Monthly Archives: August 2014

TC: Thank you very much, your highness King Willem-Alexander of the House of Orange, for inviting me to interview you during your Sunday morning breakfast. I hope that your highness has had a good night’s sleep and enjoys the excellent coffee as well as I do ?

Queen Maxima

King: Well, thank you for interviewing me. Yes, I slept well, and my wife, Queen Maxima, too. Please note that I always mention her in interviews, not only to please her, but people tend to know me as “the husband of Queen Maxima”, so that it is useful to make clear that I really am who I am saying who I am.

TC: Yes, I was going to ask about her too. Unfortunately she isn’t here for the interview, but I would like to compliment her with her presentation yesterday on occasion of 200 years of the Kingdom of the Netherlands. She was elegant as always but it struck me also how slender she looked.

King: Elegant and slender. Yes, I can remember that. I will pass on the compliment.


TC: I hate to spoil a good breakfast – did you try those sausages ? Oh, yes, you have this every morning – but, of those 200 years you have been king now for 1.5 years, and there has already been this MH17 disaster killing 283 passengers and 15 crew members, of which 193 Dutch people. One victim was senator Willem Witteveen, son of Johannes Witteveen who we once featured in this weblog with his important analysis on the economic crisis. Also Willem’s wife and daughter were in that plane. The elder Witteveen is devastated and will likely not speak in public anymore, which is sad for the economic discussion too.

King: I suffer with all the families of the victims. I or a member of the Royal Family have been to all receptions of the planes that returned with the remains of the victims. The Dutch government is doing their best to support the families, identify the remains of the victims, and prosecute those who are responsible for this.

Vladimir Putin

TC: Do you hold Vladimir Putin accountable ?

King: I am somewhat disappointed in the President of Russia. Last year November we met at a reception in the Kremlin for the Year of Dutch-Russian Friendship, and then this year in February at the Winter Games 2014 in Sochi we drank beer together. Besides, our vacation villa’s in Greece are at just walking or swimming distance. We try to be good friends and neighbours. If President Putin hadn’t supported those Russian rebels in the East then they would not have shot that plane. Putin could have exerted more influence to stop the fighting, so that our research team would have been better able to investigate the crash area.

TC: But do you hold Vladimir Putin accountable ?

King: I wished thing were that easy. The situation is complex. There is also the warring in the Middle East.

The Middle East

TC: Yes – apart from that really delicious pancake and syrup – what are your thoughts about the Middle East ? All civilized people in the world feel that they are dragged into a medieval horror show. Where is the world leadership ?

King: My impression is that history is repeating itself. The Roman Empire split up into the Eastern and Western parts, and those parts started fighting each other. General Belisarius of the Eastern Empire devasted Italy around 550 AD. When Islam came up around 600 AD, it started as a small sect, but because the Western Roman Empire had collapsed the relatively small Islamic tribes could conquer its domains in a mere 30 years. For us, when Europe and Russia are at loggerheads over the Ukraine, they forget about handling that “Islam State” that is filling the void in the Middle East.

TC: An amazing historical parallel ! What will you do about this, as King of Holland ?

Breaking News

King: I have decided that the House of Orange will claim the throne of Russia. As you know, my great-great-something-grandmother was Anna Pavlovna, so we are directly descendant from Czar Paul I of Russia. We had a family meeting and discussed the Russian problem of Putin. We observed that Russia never had a constitutional monarchy to make the common transition from feudal dictatorship to modern democracy. Hence it is sensible that a member of the House of Orange takes the throne there, and allows Russia to make the change. We will oversee that there are free democratic elections with a free press, and if Putin’s party loses, so be it.

TC: Excuse me for goggling and gurgling, sire – I am somewhat choking on this great Russian Salad. But, recouperating: You are claiming the Romanov throne ?

King: Ah, be careful here. First of all, the Romanov family is just a romantic relic. Prince Nikolai Romanovich is a good chap, but he is not a ruling king. Secondly, crownprince Charles of England has also good papers, but his eyes are on the English throne, just as prince William’s, while Andrew and Harry are too wild. Besides, the House of Hanover is much overrated. The House von Anhalt that produced Catherine the Great is too much of a good thing. History shows that the House of Orange is modest middle of the road, given that people still don’t know about the Dutch Empire. Thirdly, Maxima and I will remain in Holland, since it is much more difficult to manage a mature democracy like Holland. Besides, our vacation villa is close to Putin’s – well, I said that – and we want to remain good neighbours. Fourthly, my brother Constantijn is good in languages. Thus Constantijn and Laurentien will take the throne of Russia.

Laurentien and Constantijn

Laurentien and Constantijn, 2011 (Source: wikipedia commons)

Closing statement

TC: Thank you very much, your highness, for this exquisite breakfast, that was very nourishing for both body and mind. I hope to hear soon about your ideas of how to realize this ambitious plan. Could you also give me Constantijn’s telephone number, for his view on this ?

King: Just dial any number and ask for him. At least, that is how my phone works. You should try to get one like that.

Sharp readers will have observed that Vladimir Putin of Russia closely follows the suggestions in this weblog. After the last weblog discussion of To invade or not to invade ?” we now see the “Alea iacta est” with Russian tanks crossing the Ukrainean border.

Putin’s dilemma reminded of Shakespeare and the Danish prince Hamlet: “To be or not to be ?”  We shouldn’t be surprised that we got a response from Peter Harremoës from Denmark as well.

On the issue of taking a loss, be it the Crimea or now larger parts of the Ukraine, or children losing their fingers in Iraq-Syria or Israel-Gaza, but rather mathematically more general in the form of the subtraction of numbers in arithmetic, and thus the creation of negative numbers, Harremoës has developed a creative new approach that might stop the combatants in amazement. His 2000 article might stop you too, since it still is in Danish, and Google Translate still isn’t perfect. Harremoës mentions that he considers an extension in English at some time, so let us keep our fingers crossed till then – while we still have those.

In the mean time I would like to take advantage of some minor points on subtraction, partly relying on Peter’s article and thanking him for some additional explanation too.

Namely, in the last weblog discussion on confusing math in elementary school I stated that it is important to distinguish the operator minus from the sign min. Peter referred to a – (-b) and commented that problems of subtraction better be transformed into addition, and that subtraction can be seen on an abstract level as much more complex (or mathematically simple) than commonly thought.

One of his proposals is to create a separate symbol for -1 without the explicit showing of the min-sign. He took an example from history in which 1-with-a-dot-on-top already stood for -1. I have wondered about this, and would suggest to take a symbol that is available on the keyboard without much ado, where we e.g. already have i = Sqrt[-1].

A-ha ! Doesn’t the reader hear the penny drop ? Let us take i = quarter turn, H = half turn =   = -1, then i³ = H i = – = 3 / 4 turn =  three per four turn, and H H = full turn = 1. It would appear that H best be pronounced as ‘eta’, both for international exchange, and in sympathy for German teachers who would otherwise have to pronounce H H as ‘haha’, which would form a challenge for the German sense of humour. I considered suggesting small η or h but the nice thing about H is that it has a shade of -1 in it. In elementary school we can use just the Harremoës-operator H = -1 without the complex numbers. Later in highschool when complex numbers would arise we can usefully refer to H as something that would already be known (or forgotten).

Kids can understand that a debt is an opposite from a credit, or that losing the Ukraine is opposite to winning it. Thus if a is an asset then H a is a liability of the same absolute size. Calculation of gains and losses could be done with a + H b for counting down, or H b + a for counting up. If you lose a debt, then you gain. Losing a debt H b then would be introduced as a + H (H b) = a + b.  Actually, I suppose that it would be even better to start with the absolute difference between two numbers, Δ[a, b]. A sum would be to determine that Δ[a, H b] = a + b, presuming that a and b are nonnegative integers.

Thus H would be used in the creation of the negative numbers and the introduction of subtraction, and for later remedial teaching for who didn’t get it or lost it. Peter Harremoës seems to be of the opinion that there would be no need, in principle, to introduce minus and min, but agrees that people would currently want to stick to common notions. Once the basics of H are grasped, it is no use to grind them in, since it is better to switch to minus and min that must be ground in because of that commonality.

First the min sign and the negative integers are introduced by extending the number line: -1 = H 1 , -2 = H 2, … -100 = H 100 and so on. The teacher can show that applying H means making half turns, or moving from the right to the left, or back.

Subsequently the minus operator is introduced as a – b = a + H b.

Hence there arises the exercise a – (-b) = aH b = a + H (H b) = a + 1 b = a + b.

Or the relation between minus and min: –b = 0 – b = 0 + H b = H b.

A pupil who has mastered arithmetic will do a – (-b) = a + b directly. Otherwise return to remedial teaching and practice with H again.

Arithmetic seems simplest in a positional system. Earlier, we already discussed that English better is regarded as a dialect of mathematics. A number like 15 is better pronounced as ten-five than as fifteen. A sum 15 + 36 then fluently (yes!) translates into “ten-five plus three-ten-six equals (one plus three)-ten-(five plus six), equals four-ten plus ten-one, equals five-ten-one” which is 51. Let me introduce the suggestion that children can use balloons in handwriting or brackets in typing to indicate not only the digits but also the values in the positional system.

15 + 36

Adding 15 and 36 using the positional system, with balloons or brackets

In the same manner, the positional system allows us to state 1234 = [-1][-2][-3][-4], where we might rely on H if needed.

For subtraction, the algorithm for a b is to keep that order if a ≥ b, or otherwise reverse and calculate -(b – a). But, it is useful to show pupils the following method if they forget about reversing the order. For example, 16 – 34 = 16 + [-3][-4] and the rest follows by itself.

16 - 34

Subtract 16 – 34 when forgetting to reverse the order

One might compare the above with other expositions on subtraction. An obvious source is the wikipedia article on subtraction, while google gives some pages e.g. from the UK or the USA. Some texts seem somewhat overly complex.

Originally I thought that the subtraction a b for a ≥ b would be harmless, but on close consideration there is a snake in the grass. A point is that corrections are made above the subtraction line, so that the original question is altered. In the Wikipedia example of the Austrian method the final sum doesn’t add up any more. The Wikipedia example of the American method is okay, provided that indeed 7 is replaced by 6, and 5 is replaced by 15. But this is not a proper positional notation anymore. The method also assumes that you use pen and paper, which is infeasible in a keyboard world. Below on the right there are two examples that keep the original sum intact, and that only use the working area below that original sum. One approach is to rewrite 753 = [6][15][3] and the other approach is to do the borrowing a bit later, which is faster. These methods rely on the trick of using balloons or brackets to put values and sub-calculations into a positional place. If we allow for adaptation above the minus line, then the use of H = -1 and T = 10 would work as well, without the need to dash out digits. The second column combines the American & Austrian methods with the Harremoës operator H but now treated as a digit, and using [H][T][0] = HT0 = 0.

Subtraction of 753-491, in the American manner (source Wikipedia), with comments

Subtraction of 753-491, in the American manner (source Wikipedia), with comments

Evaluating these methods, my preference is for the last column. It follows the work flow, in which the negative value is discovered by doing the steps. The method accepts negative numbers instead of creating some fear for them. A practiced pupil would not need the 2[10-4]2 line and directly jump to the answer, so that the number of lines is the same as in the first and second column. The American / Dutch method with HT0 = 0 inserted as a help line creates the suggestion as if borrowing is required before one can do the subtraction, which goes against the earlier training to be able to do such a subtraction that results into a negative value. The borrowing is only required to finalize into a final number in standard notation.

Overall, my conclusion is that the emphasis in teaching should be on the positional system. The understanding of this makes arithmetic much easier. Secondly, the Harremoës operator H indeed is useful to first understand the handling of credit and debt, before introducing the number line and the notation a b. Thirdly, in a combination of the two earlier points, this operator also appears useful into decomposing 1234 = [-1][-2][-3][-4]. I want to thank Peter again for starting all this (apart from the more advanced ideas in his article). For completeness, let me refer to the 2012 paper A child wants nice and not mean numbers, with a discussion of the pronunciation of the numbers and some more exercise on the positional system.

But these mathematical operations don’t explain that Ukraineans first lose the Ukraine but subsequently gain it once they have turned into Russians.

The problems in Russia-Ukraine, Irak-Syria and Israel-Gaza are so large since the combatants are hardly aware of the concept of fair division and sharing. Something must have gone wrong in elementary school with division and fractions. Let us see whether we can improve education, not only for future dictators but for kids in general.

English as a dialect

In 2012 I suggested that English can best be seen as a dialect of mathematics. The case back then was the pronunciation of the integers, e.g. 14 as “fourteen” (English) instead of “ten-four” (math & Chinese). The decimal positional system isn’t merely a system of recording but it contains switches in the unit of account. In this system the step from 9 to 10 means that ten becomes a new unit of account, and the step from 99 to 100 means that hundred (ten-ten) becomes a new unit of account. This relies on the ability to grasp a whole and the notion of cardinality. Having a new unit of account means that it is valid to introduce the new words “ten” and “hundred”, so that 1456 as a number differs from a pin-code with merely mentioning of the digits. When the numbers are pronounced properly then pupils will show greater awareness of these elements and become better in arithmetic – and arithmetic is crucial for division and fractions.

When education is seen as trying to plug mathematics into the mold of English as a natural language, then this is an invitation to trouble. It is better to free mathematics from this mold and teach it in its own structural language. It is a task for the teaching of English to show that it is a somewhat curious dialect.

Rank numbers

After the recent discussion of ordinal or cardinal 0, it can be mentioned that the ordinals are curiously abused in the naming of fractions. Check the pronunciation of 1/2, 1/3, 1/4, 1/5, … With number 4 = four and the rank 4th = fourth, the fraction 3/4 is pronounced as “three-fourths”. What is rank “fourth” doing in the pronunciation of 3/4 ? School kids are excused to grow confused.

Supposedly, when cutting up a cake in four parts, one can rank the pieces into the first, second, third and fourth piece. Assuming equal pieces, or fair division, then one might borrow the name of the last rank number “fourth” to say that all pieces are “a fourth”. This is inverse cardinality. Presumably, this is how natural language developed in tandem with budding mathematics. Such borrowing of terms is conceivable but not so smart to do. It is confusing.

The creation of “a fourth”, as a separate concept in the mind, also takes up attention and energy, but it doesn’t produce anything particularly useful. Malcolm Gladwell alerted us to that the Chinese language pronounces 3/4 as “out of four parts, take three”. Shorter would be “3 out of 4”. This directly mentions the parts, and there is no distracting step in-between.

For a reason discussed below we better avoid the “of” in “out of”. Thus it might be even shorter to use “3 from 4”, but a critical reader alerted me to that his might be seen as subtraction. Thus “3 out 4” seems shortest. However, there is also the issue of ratio versus rate. In a ratio the numerator and the denominator have the same dimension (say apples) while in a rate they are different (say meter per second). Thus the overall best shortest pronunciation would be “3 per 4″, which is neutral on dimensions, and actually can be used in most European languages that are used to “percent”.

This pronunciation thus facilitates direct calculation, like “one per four plus three per four gives four per four, which gives one”.

Dividing and sharing

The Dutch word for “divide” also means “share” (Google Translate). Sharing a cake tends to generate a new unit of account, namely the part. In fair division each participant gets a part of the same size, which becomes: the same part. This process focuses on the denominator and generates a larger number and not a smaller number. It actually relies on multiplication: the denominator times the new unit of account (the part) gives the original cake again. The process of sharing is rather opposite to the notion of division that gives a fraction, that maintains the old unit of account and generates a smaller number on the number line.

A fraction 3 / 4 or “three per four”, when three cakes must be shared by four future dictators, requires the pupil to establish the proportional ratio with “three cakes per four cakes” (virtually giving each a cake even though there are no four cakes but only three), and then rescale from the four hypothetical cakes down to one cake. PM. The pupil must have a good control of active versus passive voice. The relation is that “4 kids share 3 cakes” (active) and “3 cakes are being shared by 4 kids” (passive). Thus “3 per 4” or “3 out of 4” is shorthand for “3 units taken out of 4 units” (or “4 (kids) take out of 3 (cakes)”) but not for “3 (kids) take out of 4 (cakes)” (which would give 1 + 1/3 per kid, and would require a discussion of mixed numbers).

Hence it is unfortunate that the Dutch language uses the same word for both sharing and dividing. Fraction 3/4 reads in Dutch as “3 shared by 4 gives three-fourths” (“3 gedeeld door 4 geeft drie-vierde”), which thus combines the two major stumbling blocks: (a) the sharing/dividing switch in the unit of account, (b) the curious use of rank words. When 3/4 = “three per four” would be used, then the stumbling blocks disappear, and teaching could focus on the difference between the process of dividing and the result of the fractional number on the number line.

David Tall (2013) points to a related issue in the language on sharing and dividing: “The notion of a fraction is often introduced as an object, say ‘half an apple’. This works well with addition. (…) What does ‘half an apple multiplied by half an apple’ mean? (…) However, if a fraction is seen flexibly as a process, then we can speak of the process ‘half [halve] an apple’ and then take ‘a third of half an apple’ (…) the idea is often simply introduced as a rule, ‘of means multiply’, which can be totally opaque to a learner meeting the idea for the first time.” (p97) Note that Tall’s book is rather confused so that you better wait for a revised edition. He indeed does not mention above issues (a) and (b). But this latter observation on the process and result of division is correct.

The rank words thus are abused not only as nouns but also as verbs (“take a third of half of an apple”). We better translate into “(one per three) of (one per two)”, which gives “(one times one) per (three times two)”. The mathematical procedure quickly generates the result. The didactic challenge becomes to help kids understand what is involved rather than to master confused language.


Speaking about Tall and multiplication: Apparently the English pronunciation of the tables of multiplication can be wrong too. E.g. ‘two fours are eight’ refers to two groups of four, and thus implies an order, while merely ‘two times four is eight’ gives the symmetric relation in arithmetic. Said book p94 compares a table with 3 rows and 4 columns, and Tall argues: “the idea of three cats with four legs is clearly different from that of four cats with three legs. The consequence is that some educators make a distinction between 4 x 3 and 3 x 4. (…) I question whether it is a good policy to teach the difference. (…) [ reference to Piaget ] (…) So a child who has the concept of number should be able to see that 3 x 4 is the same as 4 x 3.”

An exercise in marbles

An exercise in marbles

Tall doesn’t explain this: Pierre van Hiele focuses on the distinction “concrete versus abstract”, would focus on the table, so that children would master the insight that the order does not matter for arithmetic. Once they have mastered arithmetic, they might consider “reality versus model” cases like on the cats and their legs without becoming confused by arithmetical issues hidden in those cases. Instead, Hans Freudenthal with his “realistic mathematics education” (RME) would present kids with the “reality versus model” cases (e.g. also five cups with saucers and five cups without saucers, a 3D table), and argue that this would inspire kids to re-invent arithmetic, though with some guidance (“guided re-invention”). Earlier, I wondered why Freudenthal blocked empirical research in what method works best (and my bet is on Van Hiele).


Overall, the scope for improvement is huge. It is advisable that the Parliaments of the world investigate failing math education and its research. When kids have improved skills in arithmetic and language, they would have more time and interest to participate in and understand issues of fair division. Hurray for World Peace !

PM 1. Conquest of the Plane pages 77-79 & 207-210 discuss proportions and fractions.

PM 2. See also COTP for the distinction between dynamic division y // x and standard static division y / x.

PM 3. Some say “3 over 4” for 3/4, hinting at the notation with a horizontal bar.  I wonder about that. The “3 per 4” is actually shorter for “3 taken from 4”, and this puts emphasis on what is happening rather than on the shape of the notation. An alternative is “3 out of 4” but my inclination was to avoid the “of” as this is already used for multiplication. Also, my original training has been to reserve “n over k” for the binomial coefficient (that can be taught in elementary school too). However, a reader alerted me to Knuth’s suggestion to use “n choose k” for the binomial coefficient, and that is better indeed. In that case I would tend to avoid the “over”. It was also commented that “3 from 4” sounded like subtraction: but my proposal is to adhere to “3 minus 4” for “3 – 4” as opposed to “3 plus 4” for the addition. It is just a matter to introduce plus and minus into general usage, so that it is always clear what they are. Note that we are speaking about mathematics as a language and not about English as a natural language. Also -1 would be “negative-1” or “min-1”, with the sign “min” differing from the operator “minus”.

This Sunday morning, August 17 2014, sitting in his bathtub, he hummed to himself:

“You, Vladimir Putin, President of the Russian Federation, didn’t do too badly.”

Vlad2: “But you can look at it differently too.”

Putin looked around bewilderedly.

Vlad1: “Huh ? Who is speaking to me ?”

Vlad2: “Me, Vladimir Putin, President of the Russian Federation. You know, your conscience, who started you on this path after the fall of the USSR.”

Putin relaxed. His conscience ! It was so long ago that he had this talk with himself !

Vlad1: “Hi, Vlad2 ! What a surprise ! You are trying to second-guess me again ? Please tell me what I am doing wrong. Beware, I am grateful for your advice back then, but I am not used to criticism, and if you make me angry I may send you to Siberia.”

Vlad2: “Hi, Vlad1 ! This shows again that you don’t understand what a conscience is. We never tell what a person is doing wrong. We only ask questions. Tantalising questions. Questions that cause you never to sleep again at night. Besides, we never give in to threats.”

Vlad1: “Last time I heard from you was on Chechnya and Georgia. Boy, it took me a lot of vodka and women to finally get some sleep, but I succeeded, and I got those regions fixed.”

Vlad3: “I didn’t think those women were really that swell. You had too much vodka.”

Vlad1: “I am sorry ?”

Vlad2: “Huh, who is that ?”

Vlad3: “Well, I am just Vladimir Putin, President of the Russian Federation. You suggest that there are only Vlad1 and his conscience Vlad2, but you also know that I do not follow the rules. The world is not as simple as commonly thought, and certainly not the mind of the President of Russia. And I didn’t like your women.”

Vlad4: “I fully agree to that.”

Vald0: “And I protest that Vlad3 started counting at Vald1 and not at me, Vlad0. People always forget who provided the vodka.”

Vlad1: “Okay, okay, please relax, my dear Russian Presidents. I am overjoyed that this Ukraine crisis causes me to develop such a complex personality. My next thousand years seemed lonely but now I am looking forward to your company. But just now, I want to sit in this bathtub and decide whether I will invade the Ukraine or not. Well, I shouldn’t forget that I already did, by taking back the Crimea.”

Vlad2: “Well, I started out trying to be your conscience, and I feel that I should persist, whatever those Vlad3 and Vlad (n) say. Is there any way that you are going to bring some sense into the matter ? This Ukraine issue is pestering now for more than ten years. Shouldn’t these people be allowed to get a normal life ?”

Vlad3: “You cannot hold that what I say isn’t important. My opinion is that we all should give Vlad1 an ultimatum that he should decide what to do. He has been zig-zag-ing all year, being a host to the world in Sochi and now becoming some sort of pariah. I move that we stop zig-zag-ing. Either we take our big decision now, to invade or not to invade, or we can blame Vlad1 for the next thousand years that he is another bathtub failure of which history has given us plenty already.”

The President of Russia thought at this moment of his secret admiration of the uncompromising radical Jean-Paul Marat.

“La mort de Marat”, by Jacques-Louis David, 1793. (Source: Wikipedia Commons)

Vladimir Putin, President of the Russian Federation relented: “Okay. So what are my options ? Currently, I have made veiled threats, put troops at the border, censored the media, sent hundred thousands of people running, shot a plane, imposed sanctions, boycotted Holland and their shiny tomato’s, sent out my 278 white trucks. I cannot back down without looking silly and weak. The Ukrainean army is on the verge of destroying the Russian patriots, and if I allow this to happen then I will also betray my promises to them and to the Russian public. I will not only look silly and weak but will actually be so. Thus, there is only one option: to invade, and block the Ukrainean army. The only question is: how far will I go ? Just to the current front line ? This is a somewhat silly objective for such an invasion. If I invade, there should be a real purpose. Thus I should go all the way to the Dnieper and divide the country. The risk is that the other half then joins NATO. Kiev will be lost for the Russian Federation. Mother Russia weaps for losing Kiev.”

It was Putin himself who wept. His tear dropped in bathtub and joined the salty water of his earlier tears over the loss of the USSR.

Dmitry Medvedev: “There is also another option. You could call for a referendum or national elections in Russia. You against me. Ask all Russians whether they want you and war with the West or me and some form of Peace with Honour.”

Vlad1, Vlad3, Vlad4, Vlad (n): “What in Mother Russia’s name is Medvedev doing here ?”

Medvedev: “Wow, I thought everybody knew that I am only another figment of Putin. Well, big surprise for you then – except for Vlad2, I notice. But just to be sure: one way to get out of this mess is to call for a referendum or national elections in Russia.”

Vlad2, as always quick on the uptake in moral issues: “Let me see whether I get this correctly. We invade to the current frontline just to keep our promises to the Russian patriots. We also announce a referendum whether the Russian army should continue to the Dnieper to liberate the Eastern Ukraine from the Nazi regime in Kiev. We promise to step down as president if we lose. That wouldn’t be a loss of face with a needless massacre on the battlefield but a honorable part of democracy. You, Medvedev, will be the voice of reason, and argue for Peace with Honour. If you win the referendum, you could run for president, and continue to protect our interests. Do I understand you correctly ?”

Medvedev: “Democracy is an offer that you can’t refuse. I would insist on a free press though, to explain who shot down that plane, since I intend to win.”

Putin: “Hm, … I need to stay in this bathtub a bit longer to think this over. However, there is a knock on the door. Someone seems to want to come in with some message or so …”

On the borders of the EU the cannons are barking, and it are mostly civilians and their properties that are hit, in the Ukraine, Syria & Iraq, Israel & Gaza, and Libya.

In the Dutch Parliament, economist and Christian Democrat dr. Pieter Omtzigt calls for a recognition by the Dutch government of the genocide by the “Islam State” on the Yazidi people. The world should not allow what happened in Rwanda in 1994. He said so in a talkshow of August 11, in which he also mentioned that his priorities are Dutch pensions, persecuted Christians, and his home region Twente. The talkshow of August 11 only mentions them but the persecuted Christians get his full attention in the same talkshow of July 31.

It seems unwise to regard, define and frame the problems of the Middle East in terms of Islam, Christendom and Judaism. There are actors who have an interest in such framing. Who joins this framing supports their interests.

Obviously, it are religious fanatics who have this interest. They have little to offer except their religion, and hence they grow in importance whenever it is accepted that religion would really be the issue.

There are also other powers who refer to religion but who have a more hidden agenda. Said “Islam State” is run so well, and makes such profits, that one may wonder whether the real issue is Islam, for it may just be the robbing of the possessions of others.

Thus, I would urge Pieter Omtzigt to desist from framing his objective in terms of “persecuted Christians”. In principle it are civilians who are at issue. In principle it is freedom of religion that is at issue. That someone calls himself or herself to be a Christian is not the first point of relevance, since he or she may well be a fundamentalist on the road to a new crusade. When a Christian in the Middle East loses his or her children then this is as heartbreaking as when a Sunnite or Shiite loses his or her children. I am pretty sure that Pieter Omtzigt will agree on the latter, but he doesn’t recognize yet that he is inconsistent by putting such an emphasis on Christians. He is opening Pandora’s Box, and apparently invites Dutch Parliament to join him in doing so.

Pandora 1879

Pandora, by Dante Gabriel Rossetti, 1879 (Source: Wikimedia Commons)

Dr. Omtzigt wrote a thesis in econometrics. He is one of the few Dutch Parlementarians who has some grasp of economics and finance. He is not only on the Parliamentary Commission for Foreign Affairs but also on Commissions on Economics and Finance. For years there have been the “Economists for Peace”, now EPSEU and EPSUSA (formerly ECAAR). They study the costs of the Arms Race, or, for example, the costs of the political lies that resulted in the Iraq War. Remember that it were “re-born” Christians George W. Bush and Tony Blair who used lies to cause the Iraq War. The least that we could ask of economist Omtzigt is to look at what EPSEU has to say.

In my analysis the greatest contribution to peace will come from the solution of unemployment. Young people who have no future now and who experience the seduction of security in religion, would have better options if the economy would allow them to find a job, get economic security and the prospect to start a family. Unemployment is not a natural phenomenon, like a vulcano, that we have no control over. Unemployment is man-made, and a failure in policy. Unemployment both in the EU and the USA has been caused by neoliberal economists who adhere to dogma’s that economic science proves to be incorrect.

Pieter Omtzigt thus actually has a co-responsibility for those wars and those persecuted Christians. He should be in the seat of the accused rather than in the seat of the DA or judge.

Like the EU and the USA, Holland has a perverse system of taxation. The system in Holland is even more perverse, not only due to the larger Dutch welfare state but also because of some political pecularities. In 1997 minister of finance Gerrit Zalm (VVD) (now CEO of the ABN-AMRO bank) and secretary of finance Willem Vermeend (PvdA) lied to Dutch Parliament, when they proposed to replace tax exemption by a tax credit. Effectively they made it politically more difficult to tackle low wage unemployment. In 2013 there was a row about Bulgarian fraudsters who milked the Dutch tax credit system. The case is described in this paper: Economics as victim between lawyers and mathematics (2013).

Now, if fellow-economist Pieter Omtzigt just would take the time to study that paper, and ask questions about what would not be clear to him, then the chance for world peace could improve.

PM. Etgar Keret rightly observes that people tend to expect peace to “descend from above” while the better perspective is that you have to work hard on compromise (Los Angeles Times, July 14 2014).

Blackboards have mostly been removed from classrooms but they are still used in cafés and on terrasses.

Blackboard at The Lantern cafe

Blackboard at The Lantern cafe

Roefie Hueting (1929) is performing this Summer with also his own repertoire of jazz on the piano, on Friday evenings at The Lantern, The Hague. You can listen to “Blues for Bessie“. In the past he played with his Down Town Jazz Band in the big halls in Holland.

Hueting at the piano

Roefie Hueting at the piano in The Lantern, The Hague, July 25 2014

Hueting is also an economist, and as he has developed at least two talents, Peter van Bergeijk takes him into consideration as a “double talent”, though still has to report what he thinks about the music.

As an economist, Hueting first worked on employment and then switched to nature and the environment. At CBS Statistics Netherlands he developed the statistics on the latter. The Dutch NAMEA and UNstat SEEA are based to a large extent upon the work by Hueting, and he received the UN Global 500 Award in 1994. Hueting developed the notion of environmentally Sustainable National Income (eSNI), see his website.

In the economic theory of social welfare, it is accepted that it is rather impossible to find the (Bergson-Samuelson) Social Welfare Function (SWF) but, assuming optimality, society will choose the optimal point on the Production Possibility Curve (PPC), and the tangent of both functions at that spot will give an income measure, called “national income” (NI). A statistical observation of actual incomes and expenditures and market prices should allow, under assumption of optimality, an estimate of that tangent, and thereby provide an estimate of social welfare. Comparison over time would provide an indication of progress or regress. The relative change of real national income is generally called “economic growth”.

At least, that was the theory around 1934. Since then population growth has generated the new scarcity of the environment. While clean water was free before, it now must be produced at a cost. While CO2 was no worry before, we now have to raise the dikes and build stronger houses to withstand the storms. Curiously, all these costs are included in NI as higher income from more work, instead of being deducted as costs. For this, Hueting proposes a measure of NI exclusive of such asymmetrical bookings, NI-ex-asyms.

Subsequently, there is the notion of environmental sustainability. The present generation has the choice of consuming present resources in the manner of expletion, or, to save present resources so that future generations can also use them at a similar sustainable rate. It is the preference of the current generation that counts. The preferences of the future generations do not enter the discussion since those do not exist yet. Curiously, the preferences of our current generation cannot really be measured, because of all kinds of prisoners’ dilemma’s, lack of communication, failures in democratic processes (check voting theory for example), and so on. There are political statements about environmental sustainability, but those are seldomly implemented. But, given those pious statements and the widespread real worry about the future for our children and grandchildren, statisticians can make an assumption about a social preference for environmental sustainability. Making this assumption compares to making the assumption above on optimality. Both NI and eSNI are based upon assumptions, either on optimality w.r.t. NI or on sustainability w.r.t. eSNI. Finally, given the condition of sustainability, standards can be determined, and a model can be run on which the condition of sustainability is imposed on the current economy. The economy would adapt to a different mode, and the generated national income would be the measure of eSNI.

The relation between NI and eSNI can be clarified with a diagram and a table. Let us start with the table. One of the crucial elements in this discussion is uncertainty and how to deal with it. The current measures of NI and “economic growth” might be regarded as first approaches to measuring social welfare in 1934, but nowadays we would like to include more elements, like the disutility of unemployment, the stress of mortgages, economic inequality, and such. NI is a highly uncertain measure for economic welfare if the true preference is for sustainability. You could supplement NI with environmental indicators only, but those provide only data and no information, since you would also need sustainability standards and some form of aggregation to relate the indicators to NI itself. Overall, the publication of both NI and eSNI would be the best approach.

Uncertainty in the measurement National welfare Environmental sustainability
1934 National Income (NI) acceptable as a first approach, assumes optimality
2014 National Income (NI) unacceptable, requires amendment for unemployment and so on uncertainty so large that it fails
2014 Environmental indicators data but no information: incomplete standards and no aggregation
2014 environmentally Sustainable National Income (eSNI) comparable to NI, subject to the condition of a societal preference for sustainability the uncertainty is described, there are more complete standards, aggregation allows comparison to NI

The diagram gives a conventional graph with convex Production Possibility Curves (PPC) and concave Social Welfare Functions (SWF). There are two sets of these, one with unsustainable reality (U) with illusionary high income (NI), and another with sustainable target (S) with realistic income (eSNI). The axes are two environmental indicators: (1) the environmental function that allows the use of CO2, (2) the environmental function that allows the use of fresh water. Note that the axes thus must be read somewhat inversely. The production of more fresh water causes an increase in the output of CO2 and thus reduces the function value for its availability.

Comparison of NI (high) and eSNI (low)

Comparison of NI (high) and eSNI (low). The axes give availability of environmental functions. Red arrows: sustainable standards. Black arrows: current unsustainable use.

The diagram comes with these observations: (a) SWF-S is the social welfare function under the condition of a social preference for environmental sustainability. Hence the PPC is selected that satisfies the standards of sustainability (the red arrows). The tangent provides the eSNI measure. (b) SWF-? indicates some kind of implied SWF when we consider only the observed uses of the functions (the black arrows). If the indicators are given only by themselves then they provide relatively little information, but in this present setup we assume that they are aggregated into the NI measure. Still, it is not clear what this SWF-? and its NI mean. The assumption of optimality is only an assumption, and quite unrealistic. But given its conventional use, it seems best to have both NI and eSNI available. Of course, eSNI is work in progress. Here is a discussion of what can be done for new research. But if we want to deal with the uncertainties then we need more and not less research.

Calculating eSNI would cost only 1/4 % of the budget of CBS Statistics Netherlands (see here). The reason is that the ground material already is available so that only a model must be run. This is cheap for an important figure that could dramatically affect economic policy.

CBS Statistics Netherlands has a leading world position in the statistics on economics and the environment. They got that position because of the work by Hueting. In the past, Hueting had full support from Nobel Prize winner Jan Tinbergen (1903-1994). The last 20 years have shown a gradual reduction of attention so that it is close to zero now. See Hueting’s website and my paper The Old Man and the SNI. For further readeing, see my (draft) book on economics and ecological survival.

National accounting was started by economists and supported by the government because it was applied economics, embedded in theory. In the past national accounting was guided by economic theorists like Keynes, Hicks, Frisch, Tinbergen, Meade, Stone, Kuznets, Samuelson. Obviously, Hueting belongs in that list of distinguished authors. But CBS apparently doesn’t know what it is doing anymore. It has turned the project into accounting without theory. It is no longer applied economics but accounting for multiple purposes such as legal reasons and tax collection. CBS Statistics Netherlands prefers to publish only NI and various environmental indicators. The table and diagram above show that this produces data and not information, while the 1934 idea of NI is increasingly disinformative for our current policy objectives and possibilities. Sic transit gloria.

NB. Frits Bos (formerly CBS and now CPB) wrote a thesis and an ongoing series of articles on national accounting, calling for a greater awareness on theory, but his exception rather proves the rule, while he apparently still doesn’t delve into Hueting’s seminal contribution.) Importantly, Henk van Tuinen (vice-director at CBS till 2003) has this important article in the open access Journal of Official Statistics (JOS), Vol.25, No.4, 2009. pp. 431–465, in which he agrees with the reduced relevance of NI, and in which he explicitly mentions eSNI as a scope for the future, see sections 3.5, 6.3 and 7.2. The key stumbling block for Van Tuinen is the uncertainty in the assumptions attached to it (page 451). This essentially reduces to the question whether eSNI is a proper “statistical measure” or not. My suggestion is that statistics also deals with uncertainty, and that eSNI clarifies the uncertainty that remains a mystery in the other measures.

But Hueting can be happy that he still has one other talent, and a good one too: music.

Peter Harremoës wondered whether zero is a natural number, arXiv:1102.0418v1 [math.HO]. He starts out by admitting that it is a matter of definition, but then proceeds with the issue of ordinal and cardinal numbers, so perhaps I should rephrase his true question as I did now in my own title above or to the left. A google on “Is zero a natural number ?” and “Is 0 a natural number ?” generates some 15,000 hits. A bit to my amazement there are more people pondering the question (though close to 0.0% of mankind in statistical approximation (in writing and not reading)).

Harremoës approaches the issue from a set-theoretic point of view, though visits Kindergarten with some nice observations. My focus is didactics, and thus I will begin with Kindergarten.

Kids learn the sequence S = {1, 2, 3, … }, and tally off their fingers. When they count the elements of a set A (apples), they bring the elements in A in a one-to-one relation with the elements in S, and also in that order {1, 2, 3, … }. The last element counted gives the total number of elements in A.

For terminology, an ordered sequence gives an ordinal measure, while the total gives the cardinal measure for the number of elements in a set.

The kids learn also the sequence O = {1st, 2nd, 3rd, ….}. Now this is interesting ! It is rather this list O that gives the ordinals ! Rather than saying “This is apple one, this is apple two, this is ….” they learn to say “This is the first apple, this is the second apple, this is …”.

When counting elements in a set A then it does not matter in which order the elements are put – and the cardinal number has the property that it has the same value in whatever order its elements are put. But, for O, the order must follow the order of the elements of the set A that is being considered.

Thus, kids first learn ordinals and cardinals in a mixed manner with S. Then the ordinals are created separately in O. Then S becomes important for the cardinals. The introduction of O requires a bit of unlearning alongside learning.

Set A is counted using ordered {1, 2, 3, …} Order in A is not relevant Order in A is relevant
Counting (process) (“Order some or all.”) {1, 2, 3, ….} for first training {1st, 2nd, 3rd …}
Cardinal (result) (“How many elements are there ?) {1, 2, 3, ….} {1, 2, 3, ….}

With this established, I think that I must object to the use of the idea that “ordinal numbers” and “cardinal numbers” would be separate sorts of numbers. We only have the numbers S. The “cardinal number of a set A” is just idiom to identify the number of elements, but this does not suggest that “cardinal number” is a specific kind of number (like rational number or complex  number).  It is less common to speak about the “ordinal number of an element”. While “What number are you in line ?” indicates such ordering, still you are not a number, and “ordinal number” is not a special number but merely S applied to ordering.

Thus “Is zero an ordinal or cardinal number ?” is a nonsensical question. Zero can be the value of the cardinal number of a set. Whether you start counting with 0 is an issue of convenience, and probably not practical in Kindergarten (but this needs testing). The true issue at hand is not quite arithmetic but actually part of the theory of measurement, with nominal, ordinal, interval and ratio scales.

When you have a list of elements, it is not so practical to start the labeling with 0, since the rank numbers might become adjectives that differ from the proper ranks. For example, Pierre van Hiele, in his masterly exposition on didactics, labeled his levels of understanding by starting at a base or zero level, counting on to level 4. In terms of rank, the first level would be level 0. However, the tendency would be to associated “level 3” with “the third level”, with “third” the adjective of “three”. It appears difficult to suppress that tendency. Hence it is better to start lists with label 1. (In inverted manner, the calendar has no year 0 but it has a first year.)

Note that Dutch has different words for number (“getal”, as in the list of natural numbers, or the pure decimal system, old-English “tale”) and cardinal number (“aantal”, the number of elements, English “tally”). Teaching in Dutch is a bit easier than in English. See Google Translate on this. In etymology we can find a curious connection of counting with  speech itself. To give an account or reckoning may consist of a story or a list of numbers: “Origin of tale: (Webster) Middle English ; from Old English talu, speech, number, akin to German zahl, number, Dutch taal, speech ; from Indo-European base an unverified form del-, to aim, reckon, trick from source Classical Greek dolos, Classical Latin dolus, guile, artifice”. In Dutch the subtle distinction is between “tellen” (to count) and “vertellen” (to tell).

Let us consider kids in Syria, Israel, Gaza or Ukraine who have N[10] = {0, 1, 2, 3, …, 10} fingers left. Since the places of these fingers on their hands or the way how these fingers are ordered doesn’t matter, the natural point of view is that the numbers are cardinal values (“how many left ?”). There also arises a natural order that one set is larger than another, and with thus cardinal values 0 < 1 < 2 < 3 … < 10.

The Egyptians already had a symbol for “none”. They didn’t regard it as a number though. However once you set up a system of arithmetic then it becomes convenient to regard 0 as a number, so that 1 – 1 = 0. Slowly language adapts to that use. Given the naturalness of the question for kids to ask “How many fingers do I have left ?”, thus with the focus on cardinality, and given that the answer may also be “zero”, it is more reasonable to include 0 in the list of numbers that are regarded as natural, giving = {0, 1, 2, … }.

However, when you know that a set is non-empty, then you can use N+ = S for the positive integers. Hopefully the little ones still have a thumb to suck on, to fall asleep, and be ready for another day of counting.