Jeroen Dijsselbloem on money and math

I wasn’t going to join the hype in the European media about eurogroup president Jeroen Dijsselbloem. The hype is only about money and that isn’t so interesting, contrary to what the nice people at the Financial Times tell you. However, it so happens that Jeroen Dijsselbloem also has been the chair of a Dutch Parliamentary Inquiry Commmittee on Innovations in Education in 2007-2008. Now we are talking. For this brings us to the issue of the education in mathematics.

The European media hype is about whether Dijsselbloem knew what a ‘template’ is. He says on Dutch television that he didn’t know the word. He did however reply to a question containing it, see the FT Reuters transcript. Apparently the political distinction is now being made between an ‘approach’ and a ‘template’. We may figure that Dijsselbloem is a sensible and intelligent person who gets the drift of a question, so we can forgive him for not responding: “Can you explain what you mean by a template ?” It is actually not so nice of the reporters at FT Reuters to make such a fuss about this. They are just as guilty in this blame-game, for after Dijsselbloem’s reply they didn’t ask for confirmation: “So this will be the template, just to be sure that we will not quote you in a wrong manner ?” The real fuss is the command of English and the state of the Dutch system of education.

The European media hype is also about whether Dijsselbloem over-enthousiastically took the Dutch approach to SNS-Reaal-bank and Cyprus as the future approach for the Eurozone (if we allow for that word). It may be that he overplayed his position as president and that other members have different thoughts. This may indeed be the case. The other members may have shown polite interest in what Dijsselbloem has been explaining about his ideas, and it may be that Dijsselbloem mistook this for agreement. Deep in the hype, harsh words may have been spoken, but, as diplomats tend to do, internally, far removed from the spotlights. Well, every Dutch(wo)man has to learn that Europe isn’t just a ‘big Holland’. We all remember the difficulty that Wim Duisenberg had in 2000 after the introduction of the euro and the questions about the exchange rate policy.

Nevertheless, the EU is setting up a Banking Union with a European Banking Authority. On that single webpage we see words like ‘supervision’, ‘regulation’, ‘mechanism’, ‘vision’ and ‘roadmap’, but we do not see the word ‘approach’ yet. Downloading the Communication, we see that the Banking Union distinguishes ‘the most significant European systemically important banks’ and the ‘others’. The first will have to be saved at all cost, the others will be allowed to implode. Thus Dijsselbloem quite accurately warns us to put our money in a system bank, unless we are risk-prone and like a little bit of higher interest plus the thrill of a possible collapse. The warning is also that each bank in trouble should quicky join a system bank so that it will be saved.

I might sound a bit sarcastic but in reality I am referring to my earlier Economic Plan for Europe and the suggestion for a new EMU treaty, plus an additional analysis that I hope to be able to put on the web soon. [Addendum April 3: it is here now.]

Now the interesting part. For Dutch education, we find a similar juggling of words. The Dijsselbloem Committee in 2008 distinguishes ‘what’ from ‘how’. Parliament decides what will be taught at school, and the teaching community decides how it will be taught. The Committee observes that this rule had been violated in the past, with various ‘innovations in education’, that Parliament loved but teachers abhorred, and that caused Dutch education to go down the drain.

The Committee didn’t investigate how it came about, that Parliament loved ‘innovations’ that the teachers abhorred. Apparently Dijsselbloem takes it for granted that Parliament doesn’t listen to teachers. Indeed, after the Committee report was declared a success, Parliament decided that highschool graduation should include a test on basic numerical skills, and a fail would even block graduation. Parliament thinks that this is a ‘what’ but actually it is a ‘how’. Learning to count is for young children and not for teenagers. Elementary schools should provide for those basic numerical skills, but they are failing to do so, both because of ‘innovations’ and because of elementary school teachers who have insufficient numerical skills themselves. Apparently Parliament wants to fix this by shifting the burden onto the higher level. Dutch readers with a strong heart and love for horror shows would want to read Jaap de Jonge “Opkomst en ondergang van de rekentoets” (Rise and Fall of the Numerical Test), March 2013, Euclides 88/5 p224-225. Unfortunately, that magazine of the Dutch Association of Teachers of Mathematics tends to keep important information behind a pay-wall.

Now to the Grand Finale. My point is that Parliament has decided that schools must teach math, but the teachers do not deliver math, but something that they call ‘math’. Mathematicians are trained for abstraction, but in class they meet real live students, and they resolve their cognitive dissonance by clinging to a tradition that has grown over the ages, but that isn’t targetted at proper didactics. See my books Elegance with Substance 2009 and Conquest of the Plane 2011. Thus the Dijsselbloem distinction at first seems to have some merit, but breaks down when Parliament refuses to check whether it really gets what it intends to get. The distinction between ‘what’ and ‘how’ is somewhat illusory, if the people responsible for the ‘how’ destroy the ‘what’.

The only solution is that Parliament starts paying attention to teachers rather than the bureaucrats and lobbyists. The only solution lies in an open atmosphere, where people can speak freely and frankly, where we treat people and ideas with respect, and where we judge issues on their merit.

More on this in my paper What a mathematician might wish to know about my work, March 2013.

A disciplinary board for mathematicians

You can read this best while listening to the beautiful Eleftheria Arvanitaki and her rousing Metrisa on YouTube. I don’t know what she actually sings about, and I don’t want to know, but you can sense the thunder clouds forming on the horizon, with already some flashes of lightning.

The Italian election result feels like that too. A deadlock, commentators say. Europe is in crisis again, they hold. Perhaps it is a blessing in disguise. Belgium needed 18 months to create the Elio di Rupo government, and many thought that the country actually did well and got a much needed period of rest.

There is discussion about a grand coalition of leftist Bersani and rightist Berlusconi, or perhaps a minority government but supported by Grillo. Grillo wants to judge proposals on their merits separately and refuses to bargain about a full programme. In these political analyses the habit of thinking in majority governments wreaks havoc again. According to this view you team up to kick out a minority. Why would it not be possible to have an inclusive government, in which (most of) all parties partake ? It is a good idea indeed to judge proposals on their merits, to keep out pork barrel, but still with the intention that all have to live in the same country.

We see an example in this duet with Arvanitaki. The somewhat aged beauty who sings Na Na and who keeps breaking her earthenware bells is Haris Alexiou, who has produced wonderful albums and who performed in great concerts. Listen to her stately singing, the vivacious apopse with images from her younger years, and the touching oles, a classic. Or Ximeroni ! (Unfortunately with bad sound quality.)

I admit, illustrating Italian election results by Greek singers may come across as farfetched, but the association actually is rather sound, because co-operation is also a form of art. Many people regard democracy and elections as sport, and they only notice the winner. In reality, those winners and their supporters may be the barbarians, lost to culture and civilisation.

Some suggest to break the deadlock in Italy with new elections. In that case we might see what I have been advising for some years now: (1) governments that mirror parliament, (2) annual elections. This gives voters more power and still forces politicians to co-operate. Two other crucial advices: (3) select the prime minister with a Borda Fixed Point method, so that he or she has broad support and still can function in impartial manner above the parties (see this application to Holland), (4) create an Economic Supreme Court that watches over the quality of information. These four elements improve the responsiveness to popular sentiment without turning into populism, and they increase the quality without turning into technocracy.

Notice that there is a fundamental problem here. Election methods normally are a disaster. In the US election between Bush, Gore and Nader the winner was Bush while Gore would have beaten each of the others in pairwise comparisons. In the French election between Chirac, Jospin and LePen the winner was Chirac, while Jospin would have beaten each of the others in pairwise comparisons. Lawyers who write electoral laws tend not to understand much of mathematics, and then ask advice from mathematicians, who however create math from thin air and apply it to reality without understanding reality. Democracy disappears in the ravine between alpha’s and beta’s, the Two Cultures of C.P. Snow.

Italy has developed a complex system to allocate seats with the intention to enhance stability. That system now seems to enhance instability. It might be that it works out okay, as we hope above, but that would be by chance or wisdom, and the electoral system remains a disaster.

Of the many people who have been sleeping, a great responsibility falls on the politcians who voted this system into action. There is also a responsibility for the mathematicians who have been advising in the background. The fundamental problem is they can help to design systems, but run away from criticism, do not acknowledge error, and thus do not learn from mistakes. With their structurally erroneous advices these mathematicians destroy huge democracies.

We need a disciplinary board for mathematicians. When a medical doctor gives a wrong advice then there is such a board. A mathematician who doesn’t study reality but still advises that abstract notions apply to reality, is condemnable in the same manner.

In a short Dutch article Pas op met wiskunde over verkiezingen I explain the issue at a level for highschool students. Its appendix also contains a list of some 10 mathematicians who run away from criticism on their work on voting and democracy. In 1990 I observed that serious errors were being made, and the list gives that experience of denial since then. The list contains only mathematicians who are supposed to have an ethic of ‘definition, theorem, proof’ and who sin against that, even when the error is pointed out to them. It is no use to make a list of economists and political scientists who repeat the errors of the mathematicians, since that list would be much longer, and they would tend to refer to the mathematicians anyway (as if that would be proper).

The story turns into horror. I do not know whether I should refer to the dancing and waving of Eleni Bitali and her song about her life (zoi mou). Beware: first it seems as if she is the blond lady but later the camera switches position and it appears that she is the lady with red hair. I offered above Dutch article to the journal of the Dutch Association for Mathematics Nieuw Archief voor Wiskunde (NAW). One would think that the editors would be delighted with a short exposition of the major errors by mathematicians on voting theory and democracy. At that, a discussion that high school students should be able to understand, and that reviews which mathematician better corrects which misunderstanding. One would expect that the editors would desire to advance better mathematics. But no. Editor Barry Koren of the University of Leiden answers that he has studied the short article, fails to understand it, doesn’t specify what passage he doesn’t understand, rejects the paper and closes the discussion, in one grand sweep. I have included his name on the list of failing mathematicians because of this event, though as far as I know he hasn’t written on voting theory. But the horror is that this concerns criticism on the math profession and that a journal blocks that criticism.

Perhaps professor Koren of Leiden didn’t understand that the article was targetted at a level of exposition for highschool students, though it was explained to him. Perhaps he mistook the easy language with sloppy thinking. Perhaps he wanted to see complex mathematics though he could have found these in the references. We can imagine various misunderstandings. The fundamental point is that he presents a closed mind. An econometrician is not allowed to criticize mathematicians when they don’t study reality but still give advice on that.

Do the mathematicians fail only on democracy and election methods ? No, they do so too in the education of mathematics, when they have been trained for abstract thought and suddenly encounter real life pupils. They do so too when they are ‘rocket scientists’ and develop financial products that do not account for real risks. They do so too in the study of logic when they exclude nonsense while that is the most nonsensical thing to do. I only mention areas that I have studied myself and where I have established this. Perhaps other people have other examples.

My advice for a disciplinary board for mathematicians thus is dead serious.

We end with Eleftheria Arvanitaki and a sirtaki in the studio. Listen especially from minute 37 onwards, when the guests have unpacked their presents, and Eleftheria enchants all hearts, with all eyes becoming watery and proud men holding on to their sigaret, and with Haris Alexiou in full rapture.

When Greek singers and their musicians would travel over Europe and would teach us to sing and dance then they have another export product with great potential, alongside with those earlier ideas about democracy and mathematics.

Wikipedia acrobatics

A core argument of this weblog is that the checks and balances of the democratic model of Trias Politica fail and that we need an extension with an Economic Supreme Court (ESC) into a Tessera Politica. A government budget tends to be based on forecasts and it is better that those are scientific and hence independent. Economic scientists will forecast what the politicians will do in the future. All kinds of political promises are made, but will they be kept ? What value is a budget, voters will wonder, when it is based upon rosy promises and without scientific scrutiny ? Independence is not enough, the ESC requires the scientific ethic, and be open to society and fellow economic scientists.

This argument causes that this weblog is interested in democracy and in what the public understands about democracy. A friend asked me what I thought about Wikipedia entries on democracy, since that is a source that people tend to refer to with increasing frequency. We actually see some acrobatics here. The subject curves in on itself, like a snake dancer who is able to hold her head by her feet, forming a full circle.

Wikipedia is made by volunteers who apply some notions of democracy themselves to settle differences in approaches. Does the quality of Wikipedia improve with internal democracy ? Have its editors a sound understanding of democracy that is also reflected in what the encyclopedia states about the subject ? Or, do the editors follow what has been written – what they have written themselves ? Might it happen that Wikipedia publishes a wrong analysis on democracy, and that its editors behave in dictatorial fashion ?

Unfortunately, the latter is true: Wikipedia publishes a wrong analysis on democracy, and its editors behave in dictatorial fashion. Wikipedia has been misleading its readers since 2006 because of scientifically unacceptable conduct of its members, and internal rules that allow this. Wikipedia doesn’t show sufficient respect for science, which would be a key requirement for democracy (unless you follow the Trias Politica model where politicians can manipulate science).

The following quotes are from Wikipedia Februari 17 2013 on the entry of Arrow’s impossibility theorem. First note that the article presents a complex mathematical proof. This is needlessly complex. The issue is essentially simpler. Kenneth Arrow gives a general statement, that would apply for all kinds of preferences and situations. Hence it suffices to give a single counterexample to decide to an impossibility. See e.g. the counterexample by Donald Saari, that I copied in DRGTPE at Project Gutenberg.

Thus we arrive at an analysis that most citizens of a democracy could understand: (1) Arrow presents five conditions that would apply to collective decision making in a democracy, (2) There is a contradiction. (3) Thus those five conditions cannot hold all at the same time.

The above can be called “Arrow’s theorem” and it stands (though see below). The confusion starts from that Arrow suggested that the conditions would be “reasonable” and “morally desirable”. This inserts notions of rationality and morality that give a high weight to the discussion. Arrow argues: we must become irrational or immoral if we want to achieve collective decision making, and this will not be “perfect” democracy.

My book “Voting Theory for Democracy” (VTFD) explains that Arrow makes some crucial mistakes here. VTFD is the only book in the world that explains the situation properly. The book turns those “non-mathematical” qualifications “reasonable” and “moral desirable” into mathematics too, such that it casts doubt on the mathematical result.

(a) Reasonable means at least consistent, but his axioms are not consistent. Hence the axioms cannot be called reasonable.

(b) Morality holds that you cannot be obliged to do the impossible. Hence his axioms cannot be morally desirable.

(c) Arrow’s Theorem by their generality would also concern preferences on constitutions. This is a form of self-reference, that his axioms also apply to themselves. Can people have preferences on constitutions ? Yes. The analysis is complete if it covers this intended interpretation. Arrow assumes rational agents but no rational agent would accept his inconsistent axioms. Apparently Arrow’s analysis is incomplete or inconsistent.

Now, where does Wikipedia become misleading ?

(1) Wikipedia-quote: “Although Arrow’s theorem is a mathematical result, it is often expressed in a non-mathematical way with a statement such as “No voting method is fair,” “Every ranked voting method is flawed,” or “The only voting method that isn’t flawed is a dictatorship”. These statements are simplifications of Arrow’s result which are not universally considered to be true. What Arrow’s theorem does state is that a deterministic preferential voting mechanism – that is, one where a preference order is the only information in a vote, and any possible set of votes gives a unique result – cannot comply with all of the conditions given above simultaneously.”

In itself a rather nice synopsis of the situation, except for the points (a) to (c) above. If you assume that Arrow’s axioms would need to be complete with respect to the intended interpretation (are self-referential with respect to constitutions too), then they appear incomplete or inconsistent. 

(2) Wikipedia-quote: “the Gibbard–Satterthwaite theorem still does: no system is fully strategy-free, so the informal dictum that “no voting system is perfect” still has a mathematical basis.”

Here is the same sillyness about “perfection” without a definition about what that would be. Would Arrow’s axioms be the criterion of “perfection”, while we know that they are inconsistent ? If there is no “perfection”, are we to allow people to argue for dictatorship or ”let’s accept corruption, since democracy isn’t perfect anyway” ?

Conclusions: (1) Arrow does not study democracy but only a mathematical model, (2) Arrow uses characterisations about that model that cannot be maintained, (3) Arrow breeds cynicism about democracy, (4) many other mathematicians are parrotting this, spreading cynicism about democracy, like speaking about imperfection or even calling for dictatorial mechanisms, (5) Wikipedia neglects the better analysis is VTFD.

The other dismal point is that Wikipedia can show little respect for science and can use dictatorial methods. In 2006 I noticed that an editor had inserted a Wikipedia entry on my suggestion for a Borda Fixed Point voting mechanism. The entry was erroneous at some points, so I took the liberty to correct the Wikipedia entry. In the process I also improved some points on the page on Arrow’s theorem that applies to it. A student in computer science at MIT thought that he had a better understanding of the situation, but was unwilling to show this with logical argument and decent behaviour. See here what followed in terms of gang-rape and witch-hunting. The editors at Wikipedia did not appreciate that I regarded the professors of the student as more relevant for student education than the editors themselves. See here what another student wrote, misleadingly, about the affair. This second student, Joseph Lorenzo Hall now in 2013 has completed his Ph.D. thesis and has become a staff member at the Center for Democracy & Technology. We may wonder what he tells his colleagues about what democracy actually is and how this can be programmed so that we can all benefit from it.

PM 1. The best defence of Wikipedia might be that they base their information on science but that there has been censorship in Holland. But in a case like this you can still think for yourself, and spend some time on the arguments that discussants have given. It helps when you have studied the subject so that you can understand arguments. The subject is democracy and not just a mathematical model.

PM 2. The directorate of the Dutch Central Planning Bureau (CPB) rejected my economic analysis on unemployment and the social welfare function also by referring to Arrow’s Theorem on the impossibility of fair social choice. In response I looked at Arrow’s analysis and wrote a paper that rejected it. However, the CPB directorate did not want to discuss and publish my analysis on Arrow’s theorem either. I have looked for support from outside mathematicians on the analysis in my paper. My position in this discussion has been rather weak since mathematicians refused to look into it or came up with silly remarks and did not respond adequately when I pointed out their own errors. (Dutch readers may look at a summary here.) Resolution of this issue could be very important for understanding my position, and the resolution of the issue of unemployment and social welfare. Yes, economists fail here too, also in the fact that they follow failing mathematicians. Overall, my best advice now is to boycott Holland till the country understands that it has to stop censoring science. Perhaps Wikipedia can write a nice article about that advice to boycott Holland ?

Pyramids and the meter

Belgian television showed the film The Revelation of the Pyramids. It contains an intriguing suggestion for a mathematical relationship. Let us debunk it, though keep the intrigue.

I have three reasons to look into this. The first reason is the earlier weblog on the use of ‘archi’ Θ = 2 π = 6.283185307… rather than π as the key mathematical concept for the measurement of the circle. Other people suggest ‘tau’ τ but that looks too much like the radius r and thus will cause much confusion in the classroom. The second reason is the earlier weblog on the mathematics of Jesus. Since the holy family fled to Egypt there is ample reason to look what was happening there. The third reason is that the film suggests that there was an ancient advanced civilisation. Since we may all be disappointed about how we ourselves are doing as a civilisation, it would be great when we could discover that others in the past have been doing much better.

We will also use ‘phi’ φ = 1.618033989… or the golden ratio. This has the property that φ² = 1 + φ, or alternatively that φ = 1 / φ + 1. It allows a particular interesting application of the Pythagorean Theorem. A right angled triangle with base a = 1 and height b = √φ generates a hypothenusa of c = √ (a² + b²) = √(1 + φ) = √ φ² = φ. The associated square has the surface φ², and by using a circle of radius φ we can find that same value in the length of the interval 1 + φ. It appears that these dimensions have been used in the pyramid of Cheops. To measure length the Egyptians used the ell or the (royal) cubit of approximately 0.5236 meters (wikipedia: between 52.3 and 52.9 cm). The pyramid of Cheops has a height of 280 cubits and a full base of 440 cubits. That shape however consists of two right angled triangles. The proper triangle has a base of 220 cubits. The ratio is 280 / 220 = 14 / 11. It so happens that 11 * √φ = 13.99221614… ≈ 14. Thus the Egyptians chose a ratio in integer numbers that closely matches the real value of the golden ratio.

 The film The Revelation of the Pyramids now presents the startling equation:

π = 0.5236 + φ²    or      π = cubit + φ²

Startling about this is that π and φ are pure numbers while the length of the cubit only makes sense when everything is expressed by using the meter as the standard length. The pure numbers π and φ come about as ratio’s and thus by dividing lengths so that they do not depend upon any choice of measurement standard. But the value of the cubit changes if we switch from meters to feet and inches.

A first step is to check for accuracy. We find that π – φ² = 0.5235586648… Thus the relation only holds by approximation, though the accuracy is eery.

A second step is to divide both sides by the cubit, or rather by the pure value π – φ². Then we find:

π / (π – φ²) = 1 + φ² / (π – φ²)

6.000459671… = 1 + 5.000459671…

There we are.

Do you see it ? Well, it took me some moments to find the proper sequence of explaning, so let us follow these steps.

A major point is that the use of π has been playing a misleading role in this analysis. It gives only a half circle and it is much better to use Θ and the whole circle.

The first point is the surprise that φ² / (π – φ²) = 5.000459671… Reworked, we get:

φ² / Θ ≈ 5 / 12

What is to say about that ? Well, it apparently is a mathematical property, like 11 * √φ ≈ 14. Sometimes mathematical numbers with complex properties and long decimal expansions can get close to ratio’s of specific integer values. This may be surprising, but it is a mathematical surprise. It cannot be a base for concluding that the ancient Egyptians knew about the decimal expansions of these numbers and their particular ratio. Once you decide to build a pyramid using the ratio of 14 / 11 since it is pleasing to the eye and with structural stability, then you are stuck with the implied mathematics, but that does not imply that you know more about the implied mathematics.

Secondly, let us assume that the Egyptians had their ell or cubit as an arbitrary length (based upon the human body). They also divided the year in 12 months and day and night in 12 hours each. Thus for them it makes sense to measure the circumference of a circle by 12 cubits, like we still do in our clocks. Of these 12 pieces of a pie, six can be allocated to π, five to φ², and then one remains (all with a proportionality factor).

The radius r of that circle follows from Θ r = 12 cubit, giving r = 1.909859317… cubit ≈ 1.91 cubit. For the Egyptians there was nothing special about that number for that radius. The film shows that the capstone of the pyramid would have this side. That is not inconceivable given this geometry.

It is only for us, who have adopted the meter (rather than feet and inches), that a sense of wonder arises. We are in the realm of approximation now, where cubits as straight lengths approximate the arcs. With a cubit of standardised value of 0.5236 meters, that radius is r = 1.000002338… meters. Alternatively put, if we take a circle with radius 1 meter then the 12-side regular polygon gives us the Egyptian (royal) unit of measurement, namely via Θ r = 12 cubit or one cubit = Θ / 12 = 0.5235987756… (using the arcs rather than the sides of the polygon).

To understand what is happening here requires us to look into the history about the selection of the meter as the European standard of measurement. Officially, the French Academy decided in 1791 that a meter was to be one ten-millionth of the distance from the Earth’s equator to the North Pole (at sea level) (wikipedia). The expedition by Napoleon to Egypt took place in 1798-1801, thus later, and the results of the new Egyptology will not have been available immediately. From this we may tend to infer that the ancient Egyptians knew about the size of the Earth and reasoned like the French. It seems more reasonable to think differently. To start with, it is already curious to take something that is difficult to measure, such as the distance from the Earth’s equator to the North Pole, to define a standard. It seems more reasonable to assume that there were already circulating measures and that the story about the equator was only an embellishment. Apparently the circle with a circumference of 12 ells had been surviving over the ages and still made it into the discussion.

But the film then should be about what happened in France and not about mysteries in ancient Egypt.

NB. There is ample discussion about the measurements of the pyramid. The top is missing so we can only guess what the Egyptians intended. See the original Petrie measurements (base 9068 and height 5776 +/- 7 inches) and this discussion with drawings. Indeed, if the base is 220 cubits and the Egyptians had a precise estimate of  √φ then the height would be 279.8443229 cubits, which is only a 0.06% of the whole height or one finger of a cubit short of 280. Because of this uncertainty, we cannot infer on these grounds that the Egyptians didn’t have a precise estimate of φ. It are other documents that show us that there were severe limits to their number system. We can neither infer that they were aware of the implication that φ² = 1 + φ, We can observe however that they used geometry and architecture that closely matches these results. See the website by Gary Meisner for how you can create your own golden ratio paper pyramid.

PM. Sir Flinders Petrie (1853-1942) suggests that the basic inspiration lies in the circle rather than in the golden ratio. A circle with radius 7 has a circumference of 7 Θ ≈ 7 * 44 / 7 = 44, using the approximation π ≈ 22 / 7. This 44 gives a square with sides 11. Thus we find the numbers 14 and 11 again.

The argument then is that the Great Pyramid expresses Θ ≈ 4 * 440 / 280 = 44 / 7, and that the golden ratio is only a by-product. If this is the case then this knowledge about Θ has been kept secret or has been lost since later documents apparently don’t mention it. It is a bit curious how that knowledge can get lost when that very same pyramid is standing in front of you. Mankind however has achieved greater mysteries. Note that there is no quick transformation into φ² / Θ ≈ 5 / 12. Via Pythagoras φ² ≈ 1 + (14 / 11)² = 317 / 121 and now φ² ≈ 5 / 12 * 44 / 7 = 55 / 21. For us these are approximations only but for the Egyptians it sufficed that the construction worked. The Petrie approach to start with the circle and 14 / 11 ratio seems simplest indeed. Still, the builders will not have been insensitive to the lure of the golden ratio, and it is remarkable that they have hit upon this very shape.

The book “The simple mathematics of Jesus”

As a student in highschool in 1970 I wanted to study archeology because I found mankind’s past fascinating. I read thick books on the history of Russia and China as well, for which reading I am still very grateful since some such knowledge still is very helpful in understanding the world. However, at that time there were tv broadcasts about Biafra and the death of innocent people and children. Since politics is determined so much by economics – at least, this is what I learned from those books on Russia and China – I decided to study econometrics instead. Now I have developed my analysis in DRGTPE that can help prevent new Biafra’s. There is little that I can do about this anymore. It are my fellow economists who have to study my analysis, and it are the public and the political powers that have to decide whether they want to adapt the Trias Politica of their nation with an Economic Supreme Court. Fortunately I forecasted the crisis and forecast more crises to come, so there is some likelihood that circumstances will allow rationality and kindness to prevail. With unemployment and poverty essentially solved, I have more moral freedom to pursue other interests. Archeology popped up again.

Sometimes a person can surprise himself. This happened this year with me on the subject of both mathematics and religion. The ordeal started with the earlier weblog entry Crucifiction, deliberately spelled in that manner since the crucifixion might be fiction. At first I thought that the subject had taken too much time already, but then it hit me: the subject lends itself magnificently as a contribution to the education of mathematics. My new book The simple mathematics of Jesus proves the point (and corrects that earlier weblog entry Crucifiction).

(1) In the didactics of mathematics there are the levels of understanding discovered by Pierre van Hiele and Dina Geldof. Line and circle are abstract mathematical notions but pupils will first develop a less abstract understanding such as walking straight or riding a merry-go-round. In the same way the divine can be understood as an abstract notion that people first learn at lower levels of abstraction. Indeed, mathematics can be applied to all kinds of subjects, space, numbers, physics, and it can be enlightening to apply it to the story of Jesus.

(2) Our civilisation developed with religion and mathematics in tandem or in tango. Early religion was linked up with astrology. This predecessor of astronomy still required mathematical talent. The zodiac is the early calendar required for agriculture and the seasons of sowing and reaping. The sun, moon and five known planets were worshipped as gods and goddesses. The early mathematics of the Old Testament was dogmatic and part of mathematical autism transferred into religious intolerance. Levels of misunderstanding contributed to religious wars. Only with Thales and Euclid there arose the liberty of posing axioms and the notion that a proof is required, and that only your own mind can force you to accept a proof. In the New Testament (Matthew 22:36-40) Jesus indeed reduces the Ten Commandments of Moses to the mathematically sufficient two demands to hallow God and love thy neighbour as yourself. Unfortunately both religion and mathematics still suffer from the ancient culture of dogmatism and haven’t amply adjusted to the discovered freedom yet. Also, that the Bible relies upon astrology was considered a strong point up to the middle ages since that was the best that the ‘science’ of the period had to offer. Since then astronomy has taken over and religion hasn’t been able to follow suit.

(3) Mathematics doesn’t only apply to the development of the calendar and astronomy. There is also the mathematics of information theory, text analysis, the study of patterns in story telling and theatre. Writing and reading were a problem in the past - at least more than in our times – and the astrologers relied on stories to relay their findings. To be remembered and retold, a story had to be good. The astrologers may have been masters in encoding information into stories that capture the imagination. Jesus is born in the sign of Capricorn. The goat becomes the scapegoat that carries the sin of the world, and that is slaughtered later as the lamb of god to redeem us from that sin. The agricultural season in the Middle East differs from that of Northern Europe. Sowing of barley and wheat takes place in November and December, or in the astrological sign or house (= Beth) of bread (= Lehiem). We also see the ass and ox that pull the plough. The harvest begins at Easter, and Jesus then is the human sacrifice required for a good crop. Analysis of such themes suggest that Jesus was not a historical figure but came into being as a theological concept, similar to how Sherlock Holmes has become the quintessential detective. People have to decide for themselves naturally but it is enlightening to consider the evidence.

Composing The simple mathematics of Jesus has been an adventure. Since 1970 I have been keeping notes in the margin and such insights coalesced now. 5000 years of history are not easy to handle. So I put the evidence in “panels” of about a page each, and the reader can evaluate the steps. The whole allows the reader to link up mathematics with history, philosophy and religion. Standard histories tell that Pythagoras and Plato went to Egypt to study, but we see only few discussions that Plato’s Academy was actually a Pythagorean School, and that centuries later the Pythagoreans and Platonic philosophers in Alexandria in Egypt had a great influence on the development of Christianity. It is enlightening for an understanding of our civilization to join up these bits. If you get to read the book: brace yourself, and enjoy !

The book also relates to the current economic crisis. Does neighbourly love have a future ? For this, my economic analysis applies again. In economics everything hangs together, and I can refer to my other earlier weblog entry on the high treason of the high priests …

English as a dialect of mathematics

The West writes and reads text from the left to the right while Hindu-Arabic numbers are from the right to the left. Thus 14 is “fourteen”. English switches order from 21, to “twenty·one”, while Dutch still has “een en twintig” and so on till 100.

There exists an alternative number system that satisfies didactic clarity so that pupils could learn arithmetic rather quickly. This uses the language of mathematics. The translation to English would be a mere matter of learning another dialect, which cannot be a burden in any way also given the small set of words and concepts. For example 59 can be “five·ten·nine” where English as a dialect has “fifty·nine”.

Perhaps the English and American reluctance to learn other languages and accept dialects is a larger bottleneck than possible doubts about the didactic advantages. The key notion thus is to regard English as a dialect indeed, and extend lessons on arithmetic with clarification of the dialect.

The issue came to my attention by Gladwell (2008:228):

“Ask an English-speaking seven-year-old to add thirty·seven plus twenty·two in her head, and she has to convert the words to numbers (37 + 22). Only then can she do the math: 2 plus 7 is 9 and 30 plus 20 is 50, which makes 59. Ask an Asian child to add three-tens-seven and two-tens-two, and then the necessary equation is right there, embedded in the sentence. No number translation is necessary: It’s five-tens-nine.”

There is not only the notation of 59 and the pronunciation, but also the notation of the pronunciation. Instead of “five-tens-nine” a better notation is “five·ten·nine”, thus no “tens” and thus the use of a high dot. The hyphen is unattractive since it is too similar to subtraction. The dot is not pronounced, like the hyphen or comma.

The choice derives from mental working space. Gladwell (2008:228): “(…) we store digits in a memory loop that runs for about two seconds.” English numbers are cumbersome to store. He quotes Stanislas Dehaene: “(…) the prize for efficacy goes to the Cantonese dialect of Chinese, whose brevity grants residents of Hong Kong a rocketing memory span of about 10 digits.” The quick fix is to use Cantonese internationally, yet this will meet with some bottlenecks.

This paper contains the longer discussion. The Appendix contains a stylized presentation for six-year olds. This is not intended for actual use in class but contains the framework for starting to think about that. It was written at the occasion of my son M.’s sixth birthday.

PM. Dehaene has also this useful quote here: “A lot of conceptual difficulties could be clarified if mathematicians and theoretical physicists paid more attention to the basic distinction between model and reality, a concept familiar to biologists. ” Mutatis mutandis for economics.

Robbert Dijkgraaf as Darth Vader in Princeton

Darth Vader is iconic and doesn’t need explanation.

People feel at ease though when the obvious is being explained so let me make you feel at ease. The universe is filled with a Force that Jedi knights apply to the Good. Anakin is very sensitive to the Force and becomes the chief Jedi, but he appears to have essential weaknesses and the Force corrupts him. He turns into Darth Vader and becomes the chief villain of the evil emperor.

Robbert Dijkgraaf may become iconic if he really turns into Darth Vader. He is departing as President of the Dutch Academy of Sciences (KNAW) and becomes director of the Institute of Advanced Study (IAS) in Princeton.

When Alfred Einstein fled from Nazi Germany he came to IAS.

Also Kurt Gödel settled down there, and walked with Einstein.

I have my doubts about these two gentlemen. My book Conquest of the Plane section 14.2 discusses that when space has been defined then we lose freedom to redefine it again. The human definition of space is Euclidean. We imagine non-Euclidean geometry (say on a sphere) within Euclidean definitions. Einstein’s mathematical manipulation of space is actually a way to deal with measurement problems, and it is not warranted that this is a good way. My book A Logic of Exceptions chapter 9 shows that reasonable assumptions cause the Gödeliar to collapse to the Liar paradox that can be solved with three-valued logic, so that Gödel’s conclusions only apply to artificially weak systems.

These reservations about these two gentlemen must be seen in perspective. We might criticize Aristotle that he hasn’t developed differential calculus so that his books have less relevance today. But this is silly. Aristotle is one of the great geniuses of mankind so we must appreciate his results within the framework of his time. Perhaps the same holds about Einstein. He died only recently so we may need some more centuries to get a proper perspective. At least I do since I don’t claim to understand modern physics at this moment. About Gödel there is more room for doubt since it is at least as likely that he was a deluded mathematician. But of course I may be wrong. Anyhow, these two gentlemen have greatly contributed to the high standing of IAS in academic culture.

We can identify the Force with Science. Thus scientists can use Science in the service of the Good. Subsequently there is Robbert Dijkgraaf (Anakin) who is very sensitive to Science and he got elected to be President of KNAW (becoming the chief Jedi).

I wrote him a message about the censorship of science by the Dutch Central Planning Bureau (CPB). Dijkgraaf did not respond and didn’t do anything. Legally, the CPB does not reside under KNAW but under the Ministry of Economic Affairs. Nevertheless, KNAW has a role in Dutch society to watch over the integrity of science and it is not unreasonable to expect that Dijkgraaf should step in, and urge the government to deal with the case of integrity of science at CPB. None of this.

Thus in my experience Dijkgraaf has failed as a scientist and President of KNAW. Integrity hasn’t been defended where it mattered most. Dijkgraaf opted for his own agenda. He has the essential weaknesses of vanity and fear (for fear). He paraded all over Dutch television promoting science but it was a farce because he allowed the government to destroy the integrity.

The IAS loved the show and loved to have Dijkgraaf. Has it turned into the Evil Empire ? The IAS exists by cherry-picking apparent winners and then advertising how good they all are. It is not clear whether this breeds a critical scientific attitude. The Institute states: “For 2012-2013 the theme will be Economics and Politics”. They would need my book DRGTPE on the failure of the Trias Politica and the necessity of an Economic Supreme Court. But it is unlikely that they use it because of the censorship in Holland. Will IAS criticize Dijkgraaf’s performance in Holland ? IAS will put out corrupted science.

PS. The IAS Institute Letter Fall 2011 announces the appointment of Dijkgraaf and also discusses the “continuum hypothesis”. Trained logicians have brainwashed themselves into irrationality, and are like a sect worthy of antropological research. See also my papers Contra Cantor Pro Occam and Neoclassical Mathematics for the Schools.

PPS. I wrote the editor of the IAS Institute Letter about all this but haven’t gotten an answer.

Archimedes revisited

My proposal to use the name “Archimedes” for Θ = 6.283… got a reply from Peter Harremoës from Denmark. Peter argues that engineers and artisans in Archimedes’ time found it more efficient to measure circles by their diameter d and not with the radius r = d / 2, so that Archimedes calculated π = Θ r / d = Θ / 2 = 3.141…. Hence the latter number is called Archimedes’ number, historically. Peter discovered that the Persian mathematician Jamshid al-Kashi in 1424 apparently was the first to use 6.283… as a separate number. Hence Peter suggests to use Al-Kashi’s constant τ, where he also adopts the symbol tau as do Robert Palais, Michael Hartl and Vi Hart as shown on my proposal page.

Bear with me. I have been aware of Archimedes’ historical position, see the proposal text indeed. The point is that there is only one mathematical constant. The values 6.283… and 3.141… are mere transforms of the same constant. Thus we should select only one name. Moreover, Θ / 2 would be vocalized as “one half Archimedes” such that Θ is a unit of account and not just a number discovered by some person.

It may be fun to say that Isaac Newton discovered one Newton and Alessandro Volta discovered one Volt while Archimedes discovered only one half Archimedes, but that would stretch what we mean by a mathematical constant. Archimedes really was the first to determine the mathematically correct way to catch that mathematical constant. So, there is no conflict between using the Archimedes as the unit of account and accepting that 3.141… was historically seen as Archimedes’ number.

Subsequently, Archimedes’ reasoning was didactical, since he adopted the common usage in his day of the diameter. We have switched to the radius so let us switch consistently. Perhaps Al Kashi was instrumental in that switch but he was aware of Archimedes’ important discovery and I like to think that he would agree that Archimedes receives all honour.

I have really thought deeply about tau. I really don’t mind what is actually chosen as long as it works best in education. I considered tau independently from the others but rejected it because it looks too much like r. The capital theta looks nicely like a cirle. The little mark in the center is not a slash like for the diameter or crosssection Ø. My proposal is that we research what works empirically best in education.

It might be a nice idea to put the choices up for an opinion poll. The true vote would be to use either current π or one of the alternatives for 2 π. But this vote would be biased when there is a difference in opinion about what that alternative will be. A vote now cannot be decisive since it is a matter of empirical research. However, voters can have an opinion about what should be tested in that research, or have a forecast about what would work best, at least for themselves. Thus, an opinion poll can be somewhat informative.

See this page for the vote.

You may also include this script in your webpage, but I haven’t found out yet how to do that in this blog… <script type=”text/javascript” src=”http://www.easypolls.net/ext/scripts/emPoll.js?p=4f5619a1c2e1b0e4901bc494″></script&gt;

Mathematical constant Archimedes = Θ = 2 π = 6.2831853…

My book Conquest of the Plane (COTP) uses Θ = 2 π = 6.2831853…. My proposal in supplement to COTP is to use the name “Archimedes” for this particular symbol (“capital theta” with such assigned value). It will be a new mathematical constant.

One Archimedes thus is the circumference of a circle with radius 1. Another relevant format is 1 / Θ = 0.15915494… When you take a circle with a radius of about 16 cm then the circumference will be about 1 meter. A circle with radius r has circumference C = r Θ and surface S = r ² Θ / 2.

In wikipedia (today 2012-02-17) we can read that π is already called “Archimedes’ constant”. However, we commonly speak about “pi” and not about “Archimedes”. Thus the name is free to use as the name of Θ.

There is some momentum in the USA to use tau, thus τ = 2 π. Bob Palais (2001) originated the idea but used an own new symbol (pi with three legs like m), Peter Harremoes and Michael Hartl convinced him to use tau, and Vi Hart has a presentation on YouTube. One argument is that tau refers to “turn” or Greek “tornos”.

However, turns are counted along the unit circumference cirkel C = 1 and not along the unit (radius) circle r = 1. Thus this association of tau would be confusing. Also, there is not much difference in writing r or τ. This can create a lot of confusion in handwriting, doing homework or checking exams.

Independently from Bob Palais I also came up with the idea that 2 π is the proper unit of account. Looking at the various symbols available on the keyboard I rejected tau because of the similarity to r, and settled for Θ since it neatly looks like a circle. I wasn’t quite happy with its uninformative name Theta but we had that also with pi or “meter”. Vi Hart pointed out that lower case theta is often used for angles which causes the problem of “theta Theta”. This disappears when we use “Archimedes”.

The proposal is to take the plane itself as the unit of account for angles. We know how to cut up a pie in those pointy bits radiating from the center, and we can do the same with the whole plane, getting a half plane, a quarter plane, etcetera. All those pointy bits add up to 1 plane. When we make circles we can find one with a circumference of 1 by which we can measure the angles. Comparing circles, the Archimedes unit shows up as a proportionality factor.

We need empirical tests whether this indeed works out better for students.

Unit circumference circle = Angular circle	Unit radius circle = Unit circle
C = 1 r = 1 / Θ angles α, β functions Xur and Yur	C = Θ r = 1 arcs φ = α Θ and ψ = β Θ functions Cos and Sin

See COTP page 41. Here Xur[α] = Cos[α Θ]. Angles can be measured by arcs or possibly be identified by them. It helps to separate the notions somewhat by putting emphasis on angles on the angular circle and arcs on the unit circle.

Gowers and the boycott of Elsevier

My original advice to boycott Holland came with the exception of communication, like websites, newspapers and publishers, since a nation needs to be able to discuss what to do when it is facing a boycott.

Now mathematician Timothy Gowers complained about the high cost and oligopolistic practices of Reed Elsevier and this caused a call for a boycott by scientists to stop contributing to journals published by them: http://thecostofknowledge.com/ 

Thus, these are two entirely different things.

Interestingly, Gowers also appears to be thinking about different ways to score contributions to science other than merely counting publications weighted by journal factor. My own idea since the beginning of the internet was that it would be sufficient to use that internet. A later proposal of more than 10 years ago was to use the Elo or Rasch rating as used in chess, see for example chapter 7 in Voting Theory for Democracy. The problem of course is to establish what would constitute a “match” since scientific papers don’t have the clarity of a chess match. But if scientists put their work on their websites and link to work of others then Google by itself would give a basic ranking (that is what Google does). So I have been looking with amazement at scientific journals from the very beginning of the internet.

There is actually a good example. In this weblog professor Gowers explains the distinction between countable and uncountable sets.  It happens that I wrote the paper “Contra Cantor Pro Occam” with the new definition of “bijection by abstraction” (also labelled as ”bijection in the limit”) with the consequence that the set of natural numbers and the set of real numbers are “equally large”. This reduces infinity to the two notions that Aristotle already gave – potential infinity (natural numbers) and actual infinity (the continuum) – but as mere transforms or different orderings of each other. The happy consequence is that we are saved from Cantor’s “transfinites” and the squandered research funds on these illusions. Another consequence is that we can discuss the real numbers in highschool mathematics without feeling that we are leaving out something important. The paper has been rejected by the Dutch journal “Nieuw Archief voor Wiskunde” as “interesting but not for mathematics specialists” and “too long for a journal”. They should have said: “interesting but try to make it shorter for specialists who don’t need the introductions”. Now, if professor Gowers would look into the paper and put a review on the web, then we would have “peer review on the spot” and we are off to a “new way of publishing” (namely to use the internet).