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

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)

Mathematics professor Edward Frenkel wrote an opinion Don’t Let Economists and Politicians Hack Your Math. Of course kids need to learn algebra, Slate Feb. 8 2013. This appears partly a response to Andrew Hacker,Is Algebra Necessary?“, The New York Times, July 28, 2012. Frenkel also refers to Matthew Yglesias, “CPI Unchained. The sneaky plan to cut Social Security and raise taxes by changing how inflation is calculated“, Slate December 30 2012, where CPI = Consumer Price index.

Let me respond as both an econometrician and teacher of mathematics.

First some facts

  • In Holland, kids are allowed to graduate from middle school without algebra. The system allows for various competences. Holland isn’t perfect, though. Apparently, in the USA you either learn algebra or you drop out. If there are no alternatives then this is a rather sick approach, and then Hacker is right. See this Journal of Humanistic Mathematics that discusses the wider ranges of competences.
  • Algebra builds upon arithmetic. Thus the source of the problems in the USA may be elementary school. See for example here.
  • Hacker has a nice example:
    “But there’s no evidence that being able to prove (x² + y²)² = (x² – y²)² + (2xy)² leads to more credible political opinions or social analysis.”
    The example is quite nice (find the trick), and again Hacker is right that this kind of proficiency has limited value. We should have people who can do this. For others it might suffice that they know how to handle and understand a computer algebra package like Mathematica or open source Sage.
  • Hacker has another nice example:
    “It could, for example, teach students how the Consumer Price Index is computed, what is included and how each item in the index is weighted — and include discussion about which items should be included and what weights they should be given.”
    Below I will give an example of what would be useful to people to understand.
  • The issue is mathematics education and not merely mathematics. Hacker didn’t study mathematics education but neither did Frenkel. The opinion pieces are a bit like priests discussing marriage.
    People are entitled to some opinion as to how competent professionals in areas of medicine or computers should be (“deliver what you advertise”), but it is a different issue to translate such levels of competence to the educational programme to get there.
    It is quite possible that Hacker would be very happy with algebra in schools if the system of education would be improved such that having algebra didn’t cause those dropouts, and generated the overall competence that he desires. Thus we should rather not discuss algebra but the quality of mathematics education – which quality is awful.
  • Frenkel, without his study of mathematics education, clearly misses the latter point, and starts defending the beauty and relevance of algebra. He is a typical mathematics professor who contributes to the dismal state of mathematics education by failing to understand that education is an empirical issue and not a playground for abstraction. Frenkel also abuses the CPI issue to scare the wits out of you, see below.

Thus, the sensible position has already been formulated in my book “Elegance with Substance(2009). Blame the mathematicians and the Slate and The New York Times journalists for not paying attention.

PM. Miles Kimball & Noah Smith remind us that the ability to do math is widespread, and that it is a myth that genes cause that there are people who can do math and people who cannot do math. They rightly point out that this myth causes needless suffering and must be gotten rid of. They refer to Hacker’s article as carrying the risk of furthering that myth. I don’t think that Hacker’s article and proposal carry that risk. While we should  work hard to improve math education and eliminate that myth, it remains that at some point in life a test will determine whether a pupil has mastered the subject or not. Rather than forcing the issue, it can be better for the pupil to let him or her proceed with the other competences and joys. Educational paths shouldn’t fix life thereafter but allow for flexibility later on too.

A quantity (volume) and price index

The following method has been in use at the Dutch Central Planning Bureau (CPB) for perhaps 60 years and provides a sensible and consistent way to navigate in complex waters. It uses the Laspeyres volume and the Paasche price indices.

  • Advantages of this are:
    • Volume and price changes add up to value changes.
    • The weights for volumes and prices are adjusted over time.
    • The choice of a different base period (for chaining) does not lead to different values for the aggregate data.
  • A (“disadvantage”) consequence however is that the aggregate chained volume index differs from the sum of the disaggregate chained volume indices. This however cannot be seen as a “disadvantage” since there is no reason why that sum should hold.

Consider a fictitious example of indices for the national old age pension (social security) for the age 65+. Suppose that the pension allows for (1) bread, (2) coffee and (3) some health insurance. This is called the “basket”. Then we calculate: (4) the aggregate volume and price indices. The outlays change from year to year, partly because of entrants and departures, and partly because of changing tastes and reactions to price changes.

We will use these symbols for a fictitious change from 2013 to 2014:

Price, volume and value, symbols for level and absolute and relative changes

Price, volume and value, symbols for level and absolute and relative changes

Let us assume that an average pensioner eats 100 loaves of bread per annum, at EUR 1 per loaf (wholesale price) and that this quantity changes by 5%. Let the price remain constant. This gives us the first line of the table below. Expenditure (wn) changes from EUR 100 to 105.

In the second line we record that 450 cups of coffee were consumed in 2013 at EUR 1.3 per cup, thus at a total expenditure (wn) of EUR 585. Due to a VAT change affecting coffee only, the price rises by 7.7% to EUR 1.5 and hence we see a drop in consumption of 4.4% to 430 cups of coffee per year. Expenditure on this item changes from EUR 585 to EUR 602.

In the third line we assume that the pensioner has medical attention for one hour per week, thus 52 hours per annum, at the cost of EUR 200 per hour. Thus the expenditure in 2013 is EUR 10400. In 2014 due to aging, the average pensioner requires 53 hours per year, and the medical cost rises to EUR 210 per hour (5%). The outlays from medical insurance rises to EUR 11130 per annum.

Combining these data, the total expenditure per pensioner is EUR 11085 in 2013 and rises to EUR 11837 in 2014. The “aggregate volume index” (2013 = 100) is 101.6%. The “aggregate price index” is 105.1%. The “aggregate value index” can be found by properly multiplying these, and is consistent with calculating 11837 / 11085 * 100.

Line 4 gives the aggregate value, volume and price of lines 1 - 3

Line 4 gives the aggregate of lines 1 – 3, in value, volume and price

(Calculated with c = ChainIndexTable[{{100., 105.}, {450., 430.} , {52., 53.}}, {{1., 1.}, {1.30, 1.40}, {200., 210.}}] in The Economics Pack.)

The method uses only arithmetic but the highschool student has to understand quite a lot of concepts: value, volume, price, level, absolute and percentage change, averaging, using different weights. The student can show insight in that the figures make some sense (a price rise causes a quantity drop), and that the aggregate outcome is dominated by the large component. The student will also see that when more of such calculations are combined (say pensioners, families with and without children, singles; with subaggregates in the provinces or states) then the system remains consistent, so that the method of aggregation does not introduce curious distortions.

The aggregate volume and price changes for baskets of single years can be “chained” by multiplying the annual results. The aggregate price index in 2014 with base (2010 = 100) would be: 100 * (1 + pp[2011] / 100) * … * (1 + pp[2014] / 100). The advantage of this chaining is that the basket is not fixed but follows actual expenditure.

A criticism is: since above VAT rise reduces real spending power, then there is more weight upon less real consumption. The above allows pensioners a rise of expenditure from EUR 11085 to EUR 11837, but “adequate compensation” should require EUR 26 more per person because of their induced loss of their cups of coffee. However, is this statistics or politics ? In terms of policy, giving EUR 26 more does not guarantee that this is spent on coffee. In terms of statistics, there is a choice of using weights of the baskets of either 2013 or 2014. The method of using Layspeyers volumes and Paasche prices allows to calculate the result for 2014 using the weights from 2014.

The role of statisticians is to also inform policy makers about the observed changes in the basket. The choice of the basket is politics.

Thus, one would agree with Hacker that it would be useful that this method would be taught in highschool in Holland, so that students know what their CPB is doing. Similarly for the USA. Yglesias refers to Jill Leyland (2011) for the UK.

In Holland, Bert Balk has pointed to the properties of the Divisia – Törnqvist index but hasn’t convinced others, or me, yet that its modestly larger computational complexity generates better results (and other criteria to select than the above). It is getting to be used more often in productivity measurement, but note that hours worked and wage inflation can also be aggregated in above manner by distinguishing levels of productivity and associated wages (and it is curious that this often isn’t done).

Frenkel’s embrace of conspiracy theory

The issue is also discussed in this Wikipedia article that at the time of writing this is not yet sabotaged by MIT students and their formulas. Yglesias referred to above observes:

“(..) the point is that there’s no unique right or wrong answer (…) for how to treat product shifting, and its impact on individuals’ welfare will vary enormously. This matters a great deal for Social Security, however, because benefit levels are adjusted upward each year in line with inflation. If Congress decides that chained index is the “right” measure of inflation, benefit levels will be lower than currently predicted and the deficit will go down.”

The answer is of course that social security should not be indexed on only inflation but on the general rise in welfare. (In the above: on wn. This should also hold for tax brackets.)  If there is a rise in productivity, then let also the elderly benefit from it. If everyone has a computer then let also senior citizens be allowed to get one. Thus, we have an easy issue of policy making here.

This is also what Yglesias concludes:

“The central point is that it’s politics all the way down. (…) glib talk about apples and “better” inflation calculations masks a policy that if done sloppily could be quite damaging to the low-income elderly.”

However, mathematics professor Edward Frenkel claims that you can only understand the issue when you have learned algebra, while preferably you should also know gauge theory. In his opinion (his is an opinon piece) there is also a political conspiracy, supported by economists, to exploit your lack of understanding of mathematics, and keeping you there.

(1) Frenkel’s major claim:

“Is economics being used as science or as after-the-fact justification, much like economic statistics were manipulated in the Soviet Union? More importantly, is anyone paying attention? Are we willing to give government agents a free hand to keep changing this all-important formula [for CPI / TC] whenever it suits their political needs, simply because they think we won’t get the math?

(…) What seems to be completely lost on Hacker and authors of similar proposals is that the calculation of the CPI, as well as other evidence-based statistics, is in fact a difficult mathematical problem, which requires deep knowledge of all major branches of mathematics including … advanced algebra. Whether we like it or not, calculating CPI necessarily involves some abstract, arcane body of math. [Frenkel doesn't seem to be aware of above practical solution. / TC]

(…) The inflation index must account for this, so we have to find a way to compare the baskets today and a year ago. This turns out to be a hard mathematical problem that has perplexed economists for more than a century and still hasn’t been completely solved. But even to begin talking about this problem, we need a language that would enable us to operate with symbolic quantities representing baskets and prices—and that’s the language of algebra! [Frenkel confuses above simple arithmetic and algebra with the somewhat more involved algebra that Hacker was speaking about. / TC]

(..) As Weatherall explains in his book, to implement a true cost-of-living index, one actually has to use the so-called “gauge theory.” This mathematics is at the foundation of a unified physical theory of three forces of nature: electromagnetism, the strong nuclear force, and the weak nuclear force. (Many Nobel Prizes have been awarded for the development of this unified theory; it was also used to predict the Higgs boson, the elusive elementary particle recently discovered at the Large Hadron Collider under Geneva.) 

(…) So that’s where we find ourselves today: Politicians are still eager to exploit backdoor mathematical formulas for their political needs, economists are still willing to play along, and no one seems to care about finding a scientifically sound solution to the inflation index problem using adequate mathematics. And the public—well, very few people are paying attention. And if we follow Hacker’s prescriptions and further dumb down our math education, there won’t be anyone left to understand what’s happening behind closed doors.” [Well, there is a huge economic literature on indexation. / TC]

(2) Frenkel alarms us – a call to arms – shifting the blame to politicians and economists instead seeing his own fault:

“Now is the time not to reduce math curriculum at schools, but to expand it, taking advantage of new tools in education: computers, iPads, the wider dissemination of knowledge through the Internet. Kids become computer literate much earlier these days, and they can now learn mathematical concepts faster and more efficiently than any previous generation. But they have to be pointed in the right direction by teachers who inspire them to think big. This can only be achieved if math is not treated as a chore and teachers are not forced to spend countless hours in preparation for standardized tests. Math professionals also have a role to play: Schools should invite them to help teachers unlock the infinite possibilities of mathematics to students, to show how a mathematical formula can be useful in the real world and also be elegant and beautiful, like a painting, a poem, or a piece of music.”

It is hard to see a political conspiracy to block the education of mathematics. Kids and students already spent an amazing number of hours on arithmetic and other mathematics. It are the mathematicians and the abstractly trained teachers of mathematics who are the spoil-sports. They do not see that education is an empirical issue, and they insert their hobbies and traditions into their allotted time, without wondering whether they shouldn’t pay attention to their students. As said, Frenkel hasn’t studied education, so he is in breach of the integrity of science, claiming expertise that he doesn’t have.

(3) Frenkel claims a conflict of competence / interest for Hacker:

“In his book, Weatherall made an admirable effort to start a serious conversation about the need for a new mathematical theory of the CPI. But guess who reviewed this book in the New York Review of Books? Andrew “we don’t need no algebra” Hacker! There is nothing wrong with healthy debate; it can only be encouraged. But something is wrong when an opinionated individual who has demonstrated total ignorance of a subject matter gets called on over and over again as an expert on that subject.”

I haven’t read that review yet, but it seems to me that Frenkel wrongly disqualifies Hacker. The label “Hack” in Frenkel’s opinion piece would be like someone to write about “A mathematical proof of conspiracy by Dr. Frenkelstein”.

(4) Frenkel claims that there was already unscientific manipulation in the Boskin commission 1996:

But what most people don’t realize is that something similar had already happened in the past. A new book, The Physics of Wall Street by James Weatherall, tells that story: In 1996, five economists, known as the Boskin Commission, were tasked with saving the government $1 trillion. They observed that if the CPI were lowered by 1.1 percent, then a $1 trillion could indeed be saved over the coming decade. So what did they do? They proposed a way to alter the formula that would lower the CPI by exactly that amount!

(…) The fact that gauge theory also underlies economics was a groundbreaking discovery made by the economist Pia Malaney and mathematical physicist Eric Weinstein around the time of the Boskin Commission. Malaney, who was at the time an economics Ph.D. student at Harvard, tried to convey the importance of this theory for the index problem to the Harvard professor Dale Jorgenson, one of the members of the Boskin Commission, but to no avail. In fact, Jorgenson responded by throwing her out of his office. Only recently, George Soros’ Institute for New Economic Thinking finally gave Malaney and Weinstein long overdue recognition and is supporting their research. But their work still remains largely ignored by economists.

Mark Thoma had some comments on this in 2006 and quotes BLS in 2008. I am not tempted to follow this up here. I suppose that there is no conflict of interest in Frenkel and Weinstein appearing jointly at the Speyer Legacy school, both apparently without a background in the education in mathematics.

Overall, I am happy to include this subject in a parliamentary enquiry on unemployment and/or another such enquiry on the dismal state of mathematics education.

Mathematics professor Edward Frenkel is in a state of denial w.r.t. the reponsibility of mathematicians for the economic crisis since August 2007. I know of only one mathematician by name who warned before the crisis developed, and that is Paul Embrechts of ETH. Compared to him, there were various economists, read Dirk Bezemer on “No one saw this coming” (2009), but see also my protest w.r.t. some self-serving errors by Bezemer (including my irritation that he doesn’t look into my warning, since I am still warning about more things).

“We have to realize the power of mathematics. By now it’s well-understood that the global economic crisis was caused, in part, by misuse of mathematical models. People who understood those models were actually sounding the alarm. It was the executives who had the power, who were the decision-makers, who did not understand how these formulas functioned. Their logic was: “Well, while these things work, we’re making profits.”” (Frenkel in Slate 2013)

Thus, Frenkel denies and misrepresents the role of the mathematicians and “rocket scientists” who “understood those models”.

A key point is that mathematicians are trained for abstraction. Thus, they are oblivious to the risks in the real world, as they are oblivious to the empirical aspects in the education in mathematics. See my book Elegance with Substance (2009) that makes those points, and proves them too, with key cases from didactics and with an analysis of the political economy of the mathematics industry.

These statements by Frenkel are laudable:

“I would not tell any scientist to stop his or her research because it might have some possible evil applications. But once you discover that it does have these applications, I think it’s also your responsibility to do whatever you can to prevent the discovery from being used for evil purposes. [This seems to be formulated somewhat crookedly / TC]

(…) Mathematical power is not the power of a bomb. You cannot see its effect as immediately as Hiroshima and Nagasaki. But a formula can be just as powerful in terms of controlling our lives. It can alter the course of history; it can affect millions of people.

I think we mathematicians are a little bit behind the curve. We are not fully aware of the Frankenstein that we may have already created or could create. I think that’s another aspect of this responsibility of mathematicians to take a more public role—to educate the public by giving them access to the beauty and power of mathematics.” (Frenkel in Slate)

The latter, education, is precisely the answer of Elegance with Substance too. However, Frenkel is not aware of the conundrum: education is an empirical issue, and mathematicians are trained to think abstractly, and thus mathematicians should not the ones to “educate the public”.

Thus professor Edward Frenkel is another deluded and abstract thinking mathematician, who is in denial of the true guilt of mathematics: (a) for the economic crisis, (b) for the sorry state of the education of mathematics, (c) for the sorry state of the education of Edward Frenkel himself.

Obviously, our deluded professor wants the mathematics industry to lift itself from the current morass, as Baron von Münchhausen so famously did. Alas, mathematicians will not be able to do so. They hold society at ransom, just to pursue their own delusions. My advice is that each nation lets its parliament investigate the issue.

Baron von Muenchhausen, by Oskar Herrfurth (Source: wikimedia commons)

Baron von Muenchhausen, by Oskar Herrfurth (Source: wikimedia commons)

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