) is defined, and it is also defined at some points outside the domain of f(z). That is, the function g(z) is called the analytic continuation of f(z) within so restricted conditions. However, most infinite series only converge for some values of z, and it would be nice if we could get functions to be defined in more places. The analytic continuation of a function can define values for a function outside of the area where its infinite series definition converges. We can say 1+2+3...=-1/12 by retrofitting the analytic continuation of a function to its original infinite series definition,z.
Going to the zeta Riemann function more concretely This infinite series doesn't converge when s=-1, but you can see that when we put in s=-1, we get 1+2+3…. The Riemann zeta function is the analytic continuation of this function to the whole complex plane minus the point s=1. When s=-1, ζ(s)=-1/12. By sticking an equals sign between ζ(-1) and the formal infinite series that defines the function in some other parts of the complex plane, we get the statement that 1+2+3...=-1/12.
I hope that you can see how many different branches of Mathematics are implied here: numerical, complex, and algebraic topology mainly. All that is done in the literature of Mathematics with extraordinary care and where is was not so well introduced is in Physics, as I have shown.
merely those tricks which are demonstrated to be true are allowed: call them theorems. most often they are used to solve problems faster.
Not only tricks, we need the magic of mathematics!
Mathematics can only deal with quantity not the quality of the objective reality. But it is only quality that can represent the enormous variety and the phenomenology of the universe! Quantity in a limited way is useful for the dynamics of the universe where quantitative change leads to qualitative leap in the evolutionary process of everything in the universe.
Mathematics is miserably limited even in the realm of quantity, because it can deal only with very simple and gross systems of everyday life and classical mechanics to Newtonian mechanics by extending iteslf a little more using statistical methods. It fails totally even in a bit more more complex terrestrial phenomena like the biological systems, for example.
Beyond everyday life experience, i.e., in the microcosm and macrocosm, mathematics has to abandon the material realm to transcend to the realm of thought (to gain some qualitative attribute), through the unlimited extension of idealized mathematics; where the materialist basis of objective reality has to be replaced with abstract Einsteinian “continuous fields” (spacetime, Higgs, quantum etc.) - matter becomes myth, mathematics turns into mystery or magic and science turns into theology!
This proves the dialectical law that any truth (mathematical truths included) when extended beyond certain limit, either turns into its opposite (myth) or becomes an absurdity. The worshipers of idealized mathematics especially in modern theoretical physics set no limit to their trades and can very easily roam around from the microcosm to the macrocosm riding on the magic carpet of mathematics just to satisfy their fantasies and become great “scientists” in the bargain!
Dear Daniel,
>
your explanation has been brilliant. I actually could not explain it better, although since in my studies I went through complex analysys in some exams and analitic continuation, the topic is not new for me at all and I knew exactly what it was about.
If I use analitic continuation, which is has been considered a lawful practice, I get some results and this is undisputable. But at the same time in this case the result clashes with classical convergence of series.
So it has to be said that the Zeta function allows us to replace divergent series with a number but it is not true that such series is equal that number. Such *substituition* allowed us to get rid of infinties, otherwise not treatable and give verified results.
Since it is never true that a infinte series is equal to a negative number, otherwise I would have to rewrite all the series convergence from Gauss, we have to find why the Zeta function transformation, or RiemannZeta transform works, or gives the right answers to the problem of infinities. What is behind it?
It seems that for the most it is not the case to ask themselves such question!!!
Dear Daniel
We can get a finite value for ζ(−1) just because of the following equation
-3 ζ(−1) = η(−1) = 1/4.
But η(s) only converges for any complex number with real part greater than zero. The trick is that we try to forget about the condition and forge a finite result for η(−1) by throwing away the divergent part. Please see page 35 of the attached file for the true result.
If the finite result of this ploy is used in “many different branches of Mathematics” it does not make it more defensible.
Dear Ziaedin,
I'm not sure to understand you. Obviously there is a relationship very well known among these "functions"
η (s)=(1-21-s ζ(s)
Thus in your expression (you have forgotten a minus sign in front of the 3) is
- 3 ζ(−1) = η(−1) = 1/4.
right. What is the problem? The difficulty of all these functions is the one that Stefano has said: all one infinite "sum" converges to an only number can be even so strange as the sum of natural number: the result is not another natural number and besides it is negative. Algebraic aperations among their, once the series have been well defined and converged, are without any kind of difficulty. Isn't?
If I told many branches of Mathematics is because in the faculty of Mathematics in my University these studies belong to different departments and they publish in very different reviews. For instance algebraic topology and numerical calculus.
Thanks Daniel, I added the minus sign in my past answer.
Off course the result is strange but it only should be acceptable if one does not throw away any other number, in this case infinity, or ignore the initial condition when introducing a series.
In this case we first try to find a finite value for η(−1) and then use it in
-3 ζ(−1) = η(−1)
to obtain our desired finite number for ζ(−1). But η(s) is initially defined for complex numbers with real part greater than zero. We simply ignore the condition and this is the trick.
Please see pages 12-16 and then page 35 of the attached file to the question.
Dear Ziaedin Shafiei,
THe condition of positive Re[ ] is due to the fact that in the case of a non positive Re[ ] the analitic function would present a singularity, a pole.
But since that condition regards a limited part of the plane, the analytic continuation allows us to go around such singularity and make the olomorfic function to be continued and reach the negative number -1.
https://www.youtube.com/watch?v=sD0NjbwqlYw
So what does ζ(s)=-1/12 means?
this ζ(s) (not =) 1+2+3... is not an equality, it is a correspondence which the analytic continuation allowed us to find and we found that this is effective in cancelling the infinities in Physics. It is like a transform:
1+2..+n gets "transformed" into -1/12 with the help of the analitic continuation of the Zeta function.
We are basically using something which we don't know why it works in Physics!!!! It just WORKS, that is the problem!!
And the worse is that I make an effort to see somebody humbly admitting that it would be the case to understand why it works...
Dear Ziaedin,
There are not tricks at all. What we have are a lot difficulties to avoid singularities for assuring you convergence of the series. Topologically speaking the real line is always simply connecte and therefore without singularities. But if you pass to the complex plane you have \pi_1(S^1)=Z for homotopy group. Thus it is not trivial topologically and you need to restrict to one region where this happens. This is the problem but this can be always done properly.
But if you want to do mathematics I can put you an exercice, do analytical continuation not in the complex numbers but in the quaternion numbers (i.e. S^3 instead of S^1) (very important in magnetism because they represent the spin ). The first thing to observe is now the condition is much larger because the singularities are associated to the homotopies
\pi_3(S^3)=Z
What do you think about? What is the trick here? Please, go ahead!
Dear Stefano and Daniel
Whatever the reason for a condition we should stick to it and not abandon it if we find the going gets tough. As you mentioned this is a sign of profound weakness which should be addressed and not ignored.
DB “What we have are a lot of difficulties to avoid singularities for assuring you convergence of the series.”
If a series is divergent why we should not accept it and do not try to turn it to a convergent one?
By the way, this is how I initially wanted to ask the question but for obvious reasons thought otherwise.
“My trustworthy friend who is a QFT physicist borrowed $1 from me on 1st January. He asked for $2 on 2nd Jan, $3 on 3rd and so on until the end of that month. After getting his pay check he offered to settle his debt. He sat me down and calculated his debt as follows:
1+2+3+4+ … = -1/12
I was impressed and convinced that if I pay him just 8¢ his debt would be clear.
I was, however, crossed that he did not pay me any interest and decided to teach him a lesson. I, thus, asked him to give me $1 on 1st Feb., $2 on 2nd, $3 on 3rd and so on. After a week I thought it is enough now and sat him down and calculated my debt as follows:
1+2+3+4+ … = -1/8
I settled my debt by accepting another 12¢ from him.
I now feel terrible to profit 4¢ from a very good friend. Do you think it is morally right to keep the money or shall I pay his 4¢ back?
See the attached draft presentation material for the details of above calculations.”
Hello all,
This question is a fallacy.
There no tricks in mathematics or physics in the sense of a magician but, there are levels of interpretation, that may look wrong if one is unaware of that level -- where there is no trick at all. The question conflates these two meanings. The principle is, as said before, that sufficiently advanced technology looks like magic.
For example, it seems a trick that the product of two negative numbers is a positive number. One can prove it, carefully using the previous properties if numbers, or just think that a negative number is a complex number with an 180° degree representation, so that 180° + 180° = 360°, a positive number. Was the use of complex numbers a trick? No, it can easily be done by defining a complex number first, a higher level of technology so to say.
In the same way, mutatis mutandis, the sum of 1 + 2 + 3 + ... is not divergent when seen as the analytic continuation of the Riemann function of -1, as S(-1) = 1 + 2 + 3 + ... = -1/12, albeit the series is divergent when seen in the real number level only.
As another example, we frequently encounter the difference of two divergent series. Instead of saying that we (rightly!) cannot subtract infinity from infinity, we can work on their ratios, and find a clear, finite result.
Cheers, Ed Gerck
Dear Ed,
Such divergence is well known for showing that even if the terms tend to zero, that doesn't means convergence.
Dear Ed
The issue here is breaking the conditions we introduced into equations ourselves. One simple example is the formula n(n+1)/2 we use for the sum of partial natural numbers. If we use this for negative whole numbers, the result is not correct for any partial negative series
-1-2-3-4…-n
but according to your arguments we should change our level and accept the results.
All I try to argue is extending the principle of “stick to your conditions” to any named or nameless function.
Please see pages 43 to 45 for the trick used to propose -1/12 as the finite result for ζ(−1) .
Hello Ziaedin,
Thank you for your reply. The issue here is formal, the question is based on a play in words, where trick is based on two different meanings, which creates a fallacy. No one needs to click on more links to see that, is my observation. You may wish to revise it.
I and many others know the Riemman function well, and Ramanujan was clever in not stating the level of his results in his original letter, which would be needed in a paper, to develop curiosity.
To your reply, of “stick to your conditions” , the principles are studied in semantics, although Frege and Tarski made important contributions from mathematics. It include that "meaning depends on context". So, I can change meaning without changing any fact, by chaging the context.
In Don Quixote by Cervantes, one can see a more general plot where the importance of context is linguistically recognized and used masterfully by the author as that which gives meaning to information. The same happens in physics and mathematics, by context. If one is using the Riemann function, the result -1/12 is natural; not if one is using real sums.
Cheers, Ed Gerck
Dear Daniel,
Yes, and I saw it as a counter-example of a series that looks obviously divergent (1 +2 + 3 ...) but can be calculated in a given level, versus a series that looks possibly convergent (1 + 1/2 + 1/3 + ...) with dimishing terms, but cannot be calculated.
Cheers, Ed Gerck
Hello Ulrich,
if a German mathematician tells his friend that he found a nice Gift, the friend may not expect to see a present.
The problem is the same, words or symbols, can have any practically meaning we want assigned to them, we just have to choose the context. The symbol "=" has more than four meanings in mathematics.
Cheers, Ed Gerck
Dear Ed,
the point is that it is not possible to assign any "equality", in the sense of getting a result.
a) The Zeta function with analytic continuation states that the substitution of the natural numbers can be done with -1/12.
b) 100 years of series convergence says that such series does not give a finite number.
c) Physics chooses to use a) to cure the infinitites, and it works, genius who discovered it!!!
d) There is something behind which we don't understand, are we really sure everything is ok with our theories??
Hello Stefano and all,
Thank you for your reply, but your reply itself, practical, shows that there is no confusion in physics or maths.
But, theoretically, I agree with you, that inquiring minds may want to know why... The answer is in my reply above, to Ulrich. Context is what defines meaning to data, information in the Shannon definition, no meaning is carried by the data itself! If I ask you to pass the salt, you may rightly pass a copy of the Stategic Arms Limitation Treaty. This was treated mathematically by Frege.
In the question, which is a fallacy itself, people are conflating, e.g., "=" in complex mathematics with "="in arithmetic. The two "=" are NOT the same in meaning... even though the same data is used. For example, in arithmetic, quintic_root(1) = 1 only, but not in complex analisys! There are five values , as well-known. In arithmetically divergent series, we can use a Cesaro sum, with the same symbol "+", to achieve a convergent result ... in Cesaro terms, not arithmetic.
In a more comprehensive context, we say that "trust me" cannot induce trust, where trust is that which gives meaning to information, not more information, where trust includes context. Trust must travel in parallel channels, to be trust. These definitions have been applied directly to Internet protocols, and are used billions of time every second. Please Google "Gerck trust" for more.
Cheers, Ed Gerck
Dear Ed,
I don't agree.. the equality in the case of the zeta function analytic continuation is not to be used at all!!! What occurs is a substituition.
Instead of the "infinite sum" it is used -1/12.
It hides an arbitrary "associative property" of numbers which incidentally is respected Physically for some reasons we don't know!!
Discussion regarding the use of divergent series is really an old topic.
Please keep in mind that there are mathematical tools to keep out any trickery. Besides analytic continuation, there are various techniques that essentially define a series to be summable in a certain sense X, say if there is an algorithm related to X that assigns a finite result to the series. Often, a proof that a series or class of series is summable in sense X is quite involved.
In my opion, this approach is mathematically sound, and in many cases applicable to mathematics and also in the sciences.
I suggest that the present discussion should not be based on arguments 'ad hominem' but on facts.
Herbert H H Homeier
but it simply works in Physics, and let's say "by coincidence", until some reasonable explanation will be given...
complex maths versus real numbers: strictly order relation "" gets lost for complex numbers. (for instance)
Dear Ed and Paul
The problem is that, we cannot see any complex number in either ζ(−1) or η(−1). Moreover, for calculation of η(−1) we either use Ramanujan’s derivation (Page 29) or use an ordinary function with divergent part removed (page 35), and here lies the trick.
Many procedures In are really difficult to work them directly or rigorously. Fortunately witty tricks are very useful to get some solutions faster. Besides, in my opinion, math tricks are part of Math's beauty.
Hello all,
There are several other ways to verify the law (in maths, computer science, physics, linguistics,...) that "context can change meaning, even if the actual data does not change" (which answers the question, as a fallacy)
1. Ask a C compiler. The expression "1=1" will be an error, the right way is "1==1". There are many meanings mathematically, to "=", including comparison, which the C compiler disambiguates. Other meanings, mathematically, to "=" is left to right attribution, or the reverse. That is one of the reasons that "a=a" is different from "a=b" (see Frege, also cited above).
2. Using topology, which all mathematics, physics, and so on, must obey, comes the last reason I would like to note here, but there are others, not as basic. I can search my notes and copy if desired. It is used in information theory signal analisys of intermodulation, for example.
Cheers, Ed Gerck
Dear Gloria
Tricks might be fun, until it is exposed, but it takes away the rigorousness of mathematics. But its most damaging effect is removing its critical front which allows the spread of unsubstantiated theories.
Dear Ed
I fully accept your argument but it is not applicable to our issue in this forum. The sum of a series should not be different if we write it in two different forms. 1+2+3+4+… or ζ(−1)= 1/1-1 + 1/2-1 + 1/3-1 + 1/4-1 +…
We should also note that the series 1+1/2+1/3+1/4+... can be written as ζ(1)= 1/11 + 1/21 + 1/31 + 1/41 +… and no matter what form we write it down it still has no finite result. Why cannot we find a finite answer for the sum of this series? Only because in this case we cannot find a formula with a condition to ignore it and get a false finite value.
Hello Ziaedin,
Thank you for your request. Your inquiry should be regarded as a free visit, an opportunity to be of service, not as a squander of time. There are many who feel unfortunate in life, but the worse unfortunates are the ones who squander such free visits!
In that sense, and I hope not to squander your time as well, is very common to see it in complex analysts, although it may happen in any area (see proofs above).
You are looking at points A and B, metaphorically, and asking why is it that, at the same time, one can write A = B and A != B, where "!=" means "not equal". Is that your question? If it is, we can proceed, if not, please clarify.
Cheers, Ed Gerck
Dear Ed
With utmost respect, I have no intention to squander anybody’s time and I try to read all answers with total appreciation. Thus, please feel free to clarify your points if you wish to do so.
My question is clear; why do we need tricks in mathematics? As an example I examined two finite results, -1/12 and -1/8, for the sum of natural numbers and the use of -1/12 in physics. My concern is that mathematics has ended up allowing and supporting some misuses in other branches of science and it would not be beneficial to both.
Your answer was that my question is a fallacy as there are levels of interpretations. We can introduce a complex function, zeta, and our divergent series become a convergent one as we are in a different level of mathematics.
My answer was that we in fact use another function for this new calculation, η(−1), and that function uses a non-complex equation with a firm condition.
1/(1-x) = 1+x+x2+x3 +… |x|
Hello Ziaedin,
I was hoping for a more neutral and general framework, using points A and B, metaphorically, instead of a confusion of symbols themselves, reducing the complexity.
Let me try that. You wrote, "In other words, “=” in the same complex plane cannot have two different level of meanings." Is that, as you see it, a phrase that if proved wrong, would deny your argument?
1. If it does, then one can find many counter-examples, such as a multiple root of same complex number versus a single root of the same value, or the infinite winding number k in X + 2.k.pi, with k=0, 1, 2, ..., or the residue theorem, or Lebesgue versus Riemann integrals of the same funtion (one integrable, the other not), or the many meanings of 1=1, and so on, and end the question as a fallacy, as defined in logic:
fallacy: a failure in reasoning that renders an argument invalid.
2. If it does not, then your question could still be acceptable, albeit not based on logic.
Cheers, Ed Gerck
It is well known that Ramanujan used to simply write down the equalities that came to him intuitively (or were dictated to him by a goddess). these equalities were correct as far as he was concerned. for him it was irrelevant what they meant. these equalities were not the result of theorems and he felt no need to derive them either. he was expressing these equalities using numerals and other mathematical symbols, but it would be incorrect to say that he was using them in the same way that we are reading them. this was probably the reason that he could not prove them through derivations where he gave them the usual meaning. I cannot imagine that Ramanujan was so blind to see that the sum of infinite number of natural numbers can be anything other than infinity. if he wrote that the sum is a negative fraction, then as far as he was concerned that equality is correct. in what way or situation such an equality is valid was beyond him and irrelevant to him. we can get an idea of Ramanujan's work as follows: using the English alphabets I write my name which is Arabic. if an Englishman reads my name and does not know that it is an Arabic name written using English alphabets then he will be completely lost. so also, we need to find what the equalities Ramanujan wrote using the known mathematical symbols and numerals mean. unfortunately, he did not write a decoding book to help us understand his work. he most likely was not aware that his equalities would be read in the usual way, just as I do not have to tell people that I am using English alphabets to write my Arabic name. I hope this perspective of Ramanujan's work will clarify what he had done and how we should approach it. thanks.
Dear Mustafa
It is well known how the value -1/12 is derived. It is based on chain of tricks. I tried to summarize the chain in the attached file to the question.
It is more difficult to see the tricks when ζ and η functions are employed. Under the banner of analytical continuation some, but not all, conditions are just ignored for some relevant function elements.
Functions 1/(1-x), x≠1 is analytical continuation of 1+x+x2+x3 +… |x|
Dear Paul
I was eagerly waiting for your promised response to my other question. Hope you have been able to find an answer to that issue too.
https://www.researchgate.net/post/Does_Michelson_and_Morley_experiment_support_length_contraction!
dear Ziaedin, I thought I have answered Your question (wasn`t it about contraction?) if so: LT is a contracting operator, that is: there must be a fixed point (Banach Theorem). but that fixed point exists merely for v=0, what (on the other hand) is not allowed, because velocity zero does not exist (RT axiom). ergo: mathematically, length contraction is not correct.
ps. if the question was another, please let me know. kind regards, Paul
dear Ulrich, You are right: it was an irony., nevertheless: if someone could programm a PC to solve problems (like chess, deep thaught, find out PI decimals, aso.) that were the best mathematical trick. once programmed a Computer can solve the (same) problems faster: again and again. I think that man can programm tricks too (statistics is an example :))
Concerning the (in)famous 1+2+3+...=-1/12 case, I'll just leave this video by Mathloger here:
https://www.youtube.com/watch?v=YuIIjLr6vUA
it's where I learned the explanation and he explains it much better than I ever could. Long story short: the relationship above is false, the -1/12 result exists but it's a much more complex result, as the operation of sum itself is defined differently for it. The rest is all magic tricks used by popular science writers and YouTube mathematicians who think they can blow your mind by proving true something that is patently false. Your mind is right to be blown since in this case the obvious answer is of course the correct one: 1+2+3+... sums to infinity.
Dear Simone
Unfortunately, this issue is not limited to Youtube mathematicians and popular science writers. The result has been used as the experimental proof for the existence of virtual particles, i.e. Casimir effect. I do not reject the experiment. The problem, in general, is legitimizing a theory by linking it to an experiment, using mathematical tricks.
Unfortunately, those who have voice/clout in physics have not been shying away from this sort of results. That is why I brought up the issue in this forum.
Dear Dr. Shafiei,
However, in our high school we had a method which we applied unofficially which was called "miscellaneous artifices". These were methods of solution whch are not generally followed. Like for example I use a miiscellaneous artifice when I teach my module on Quantitative Methods in Managerial Economics as part of the 1st. semester Managerial Economics course to MBA students of Calcutta University which is called Integrability Theory where I teach how to derive the Long Run Demand Function from a distribution of Short Run Demand Functions. This is favourite topic which I do by matching variables and something I have applied in my research and used to prove the existence of long run equilibrium from an estimated panel data asset pricing model using conintegration. This method has proved useful in even proving the P vs. NP problem and we have won the Millenium Prize and with further generalisations to field theories using Strings we may go on to win the Abel Prize and the Nemmers Prize now that the basic field of String Matching Field Theory has been testedin Riemann and derived Lie Groups. This is an example of a "miscellaneous artifice" which is forthcoming in a volume of American Journal of Modern Physics in the Higgs Physics issue (we have made a prayer for free publication which will possibly be granted) (Mallick, Hamburger & Mallick (2016, 2017, 2019). This is not a trick but neither is it a general method of proof mixing mathematics and statistics together. Thought I would share this research here.
Soumitra Kumar Mallick
for Soumitra Kumar Mallick, Nick Hamburger & Sandipan Mallick
for RHMHM School
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