Tag Archives: Russia

I am no expert on terrorism and wonder whether the supposed experts aren’t either.

Dutch historian and “expert on terrorism” Beatrice de Graaf gave a lecture on Dutch TV on March 11 (or see Utrecht University) about David Rapoport‘s four waves of terrorism (his original article).

Her main message was that people might find some comfort in the idea that waves die out. March 22 saw the bombs in Brussels.

This theory of four waves of terrorism appears to be rather silly. Below gives my common sense rejection.

De Graaf is not the only academic who regards the theory of the four waves as serious. The West is vulnerable to terrorism when its “experts on terrorism” are academics lost in theory. It is okay to sooth people not to worry too much, but intellectuals should present effective approaches rather than fairy tales.

The so-called “four waves”

Jeffrey Kaplan summarizes (and then proceeds in adding his own fifth wave) (while Dutch readers can check Edwin Ruis’s review of March 13):

“Rapoport’s theory, first published on the web before finally finding a home in a printed anthology, posited four distinct waves of modern terrorism (anarchist, nationalist, 1960s leftist, and the current religious wave). Each wave had a precipitating event, lasted about 40 years before receding, and, with some overlap, faded as another wave rose to take center stage. Most terrorist groups would gradually disappear, a few (the Irish Republican Army for example) proved more durable. Rapoport’s theory was elegant, simple, inclusive, and had a high degree of explanatory power. In short, it provides a good academic model.” (Kaplan 2008).

Jeffrey D. Simon holds (and wonders about a fifth wave too):

“David Rapoport’s “The Four Waves of Modern Terrorism” is one of the most important pieces ever written in the vast literature on terrorism (Rapoport 2004).  What Rapoport did in his classic study was take the complex phenomenon of terrorism and put it in a historical context that not only explained different periods of international terrorism, but also set forth theories and concepts that can be used to attempt to anticipate the future of terrorism.  That is no easy task.  There haven’t been many assessments and articles written about Rapoprt’s “Four Waves” theory, although this volume of papers initiates a discourse about his important thesis (See Thompson and Rasler, this volume).  Despite the numbers of scholars, policymakers, and others who have joined the field of terrorism studies after the 9/11 attacks, there does not appear to be a great deal of interest in the history of terrorism.  In today’s instant access and information-overload society, we are inundated with analyses of current affairs but pay scant attention to what we may learn from what has transpired in the past.” (J.D. Simon on the Lone Wolf, likely 2010)

I googled to find some criticism, but didn’t see much, though perhaps I didn’t google well. I noticed a critical text by Ericka Durgahee. I didn’t have time to look into this, and the following are my own common sense short remarks.

The anarchists 1880-1920

The dynasties of Hohenzollern, Romanov and Habsburg collapsed. Perhaps the anarchists didn’t really win because we don’t have anarchy now, but those anarchists were replaced by communists and fascists, and we ended up with two world wars, which isn’t quite “die out”.

Anti-colonialism 1920-1960

The anti-colonialists won. Winning isn’t quite “die out”.

Leftists 1960-1989

Leftism became impopular because of the Great Stagflation (unfavourable unemployment and inflation) and the collapse of the Berlin Wall. Young radicals were more motivated by Ronald Reagan and Margaret Thatcher.

In Germany, the police managed to isolate the Rote Armee Fraktion (RAF). In another article, Beatrice de Graaf explains how the Dutch radicals (Rode Jeugd, Krakersbeweging) lost their motivation by incompetence of the Dutch police. The Dutch police intended to adopt the tough German approach, but mismanaged this, and both radicals and the general population got the impression of an atmosphere of tolerance and dialogue. In that atmosphere, potential supporters saw no need for radicalisation, and radicals had the example of the dead-end street in Germany.

These events rather concern the transformation of European society after World War 2. There are pockets of terrorism, but there doesn’t seem much difference between RAF and other groups like IRA and ETA: except that each group requires specific attention for its idiosyncracies.

Religious terrorism 1979-now

Religious violence is of all times. There is no reason to predict that it will pass. This is no wave.

Alternative approach

Terrorists tend to be higher educated who are frustrated w.r.t. opportunities in society. They may feel sympathy with the unprivileged. They may adopt any ideology to recruit others in the resistance against the establishment. To counter this, one must look at society as a whole, create fair opportunity, and encourage people to participate. My own work contains aspects that are key to reduce terrorism.

  • Create a social welfare state that works. See DRGTPE.
  • Make democracy work. See VTFD.
  • Provide for good education, e.g. re-engineer mathematics. See EWS.
  • Let people learn how to deal with the human capacity for abstraction. See SMOJ.
Beatrice de Graaf, soothing Dutch viewers that a wave dies out

Beatrice de Graaf, soothing Dutch viewers that a wave of terrorism dies out



continued from the former entry of April 4

The turnout of 32% at the April 6 2016 Dutch Referendum on the EU-Ukraine Association Agreement does not seem convincing. The 61% No vote seems impressive over the 38% Yes vote and 1% blanks, but the low turnout gives it little weight. (PM. The electorate are 13 million voters. Turnout was 4.2 million. For the treaty were 1.6 million and against the treaty were 2.5 million.)

This advisory referendum is labelled “valid” because it passed the ceiling of 30%, but the margin is so low that Mark Rutte and his government might feel excused to still ratify the treaty anyhow.

It would take only 12% to switch the No majority into a Yes majority.

Potentially, the No voters were most motivated to turn up, and the Yes voters potentially stayed at home because: (a) the referendum was advisory only, and they have reason to believe that the government would ratify the treaty anyway, (b) the referendum would be invalid when the ceiling of 30% turnout would not be reached, (c) they might believe that they were the real majority so that the outcome would be Yes anyway. Perhaps a 12% higher turnout would only have come from Yes voters.

Thus the EU might blame Mark Rutte and his government for not having tried hard enough to increase the turnout that might have caused that switch in outcome.

However, the earlier poll by Maurice the Hond suggests that a full turnout would show 53% No. We can find this figure by weighing the views of the polled votes per party with the current polled seats per party. Indeed, De Hond also concludes that the No vote would never fall below 50%. However, it is hazy what a full turnout would be. There hardly ever is 100%. Let us take the 75% turnout of the last general elections in 2012.

Assuming proportional change between these two data points {.32, .61} and {.75, .53} gives us the following sloping line that drops to 50% at 90% turnout.

Referendum result at 32% and potential at 75%

Referendum result at 32% and potential at 75%

Conclusions are:

  • With the {.75, .53} poll data point included, the referendum result is more impressive than it at first seems. Apparently the Yes vote indeed had a losing uphill battle.
  • Mark Rutte and his goverment should have tried harder to inform the electorate about the pro’s and con’s of the treaty, if they really thought that more information would have resulted in both a higher turnout and a switch to more Yes votes.
  • Mark Rutte and his goverment might not have put in this effort, since the referendum was only advisory and might not have reached the 30% ceiling.

On the sight of it, Mark Rutte and his government have blundered.

It may also be plain old Dutch deviousness: with a ploy to play into Putin’s paranoia, and a scheme to prevent a casus belli. Remember the movie Being there“.

Yesterday I had my regular lunch with Vladimir Putin, President of the Russian Federation.

We mostly talked on how the oil price inflates corruption, and we focused on which sport he wants to enter in the next Olympic games: judo, swimming, horseback riding, shooting, or some other sport that he hasn’t shown us yet that he excels in. We curtly discussed the Dutch referendum of next Wednesday April 6 2016 on the EU – Ukraine Association Agreement.

TC: “Vlad, you have been remarkably silent on this Dutch referendum. I had expected to see the Russian Ambassador on Dutch TV campaigning for a No, but I haven’t seen him.”

VP: “Well, I want to invade the Eastern Ukraine this Spring, and this treaty is a welcome cause of war. I would be dismayed when the Dutch people would say No to the treaty and thus block it. Besides, I have asked the Ambassador to undergo a sex change operation to be more agreeable to the Dutch gays.”

TC: “When you ask the Dutch to say No, they will likely become obstinate and vote Yes. Then you have your casus belli and can also claim that you have done your best to prevent it. Or am I trying to out-smart you ?”

VP: “That would also be seen as meddling in internal national affairs. Russia doesn’t meddle. We spy or invade. I trust that the Dutch government will ratify the treaty, even when the population says No. I have promised your prime minister Mark Rutte a good job at Gazprom later on, and he seemed very happy to sit up and eat the cookie.”

TC: “Thus by July 2016 Russia will have occupied the region East of the Dnieper, and Kiev will be divided by a Wall. The EU can have its treaty with the West, and can start trying to feel happy with the mess they created.”

VP: “There is something very alluring to a permanent stalemate, like the division between North and South Korea. A permanent state of war will be very invigorating to Russia’s youth.”

TC: “Ah yes, I understand what you are thinking. Marches and parades. Battle songs. Soccer stadiums filled with ballet dancers in clockwork performances. Adrenalin is better dan meldonium. The European Union will not understand what hit them. You are brilliant and a real father for Russia.”

Vlad turned to me with a surprisingly modest expression: “Don’t give me credit. I got the idea from Garry Kasparov. Let me give you his Memo. I don’t mind when you publish it and let your Dutch friends read it too. Kasparov always loses from me in chess or politics, and he is a lousy house cleaner and waiter, but his Memo deserves some credit.”

I read the Memo in the plane, and was happy that there was plenty of Stolichnaya for the remaining three hours of the flight back home.

Memo by Garry Kasparov for Vladimir Putin, April 1 2016

Normally I try for a straight argument and end up into convolutions. Now I won’t even try.

The EU – Ukraine Association Agreement (henceforth treaty) text is here and on wikipedia (or in Dutch here and here and wiki). The wikipedia article shows that the treaty is ratified in all countries except that Holland has a referendum on April 6 2016 that might still block it.

Wikipedia actually has an article on this Dutch referendum. The Dutch referendum allows citizens only to advise the government. The government might still choose its own way – which likely is a Yes. Hence we are mostly discussing chimeras.

My main concern is the risk of war, whence I am against the treaty as it stands. Article 7 on Foreign and security policy involves the EU in a “timely and coherent manner” in the defence of Ukrainean sovereignty. The Ukraine might claim that the Crimea has been stolen by Russia, must be returned, and that the EU is legally obliged to help to get it back in “timely and coherent manner” . This is Article 7 with my emphasis:

1.   The Parties shall intensify their dialogue and cooperation and promote gradual convergence in the area of foreign and security policy, including the Common Security and Defence Policy (CSDP), and shall address in particular issues of conflict prevention and crisis management, regional stability, disarmament, non-proliferation, arms control and arms export control as well as enhanced mutually-beneficial dialogue in the field of space. Cooperation will be based on common values and mutual interests, and shall aim at increasing policy convergence and effectiveness, and promoting joint policy planning. To this end, the Parties shall make use of bilateral, international and regional fora.

2.   Ukraine, the EU and the Member States reaffirm their commitment to the principles of respect for independence, sovereignty, territorial integrity and inviolability of borders, as established in the UN Charter and the Helsinki Final Act of 1975 of the Conference on Security and Cooperation in Europe, and to promoting these principles in bilateral and multilateral relations.

3.   The Parties shall address in a timely and coherent manner the challenges to these principles at all appropriate levels of the political dialogue provided for in this Agreement, including at ministerial level.

Professor Richard Sakwa (page at Kent) advises a No. His book is here and reviews are here and here. He agrees that the Ukraine already is a sovereign nation and has the right to engage into the current treaty with the EU. The EU has the right to engage into such a treaty too. However, it would not be wise to do so. The EU is sleepwalking again, like the sleepwalkers of 1914-1918.

Sakwa was on Dutch TV on March 20 2016 explaining – in English with Dutch subtitles (minute 41) – that the EU should look at the whole region, including the position of Russia. He was challenged by the Dutch vice-prime-minister Lodewijk Asscher with the argument that including Russia in the picture would violate the notion that the Ukraine is a sovereign nation. Asscher is a lawyer and wields a legal argument in a discussion on geopolitics and war. When Hitler invaded Holland, Asscher would say that this was illegal. Asscher has been doing all kinds of silly things but he has a quiet and soft presentation that Dutch people seem to take for wisdom.

Thus the reasoning is:

  • There should be a peace agreement with both Russia and Ukraine.
  • There should be a trade agreement with both Russia and Ukraine.
  • This current Association Agreement does not satisfy these conditions. In fact, it reduces the freedom of the Ukraine to enter into trade agreements with Russia. The Ukraine becomes the outer border of the European trade area. Trade between the Ukraine and Russia will be hit, and this will cause unemployment in the Ukraine, similarly to the unemployment that occurred in Eastern Germany after the fall of the Berlin Wall (see here).
  • The EU should stop the policy that association with the EU means association with NATO.
  • The EU should use “Europe” as a term in geography and not as synonymous with EU.
  • Hence No to the current treaty.
Richard Sakwa on Dutch TV (March 20 2016)

Richard Sakwa on Dutch TV (March 20 2016, minute 41)

PM 1. Arno Wellens (in Dutch) finds it curious that Asscher was not at the discussion table, and was asked for his opinion while sitting in the audience. Wellens suggests that Sakwa anticipated a 20 minute focus on his analysis, but was surprised to find himself in a discussion with the vice-prime-minister. Perhaps this is the case. Still, it is useful to see the clash between geopolitics (Sakwa) and legalism (Asscher).

PM 2. Another English source is Mark Almond at Oxford, but perhaps a bit less outspoken as Sakwa.

PM 3. For Dutch viewers there is a nice discussion of March 27 between Jeroen Dijsselbloem (Yes) and Arjo Klamer (No). Klamer’s main concern is that the Ukraine has too much corruption, so that the EU’s neoliberal policies will benefit the oligarchs and be disastrous for the common people. This is partly correct. It is strange that Klamer doesn’t give a better economic analysis, but see here (in Dutch).

PM 4. There is some information on a EUR 11 bn Ukraine state building project financed by the EU. This is about EUR 250 per Ukrainean, or one-time 12.5% of the annual GDP per capita of EUR 2000. We may guess where this money is going to land. Perhaps it is an acceptable bribe to get better law and police in the long term. That said, there is still the issue of war. It would be foolish to pay a bribe to get involved in fighting.

Listening to Poulopoulos, O Dromos


While Europe is busy with refugee immigrants, I take a leave, since I already discussed this two years ago: Europe’s bloody border. Some people accuse this weblog of not looking at the real issues but these people don’t boycott Holland and thus are in serious need of a reality check.

Instead, my thoughts and warm feelings go out to Adriaan de Groot (1914-2006), because his work has always been relevant to me, and comes out top again. He is five years younger than Pierre van Hiele (1909-2010) and like him also studied mathematics with Gerrit Mannoury (1867-1956). De Groot got a bachelor in mathematics, switched to psychology with an MA in 1941, and got his 1946  PhD in mathematics & physics, with supervisor psychologist Geza Revesz who remarkably worked at that department. This study is: Thought and choice in chess (online). Original Dutch: Het denken van den schaker (online).

NB. Mannoury was early into semiotics, and found that “the meaning of a word is its use” quite early before Ludwig Wittgenstein (1889-1951) did. For example, immigrant has a different use than future compatriot. Hopefully one day I can say more on Mannoury.

The spelling checker alerts us to problems in this Abstract by Amsterdam University Press. There is also a distinction between a chess master and the computer game chessmaster. But it is great that the English translation is online.

“What does a chessmaster think when he prepartes [sic] his next move? How are his thoughts organized? Which methods and strategies does he use by solving his problem of choice? To answer these questions, the author did an experimental study in 1938, to which famous chessmasters participated (Alekhine, Max Euwe and Flohr). This book is still usefull [sic] for everybody who studies cognition and artificial intelligence.” (Abstract)

A.D. de Groot (1914-2006) (Wikimedia Commons)

A.D. de Groot (1914-2006) (Wikimedia)

Chess and its levels of competence

I will copy much from the obituary, by Fernand Gobet, ICGA journal 240 (?), 2006, p236-243 (?).

“De Groot’s thesis did not, however, attract the interest of chess players only; it was a harbinger of the cognitive revolution in psychology that would occur in the early sixties.Because of its strong impact on cognitive psychology, and because of the breadth of the study, de Groot’s thesis can safely be considered a classic in the field.” (Gobet p36)

Indeed, Gerald Goldin reminded us of the strong influence of behaviorism in the USA, for us: on testing students – see here. The book by De Groot helped the turn to cognitive psychology.

Fernand Gobet p239-240 summarizes the findings by De Groot (1946) and it has been input for Van Hiele (1957).

“De Groot proposed also that a player’s thinking process may be divided into four main phases: orientation, exploration, investigation, and proof. In the orientation phase, players collect relevant information and try to form a first (tentative) judgment of the position. During the exploration phase, sample variations are analysed, and, typically, the number of critical moves or plans is reduced to two. The two candidate moves are analysed in great detail during the investigation phase, which is characterised by a more in-depth search than during the exploration phase. Players strive to validate their favourite move (or plan). Note that most of the argumentation used by chess players consists of convincing themselves that one of the two variations is better than the other. Finally, the proof phase is used to recapitulate the information obtained in the analysis and to check the correctness of the argumentation. De Groot described also several chess methods used by players to reach their solution. These methods include strategic and tactical plans, ideas, and goals. Note that while they differentiate well between strong and weak players, all these methods are tied to the domain of chess. The higher-level thought and choice methods, which organise the structure of the protocol, did not differ between players of various skills.”

I don’t want to go into chess now. Some meta-comments are relevant:

  • Personally, I found it quite surprising that the Georg Rasch model (on competence in reading by Danish children) was mathematically the same as the Arpad Elo rating for chess players: Item Response Theory. It is included in my book Voting Theory for Democracy.
  • Main idea: When you are not challenged enough then you get bored, when the challenge is larger than your competence then you get stressed, and when challenge and competence match then you get flow.
  • This is also a model for competition between theories or academic papers. Currently editors select the articles of a journal – or manage the selection of those. Alternatively, researchers just put their article online. Subsequently the process of Elo rating starts. The main question is how to define the rules of the game. Facebook does it with “likes”. But this is too coarse.

One of the sickest comments in science is to say about an article: “Get it published in a peer-reviewed journal, and then I will look at it.” It is illogical and perverse, and a frequently abused lame excuse for not looking into criticism. When a scientists looks at the article directly, then this is peer-review on the spot. Not looking at it, is shifting the effort to others. For example, when you contact a scientist to report an inconsistency in his or her thesis: then this scientist should look into it, and the reply should not be: “Get …. it.”

There are also other players who discovered that it pays not to stick to the rules. See my game of chess with Vladimir Putin, and the performance of Garry Kasparov on Dutch TV. Surprisingly, in Gobet’s Obituary on page 241  we find the game of chess of 3000 BC that we discussed yesterday, and in this publication Kasparov apparently succeeded in replacing the Egyptian pieces by Staunton pieces. It still isn’t clear yet who must move. The ancient Egyptian or Putin rule is that there are no moves: just proceed at will.

De Groot Obituary, by Fernand Gobet, p241

De Groot Obituary, by Fernand Gobet 2006, p241

In retirement: Forum theory

From onset to retirement is a great leap, but it shows De Groot’s most important contribution. This is the Forum theory. Gobet summarizes:

“During his retirement, de Groot spent much of his energy on philosophical questions, most of them related to psychology. A first theme is related to the notion of truth in science. The Forum Theory , which he had been developing over thirty years, insists on the idea that science is a communal activity directed towards rational consensus. As there is no absolute truth in science, all that scientists can do is to strive for truth, that is, to strive for theories having the highest possible level of certainty. This criterion is met in the case of statements that are unanimously endorsed by all pertinent scientific experts. Such statements then are scientifically true to the best of our present knowledge. Neither are the rules for the correct way of conducting science unchangeable or indisputable. These, too, are to be discussed, and agreed upon in what de Groot calls the forum of expert opinion. A second important theme in de Groot’s reflections was a conception of unifying psychology, a field that is now split into innumerable schools. His approach to this gigantic task was to strive for agreements on the definitions of basic concepts in scientific psychology. De Groot conceded that the task of bridging methodological and terminological differences between schools will not promise any early success. However, connectibility of terminology and method is a necessary requirement for any mature scientific discipline, he argued. Working on it is a must.” (Fernand Gobet, p238)

The key book is De Groot (1982), Academie en Forum, as far as I know not translated into English. The book is great and has for example these elements:

  • Forum continues where his other book Methodologie (1961) ends (see below). Methodologie already contains the Forum (capitalized) but still ends somewhat depressing: exact results like in mathematics cannot really be gotten. The further development of Forum Theory is a positive idea, and uplifting.
  • It indeed also contains the suggestion to look into Elo-rating of research (-ers).
  • Academia discusses education research, design of structure and curriculum, selection processes for higher education (equal input of time versus equal output of quality), innovation (at that time). Design of new topics for education should be accompanied by description how those are going to be tested.
  • It rejects the triad knowledge, skill and attitude, with the argument that it is rather difficult to operationalise and test attitude; and replaces this with another scheme.
  • It highlights the sectarian character of Holland. Two persons are a church; a third causes a schism.

My book Trias Politica & Centraal Planbureau (1994) page 81 quoted from Academie en Forum p 9. Appendix A below contains that quote and a remarkably fair Google Translation of it. Please observe that one objective of this weblog is to contribute to the unification, by showing how findings are related. The suggestion of an Economic Supreme Court is also based upon a Definition & Reality Methodology that supplements De Groot’s Methodologie, and that is also required to qualify the Van Hiele theory of levels.

A bit on significance 1956

Psychologists Eric-Jan Wagenmakers and others found it useful to translate an article by De Groot (1956) on (statistical) significance. I agree that the translation is helpful. Let me immediately refer also to Ziliak & McCloskey (2006) Cult of Statistical Significance. In addition: large sample sizes may easily create statistically significant differences: but with little relevance for meaningful significance (how you want to use the results).

“Adrianus Dingeman de Groot (1914–2006) was one of the most influential Dutch psychologists. He became famous for his work “Thought and Choice in Chess”, but his main contribution was methodological — De Groot cofounded the Department of Psychological Methods at the University of Amsterdam (together with R. F. van Naerssen), founded one of the leading testing and assessment companies (CITO), and wrote the monograph “Methodology” that centers on the empirical-scientific cycle: observation–induction– deduction–testing–evaluation. Here we translate one of De Groot’s early articles, published in 1956 in the Dutch journal Nederlands Tijdschrift voor de Psychologie en Haar Grensgebieden. This article is more topical now than it was almost 60 years ago. De Groot stresses the difference between exploratory and confirmatory (“hypothesis testing”) research and argues that statistical inference is only sensible for the latter: “One ‘is allowed’ to apply statistical tests in exploratory research, just as long as one realizes that they do not have evidential impact”. De Groot may have also been one of the first psychologists to argue explicitly for preregistration of experiments and the associated plan of statistical analysis. The appendix provides annotations that connect De Groot’s arguments to the current-day debate on transparency and reproducibility in psychological science.” (Abstract by E-J. Wagenmans et al.)

Methodology 1961

De Groot wrote a classic Methodologie, (1961, 1994 online).  The recommendation in Dutch by G.J. Mellenbergh on pages v-vi is well-deserved. Apparently De Groot’s empirical cycle is appreciated by the English speaking wikipedians, but the Dutch version looks deserted. Amazon states that the English translation Methodology (1969) is out of print.

Logic, philosophy of science and methodology were an early interest of mine, see A Logic of Exceptions (1981, 2007, 2011). It also caused the Definition & Reality Methodology used in DRGTPE.

Methodology (the study, not necessarily this book) appears to very relevant – see here – when you want to understand the Van Hiele theory of levels of insight.

Psychologist Ben Wilbrink rejects Van Hiele’s theory, referring to Popper: the theory wouldn’t be falsifiable.

  • Interestingly, Popper’s approach in the philosophy of science is based upon the approach in psychology by Otto Selz (1881-1943).
  • The Dutch wikipedia text states that De Groot was inspired to his thesis on chess by work by Selz.
  • However, see this discussion that explains where Popper’s criterion of falsifiability doesn’t work.
  • A major problem with the rejection by Wilbrink is that he apparently is not interested in mathematics education research: but it is strange to do “psychology” and not look at the relevant field of application.

One reader at Amazon gave it 100% appreciation, with wonderful words like paedagogocal (kids enjoy a go-go approach) and crimonology (monologues like this weblog are a crime), while the expansion to Ayurveda comes as a surprise out of the blue.

“This book is complete, superb and perfect, therefore handy for research and practical work. It has a high abstract level. Exact for all social sciences; psychology, clinical psychology, social psychology, sociology, paedagogocal [sic] and political sciences, journalism, etc. Regarding the human aspects also for biology, medical science; neurology, psychiatry, law, crimonology [sic], general economics, languages and history. A.D. de Groot became a doctor cum laude in Math and Physics in 1946. He had a Fellowship on the Center for Advanced Study in Behavioral Sciences, Stanford, California, 1959-1960. With this book one can expand too, f.i. like Ayurveda methodology, where cures by placebo effects or self-curings are studied too.” (By JPR Petersen on December 2, 2011 on Amazon)

Gobet summarizes the conundrum of psychology that cannot observe “thinking”. I take two quotes:

“De Groot’s approach to psychology is complex. It is a subtle mixture of “hard” techniques, mathematical and statistical – do not forget that he was a professor of methodology for over 20 years – and of “softer” approaches, such as interpretative analysis of verbal protocols. In this case, the psychologist tries to understand the subjects’ behaviour at several levels, some of which are not accessible through sheer quantitative techniques. His work on chess, starting with his thesis, offers a good example of the concomitant application of these two approaches.” (p 241)

“History may prove de Groot correct after all. In the last two decades, there has been a renewal of interest in more qualitative ways of analysing chess data. There has been a revival of interest, too, in “higher descriptions” and of “global descriptions” of positions, besides the more detailed descriptions at the chunk level. Obviously, this is the level you get when you ask experts to speak about their field of expertise. And this is the level of analysis de Groot emphasised in Thought and Choice in Chess.” (p242)

De Groot 1961 overlooked the 1957 Van Hiele levels of insight

Van Hiele’s thesis’s list of literature mentions De Groot (1946) on chess and De Groot (1955) Cognitive psychology and education in geometry in the introduction to geometry. (This is my translation of the title.) Apparently the Dutch term “aanvankelijk meetkunde-onderwijs” means geometry in junior highschool. (Van Hiele also refers to Emma Castelnuovo Intuitive Geometry. Lehrer Rundbrief VII, 11. See my earlier question on history of math (ed).) (This link generates a different person, teacher A.J. de Groot.)

Given that Van Hiele’s 1957 thesis concerns levels of insight, De Groot could have taken an interest, also with the common background in math and Mannoury. Indeed, Van Hiele’s thesis is in the literature of Methodologie (my copy 1961), and it is mentioned on page 184.

The great disappointment is that:

  • De Groot doesn’t include the subtitle in the reference. The full title of Van Hiele’s 1957 thesis is, my translation: “The Issue of Insight, Demonstrated with the Insight of School Children in the Subject Matter of Geometry.”  Thus he indicates that geometry is only used for an existence proof for the levels of insight. Observe the pun: geometry itself uses demonstration as a method.
  • De Groot puts Van Hiele into the box of “geometry only”, instead of seeing that Van Hiele presented a general theory of insight, cognition, epistemology.
  • We can only suppose that De Groot was so busy with Methodologie that he wasn’t really interested in new insights coming from mathematics education.

Never teach generalities by giving only one example. Give strings of examples, so that people see the general portent. Don’t give people any opportunity to put you into the box of a single example.

This is the relevant section of De Groot (1964), Ch 6, p 184. Let me cheat in translation for a first time now, and edit the output of Google Translate. The originals are in Appendix B.

“But what are ‘educational objectives that are considered important ‘ in the case of plane geometry in the first grade of junior highschool ?

One can have rather different views on the aim of teaching geometry. One can even argue whether it is a necessary part, for example of the advanced (HBS) program. As is known it has indeed been proposed to replace plane geometry entirely by something else, for example symbolic logic or set theory. One can see the aim as limited, strictly tied to the program itself: learn to solve certain types of problems. Or one can see the aim as wide in scope, for example: learn to think, analyse a problem, learn to apply systematic methods of analysis and general methods of solution, also for other objectives (cf. e.g. BOS 1955). Or one can put emphasis on the spatial aspect: geometry as a means to develop a structured ‘spatial insight’ (e.g. VAN HIELE 1957). Or, something else again, as a way for a first encounter with a scientific, partly formalised deductive system. There are many sagacious and profound reflections on this – and few conclusions in agreement.”

St Nicholas 1965

Our hero De Groot was no saint but a mere human, and one day became ill. However, he turned his illness into a wonderful opportunity to research the issue of St Nicholas. De Groot (1965), Saint Nicholas: A Psychoanalytic Study of History and Myth, discusses:

“In this book, the fascinating St. Nicholas story is examined from a specific angle. Intrigued by the abundance of fertility and birth symbolism in folklore and legends, the author has tried his hand at a “psychoanalysis of St. Nicholas.” To this end, present-day (Dutch) folklore is traced back to medieval customs and legends and, through a partly historical, partly psychoanalytic interpretation, to pre-Christian beliefs, to Germanic and Greek gods– and in particular to the birthplace of the legend, the city of Myra. The result is an absorbing and often surprising perspective of sixteen centuries of Christian culture.” (cover text)

This is essentially the same analysis as the later one by Tony van Renterghem (1995), When Santa was a Shaman: Ancient Origins of Santa Claus & the Christmas Tree. De Groot has a tougher task as a psychologist, also thinking about semiotics, while Van Renterghem has more liberty to wonder at questions and cultural relevance.

My own proposal since 1992 is to get rid of the religious burden and male chauvism here, and speak about Kidda Claus or Claudia, with helper Jester Peter or Petra. In Dutch: Kinderklaas & Narren-Piet.

Perhaps I am overdoing this. “Santa Claus” is already different from “St. Nicholas”. The fellow in his reindeer sled is not the common religious saint. The situation in the USA may already be neutralised, except for the male chauvism.

Santa in Holland has a helper: not an elf, but a character with the traditional name of Black Peter and the appearance of the black-up Morris Dancers.

  • Protests about racism on one side and pleas to keep the tradition on the other side are at boiling point.
  • Last year the mayor of Gouda arrested 90 people in the mayhem, see my report. Of these 90, 89 have been acquitted so this indeed smells of an abuse of power, and the remaining 1 likely is a case to cover up police brutality.
  • My proposal is to use the name Jester Peter or Petra, and allow all kinds of colours (of which black might be one, but only as one of many). Keep most of the tradition but don’t use a name that plays into racist misunderstandings.

Holland has close connections with Russia, – by chess, – by the Romanovs, – by MH17, – by the current Russian boycott of Dutch agricultural products, – by the Crimea Scythes gold that was on exhibition in Amsterdam when Russia took the Crimea: with now the tantalising question whether it should be “returned” to the Ukraine or Russia. And thus also with St Nicholas, the patron saint of Great Russia. We can only hope that the boycott of Holland is successful for the right reason.

Russian icon (Wikimedia commons)

Russian icon (Wikimedia commons)


In Holland, mathematician Hans Freudenthal (1905-1990) rejected empirical methods, like statistics, because he was trained to be an abstract thinking mathematician and he knew little about empirical science anyway. He created “realistic mathematics education” (RME) based upon his own insights on what would work. He also succeeded in establishing this as the norm in Dutch math education. Well, there are plenty of reasons to be wary of abuse of statistics, but the only answer is to become a better statistician. This however was lost to the mathematics education research (MER) community.

Given the void thus created by the absence of MER, psychologists stepped in. At the academia they for example study number sense: in kindergarten and the first years of elementary school, the importance of fingers. As applied science, producing tests as paid-for products, there is CITO. At CITO, (psychological) testers design, process and analyse annual tests of pupils leaving elementary school, for language, math and other subjects, and those support advice on the subsequent school system. It is a huge achievement of De Groot to help set up such a system, that creates some standards, and that doesn’t leave kids in the jungle of well-meaning but perhaps incompetent teachers.

CITO must be praised for this too, since it is by this kind of testing that the failure of RME came to full attention. Teachers at higher and middle education were complaining that students could no longer count, but CITO turned the anecdotal complaints into facts – also eliminating the sneer to similar complaints already in antiquity.

It is however curious that:

  • These psychologists rely on their own understanding of arithmetic, and they don’t feel as if they should study MER.
  • The psychologists apparently took RME as the authorative standard of MER, and devised tests to measure RME, contrary to Freudenthal’s view.
  • Thus CITO created tests with “context sums”, in which text is used to describe situations, such that pupils must detect the underlying mathematical model, and then solve the issue and calculate some answer. See our discussion on Gerald Goldin for similar test issues in the USA.
  • CITO allows an uncontrolled experiment on children, while there would be scientific and medical rules for chimps and rabbits. There are now two competing methods in Dutch primary education: RME and Van de Craats’s “traditional method”. A scientific experiment would be stopped once it is clear what method is best, after which the best method is given to all guinea pigs. CITO just allows the mess. See my letter to CITO, in Dutch, October 18, that didn’t get a decent reply yet. CITO refers to the Inspection of Education, as the formal authority for testing elementary school kids. This is a false referral. The Inspection isn’t a scientific institute. CITO does the actual testing and claims to maintain standards of science. My question to CITO is one of scientific ethics, and they simply dodge it. In psychology it might be called cognitive dissonance.
Problematic psychological research on mathematics education

I am no psychologist, and only give my response from teaching practice and MER and econometric technique (including Jöreskog’s LISREL and latent variables).

Overall: Van Hiele opposed concrete versus abstract while Freudenthal misrepresented this as model versus reality (applied mathematics).

To test mathematical competence according to Van Hiele, you would have to test at the various levels of insight. Students would have to know mathematics before they can apply it – by level – and it is not proper to equalise understanding to applied mathematics. Mathematical competence is not a collection of fields of application. Testing Freudenthal is easier, because you can resort to situations of applied mathematics. It reduces to behaviorism again. When the chimps push the right buttons on the calculators then by definition they have mastered some skill.

These so-called cognitive psychologists have reduced to behaviorism again.

Let me mention seven problematic experiences, and correct me if I am wrong, because I am no psychologist, and they do a lot of testing about issues that I am not aware of:

  1. Research on number sense can be invalid because of inadequate handling of pronunciation of numbers, see here. It is very curious that there is no movement amongst psychologists to reform collective pronunciation of numbers. (Norway had a reform in 1950, the exception.) See my booklet A child wants nice and no mean numbers (2015) (online).
  2. Stellan Ohlsson inverts the process of learning, saying that it would go “from abstract to concrete”, but he means to say “from vague to precise”, see here. How is it possible to confuse these terms ? It is not just Ohlsson as a single person, because he is member of a community that would have discussed these issues.
  3. Psychologist Ben Wilbrink shows inadequate grasp of methodology, and doesn’t want to look into this, see here.
  4. CITO tests mixed fractions in the traditional manner, but those are didactically cumbersome. If psychologists – and especially mathematically capable psychometricians, with the journal Psychometrika founded in 1935 –  had been aware of MER then they could have protested early on that this isn’t mathematics but “mathematics” – see the discussion on the torture by Jan van de Craats, see here. (Let me refer to Van Hiele (1973) with a proposal to abolish fractions – here.)
  5. CITO tends use outcomes of sums as the indicator of achievement, and neglects the methods how the outcomes have been achieved, apart from legal rules that allow or don’t allow a calculator. (Generally, when there is a descriptive text, then it is called a “context sum” and then calculators are allowed.). Thus, kids who use traditional algorithms (e.g. long division) and kids who use RME algorithms (e.g. partial quotients), would be judged equally competent when they have the same outcomes. This is not only the equalisation of “good method but small error” and “hopelessly lost”. Such is a common feature of computerised testing and probably cannot be avoided (except by creative chunking): except by concluding that some tests shouldn’t be computerised. The true problem is that you require the traditional algorithms in arithmetic to do algebra at a later stage. Thus RME might seem to generate “competence in simplifying 165 / 7” but in fact it maims your brain for higher schooling. Hickendorff (2011), a cum laude thesis using CITO data, falls into that trap. This thesis played and plays an important role in the discussion in Holland, with its conclusion as if the traditional and RME methods would be equally effective at the end of elementary school. Such a conclusion only derives from a disregard of MER. Hickendorff explicitly states not to be competent in that field. In itself it is a sign of integrity to emphasize what you are not competent in. This clarity is much appreciated and helps us to identify the problem. For, there still is a problem. Apparently she worked in an environment that was cocooned from the notion that a researcher must develop expertise in the area of application.
  6. I presume that there are psychologists who supported RME by Freudenthal, that wasn’t empirically tested – but I am new to this world and cannot give references.
  7. I presume that there are psychologists who supported traditional mathematics education, like from Hung-Hsi Wu in the USA, that doesn’t seem to be tested empirically either. But I am new to this world and cannot quite give references. For example, John Hattie has been educated on education, and I don’t know how psychology features in that, in his part of the world.
I wouldn't want to be caught before a blackboard like that (Screenshot UChicago)

I wouldn’t want to be caught in front of a blackboard like that (Screenshot UChicago)


Loose ends we haven’t looked into:

  • Wouldn’t Van Hiele have been interested in De Groot’s Methodologie, and have contacted him on that ?
  • What about the role of Hans Freudenthal (1905-1990) in all of this ?
  • Who are the psychologists supporting Jan van de Craats anno 2015 ?
  • When will this misery ever end ?


Appendix A: De Groot Academie en Forum (1982:9) quoted by Trias Politica & Centraal Planbureau (1994:81)


Google Translate actually does a reasonably fair job, for a computer programme.

“(…) is a democratic polity necessary, but not sufficient. For a fruitful development of the politically sensitive social sciences in particular is also needed: a government that understands well its main task of science policy; namely the duty, even in those subject areas, the tradition of critical inquiry, rational discussion and strive for objective judgment – in short – to encourage support, protect the culture Forum. Perhaps a modern democracy was not a triad but with tetras [tessera? / TC] politician must be equipped with the fourth independent power of science. One might think of a corresponding Supreme Court, which is common in severe cases, the government can condemn political prostitution of research, for abuse of expressions such as ‘scientifically proven that …’ and scientifically irresponsible applications. It seems in principle a good idea, at least – in the current situation – a beautiful pipe dream. Realized or not, the idea is dictated by a certainly legitimate concerns about the socio-political climate is such that the last decades has developed in the Netherlands. It seems that the public respect for rationality and integrity, for (better) understanding, and (better) intellectual performance in general, there is no greater on has become. The common aim leveling work a short-sighted anti-intellectualism in the hand; the continued politicization of scientific research standard and rationally decidable problems, not only undermines the (gamma) science but also adversely affects the quality of our entire culture.” (Google Translate)

Dutch original:

“(…) is een democratisch staatsbestel nodig, maar niet voldoende. Voor een vruchtbare ontwikkeling van de politiek zo gevoelige gamma-wetenschappen in het bijzonder is tevens nodig: een overheid die haar voornaamste taak van wetenschapsbeleid goed verstaat; namelijk de taak om, ook op die wetenschapsgebieden, de traditie van kritisch onderzoek, rationele discussie en streven naar objectieve oordeelsvorming – kortom: de Forum-cultuur – te steunen, te bevorderen, te beschermen. Misschien zou een moderne democratie niet met een trias maar met een tetras [tessera ? / TC] politica toegerust moeten worden, met als vierde onafhankelijke macht die van de wetenschap. Men zou kunnen denken aan een bijbehorende Hoge Raad, die in voorkomende ernstige gevallen de overheid kan veroordelen voor politieke prostitutie van onderzoek, voor misbruik van uitdrukkingen als ‘wetenschappelijk is aangetoond dat …’ en voor wetenschappelijk onverantwoorde toepassingen. Het lijkt in principe een goed idee, althans – in de huidige situatie – een mooi luchtkasteel. Realiseerbaar of niet, de gedachte wordt ingegeven door een wel degelijk gegronde bezorgdheid over het sociaal-politieke klimaat zoals zich dat de laatste decaden in Nederland heeft ontwikkeld. Het ziet ernaar uit dat het publieke respect voor rationaliteit en integriteit, voor (beter) inzicht, en voor (betere) intellectuele prestaties in het algemeen, er niet groter op is geworden. Het gangbare nivelleringsstreven werkt een kortzichtig anti-intellectualisme in de hand; de voortdurende politisering ook van wetenschappelijk onderzoekbare en rationeel beslisbare problemen, ondermijnt niet alleen de (gamma-) wetenschap maar schaadt ook de kwaliteit van onze hele cultuur.”  (A.D. de Groot, “Academie en Forum”, Boom, 1982, p9)

Appendix B: De Groot (1961) on Van Hiele (1957)

De Groot (1961, 1964, Ch 6 paragraph 2.2, page 184) (Source DBNL):

Google Translate:

“But what are” considered important educational objectives’ in the case of plane geometry in first class?

One can aim of geometry teaching in general look very different. One can even argue about whether it is a necessary part, for example of the HBS program; as is known has been proposed to replace the plane geometry entirely by something else, such as symbolic logic and set theory. One can see the target limited, strictly tied to the program itself: certain types learn to solve problems; or one can see the large, for example, learn to think, analyze a problem, systematic thinking methods and general solution methods learn to apply also for other purposes (cf. eg forest in 1955.). Or one can focus on the spatial aspect geometry as a means to develop a structured ‘spatial awareness’ (eg from hiele 1957); or different, as a means of an introduction to a science, partly formalized deductive system. There are about many sagacious and profound reflections – and few corresponding conclusions.”

Dutch original:

“Maar wat zijn de ‘belangrijk geachte onderwijs-doelstellingen’ in het geval van de vlakke meetkunde in de eerste klasse?

Men kan het doel van meetkunde-onderwijs in het algemeen zeer verschillend zien. Men kan zelfs twisten over de vraag of het een noodzakelijk onderdeel is, bijvoorbeeld van het H.B.S.-programma; zoals bekend is wel voorgesteld de vlakke meetkunde geheel te vervangen door iets anders, bijvoorbeeld symbolische logica of verzamelingsleer. Men kan het doel beperkt zien, strikt gebonden aan het programma zelf: bepaalde typen vraagstukken leren oplossen; of men kan het ruim zien, bijvoorbeeld: leren denken, een probleem analyseren, systematisch denkmethoden en algemene oplossingsmethoden leren toepassen, òòk voor andere doeleinden (vgl. b.v. bos 1955). Of men kan het accent leggen op het ruimtelijke aspect: meetkunde als middel tot ontwikkeling van een gestructureerd ‘ruimtelijk inzicht’ (b.v. van hiele 1957); of, weer anders, als middel tot een eerste kennismaking met een wetenschappelijk, gedeeltelijk geformaliseerd deductief systeem. Er bestaan hierover veel schrandere en diepe beschouwingen – en maar weinig overeenstemmende conclusies.”

There is Good news and there is Bad news. The Good news is that while Bashar al-Assad flew in secrecy to Moscow to meet with Putin – of which photo’s were released when he was safe at home again – that also his enemy IS leader and self-acclaimed “Caliph Ibrahim” flew in secrecy to Moscow to see Putin – though no photo’s were released. When asked, Putin will invoke plausible deniability.

The President of the Russian Federation will also show his intense irritation about questions on this – a dead give-away. I happened to be present at the occasion because of my monthly chat with him, so let me help him by reporting on the main details.

The Bad news is that self-acclaimed “Caliph Ibrahim” brought along an ancient game of chess as a present for Putin, and then destroyed it before his eyes, throwing it to shambles and beheading the pieces.

Vlad: “That was a gift ! You were going to give it to me ! You completely ruined it !”

sa”CI”: “Let this be a lesson for you !”

Vlad: “That was an ancient game of chess ! A priceless artifact !”

sa”CI”: “Yes, indeed. We found it when we ransacked the Mosul Museum. It belonged to Egyptian Pharao Ramses I when he fought the war at Armageddon around in 3000 BC. It was his gift for his ally from Babylon, Nebukadnezar the Terrible.”

Vlad: “And you barbarian just smashed this ?! I must ask Garry Kasparov to glue those heads on again. He can be smart, you know ? He won’t glue a pawn’s head onto a horse. Though these Egyptian pieces look funny.”

sa”CI”: “That isn’t a horse but a crocodile. And what you call a pawn is a scorpion.”

Queen Nefertari at the board (Source: wikimedia commons)

Queen Nefertari at the board (Source: wikimedia commons)

Vlad: “Well, in that case I am happy that you cut those heads off. Was that your lesson for me ?”

sa”CI”: “Not really. See those pieces that look like penises ? We cannot tolerate pornography.”

Vlad: “And now I am unhappy again. You Cut Those Off.”

The strong man of Russia shuddered.

sa”CI”: “I said that Ramses gave it as a present to his ally Nebukadnezar the Terrible. You must know the full story. They fought together against Alexander the Great and his Hittites, and when they had won the war, then Ramses put Nebukadnezar in a cellar, and gave him this game of chess, so that he and Alexander the Great could pass the time by trying to take each other’s penises.”

Vlad – flushing: “Ah. Oh. So. Ah. Yes. You have got me completely freaked now, I must admit. Am I really getting grateful that you destroyed that … priceless artifact ?”

sa”CI”: “There is no need to thank me. I enjoyed doing it. I only brought it along to show something about the Middle East. We have been at each other’s throats since the beginning of civilisation. Actually, civilisation began because we have been at each other’s throats. Don’t think that you can win. Don’t think that you can invent any new power ploy that we haven’t practiced to perfection for millennia already. Just give us weapons. Then I will give you my advice.”

Courtesy of

Courtesy of

Post scriptum

Below is the ancient chess game between Nebukadnezar the Terrible and Alexander the Great from around 3000 BC, restored by Garry Kasparov. The crocodiles were too much damaged and have been replaced by Staunton horses. On insistence by Putin the penises have been replaced by the sun symbol, now the king. Because of his mortal fear of scorpions, Kasparov replaced those by penises again, but Putin selected the Staunton pawns again. The sitting figure is the queen, scarabs are rooks, and three feathers are the bishop. It has been lost in history who took what side and whose turn it was, and why they didn’t finish the game. President Putin’s main goal now is to get peace in the Middle East so that he can start digging to find out.

Ancient game of chess, restored

Ancient game of chess, restored

The earlier discussion on Stellan Ohlsson brought up the issue of abstraction. It appears useful to say a bit more on terminology.

An unfortunate confusion at wikipedia

Wikipedia – no source but a portal – on abstraction creates a confusion:

  1. Correct is: “Conceptual abstractions may be formed by filtering the information content of a concept or an observable phenomenon, selecting only the aspects which are relevant for a particular purpose.” Thus there is a distinction between abstract and concrete.
  2. Confused is: “For example, abstracting a leather soccer ball to the more general idea of a ball selects only the information on general ball attributes and behavior, eliminating the other characteristics of that particular ball.” However, the distinction between abstract and concrete is something else than the distinction between general and particular.
  3. Hopelessly confused is: “Abstraction involves induction of ideas or the synthesis of particular facts into one general theory about something. (…) Bacon used and promoted induction as an abstraction tool, and it countered the ancient deductive-thinking approach that had dominated the intellectual world since the times of Greek philosophers like Thales, Anaximander, and Aristotle.” This is hopelessly confused since abstraction and generalisation (with possible induction) are quite different. (And please correct for what Bacon suggested.)

A way to resolve such confusion is to put the categories in a table and look for examples for the separate cells. This is done in the table below.

In the last row, the football itself would be a particular object, but the first statement refers to the abstract notion of roundness. Mathematically only an abstract circle can be abstractly round, but the statement is not fully mathematical. To make the statement concrete, we can refer to statistical measurements, like the FIFA standards.

The general statement All people are mortal comes with the particular Socrates is mortal. One can make the issue more concrete by referring to say the people currently alive. When Larry Page would succeed in transferring his mind onto the Google supercomputer network, we may start a philosophical or legal discussion whether he still lives. Mutatis mutandis for Vladimir Putin, who seems to hope that his collaboration with China will give him access to the Chinese supercomputers.

Category (mistake) Abstract Concrete
General The general theory of relativity All people living on Earth in 2015 are mortal
Particular The football that I hold is round The football satisfies FIFA standards
The complex relation between abstract and general

The former table obscures that the relation between abstract and general still causes some questions. Science (Σ) and philosophy (Φ) strive to find universal theories – indeed, a new word in this discussion. Science also strives to get the facts right, which means focusing on details. However, such details basically relate to those universals.

The following table looks at theories (Θ) only. The labels in the cells are used in the subsequent discussion.

The suggestion is that general theories tend to move into the abstract direction, so that they become universal by (abstract) definition. Thus universal is another word for abstract definition.

A definition can be nonsensical, but Σ strives to eliminate the nonsense, and officially Φ has the same objective. A sensible definition can be relevant or not, depending upon your modeling target.

(Θ) Aspects of scientific theories (Σ) Science (Φ) Philosophy
(A) Abstract definition (developed mathematically or not) (AΣ) Empirical theory. For example law of conservation of energy, economics Y = C + S, Van Hiele levels of insight (AΦ) Metaphysics
(G) General (GΣ) Statistics (GΦ) Problem of induction
(R) Relation between (A) and (G) (RΣ) (a) Standards per field,
(b) Statistical testing of GΣ,
(c) Definition & Reality practice
(RΦ) (a) Traditional epistemology,
(b) Popper,
(c) Definition & Reality theory

Let us redo some of the definitions that we hoped to see at wikipedia but didn’t find there.

Abstraction is to leave out elements. Abstractions may be developed as models for the relevant branch of science. The Van Hiele levels of insight show how understanding can grow.

A general theory applies to more cases, and intends to enumerate them. Albert Einstein distinguished the special and the general theory of relativity. Inspired by this approach, John Maynard Keynes‘s General Theory provides an umbrella for classical equilibrium (theory of clearing markets) and expectational equilibrium (confirmation of expectations doesn’t generate information for change, causing the question of dynamic stability). This General Theory does not integrate the two cases, but merely distinguishes statics and its comparative statics from dynamics as different approaches to discuss economic developments.

Abstraction (A) is clearly different from enumeration (G). It is not impossible that the enumeration concerns items that are abstract themselves again. But it suffices to assume that this need not be the case. A general theory may concern the enumeration of many particular cases. It would be statistics (GΣ) to collect all these cases, and there arises the problem of induction (GΦ) whether all swans indeed will be white.

Having both A and G causes the question how they relate to each other. This question is studied by R.

This used to be discussed by traditional epistemology (RΦ(a)). An example is Aristotle. If I understand Aristotle correctly, he used the term physics for the issues of observations (GΣ) and metaphysics for theory (AΦ & GΦ). I presume that Aristotle was not quite unaware of the special status of AΣ, but I don’t know whether he said anything on this.

Some RΦ(a) neglect Σ and only look at the relation between GΦ and AΦ. It is the price of specialisation.

Specialisation in focus is also by statistical testing (RΣ(b)) that only looks at statistical formulations of general theories (GΣ).

The falsification theory by Karl Popper may be seen as a philosophical translation (RΦ(b)) of this statistical approach (RΣ(b)). Only those theories can receive Popper’s label “scientific” that are formulated in such manner that they can be falsified. A black swan will negate the theory that all swans are white. (1) One of Popper’s problems is the issue of measurement error, encountered in RΣ(b), with the question how one is to determine sample size and level of confidence. Philosophy may only be relevant if it becomes statistics again. (2) A second problem for Popper is that AΣ is commonly seen as scientific, and that only their relevance can be falsified. Conservation of energy might be relevant for Keynes’s theory, but not necessarily conversely.

The Definition & Reality methodology consists of theory (RΦ(c)) and practice (RΣ(c)). The practice is that scientists strive to move from the particular to AΣ. The theory is why and how. A possible intermediate stage is G but at times direct abstraction from concreteness might work too. See the discussion on Stellan Ohlsson again.


Apparently there exist some confusing notions about abstraction. These can however be clarified, see the above.

The Van Hiele theory of levels of insight is a major way to understand how abstraction works.

Paradoxically, his theory is maltreated by some researchers who don’t understand how abstraction works. It might be that they first must appreciate the theory before they can appreciate it.

The discussion of Putin’s proof gave me an email from Alexis Tsipras, who just resigned as prime minister of Greece and is busy with the general elections of September 20 soon. Rather than reporting on it, I might as well fully quote it.

To: Thomas
Subject: My proof of Fermat’s Last Theorem
Date: Fri, 28 Aug 2015 11:58:03 +0100
Google Unique Message Identifier: 23DFGA@671

Dear Thomas,

Thank you very much for your discussion of President Putin’s proof when he was a youngster of Fermat’s Last Theorem. I know his mother Vera Putina very well. The Putin family has a vacation home here in Greece, and she can stay there on the condition that she immediately leaves when Putin himself comes down. She has shown me his proof too. I can only agree with your conclusion that it shows how smart President Putin was when he was young.

Putin’s proof inspired me to find a proof too. I am sometimes exhausted by the tough negotiations with the European Heads of State and Government, if not with members of my own party. Thus I resort often to a sanatorium for recuperation. Thinking about such issues like Fermat’s Last Theorem helps to clear my mind from mundane thoughts. I was very happy last Spring to indeed find a much shorter and more elegant proof. 

For the theorem and notation I refer to your weblog. My proof goes as follows.

Theorem. No positive integers n, a, b and can satisfy an + bn = cn for n > 2.

Proof. (Alexis Tsipras, April 31 2015)

Let us assume that an + bn = cn holds, and derive a contradiction.

There are two possibilities: (1) n is even, or (2) n is uneven.

(1) If n is even, then we can write A = an/2 and B = bn/2 and C = cn/2 such that A, B and C are still integers. Then we get the following equation:

A2 + B2 = C2

This equation satisfies the condition that n = 2, and thus it doesn’t satisfy the condition n > 2.

(2) If n is uneven, then we can write A = a(n-1)/2 and B = b(n-1)/2 and C = c(n-1)/2 such that A, B and C are still integers. Then we get the following equation:

a A2 + b B2 = c C2

This equation does not satisfy the form of an + bn = cn so that it falls outside of Fermat’s Last Theorem.

In both cases the conditions of the theorem are no longer satisfied. We thus reject the hypothesis that an + bn = cn holds. Q.E.D.

This is much shorter that President Putin’s proof. And, I prove it while he only came close. I have been hesitating to tell him, fearing that he might become jealous, and be no longer willing to support Greece as he does in these difficult times for my country. Now that you have confirmed how wonderful his proof at only age 12 was, I feel more assured. Will you please publish this proof of mine too, like you did with President Putin’s proof ? I have put my best efforts in this proof, just like at the negotiations with the European Heads of State and Government. Thus I hope that it will be equally convincing, if not more.

After the next elections I will probably be exhausted again. I would like to work on another problem then. Do you have any suggestions ?

Sincerely yours,


Fermat and Tsipras (source: wikimedia commons)

Fermat and Tsipras (source: wikimedia commons)