Tomáš Sedláček (1977) (henceforth without accents) will be giving the 32nd Van der Leeuw lecture in Groningen, November 7 2014. The title of the lecture isEconomics as an Unorchestrated Orchestrator“. This reminds of Adam Smith’s Invisible Hand or modern-day recourse to The Great Divinator of “the financial markets”. Since the lecture is held in the Groninger Martini church the religious notion that God himself creates order comes to mind as well. In this RSA video Sedlacek refers to economics as the modern religion indeed. However, his lecture in Groningen will be refereed only by professor Barbara Baarsma (1969), CEO of SEO in Amsterdam.

Actually, the Foundation that organizes the lecture has as its main purpose to use that church for other cultural or social events rather than the dwindling religious services. Especially when the heating costs in November must be bridged before the uptake around Christmas, it is useful to organise some event to get people into the building. Since the building concerns a church, they found a theologian to name the lecture series after, even though Gerardus van der Leeuw (1890-1950) isn’t so remarkable compared to other Groningers Daniel Bernoulli (mathematician), Johan Huizinga (historian), Heike Kamerlingh Onnes (discoverer of superconductivity) or Hendrik Willem Mesdag (painter). Every human being is important and should be remembered however, so we can only hope that more cities take the opportunity to dedicate their lectures to those who are in danger of being forgotten especially when the weather turns cold.

Announcement 32nd Van der Leeuw Lezing (Source: screenshot website)

Announcement 32nd Van der Leeuw Lezing (Source: screenshot website)

That Sedlacek gives the lecture fits the confusion of location and purpose. Sedlacek does history and philosophy but uses the label of economics. Listeners in a church and partaking in a non-religious event should not mind another and lesser distortion. Unless we have returned to the historical situation that everything is religion anyway.

Sedlacek is known internationally for his 2011 book Economics of Good and Evil: The Quest for Economic Meaning from Gilgamesh to Wall Street, see this review by Samuel Brittan in the FT. The book is his thesis that was rejected by the Charles University, and his website mentions that he is still registered there as a Ph.D. student. I haven’t read that book but have read some reviews and watched also this video recorded in Amsterdam June 11 2013.

I know about Evil, since I wrote about the pure evil of the basic income. I know about Good since I wrote The simple mathematics of Jesus (2012). I know about Economics, see the About page. I still don’t know whether Sedlacek’s book is good or evil but it doesn’t look like economics to me, whatever Deirdre McCloskey says about it. A term used is “meta-economics” but that might be comparable to sociology perhaps. I settle for “history and philosophy while trying to focus on economic thought”.

The publisher “describes” the book as:

“Tomas Sedlacek has shaken the study of economics as few ever have. Named one of the “Young Guns” and one of the “five hot minds in economics” by the Yale Economic Review, he serves on the National Economic Council in Prague, where his provocative writing has achieved bestseller status. How has he done it? By arguing a simple, almost heretical proposition: economics is ultimately about good and evil.

[Comment by TC: Surely, since economics is not about good and evil, it is ground-shaking to turn economics into theology indeed. Doing so is not heretical but quite fitting in church. It is quite a miracle: to be at an economics department, stop doing economics, but convince other people that you are still doing economics. As people can believe that Jesus walked on water, they can also believe that you are doing economics. The same miracle was performed by mathematicians who said that they were doing economics but in fact continued doing mathematics.]

In The Economics of Good and Evil, Sedlacek radically rethinks his field, challenging our assumptions about the world. Economics is touted as a science, a value-free mathematical inquiry, he writes, but it’s actually a cultural phenomenon, a product of our civilization. It began within philosophy–Adam Smith himself not only wrote The Wealth of Nations, but also The Theory of Moral Sentiments–and economics, as Sedlacek shows, is woven out of history, myth, religion, and ethics.

[Comment by TC: Economics as a science ought to be value-free, but its application is in society and thus its application is immersed in values. Yes, there have been and still are many influences on the development on economic thought, but that does not take away that former distinction.]

“Even the most sophisticated mathematical model,” Sedlacek writes, “is, de facto, a story, a parable, our effort to (rationally) grasp the world around us.”

[Comment by TC: There is nothing new in this, that a mathematical model can be seen as a story or parable - except that it would tend to be consistent and more precise. So what is the point ? Can philosophy be set equal to mathematics, since both are "just stories" ?]

Economics not only describes the world, but establishes normative standards, identifying ideal conditions. Science, he claims, is a system of beliefs to which we are committed. To grasp the beliefs underlying economics, he breaks out of the field’s confines with a tour de force exploration of economic thinking, broadly defined, over the millennia. He ranges from the epic of Gilgamesh and the Old Testament to the emergence of Christianity, from Descartes and Adam Smith to the consumerism in Fight Club. Throughout, he asks searching meta-economic questions: What is the meaning and the point of economics? Can we do ethically all that we can do technically? Does it pay to be good?

[Comment by TC: (1) Economics does not establish normative standards. Economics enlightens such choices. Check e.g. Pareto Optimality: Economic models don't impose this but elucidate the notion. (2) The latter quoted questions are useful for the talk between an economic scientist and a policy maker. (3) The inner value of economics lies in increased knowledge, as for any science. Like pure number theory in mathematics. (4) The outer value of economics lies in its application. Like using number theory for cryptography for secure bank accounts.]

Placing the wisdom of philosophers and poets over strict mathematical models of human behavior, Sedlacek’s groundbreaking work promises to change the way we calculate economic value.”

[Comment by TC: If philosophers and poets can do without bread and butter, they can be excluded from the economic calculation, and we indeed have something novel. Overall though, economics was developed to get away from those unscientific story-tellers.]

Sedlacek in Dutch VPRO "Tegenlicht" program, June 11 2013 (Source: screenshot)

Sedlacek in Dutch VPRO “Tegenlicht” program, June 11 2013 (Source: screenshot)

Let us conclude with the following points:

  1. Dutch VPRO and professor Baarsma do not report about the censorship of economic science by the directorate of the Dutch Central Planning Bureau since 1990.
  2. Dutch VPRO and professor Baarsma do pay attention to Tomas Sedlacek’s story that isn’t economics and that is at points unscientific.
  3. We can enjoy various points in Sedlacek’s tale. The history of economic thought and its precursors is interesting and it would require a worse author to destroy this. For example the analogy between Christianity and the calculation of sin and redemption is nice. Hopefully he included the invention of Purgatory for the collectors of interest too. But the book should be rewritten before it can be advised.
  4. Check my books DRGTPE and SMOJ referred to above, for the full story on getting an Economic Supreme Court, for a better orchestra.

PM. Since Sedlacek is from the Czech Republic and advised Vaclav Havel, he might take an interest in the point that my analysis in 1990 originated from the Fall of the Berlin Wall in 1989, and was targetted at handling the economic fall out, see this text. The history of Eastern Europe and Russia would have looked quite different when the directorate of the Dutch CPB had respected science – or others in the surrounding had made a correction.

It turns out that Cressida Cowell has been writing her dragon books for years and that the box offices of the films are approaching $1 billion, while this weblog was rather oblivious of that. I only noticed these elementary facts from watching with my youngest son and hugely enjoying How to train your dragon 1 on television and How to train your dragon 2 in the theatre. See the official website for the trailers. The films establish that mankind is destined to fly.

While the Harry Potter films were never convincing with their crude suggestion of broomsticks and overall tendency to neglect humour, the relationship of Hiccup and Toothless not only engages us, reminding of other stories of boy & horse or boy & dolphin, but also makes us want to fly along, dive, loop, plunge, and what you have, and share this close bonding of body and mind. Now that Google is developing robots, the obvious suggestion is to concentrate on flying robots and then notably in the form of dragons, so that the phantasy is just a premonition.

How to Train Your Dragon - And flying it (Source: Trailer Screenshot)

How to Train Your Draghi – And flying it (Source: Trailer Screenshot)

That future is already with us, in that Mario Draghi, the president of the European Central Bank, is yet untrained, and takes us and the world economy diving, looping, plunging, and what you have.

Europe needs someone like Hiccup who neglects danger and is convinced that feeding Draghi some fish will gain the monster’s trust, so that it will let itself be put into a harness and be controlled.

How to Train Your Dragon (Source: Trailer Screenshot)

How to Train Your Draghi – Using a big fish (Source: Trailer Screenshot)

Angela Merkel (in the film called “Astrid”) finds the house on fire and gets a bucket to extinguish it. Obviously, she will not succeed.

Angela Merkel discovers that the house is on fire and picks up a bucket to extinguish it (Source: Trailer Screenshot)

Angela Merkel fnds the house on fire and gets a bucket to extinguish it (Source: Trailer Screenshot)

Read more about this story in these links:

(A) Thomas Colignatus,An Economic Supreme Court“, a piece in the Royal Economic Society Newsletter, issue 167, October 2014, see above “About“.

(B) Still in the dark, not seeing the evidence:

This weblog warned about Frans Timmermans who is intended to play a key role in the new EU Commission. After this warning I deemed it wiser to be silent on him and focus on the math, statements, book, film and interviews by mathematics professor Edward Frenkel of the University of California at Berkeley.

This Friday Timmermans – henceforth T, rather than FT since that is the Financial Times - passed his responsibilities as Foreign Secretary of the Kingdom of the Netherlands on to Bert Koenders who ought to be able to do a better job. It remains amazing that policy can depend upon the Department of Personnel so much.

Let me explain that T has committed social suicide in Holland. He essentially has no political credibility anymore on the world stage. When I meet Jean-Claude Juncker this weekend at one of the farewell parties for José Manuel Durão Barroso then I can explain the situation to him over a few good drinks. Hopefully Jeroen Dijsselbloem will not be there to tell Juncker that he shouldn’t be drinking. This issue would be hard to swallow for most people.

The sentiment in Holland has turned against T and this is indicated, apart from my verdict, by:

  1. The satyrical column “the pin” prints a “speech” by “T” in which he lauds himself.
  2. The site Joop.nl prints an exchange on twitter, in which (a) someone criticizes T at his farewell party, and (b) T reacts remarkably rudely, shooting himself in the foot as a diplomat. Joop.nl holds that T is a “walking mine field between citizen Frans and minister Timmermans”.
  3. Columnist Bas Heijne in the leading newspaper in Holland criticizes T on his mentioning of the oxygen mask: C’est pire qu’un crime, c’est une faute. (It is worse than a crime, it is a mistake.)
  4. Columnist Jonathan van het Reve in a column of October 11 (Vonk p13) in the 2nd newspaper precisely states what I thought myself too, when I noticed that oxygen mask incident.

Let me refer to the BBC report on the T – oxygen mask incident. I hope that Jonathan van het Reve doesn’t mind that I relate his analysis that was precisely mine too:

  1. His tears at the UN Security Council did not achieve anything but turned him into an international celebrity.
  2. When Jeroen Pauw questioned his inconsistency that the explosion was instantaneous and that the passengers and crew had suffered, T was irritated and defended himself with the mask.
  3. But he may well have abused the happenstance that such masks fly around and may land around some passenger’s neck. This aspect of the research has not been completed yet.
  4. T gave priority to his irritation and public standing and new status as a celebrity, above the families of the victims and the political impact.
  5. Families now worry that their relatives have been suffering the 10 km drop.
  6. Commentators in Russia hold that there wasn’t an instant explosion but machine guns from an Ukrainean fighter plane.
  7. T is narcissistic and will meet destiny at one time. Well, in addition to that, my own point is that he comes from Limburg, alike Wilders, see my earlier warning. This Catholic province has an inferiority complex because of 300 years of domination by the Protestant provinces of Holland. We shouldn’t label all people from Limburg or from anywhere in the world, but we should take heed of the facts, and ask the Department of Personnel to do so too.
Frans Timmermans versus oxygen masks (Source: Wikimedia commons)

Frans Timmermans versus oxygen masks (Source: Wikimedia commons)

In the 9-minute Numberphile interview Why do people hate mathematics? – see yesterday’s discussion – professor of mathematics Edward Frenkel states, in minute three:

“Georg Cantor said: “The essence of mathematics lies in its freedom.” But I would like to augment this with the following: Where there is no mathematics there is no freedom. So mathematics is essential to our freedom, to the functioning of our democracy. (…) Our ignorance can be misused by the powers that be. And for us … as citizens in this Brave New World … we have to be more aware of mathematics, we have to know and appreciate its power – to do good but also to do ill.” (Edward Frenkel, Jan 19 2014)

We can only applaud this. In my Elegance with Substance (EWS)(2009):

“Mathematics is a great liberating force. No dictator forces you to accept the truth of the Pythagorean Theorem. You are free to check it for yourself. You may even object to its assumptions and invent non-Euclidean geometry. Mathematical reasoning is all about ideas and deductions and about how far your free mind will get you – which is amazingly far. But you have to be aware of reality if you say something about reality.” (EWS p9)

“Democracy is an important concept. The mathematics of voting is somewhat complex. It would be beneficial for society when its citizens understand more about the mathematics behind election results. Students in the USA have a Government class where such aspects can be indicated. Political Science as a subject has not reached highschool in general. Much can be said in favour of including the subject in economics, since the aggregation of preferences into a social welfare function is a topic of Political Economy. See page 59 and Colignatus (2007b) Voting theory for democracy (VTFD) for details and other references. Most economists will be unfamiliar with the topic and its mathematics though and thus it may well be practical to include it in the mathematics programme.” (EWS p48)

However, let us also look at key criticism:

  1. Mathematician Kenneth Arrow presented his “impossibility theorem” in his 1951 thesis. It holds, in his own words: “there is no social choice mechanism which satisfies a number of reasonable conditions”  Palgrave (1988:125) and quoted in Voting Theory for Democracy (VTFD)(2014) 4th edition p240. Thus collective choice would require us to be unreasonable. Mathematician Arrow continued in economics and got the Nobel Prize in economics for this and other work.
  2. Mathematicians, political scientists and economists have tried since 1950 to debunk Arrow’s result, but did not find real solutions. These areas of science have become a force against democracy. Collective choice would require us to be unreasonable, and this would be scientifically proven.
  3. When I showed in 1990 that Arrow’s words do not fit his mathematics, and a bit later that his result was either inconsistent or incomplete, hell broke out. My paper was suppressed from discussion and publication. A mathematician who was supposed to review VTFD (3rd edition) started slandering. See the journal Voting Matters (April 2013). See my point however that there is a distinction between “voting” (counting ballots) and “deciding”. And see VTFD for the more involved presentation (starting with matricola).
  4. It has been impossible to find someone in Holland to discuss this issue rationally. Here is a report in English on a working group in social choice theory. Here is a page in Dutch. On a website for highschool students, Kennislink.nl, deluded mathematician Vincent van der Noort, who did not properly study the issue, claims that “democracy isn’t entirely fair“, thus encouraging highschool students to use their ellbows. The editors refuse to correct this falsehood and selective use of sources (or mystery, since Vincent doesn’t define fairness).

I suppose that professor Frenkel discusses democracy in general, without thinking specifically about Arrow’s “Theorem”. Perhaps he doesn’t know about it, and would be surprised that it would be “mathematically proven” that some degree of dictatorship would be necessary. However, to some extent we can agree with him. Good education in mathematics will do wonders for liberty and democracy. But, my point again: the definition of “good education in mathematics” is subtle. See these quotes from EWS too:

“With respect to logic and democracy, Colignatus (2007ab, 2008b), updated from 1981 / 1990, considers statements by mathematicians that have been accepted throughout academia and subsequently society on the basis of mathematical authority. It appears however that those statements mix up true mathematical results with interpretations about reality. When these interpretations are modelled mathematically, the statements reduce to falsehoods. Society has been awfully off-track on basic notions of logic, civic discourse and democracy. Even in 2007, mathematicians working on voting theory wrote a Letter to the governments of the EU member states advising the use of the Penrose Square Root Weights (PSRW) for the EU Council of Ministers. See Colignatus (2007c) on their statistical inadequacy and their misrepresentation of both morality and reality.

Over the millennia a tradition and culture of mathematics has grown that conditions mathematicians to, well, what mathematicians do. Which is not empirical analysis. Psychology will play a role too in the filtering out of those students who will later become mathematicians. After graduation, mathematicians either have a tenure track in (pure) mathematics or they are absorbed into other fields such as physics, economics of psychology. They tend to take along their basic training and then try to become empirical scientists.

The result is comparable to what happens when mathematicians become educators in mathematics. They succeed easily in replicating the conditioning and in the filtering out of new recruits who adapt to the treatment. For other pupils it is hard pounding.” (EWS p10)

PM. See where Georg Cantor went wrong: Contra Cantor Pro Occam (2012, 2013).

In 2009 I wrote Elegance with Substance (EWS), discussing both better education in mathematics and the political economy of the mathematics industry. See the available PDF. Check also Steven Krantz Through a Glass Darkly at arXiv 2008.

The dismal state of mathematics education is generally acknowledged, essentially since Sputnik 1957. People have tried all kinds of solutions. Why do those solutions not work ?

The answer: because of barking up the wrong tree. The finding in EWS is:

  1. Mathematicians are trained to think abstractly.
  2. Education is an empirical issue.
  3. The courses for becoming a math teacher don’t undo what has gone wrong before.
  4. When abstract thinking math teachers meet real life students, those math teachers solve their cognitive dissonance by sticking to tradition: “School Mathematics” (SM).
  5. School mathematics isn’t clear but collects the confusions and wreckages of math history.
  6. Thus we need to re-engineer math education and reorganise the mathematics industry. One idea is that education would use the form of the Medical School: both practice and research.

EWS contains various examples where traditional math is crooked instead of clear. One example is that “two and a half” means addition and should be denoted as 2 + 1/2, but is denoted as multiplication or “two times a half” or 2½.

2009 + 5 = 2014

Now five years later in 2014, this explanation can be enhanced by including:

  1. There is a collective failure w.r.t. the integrity of science, in that Research Mathematicians step outside of their field of expertise (RM) and make all kinds of unwarranted claims about Education in Mathematics and its research (EM). This aggravates the observation above that the conventional EM is lopsided to SM.
  2. It is also a breach of research integrity that the warning in EWS is not responded to. When it is shown that the brakes of some kind of car don’t work properly, it should be recalled – and the same for EM.
  3. This especially holds in Holland. In Holland there is even explicit fraud in EM
  4. For the UK there is some worry, see my 2014 paper Pierre van Hiele and David Tall: Getting the facts right.
  5. For the USA there is now the worry concerning professor Edward Frenkel.

Pierre van Hiele (1909-2010) was the greatest analyst on mathematics education of the last century, with his main thesis in 1957, coincidentally with Sputnik. However, his analysis was maltreated by Hans Freudenthal (1905-1990), who stole Van Hiele’s ideas but also corrupted those – partly claiming his “own” version but without proper reference. Van Hiele looked at the angle of abstract versus concrete, while Freudenthal turned this into model versus reality, which is didactically rather absurd, but which apparently appealed to policy makers after Sputnik 1957. Holland now has a 95% dominant “Freudenthal Institute” that rather should be called the “Freudenthal Head in the Clouds “Realistic Mathematics” Institute”. Apparently, the Dutch RM and EM community is unable to resolve the issue. Internationally, IMU / ICMI (see my letter) has a “Freudenthal Medal” honoring the fraudster.

A leading analyst in the UK is David Tall (b. 1941) who rediscovered the importance of the Van Hiele analysis, but erroneously thinks that Van Hiele was not aware of what he was doing, so that Tall claims the discovery for himself. Part of Tall’s misunderstanding of the situation is the consequence of Freudenthal’s abuse of Van Hiele. Professor Tall should however quickly bring out a revised 2nd edition of his 2013 book to set the record straight.

From Russia with math and confusion

I have discussed some of Frenkel’s ideas. As he hasn’t studied math education empirically, he is not qualified to judge, but he follows the RM arrogance to think that he is. Well, hasn’t he passed through the educational system himself ? Isn’t he teaching math majors now ? These are hard fallacies to crack.

Numberphile has a 9-minute interview with Frenkel, asking him: Why do people hate mathematics?”  I leave it as an exercise to the viewer to identify the amazing number of delusions and fallacies that Frenkel mentions in this short time. Perhaps shortness invites imprecision. However, check this weblog’s texts of the last week, and see that these delusions and fallacies are systematic. Just to be sure: debunking those delusions and fallacies may not be easy. If it were easy, the state of math education would not be as dismal as it is now.

To help you getting on the way, check some of these delusons or fallacies:

  • The beauty of art is abused again. Math education would teach you painting fences but not the appreciation of the great results of mathematics. To some extent one can agree. Math history and some encyclopedia of math are very useful to have. But art education is not intended to get people to make masterpieces. Mathematics education is intended to help students develop their understanding and competence. These are different settings.
  • Frenkel claims that everything is based upon the language of mathematics. “In a way one can survive without art. No one can survive without mathematics.” Since abstraction means leaving out aspects, it should not surprise that if you start with the world and then abstract from it, then your results may indeed be relevant for “everything”. But you cannot infer from such an abstract position that people should love their math education.
  • He again is in denial of the role of mathematics in causing the economic crisis.
  • The problem is often stated in the terms of “people hate mathematics” in a manner that is not linked to mathematics education. As if there are two kinds of  people, mathematicians and other – the elite versus the peasants. But the true problem is mathematics education. Math teachers have their students for some 12 years as their captive audience, and manage to turn human innate interest into said hate. By stating the problem in terms of some vague “general audience” it becomes easier to run away from the responsibility staring you in the face, and the destruction of human lives going on in the classrooms around the world.

Taking a blame without any consequence

There is no doubt that Frenkel respects education – though it is from personal experience and without empirical research of a national curriculum:

“Now that I’ve had students of my own, I appreciate even more what (… my teachers …. have …) done for me. It’s hard work being a teacher! I guess in many ways it’s like having children. You have to sacrifice a lot, not asking for anything in return. Of course, the rewards can also be tremendous. But how do you decide in which direction to point students, when to give them a helping hand and when to throw them in deep waters and let them learn to swim on their own? This is art. No one can teach you how to do this.” (“Love & Math p129)

The major point is this: Asked who is to blame for the dismal appreciation for mathematics (minute five) he offers himself as the scape-goat:

“If I really were to assign the blame, … I would assign the blame to myself. And my colleagues, professional mathematicians. We don’t do nearly enough, in exposing these ideas to the public.”

Okay, so, Frenkel takes the blame. But there is no consequence. No reduction in salary. No prison term – with use of the library to start studying mathematics education. Just the burden to go out into the public and become a media star by comparing mathematics to Van Gogh, Picasso, and what other artist that can be abused and intimidated into an admiration for mathematics that they don’t understand but generally hate.

In minute six he says that the math teachers are not to blame. “They are overworked and underpaid” and “products of the same flawed system”. Thus, the idea that grown-ups should take responsibility for what they are doing, and that professional educators have an ethic to live up to, is flushed down the drain. Jesus absolves the sins of those who believe in him. The topic of discussion is reduced to “beauty”. This will generally concern topics that require an advanced university degree to understand – and that conventionally are presented in a confused manner to the general public (see yesterday).

About the improvement of education, Numberphile properly aks (minute seven-and-a-half): “Why has that not happened ? It seems so obvious. What you said is not like a huge conceptual link. Why isn’t it not already happened ?”

Since he has no clue about empirical science, the world turns into a conspiracy:

“Sometimes I am wondering myself why it hasn’t already happened. It is almost like a conspiracy. I mean, honestly. It is almost like there is this system of mirrors that has been created which distorts reality, that does not allow people to see what is out there.”

His closing statement turns failure on scientific integrity, fraud and dismal negligence into “irony”:

“This is the coolest stuff in the world. And yet everyone hates it. Isn’t it ironic ?”

Left: Dali's "Crucifixion" on a hypercube. Right: Edward Frenkel teaching (Source: wikipedia commons, Dali, Eget værk, Søren Fuglede Jørgensen)

Left: “Crucifixion” on a hypercube, Salvador Dali. Right: Edward Frenkel teaching (Source: wikipedia commons, Dali, Eget værk, Søren Fuglede Jørgensen)

PM. The link of Jesus to a scape-goat is no coincidence. December 25 falls in the sign of Capricorn and Jesus was sacrificed as the Lamb of God. See The simple mathematics of Jesus for a discussion that the Bible is an astrological book – and, if you didn’t know, that astrology isn’t science.

There is a curious argument that 1 + 2 + 3 + 4 + … = -1 / 12  (New York Times February 3 2014).

Some pronounce this as “minus one over twelve” but this weblog proposes “min per ten-two” or “negative per ten-two”. On occasion we employ H = -1, to be pronounced as “eta”. Thus “eta per ten-two” is okay as well. We can also use 1 / 12 = 12H, pronounced as “per ten-two”. (The Germans would pronounce H as “Ha” and we would not want them to be laughing all the time.)

The NY Times article and Numberphile video was debunked by other mathematicians and physicists on the internet, see some links below. However, this weblog looks at issues from the angles of both econometrics and the education of mathematics. From these angles we find:

  1. The article and video do not satisfy the conditions of didactics.
  2. There appears to be a large mathematical industry to confuse people.

Mathematics professor Edward Frenkel is part of the mêlée. He is quoted in above article (and can be heard in some video’s saying similar things):

“This calculation is one of the best-kept secrets in math.”
“No one on the outside knows about it.”

The article states:

In modern terms, Dr. Frenkel explained, the gist of the calculations can be interpreted as saying that the infinite sum has three separate parts: one of which blows up when you go to infinity, one of which goes to zero, and minus 1/12. The infinite term, he said, just gets thrown away.

The latter is rather curious. Why are you allowed to throw infinity away ? If you take something from infinity before you throw infinity away, why would you select -12H and not something else ?

Let us consider the situation, and start with Grandi’s Series. Personally, I was reminded about an approximation to -12H found last year, but since it is only an approximation this comment has been put into Appendix A.

An unwarranted deduction

In Numberphile, Thomsons’s Lamp, there is this video discussion about “Grandi’s SeriesG. That discussion (and on wikipedia retrieved today, see Appendix B) is unwarranted. The proper deduction is:

G = 1 – 1 + 1 – 1 + …. = (1 + 1 + 1 + ….) – (1 + 1 + 1 + …) = ∞ – ∞ = undefined

It is an altogether different question that we can look at the average of the series of partial sums. The Lamp mentions this (to their credit) but uses the same plus-sign which is unwarranted. We should use a different plus sign. Then we find:

G’ = 1 ⊕ H ⊕ 1 ⊕ H ⊕ ….  = 1 – G’    so that   G’ = 2H

Partial sums of G:  1, 0, 1, 0, 1, 0, ….

Summing (again !) those into a series: 1 + 0 + 1 + 0 + 1 + ….

Averaged series G’:   1 / 1,   (1 + 0) / 2 = 2H,   (1 + 0 + 1) / 3 = 2/3,   2 / 4 = 2H, …

The mystery completely disappears.

Divergent series can be operated upon, with differences, sums, averages, until you find something that converges. You might use this to catalog them.

That Lamp video discusses turning on and off an actual lamp, in ever smaller fractions 2^(-n) of a minute, starting at zero, such that the process should stop after two minutes (we can calculate that period mathematically): and then the question is whether the lamp is on or off. This is a badly defined problem. It is the same as the Zeno paradox of Achilles and the hare. A mathematical story using terms from physics doesn’t make it proper physics.

A string theory mystery

I am no physicist and know nothing about string theory, but am a bit perplexed when this other Numberphile video shows that page 22 for 1 + 2 + 3 + 4 + … ⇒ -1 / 12. Note the arrow rather than the equality sign. It remains a question: are they really taking the limit ? Hopefully the deduction in string theory is more to the point than the deduction given in the video. The deduction in that video clearly is not sound. It uses G = 2H but we have shown that only G’ = 2H. Indeed, see below for some links to physics websites that show that the video is crooked.

Page 22 of Joseph Polchinski, “String Theory" (Source: Numberphile video)

Page 22 of Joseph Polchinski, “String Theory” (Source: Numberphile video)

The Numberphile video uses three series. Confusingly it uses the normal plus sign but let us consider the idea that these would concern averages of a series of partial sums (with ⊕ instead of +). Series S1 = G and S2 is another form of ∞ – ∞ = undefined.

Read (+) and (+ H) instead of plus and minus (Source: Numberphile video)

Read (+) and (+ H) instead of plus and minus (Source: Numberphile video)

Let us repeat above procedure for S. Since there are no negative values involved, the series merely explodes, and obviously the outcome cannot be negative.

S’ = 1 ⊕ 2 ⊕ 3 ⊕ 4 + ….  

Partial sums of S:  1, 3, 6, 10, 15, ….

Summing (again !) those into a series: 1 + 3 + 6 + 10 + 15 +  ….

Averaged series S’:   1 / 1,   (1 + 3) 2H = 2 ,   10 3H = 3 + 3H,   16 4H = 4  ….

The Numberphile team has a longer video on the sum of the natural numbers that uses the Euler-Riemann Zeta function to argue their point, supposedly in “proper fashion”. However, they do not discuss the paradoxes here, and thus leave the reader confused. For example, they also refer to the basic geometric series, differentiate this, and then substitute r = -1 to create S2 (calling this “analytic continuation”), but, if the original geometric series is undefined for r = -1 (and then actually generates the Grandi Series again): why do you think that you can do this ?

Geometric series converges for -1 < r < 1 (Source: wikipedia)

Geometric series converges for -1 < r < 1 (Source: wikipedia)

See some physics links

The 1 + 2 + … = -1/12 video got 1.5 million hits and a fair amount of reactions from physicists. Their point is that Riemann and they are doing their job. See Steven Corneliussen in Physics Today and Phil Plait at Slate, for example. Plait has this quote from Jordan Ellenberg:

“It’s not quite right to describe what the video does as “proving” that 1 + 2 + 3 + 4 + …. = -1/12. When we ask “what is the value of the infinite sum,” we’ve made a mistake before we even answer! Infinite sums don’t have values until we assign them a value, and there are different protocols for doing that. We should be asking not what IS the value, but what should we define the value to be? There are different protocols, each with their own strengths and weaknesses. The protocol you learn in calculus class, involving limits, would decline to assign any value at all to the sum in the video.  A different protocol assigns it the value -1/12. Neither answer is more correct than the other.”

This is not entirely correct. Once you have defined “addition” and “equals” then you are stuck with it. Yes, you are free to find another protocol, but, beware of using “addition” and “equals” in general publications and education in another sense than people understand, because then your create confusion.

It seems to me that Physics Buzz is the most enlightening on what the real intention is.

Some nice quotes however

However, to soften our conclusion, the NY Times article by Dennis Overbye provides some nice quotes:

The problem with infinity is that you can’t stop. You never get there. It’s more of a journey than a destination.

Niels Henrik Abel, whose notion of an Abel sum plays a role here, once wrote, “The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever.”

Of course there is also Wigner again:

To him and others, this is just another example of what the eminent physicist Eugene Wigner called the “unreasonable effectiveness of mathematics.” Why should such woolly and abstract concepts as zeta functions or imaginary numbers, the products of a chess game in our minds, have such relevance in describing the world?

However, as mathematics = abstraction, and abstraction = leaving out aspects, it should not be surprising that if you start with the world and leave things out then you still have something. See here for complex numbers, and check the steps in turning around a circle:

1,  i,  H = i²,  H i, 1  (start at 1 = {1, 0}, quarter turn, half turn, three-quarters turn, back to 1)

Appendix A:   φ² / Θ ≈ 5 / 12

Remember that we found that φ2 / Θ ≈ 5 / 12 with an error of 6 per million, where ‘phi’ φ = 1.618033989… is the golden ratio, and where ‘archi’ Θ = 2 π = 6.283185307….

φ2 / Θ   =   0.416673050492137…

5  / 12   =    0.4166666…

φ2 / Θ  –  5 / 12  =  0.00000638382547060161…

φ2 / Θ  –  2H   ≈    -12H                                                    (with the same error)

Thus, the suggestion is that when some physics formula generates the number -12H, look whether this kind of thing might be involved. We came upon this from an application. The relation holds by approximation only, however, and might be abused again to confuse people.

Appendix B:  Wikipedia 2014-10-15 on Grandi’s series contributes to confusion

Wikipedia's discussion today on Grandi's series (Source: wikipedia)

Wikipedia’s discussion today on Grandi’s series (Source: wikipedia)

The complex number i = √has a danger that some people may not be aware of. We use H = -1, see here.

For, consider:

-1 = i²  = (H) (H) = (H H) = 1 = 1

Professor of mathematics Edward Frenkel states in his book, intended for the general audience, and thus giving false information to that general audience:

“Note that it is customary to denote √-1 by i (for “imaginary”), but I chose not to do this to emphasize the algebraic meaning of this number: it really is just a square root of -1, nothing more and nothing less. It is just as concrete as the square root of 2. There is nothing mysterious about it.” (E. Frenkel, “Love & Math”, p101-102)

Observe the factual error and the error in didactics:

  1. The factual error is to say that the symbol √ has the same meaning in √-1 as in √2.
  2. Didactically, it is writing that conveys the algebraic meaning better, not writing √-1.

It took William Rowan Hamilton (1805-1865), the hero of Irish mathematics, a major part of his time to discover that = {0, 1}, i.e. the point in the two-dimensional plane where x = 0 and y = 1. Stepping into another dimension is not the same as staying in the same dimension. If you treat those at the same then you get above deduction that -1 = 1. The conclusion is that i is an operator and not a common number. The step (√H) (√H) = (H H) is forbidden since it concerns an operator, with a different rule for √. We can only call i a “(complex) number” if we adapt the notion of “number” to include it.

Let us look a bit more at the reason why i was mysterious and imaginary. Consider the quadratic equation, and let us “complete the square” on the left hand side

a x²  + b x + c = 0                                                  (formula for a vertical parabola)

x²  + b aH x       = – c aH                 (bring c to the right and multiply by aH= 1 / a)

b aH 2H) ²  = (b aH 2H) ²  – c aH                                    (using  2H 2H =  1)

+ b aH 2 =  ±  √ ((b aH 2H) ²  – c aH )                 (discriminant       

=  aH 2H   (- b  ±  √ (4 a c ))                                      (the quadratic formula)

From wikipedia: this formula covering all cases was found by Simon Stevin in 1594, who also gave us the decimal dot. The present form was given by Descartes in 1637. In the past people were calculating every step. Having the final formula allows you to reduce the actual number of calculations you have to do.

There will be an intersection with the horizontal axis (above equation has a root) only if D ≥ 0. Otherwise there is no intersection.

It is an option to interprete i = √H as a number too. In that case the problem is redefined to have existed in the complex plane all along, and then there is always a solution. This explains where the mystery comes from: you have to grow aware that your original problem was not one-dimensional but two-dimensional.

Frenkel’s approach “there is nothing mysterious about it” kills this last insight. He claims to draw you to the beauty of mathematics, comparable to masterpieces of art, but at the same time he says that you should not be worried since it is as common as bread and butter. There is a difference between admiring a masterpiece and making one yourself. The professor is seriously confused. It is better that students understand the quadratic equation and the complex plane, and then admire their own understanding too.

Parabolic jump (Source: Jarek Tuszynski, wikimedia commons)

Parabolic jump (Source: Jarek Tuszynski, wikimedia commons)

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