Mathematicians can be seen as lawyers of space and number.

Euclid wrote about 300 BC. Much earlier, Hammurabi wrote his legal code around about 1792-1749 BC. It is an interpretation of history: Hammurabi might have invented all of his laws out of thin air, but it is more likely that he collected the laws of his region and brought some order into this. Euclid applied that idea to what had been developed about geometry. The key notions were caught in definitions and axioms, and the rest was derived. This fits the notion that Pierre de Fermat (1607-1665) started as a lawyer too.

Left Codex Hammurabi, right a piece of Euclid 100 AD. Wikimedia commons

Left: codex Hammurabi. Right: a piece of Euclid 100 AD. Wikimedia commons

The two cultures: science and the humanities

In Dutch mathematics education there is a difference between A (alpha) and B (beta) mathematics. B would be “real” math and prepare for science. A would provide what future lawyers can manage.

In the English speaking world, there is C.P. Snow who argued about the “two cultures“, and the gap between science and the humanities. A key question is whether this gap can be bridged.

In this weblog, I already mentioned the G (gamma) sciences, like econometrics that combines economics (humanities) with scientific standards (mathematical models and statistics). Thus the gap can be bridged, but perhaps not by all people. It may require some studying. Many people will not study because they may arrogantly believe that A or B is enough (proof: they got their diploma).

Left and right hemisphere of the brain

Another piece of the story is that the left and right hemispheres of the brain might specialise. There appears to be a great neuroplasticity (Norman Doidge), yet overall some specialisation makes sense. The idea of language and number on the left hemisphere and vision on the right hemisphere might still make some sense.

“Broad generalizations are often made in popular psychology about certain functions (e.g. logic, creativity) being lateralized, that is, located in the right or left side of the brain. These claims are often inaccurate, as most brain functions are actually distributed across both hemispheres. (…) The best example of an established lateralization is that of Broca’s and Wernicke’s Areas (language) where both are often found exclusively on the left hemisphere. These areas frequently correspond to handedness however, meaning the localization of these areas is regularly found on the hemisphere opposite to the dominant hand. (…) Linear reasoning functions of language such as grammar and word production are often lateralized to the left hemisphere of the brain.” (Wikipedia, a portal and no source)

For elementary school we would not want kids to specialise in functions, and encourage the use of neuroplasticity to develop more functions.

Pierre Krijbolder (1920-2004) suggested that there is a cultural difference between the Middle East (Jews), with an emphasis on language – shepherds guarding for predators at night – and the Indo-Europeans (Greeks), with an emphasis on vision – hunters taking advantage of the light of day. Si non e vero, e ben trovato.

There must have been at least two waves by Indo-Europeans into the Middle-East. The first one brought the horse and chariot to Egypt. The second one was by Alexander (356-323 BC) who founded Alexandria, where Euclid might have gotten the assignment to write an overview of the geometric knowledge of the Egyptians, like Manetho got to write a historical overview.

Chariot spread 2000 BC. (Source: D. Bachmann, wikimedia commons)

Chariot spread 2000 BC. (Source: D. Bachmann, wikimedia commons)

It doesn’t actually matter where these specialisations can be found in the brain. It suffices to observe that people can differ in talents: lawyers would deal much with language, and for space you might turn to mathematicians.

Pierre van Hiele (1909-2010) presents a paradox

The Van Hiele levels of insight are a key advance in epistemology, for they indicate that human understanding itself is subjected to some structure. The basic level concerns experience and the direct language about this. The next level concerns the recognition of properties. Another level is the recognition of relations between these properties, and the informal deductions about these. The highest level is formalisation, with axiomatics and formal deduction. The actual number of levels depends upon your application, but the base remains in experience and the top remains in axiomatics.

Learning goes from concrete to abstract, and from vague to precise.

Thus, Euclid with his axiomatic approach would be at the highest level of understanding.

We arrive at a paradox.

The axiomatic approach is basically a legal approach. We start with some rules, and via substitution and reasoning we arrive at other rules. This is what lawyers can do well. Thus: lawyers might be the best mathematicians. They might forget about the intermediate levels, they might discard the a-do about space, and jump directly to the highest Van Hiele level.

A paradox but no contradiction

A  paradox is only a seeming contradiction. The latter paradox gives a true description in itself. It is quite imaginable that a lawyer – like a computer – runs the algorithms and finds “the proper answer”.

However, for proper mathematics one must be able to switch between modes. At the highest Van Hiele level, one must have an awareness of applications, and be able to translate the axioms, theorems and derivations into the intended interpretation. In many cases this may involve space.

Just to be sure: the Van Hiele levels present conceptual divides. At each level, the languages differ. The words might be the same but the meanings are different. This also causes the distinction between teacher-language and student-language. Often students are much helped by explanations by their fellow students. It is at the level-jump, when the coin drops, that meanings of words change, and that one can no longer imagine that one didn’t see it before.

Thus it would be a wrong statement to say that the highest Van Hiele level must have command of all the lowest levels. The disctinction between lawyers and mathematicians is not that the latter have command of all levels. Mathematicians cannot have command of all levels because they have arrived at the highest level, and this means that they must have forgotten about the earlier levels (when they were young and innocent). The distinction between lawyers (math A) and mathematicians (math B) is different. Lawyers would understand the axiomatic approach (from constitutional law to common law) but mathematicians would understand what is involved in specific axiomatic systems.

Example 1

I came to the above by thinking about the following problem. This problem was presented as an example of a so-called “mathematical think-activity” (MTA). The MTA are a new fad and horror in Dutch mathematics education. First try to solve the problem and then continue reading.

2016-10-26-petervanwijk-smaller-englishDiscussion of example 1

The drawing invites you to make two assumptions: (1) the round shape is a circle, (2) the vertical x meets the horizontal x in the middle. However, why would this be so ? You might argue that r = 6 suggests the use of a circle, but perhaps this still might be an ellipse.

In traditional math ed (say around 1950), making such assumptions would cost you points. In fact, the question would be considered insoluble. No question would be presented to you in this manner.

In traditional math, the rule would be that the proper question and answer would consist of text, and that drawings only support the workflow. Also, the particular calculation with = 6 would not be interesting. Thus, a traditional presentation would have been (and also observe the dashes):

2016-10-26-petervanwijk-smaller-english-altA quick observation is that there are three endpoints, and it is a theorem that there is always a circle through three points. So the actual question is to prove this theorem, and you are being helped with a special case.

Given that you solved the problem above, we need not look into the solution for this case.

The reason for giving this example is: In mathematics, text has a key role, like in legal documents for lawyers. Since mathematicians are lawyers of space and number, they can cheat by using supporting drawings, tables and formulas. But definitions, theorems and proofs are in text (formulas).

(Potentially lawyers also make diagrams of complex cases, as you can see in movies sometimes. But I don’t know whether there are particular methods here.)

Example 2

The second example is the discussion from yesterday.

In text it is easy to say that a line has no holes. However, when you start thinking about this, then you must define what such a hole might be. If a hole doesn’t belong to the line, what does it belong to then ? How would you know when you would pass a hole ? Might you not be stepping over holes all the time without noticing ?

These are questions that lawyers would enjoy. They are relevant for math B but can also be discussed in math A.

See the discussion of yesterday, and check that the main steps should be acceptable for lawyers, i.e. math A.

These students should be able to master the symbolism of predicate logic, since this is only another language and a reformulation of common text.


Thus, a suggestion is that students in math A should be able to do more, when better use is made of the legal format.

Perhaps more students, now doing A, might also do B, if their learning style is better supported.

(Perhaps the B students would start complaining about more text though. Would there still be the same question, when only the format of presentation differs ? Thus a conclusion can also cause more questions. See also this earlier discussion about schools potentially manipulating their success scores by causing student underperformance.)







The standard treatment of continuity in mathematics textbooks in schools tends to be a bit crooked.

  • The continuum is first assumed, but it is not stated what is assumed.
  • For the real line, the lack of holes is a key property of continuity, but it is called by a word that students might have no affinity with (“completeness” rather than “wholeness”).
  • When continuity is actually discussed in analysis (if at all), then this concerns the continuity of functions, which is rather a different subject.
  • A discussion of the continuum brings us to topology, but do we really need to start with topology before we can do analysis ? Do you want to start your junior highschool class by stating: “In the mathematical field of point-set topology, a continuum (plural: “continua”) is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space.” ?

Our research question for today is: What might be a more logical exposition ? (Didactics would be second phase.)

I will not be telling anything new here, but some students might benefit from the more explicit and straightforward discussion.

Continuity as a primitive notion that cannot be defined

The basic notion of continuity is the real line. One might also think about 3D space or time. L.E.J. Brouwer wouldn’t trust space (Euclidean or non-Euclidean ?) and take time as his intuition, and hence speak about “intuitionism”. My impression is that space is more easy to communicate about (measuring rods are easier to make than clocks), whence I adopt the real line.

Definition w.r.t. human experience: Continuity is a primitive notion, that you might grasp by considering a line (section) in space.

Definition for mathematics: The set of real numbers R can be defined in a particular way. Personally I prefer the method by Timothy Gowers to develop the real numbers as infinite decimals.

Once the real numbers have been defined, then we can say that they also satisfy the notion of continuity. Thus, continuity is either a human experience or defined as the real numbers.

Once we have done this, then we can find the “Cantor-Dedekind Axiom“:

“In mathematical logic, the phrase Cantor–Dedekind axiom has been used to describe the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states that there is a one-to-one correspondence between real numbers and points on a line.” (PM. This wikipedia page should link directly to Tarski’s axioms for geometry.)

I find the term “axiom” a bit problematic.

  • Given the two properties mentioned above, this isomorphism rather expresses human experience from modeling practice. The real numbers would be a model for the line (section) in actual space.
  • Likely though, the identification of R with space is best seen as a definition of what we regard as Euclidean space. It is a question whether actual space would be Euclidean. It is a question whether we can actually imagine space being non-Euclidean, since we imagine e.g. a sphere still in 3D Euclidean space.

But the isomorphism is explained rather easily – and for didactics we would likely begin with this. For the numbers we can look at a development of binary decimals between 0 and 1. The next decimal is 0 or 1, and again, and so on. For space we can make a cut and have left and right parts, make a cut again with new left and right parts, and so on. Thus this is the same structure. But these also are different realms: numbers and space. Thus it is not quite an identity but an isomorphism. Interestingly: when cutting in this manner, we will never meet a hole.

Continuity can be explained only for subsets

Subsequently, the key properties of continuity can be formulated w.r.t. subsets S of R, rather than w.r.t. R itself.

Definition: A subset S of R is called continuous, if between two elements (values) in S there is always another one (i.e. it is dense), and when there are no holes. Or in formulas:

(a) For each x, y in SR with x < y, there exists z in S such that x < z < y

(b) For each x, y in S R with x < y, there exists no z in R \ S such that x < z < y

The last property shows the difficulty for R. If one would want to specify that R has a hole, then one would have to specify what that hole belongs to. To some X ? What is X ? In the past, people had a problem imagining what a vacuum was: the horror vacui. For them, space could only exist if something occupied that space. Nowadays, mathematical space is understood merely as a set of co-ordinates, and the issue what physical space would be is left to physics.

Also observe that this definition essentially depends upon the fact that the real numbers have been given, i.e. the earlier section. Thus, continuity is a basic notion given for R and there is only a “proper” (explicit) definition for subsets: which definition relies on the use of R.

If you don’t assume R, you get into problems. For example, if you were to take the set of rational numbers Q rather than R, then (a-Q) could be satisfied for some S, say = [1/2, 3/2], and (b) would become:

(b-Q) For each x, y in S Q with x < y, there exists no z in Q \ S such that x < z < y

In that case, one might say that Q becomes Q-continuous, but this is not the continuity that we want, since there are elements in R still in that interval. (Contiguity comes to mind as a label, but already has some use.)

Further developments

Property (b) is called “(Dedekind) completeness“. It is true that “complete” is a proper translation of German “vollständig” (German wiki), but I would rather prefer “(Dedekind) wholeness”, since this better indicates the lack of holes. But let me admit that I am used to the phrase “completeness” as well, for the chapter of ordering, and thus my preference is weak. Perhaps it is best to speak about “completeness (wholeness)”.

Subsequently, when we forget about the reliance on R, and try for a more abstract formulation, then the notion of supremum (least upper bound) comes into play. We can look at some S independent from R, as the “linear continuum“. This is not intuitive and not feasible in junior highschool. Potentially this approach actually captures continuity in a definition, so that it isn’t just primitive, and can be defined, but (for me, yet) there is no clear connection between the notion of continuity and the property of having suprema. The switch to topology comes into play, see G.H. Moore “The emergence of open sets, closed sets, and limit points in analysis and topology” (and thanks to Dag Oskar Madsen for the reference, in a discussion about open sets that is closed now).

Continuous function

Obviously it helps to have clarity on continuity in the reals first before speaking about continuous functions.

The notion of a continuous function f uses both domain S and range f[S]. At stackexchange, many readers liked the view by Qiaochu Yuan (answer 17):

“One abstract way to think about continuity (…) is that it is about error. A function f:XY is continuous at x precisely when f(x) can be “effectively measured” in the sense that, by measuring x closely enough, we can measure f(x) to any desired precision. (…) This is an abstract formulation of one of the most basic assumptions of science: that (most of) the quantities we try to measure () depend continuously on the parameters of our experiments (…). If they didn’t, science would be effectively impossible.”

For Dutch readers, Vredenduin has a nice exposition in Euclides 1969 on the notion too, partly containing this intuition on the error too, but not so explicitly. He speaks about a small change in the domain and no dramatic change in the range, but it is more enlightening to explicitly speak about (measurement) error.  (And I would have a question on continuity of “f / g“, p14.)


The main argument is that this storyline is more straightforward for understanding continuity. All this suggests that school would benefit from a discussion of the reals. This would include issues like 0.9999…. = 1.0000…

I am supposing that junior highschool could manage the expressions of mathematical logic. The New Math tried and failed, but there should be more clarity why it failed.

PM. For completeness: there is always philosophy (and nonstandard analysis).

Horror vacui: no space without something (Wikemedia commons)

Horror vacui: no space without something (Wikimedia commons)

After my short stint as expert on national security in 2005, I now had another short stint, now as expert on mathematics education, STEM and the role of mathematics education in the whole curriculum. First Jos Tolboom of SLO had hinted that I could be invited for an expert meeting, but then this invitation didn’t actually materialise.

This is a bummer. I wrote several books and articles on the subject. One of the reasons why there is so little progress in the field is that there isn’t enough attention for my novel analysis.

The November event

This concerns the following event:

22 – 23 November 2016
CIDREE STEM (science, technology, engineering, mathematics)  Expert Meeting

Utrecht, The Netherlands
Hosted by Freudenthal Institute and SLO
Topic: The position of mathematics education and informatics education in a coherent STEM curriculum
This meeting aims to create an international overview of innovations in mathematics education and informatics education, their relationship and the coherence from the STEM perspective, with a special interest in computer based mathematics and its relation with computer science (informatics). This goal has been determined with other CIDREE members as a follow up on the expert meeting in June 2015 in Trondheim, Norway.

The announcement in Dutch is here. Three questions for the meeting are (in my translation):

  1. How can we create more coherence in STEM education overall ?
  2. How can we create a mathematics curriculum with a strong component in computer based (mathematics) education ?
  3. How can we create a curriculum for computer science (“informatica”) as part of the STEM curriculum ?

The schedule with the speakers is here. For example, Cambridge Mathematics will be present, as they also announce on their website:

Cambridge Mathematics will be represented at this event as we explore innovations in mathematics education and how we can plan for the future of the mathematics curriculum.

Key warning for STEM researchers

(1) Researchers on STEM should be aware that the researchers on STE may have little knowledge or interest in both Mathematics education (ME) and its research (MER). Every field tends to focus on itself, and coherence is secondary.

(2) A key difference is:

  • STE fields have empirics as a judge of what works. This empirical mindset is also applied to the education in these fields.
  • Mathematics is directed at non-empirics (abstraction). There is no external judge but only personal opinion. Thus mathematicians tend to regard power play and “math wars” as acceptable methods to get views accepted. (Examples of such thinking in Holland are mathematicians Jan van de Craats and Henk Broer.)
  • See this discussion about the math war between “realistic mathematics education” RME and traditional ME (TME), and the scientific alternative of neoclassical NME. Look also for the explanation that the name “Freudenthal Institute” does not convey the true meaning of the institute, and that it is better to speak about “Freudenthal Head in the Clouds Realistic Mathematics Institute” (FHCRMI). Namely, RME is like astrology or homeopathy.

(3) In combination: STE are willing victims of “realistic mathematics education” (RME) ideology. STE provide “contexts” and they apparently appreciate the interest. However, it really requires a study of ME and MER to get rid of the RME ideology and their unscientific narratives.

My qualifications

Let me mention my qualifications as expert for this topic and these questions, and observe that my books are online:

  1. I developed four books within Mathematica, a system for doing mathematics on the computer: Voting Theory for Democracy (2001), A Logic of Exceptions (2007), Conquest of the Plane (2011), and The Economics Pack. (since 1993). These books provide the coherence that the expert meeting is looking for, with text, formulas, graphs, tables, routines, programming (informatics) and interaction. (A missing element is assessment.) There is also the book Transport Science for Operations Management (2000) that has been supported by routines in Mathematica. It is likely the “not invented here” syndrome at Dutch universities that these books are not being used regularly in matricola. (There is also the breach of scientific integrity w.r.t. a “review” of COTP.) (The distinction between the popular vote for Clinton and the Electoral College for Trump might cause more attention for voting theory nowadays, but my expectation is that Dutch universities will continue to neglect VTFD.)
  2. I also discussed “Beating the software jungle”, included in Elegance with Substance (2009, 2015). This explains about the current chaos in software for education and what an effective and proper approach would be to resolve this. (The STEM researchers at this meeting might not have enough background in economics to understand the argument on market structure.)
  3. I clearly explained the failure by the Dutch organisers of the event, both SLO and the Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI), in their dealing with these issues before. Thus the organisers would know that inviting me would give scope for a discussion that goes to the heart of matters (and not beating about the bush again). Let me discuss this in the subsequent sections. (If these institutes would be scientific, then not-inviting me amounts to blocking me, since I would like to attend. Blocking me is an abuse of power, made possible by the current power void in mathematics education and its research, see here. But these institutes might also argue that they are not scientific.)
Freudenthal Head in the Clouds Realistic Mathematics Institute (FHCRMI)

I find myself repeating again. This isn’t good.

The actual argument is quite elaborate. When I would fall into the trap of using one-liners, then readers might think that I am being simplistic and that I (over-) generalise. Let me refer to Elegance with Substance, Chapter XIII and the note on p114.

A new phase in the discussion is the breach of research integrity by psychometricians at Leiden University. They actually expose the unscientific nature of RME / FHCRMI ideology but they don’t do so adequately.

Let me also refer to the abuse of so-called “21st century skills”. This label is deliberately used as a Trojan horse for re-introducing “realistic mathematics education” (RME) ideology. The true revolution is computer algebra. See here.

News on Michiel Doorman

A new element – on which I am not repeating myself – is that I have now collected my documentation about the unscientific and ideological performance by Michiel Doorman, one of the employees of FHCRMI on STEM and one of the key organisers of above event. Much of this documentation is in Dutch, but I provided an overview in English.

Two elements are relevant for STEM:

  • Doorman maltreated the new algebraic approach of the derivative. Obviously, for physics education it is a key discovery that it is a false mathematical argument that limits would be required. For the derivative, it suffices to use algebra.
  • Doorman promoted Java applets instead of computer algebra.

A google on Doorman also generated this diagram within the EU project of Mascil. I regard this as simsalabim, a phrase used in magic tricks, with flash and smoke that hide what is really happening. For example, there are “inquiring minds” and there is reference to a “collaborative classroom culture”, but these RME / FHCRMI “experts” clearly close their minds and use their elbows.


Simsalabim, taken from Doorman et al. 2014 on the European Union Mathematics and Science for Life (mascil)

SLO – Dutch expertise center on curriculum

SLO would be the Dutch national expertise center on the curriculum. It started as a foundation and initiative by researchers, and it is gradually absorbed by government regulations, with work packages and subsidies. This particular QUANGO has no longer a transparant structure. Best would be a decent government body, with accountability, but at some distance of political decision making because of the scientific base. For mathematics education, each nation should have a national organisation with a key role for teachers and researchers, and this organisation would also supervise the curriculum. For Holland my suggestion is a Simon Stevin Institute, and it would give directions to SLO (instead of SLO telling teachers what to do).

The traditional approach in pedagogy looks at the triad of student, material and teacher. In this approach, the student features with both a personality and personal development. SLO however derives from the world of “education studies” that are at a distance of the traditional development of pedagogy. “Educational studies” tend to overlook the student. In the Van den Akker diagram about learning, students who would do the learning are not mentioned themselves. They are regarded as learning machines, and the personalities of students might only be considered from contacts w.r.t. “other students”, see the “spider diagram”, at the SLO “Europe: Mathematics and Science knowledge” and SECURE project page. Potentially SLO exports this spider diagram to other countries, and foreigners cannot check whether they listen to criticism. Dutch readers / viewers will benefit from this video interview with professor emeritus in traditional pedagogy Jan Dirk Imelman. (Interviewer Ad Verbrugge is an ideologue too, but Imelman makes it a useful interview.)

Spider web by Van den Akker (SLO)

Spider web, by Van den Akker (SLO). (Replace the word “learning” with “human” or “humanity”.)

Letter to Dutch Parliament, September 27 2016, on Onderwijs2032

I wrote this letter in English to Dutch Parliament, about much of the same thing. As stated in the letter, I chose for the use of English because I wanted that OECD and CIDREE would be able to read the argumentation.

The minister of education is considering a transformation of Dutch education to “21st century skills” (“Onderwijs2032”). This would involve the abolition of traditional subjects (like physics or economics) and merging those into common labels (like nature or society). The idea is that education should increase understanding that surpasses the various subjects. This is also known as the “transfer” problem. However, the traditional approach is: one must first master a subject before one can surpass it. Thus Onderwijs2032 is created from rosy dreams.

What is crucial to know is that RME already belonged to that stream of rosy “21st century skills” way of thinking, and that it failed miserably. This is also why it is so curious that the Leiden psychometricians failed in reaching the proper analysis, even though they did show that RME claims are exaggerated. See this submission to the integrity board and this link on algebra as a troubling word.

Curiously, SLO has been supporting this Onderwijs2032 project. Though they oversee the curriculum, they did not notice that you need traditional algorithms in arithmetic in elementary school because you need those for algebra in secondary education.

News on Jos Tolboom

At SLO, Jos Tolboom (LinkedIn) is the other key organiser of the November event. He informs me that he read Elegance with Substance this last Summer and is impressed by its quality and relevance. I hope that he finds time to express this in a public statement that others can check (and that I would include on the EWS website). It is still possible that there are misunderstandings though, for we haven’t had a discussion on particulars. I also hope that he finds time to read COTP and then will protest against the abuse by Jeroen Spandaw in the journal Euclides in 2012.

Tolboom also alerted me to this CIDREE event and hinted at the possibility that he might invite me to attend. I am wondering now why he doesn’t. He gave me a reason but in my expert view not a convincing one, and lacking in respect for science. (Perhaps though he doesn’t regard SLO as a scientific organisation.)

I observed that Tolboom is giving video presentations on the new Dutch national exam on mathematics. A key element in the exam renewal are the “mathematical think-activities” (MTA). When I evaluated this MTA notion last month, I found it deficient. (1) MTA is severely confused w.r.t. didactics and testing, (2) MTA is a disguise of RME, (3) MTA is not proper mathematics (which would have a development towards deduction with definitions, theorem, proof). Another suggestion for Tolboom is to state a reply to this criticism. Hopefully, Holland finds a way to rewrite the national exam regulation.

The revolution is computer algebra

Let us return to the November event and the revolution of computer algebra. The problem with this revolution is that mathematicians are riding their hobby horses and creating chaos, e.g. by creating that software jungle. Perhaps there must be a jungle for a survival of the fittest, but in the past organised efforts like for Algol seem to have worked better.

Looking at the schedule, I don’t see a presentation that will explain that computer algebra forms the common core. Suggestions for a common core are “drawings” and “modeling”. You can check how the sessions of November 23 are geared to such a conclusion. Such a (prepared) inference would be deficient in understanding of didactics. Apparently Jos Tolboom didn’t read Elegance with Substance well enough. (Potentially, for mathematics, the “modeling” can be translated as MTA ?)

There will be a presentation on November 22nd by “Computer Based Maths” (CBM). This organisation was founded by Conrad Wolfram, and thus CBM is personally related to Wolfram Research Inc. (WRI), the makers of Mathematica, created by Stephen Wolfram. WRI apparently decided that it would not do to simply advertise the use of Mathematica. This might come across as a commercial enterprise. Conrad Wolfram set up CBM, and I suppose that they use more resources than only Mathematica (for example on assessment). See my earlier comments on computer algebra and Conrad, here.

Still, I find the situation needlessly complex and disinformative. The proper analysis is that computer algebra is the revolution, and that this provides the common core in education with the computer, not only for STEM but also for languages and the arts. There are other computer algebra implementations than Mathematica, but this remains the best system, consistently since at least 1993 when I started using it. We can allow for various implementations but as long as the same computer language is used for doing mathematics. The commercial venue by WRI is somewhat of a distraction. The creation of CBM is evidence of this too, and an admission by WRI too. It would be better that WRI is turned into a public service utility. The true question is what arguments would cause WRI to agree on this.

My impression is that it would help a great deal when the community of educators would agree that the common core can be found in mathematics itself, and, when the computer is used, in computer algebra. It is easier (also for CBM and WRI) to agree to help out when you are lauded than when your accomplishments are misunderstood and when you feel that you have to put in an effort to get recognition. If WRI would make Mathematica free for elementary and secondary education, then they can still earn their income on universities and research institutes as they are doing now.

When these CIDREE conference “experts” would arrive at the “STEM common core” of “modeling” then they do not understand didactics of mathematics and then their conference result is caused by an abuse of power by excluding a proper expert, duly signed, yours truly.

Event agenda for November 23

Event agenda for November 23

The last weblog text on open access publishing caused me to write this letter to VSNU and other bodies in Holland. VSNU is the platform at which Dutch universities collaborate.

Letter to VSNU and others on membership dues and open access publishing.

Addendum October 17 2016: There is a spreadsheet example now with rough data for mathematics education in Holland.

  • It would cost school employers an additional EUR 50 per mathematics teacher to compensate teachers for memberschip of NVvW (the Dutch association of mathematics teachers) and turn Euclides (its journal) into an open access journal.
  • Employers have already agreed to compensation. It is just that (mainly 2nd degree) teachers do not join up.
  • For publishers, schools already have contracts for access, such that a teacher only has to activate the account. Similarly, schools might see it as a contract with NVvW, and teachers only have to activate their membership. (NVvW thinks of itself as an association only and not as a publisher too.)
  • Employers cannot say what association their teachers should join. The closed shop should not be with a particular association but with a default association. The individual teacher decides whether to join an association and which one. The decision to join is only made easier via “complementary subscription” and the “activation of membership”.

Secondly, I had to think about what Timothy Gowers wrote when announcing “Discrete Analysis” while using the Scholastica platform.

(a) He uses that accepts submissions from all over the world. But this would be difficult to create for each discipline. Best is that the institute where you graduate also supports your follow-up.

(b) In that line of argument: It appears that arXiv rejects some papers because they are second-guessing universities whether you are a “true” scientist or not. Papers of mine have been rejected as if I were a crackpot. See my protest about how they can handle this. But I have a degree of econometrics from Groningen (1982) and teacher of mathematics from Leiden (2008). I am quite dismayed that arXiv starts judging on quality while they don’t have the background to judge. See also Richard Gill’s experience on a similar strange rejection, the paragraph “Quantum crackpots”.

(c) Gowers wants “quality” for his journal on “Discrete Analysis”, as if this would be a criterion for open access publishing. This is really no argument but misunderstood vanity. An editor for “peer review” should check on clarity and scientific nature, and that is it. The discussion on quality might be done at a second stage, and is a different kind of discussion. Such decisions actually are new articles, in which an editor-author may argue why some paper or analysis has quality. Gowers identifies these “judgements on quality” as “editorial comments”. Those however should also be submitted to peer review, and they generate citations (namely when such an author refers to proper sources). Gowers now generates the strange phenomenon that some people might make the inference “It was rejected by Discrete Analysis and thus it doesn’t have enough quality”, while that very topic of quality should be subject to peer review at least.

Addendum 2016-10-13: Thus, there might be three types of journals while Gowers has only one. Note the word type. With three types, there can still many different titles, also on Discrete Analysis.

(c1) A journal type “Proofreadings for Discrete Analysis”, consisting of links to submissions (abstracts and full texts) and links to the referee reports.

(c2) A journal type “Recommendations for Discrete Analysis”, with again a list of links, likely introduced by the abstracts. The meaning of this journal is that editors perform as 2nd stage proofreaders, who judge on the articles and referee reports, and provide recommendation by inclusion in the list. It would be best to specify which editor does the recommendation, since we cannot presume that all have read it. Potentially, though, editors hide in the herd to secure anonimity. Editor reports are entered as 2nd stage referee reports in the repository, both for recommended and rejected articles.

(c3) A journal type “Discussion of the recommendations for Discrete Analysis”, in which editors of (c2) given an overview clarification of their recommendations and rejections. Basically these texts have first been published in (c1), and the selection for (c3) is done by another group of editors than for (c2). Editor reports of (c3) are again included in the repository, with links to the underlying original papers.

(d) We should not confuse open-access with open-minded. I wonder whether Gowers is aware that mathematicians have a mortal fear for crackpots. Will his journal be open-minded or will it be another exercise by mathematicians to abuse others as crackpots ? An example of abuse is given in the former weblog text w.r.t. a paper on a new algebraic approach to the derivative.

Let each institute of higher education (HE) – university or college – create a working paper archive (WPA) for its alumni.

The alumni can be at the level of bachelor, master or PhD, see the Bologna Process at ECAHE for a description of the quality levels.

For example, one of the qualifications for a master’s degree is:

“have demonstrated knowledge and understanding that (…) provides a basis or opportunity for originality in developing and/or applying ideas, often within a research context;”

If a university deems someone worthy of a master’s degree then it would be logical to assume that the new master might develop some new ideas, and then it would be useful when those could be archived. Let the institute that granted the degree create such an archive for its alumni.

The alumni would not be obliged to use this archive, but having such an opportunity would be a great service to the world. Creating such an archive would serve the purposes of universities and colleges and their libraries on the recording and distribution of knowledge.

It would be better when the archive would not only contain drafts for articles submitted to peer reviewed journals, but also versions, research notes and comments, or links to those.

These higher education HEWPAs would form the basis of what would be called the “publishing process”.

  • Submission to the database would make the results public. Everyone should be able to read the submission. Potentially there might be an option to time-stamp an idea and keep it secret for a while, but such ideas should become public after the author’s death plus some years. Potentially there might be a protocol for patenting as well.
  • The copyright remains with the author. Submitted material can be read by others but can only be reused with permission for which types of contracts are available.
  • There would be monitoring of citations, comments, versions, and so on.
  • Groups of editors might form journals to serve particular interest groups. They might choose to list only the abstracts with links to the WPA, or assist the author in a new version or new layout. These would be scientific journals and hence open access. There would be no additional costs for publishing or distribution, since the costs of the WPA are paid for by institute of HE. If editing is seen as part of an academic job, then these editors are paid for the institutes as well. The commenting on work by others forms part of scientific research indeed. Trying to make your journal “the best” might be part of unavoidable vanity.
  • The archive function is intermediate between library and press. Many elements in the datebase can be printing on demand. For some publications there can be volume printing, as now happens at university presses. One would suppose that many readers would appreciate quality control by editors for expensive outlays in print.
  • Alumni can also join scientific societies and associations. When these unions are professionally relevant, then the employers would pay for the membership dues. When these dues would be used to support a journal – like the Econometric Society has Econometrica – then this means that the employers would ultimately pay for the editing (or that academia pay for their own editing).

This is how it always should have been. There is no need for “commercial scientific publishing”. The current situation with commercial scientific publishing, pay walls and even scientific associations that put their journals behind pay walls to encourage membership to cover costs, are a deviation from sound management of science.

It is true that a commercial publisher can put out advertisements and use fancy labels and claims like “this is the best journal” while a scientist is supposed to remain modest. Thus we can understand from historical reasons that we got where we are now, and that scientists washed their hands and allowed publishers to take control, but those are not good reasons.

Example 1. Dispersion over various archives and lack of some

Currently, for example, I use (1) my website, (2) this weblog, (3) EconWPA in the past and now the Munich Personal RePEc Archive (MPRA) for my economics papers, (3) I have some papers at on statistics and mathematics history and overview, (4) for time stamps I also used, (5) recently I discovered an archive for Dutch educational materials that also allows English (, (6) and there are the “official publications” with their archives.

This means that economic texts in Dutch are only archived on my own website, and not indexed etcetera. Will foreigners be able to find the English educational material ? This also means that most of my work on education and didactics of mathematics are achived only on my website, and not indexed as such.

For mathematics education there is this curious observation:

“Robert’s last recommendation is to have a preprint server for math education research. As he notes, this is a road we’ve tried to go down before and we didn’t get very far. I don’t think the problem has nearly as much to do with policy or categories of the arXiv as it does with the lack of a “preprint culture” in mathematics education. What I learned in those previous preprint discussions, and in my observations as a developing scholar, is that math educators regularly and happily share work in progress — with a select group of people. In math ed, there doesn’t seem to be widespread faith in anything like Linus’ Law,   the open source software dictum that says, “With enough eyeballs, all bugs are shallow.” I think the math wars led to a lot of distrust, and some of it is very rational. It’s safer to only share preliminary work with a few scholars who share similar methods and theoretical frameworks, and then refine the work after peer review before publication in a journal whose readership is likely to understand the work. Maybe it shouldn’t be this way, but to move forward we’re going to have to confront some of these beliefs.” (Raymond Johnson, August 2014)

Example 2. Horrible effects of paywalls

I am already member of the Dutch association of teachers of mathematics NVvW, but for access to the journals of the English ATM or US NCTM I would have additional member fee charges or reading fees, while the Dutch council on education research NRO doesn’t provide subsidies for the kind of research (PROO) that I am doing (letter). Fortunately, my work is so creative that I don’t have to rely on extensive search in the literature, but alternatively put, since I cannot do such extensive search, I focus on what I can do, which is being creative. The snag is that my creative work again hits the pay wall, when “open access” journals again require submission fees and/or transfer of copyrights. My impression is that my membership of the Dutch Royal Library should be sufficient to gain access to information all over the world. Why not ?

Better per institute than discipline

Originally, Bob Parks had an initiative EconWPA at WUStL, modeled after This worked fine, but WUStL stopped it for administrative reasons. One can imagine that it requires an investment to archive papers in a field for the whole world. I am very grateful to Bob for his efforts in the past, and grateful to MPRA to take it over.

This indeed causes the question whether it is useful to archive per discipline or per graduating institute. My impression is that it is better to have the archives at the institutes. They have granted the diploma, and the story starts from there. Over the years an author might shift into a new field but on occasion it would be useful to time-stamp the qualification for that new field.

There are “academic social media” initiatives like ResearchGate, and Mendeley, of which Gaudeamus states: “Too bad they aren’t specialized in socializing the process of publishing in scholarly journals, both to editors and authors.”

Example 3. Article on algebraic approach to the derivative

Having an alumni WPA would mean that I could submit this article to a WPA at Leiden University, where I graduated in mathematics teaching. The article explains the algebraic approach to the derivative as originated for education, and provides a bridge for research mathematicians who might be interested to see whether they can develop it further mathematically.

Currently, the article is only on my website and not at any archive where it can be indexed etcetera.

The article is in English and thus not relevant for the journal Euclides of the Dutch teachers of mathematics.

I submitted the article to “Nieuw Archief voor Wiskunde” (NAW), which is the journal of the Dutch society of mathematics, who are mostly research mathematicians (working at the academia).

The referee wrote an abusive report. While I referred to the domain of the reals, which are a Field, he or she gave a “counterexample” with a Ring. There are some more abuses. See my letter of protest.

The editors of NAW rejected the paper while referring to this referee report. I asked the editors whether they had actually read the submitted paper. I received no reply to this. When they had read the paper, then they could have seen themselves that the report was abusive, and that the referee was dysfunctional, especially when I had pointed out the abuse. This is an abysmal manner of editing.

Thus, the paper still is only at my website. If it were at a Working Paper Archive at Leiden University, then editors could be hunting for material, and discover the paper via indexes and abstract and quality of the author. I would not have to look for journals and submit it, and be exposed to such abuse, but quality editors would locate it. The world would be quite different.

A bit more on this example 3

I wrote my protest the same day when the report arrived. My response is to the point. It expresses sheer intellectual outrage. It doesn’t express the emotions that one feels when one’s work is abused, but it expresses the rational outrage that can be substantiated with arguments. The arguments in my response of October 3 suffice.

It is useful to explain a bit more about it. The discovery of the algebraic approach to the derivative dates from 2007 when I retyped “A logic of exceptions” and programmed this in Mathematica, and included a section on the “paradoxes by division by zero”. It is quite a horror show how the mathematics community in Holland has responded to this for almost 10 years now. I have a lot of praise for Richard Gill and Christiaan Boudri but am quite weary w.r.t. others.

With this present paper and referee report: had I waited with my response a few days longer, then I might have added the following. It would have made for a longer reply so it is better that I dispatched my response immediately. Yet, let us look at the additional arguments why the referee report is below standard. The referee refers to “sin x“.

2016-10-03-naw-sinI already wrote my response that the referee could have asked me before rejecting the article. I could have pointed to “Conquest of the Plane” (COTP) (2011) where this is explained, or the review of COTP by Richard Gill in NAW 2012. Those are both in the list of references, and thus the referee was too lazy to look for this, or ask this. Apparently he or she found sufficient satisfaction in having insulted me for not thinking about trigonometry as the most obvious answer for his or her state of ignorance.

However, I would deem that the admission that the algebraic approach would work for polynomials would already be a major reason for publication. The common perception is that limits are required, but if they are not required for polynomials then this is a major step ahead. The referee could have pointed out that the stated objective of the paper is to provide a bridge from education to research mathematics, and then have concluded that the further development might be something for research mathematicians. However, the referee doesn’t see how important this admission is.

However, the referee might also have realised that Sin has a Taylor expansion as a polynomial. Obviously, this expansion uses the limit of n → 0, but not the limit  Δ→ 0 as required for the derivative. My analysis on the algebraic approach to the derivative doesn’t reject limits per se, but only argues the limit Δ→ 0 is superfluous. The required information for the derivative is already in the formula for f[x], and it suffices to use algebraic methods. Thus, if the referee had had a first year course in analysis, then he or she should not have made that remark as quoted above.

Obviously, the Taylor expansion uses the derivatives themselves, and thus there is the question how this feat of Baron von Münchhausen is achieved. Again, the referee could have pointed out that the stated objective of the paper is to provide a bridge from education to research mathematics, and then have concluded that the further development might be something for research mathematicians.

However, I refer to COTP for the deduction. Richard Gill’s review suggests that there is hidden use of limits again in squeezing of values, but, there is actually an application of logic only.

Note also that the referee writes “sin x“. However, the sine is defined as the y-value on the unit circle of arc φ on that unit circle, and Cos[φ] is defined as its x-value. Thus “sin x” is gibberish as if x = φ. See my earlier suggestion for a didactic presentation of trigonometry.

A hypothesis for social psychology: an autistic fear for crackpots

The problem is rather that the world suffers from mathematicians and their fear for crackpots.

Let me quote form here, page 2:

There are ample indications that this hostile attitude is not uncommon in the world of mathematics. A mathematician wrote to me on March 7 2012:

“Once  you have irritated old-style mathematicians (…) they turn, of course, into crackpot interception mode. Start nit-picking, misunderstanding, finding real small errors, maybe some big ones, but  certainly consistently misunderstanding what you are trying to say. We all get letters and papers from crackpots who are squaring the circle, proving that Bell’s theorem is wrong, or solving the P=NP problem. (…) It’s quite a sport to show in public to your mathematical friends that these crackpots are a public nuisance. (…) You drew attention to yourself, you got attention, and now several Delft mathematicians are thoroughly enjoying a little group-crackpot-ridiculization. Bu t I could say (and in fact do) that one could say that you asked for this! Never mind. Remember Gandhi: first they ignore you, then they fight you, then you win.”

I object that I “asked for it”. The quote above concerned “Conquest of the Plane” (COTP) (2011) but the issue is the same for the current paper on the algebraic approach.

The paper that I submitted to NAW is an excellent review of both the algebraic approach and the current state of research on this. It fits the stated objectives of NAW to publish the article. The algebraic approach to the derivative a world class discovery, and it deserves to be treated with respect and be published. I never claimed that everyone should agree with everything. The ideas of publication is dissemination and does not imply that the editors agree with the analysis. They would have a useful role in checking on clarity and relevance for the readership. Competition with other articles is less relevant for digital publishing (though NAW is also on paper).

I am an empirical scientist and no psychologist. I have a bit more leeway than a psychologist to hypothesize about what is happening here.

Earlier I already applied a lay reader’s understanding of social psychology to the world of economics (here or local file). Let me now indicate this for mathematics.

My impression is that mathematicians are further up in the autistic spectrum and have difficulty in dealing with conflicting information. They know that everyone in their peer group hunts for crackpots. In the case of conflicting information they consider it their best protection to accuse others of being a crackpot, rather than confront their peers and be accused of being a crackpot too. They have a mortal fear of being associated with crackpots, and then can no longer think straight or treat someone with respect.

Addendum 2016-10-11: I now located an actual “autistic spectrum”, namely the AQ questionnaire by Simon Baron-Cohen, see also wikipedia (a portal and no source), and Telegraph 2015. Presumably as a scientist I would be scoring higher on this too, but my point would be that mathematicians are trained to look at formulas at the neglect of empirics, language and humans.

In this case mathematicians have been trained to think that limits are required for the derivative (excepting perhaps non-standard analysis). An article claiming otherwise thus provides conflicting information. For me, the referee is anonymous, but he or she is not anonymous for the editors. When they would say “You really liked that article from that crackpot ?” then the referee might feel exposed. For the referee psychological survival dictates the “crackpot-intercepting mode”, with misrepresentation, slander and neglect.

For me as a non-psychologist this is a fair description of the mathematics community as I have experienced it throughout my whole life.

There is a difference between “(research) mathematics” and “mathematics education”. Research mathematicians, like very likely the editors of NAW and their referee, focus on abstraction, and will have less experience with the empirics of education. Teachers of mathematics have to deal with real-life students, but when those teachers originally have been trained as mathematicians, then those teachers suffer cognitive dissonance, and they resort to traditional ways, that however have not been designed for didactics. The educational programme in mathematics gives ample proof that it is not didactic. This situation can only be caused by teachers who do not observe what is happening with students.

Check the evidence.

As in 2008 I advise each country to have a parliamentarian enquiry into mathematics education. For Holland, a petition is here.

Playing the ball and not the man

Obviously, this involves social psychology, and it is not fair to use the argument “ad hominem”, in which one plays the man and not the ball, like in: “He is a mathematician and thus an autist and thus cannot be trusted.” My argument is quite different. My argument observes the facts of an abusive referee report and a abysmal way of editing, relates this to earlier observations, and arrives at the inference that there must be a common factor. The suggestion that the common factor would be a low quality of my work can be rejected, check some other reviews, and, appreciate the point that this concerns mathematics education so that you can check a lot of formulas and arguments yourself. Society has a serious problem in dealing with the human quality of dealing with abstraction.

Effect of name-calling in the community of mathematics education

The hypothesis that mathematicians are higher up in the autistic spectrum, and far too easily resort to the “crackpot-interception mode” (as it euphemistically is called, as it actually involves misrepresentation and slander), fits the observation that name-calling has such a devastating effect within the community of mathematics education.

Some people have the attitude that this is just “name-calling”, and that grown-ups should be able to neglect it. I have indeed been accused of being overly sensitive to such “name-calling”. However, in the community of mathematics education, such name-calling is a “call to arms” and invitation to all to get into the “crackpot-interception mode”. I find it only sensible to protest against it, because we are dealing with a community that stops reading well once the first rock has been thrown.

The algebraic approach to the derivative was discovered and published in 2007 and the first “review” was in 2010, when Ger Limpens reviewed the book “Elegance with Substance” (EWS) (2009, 2015) in Euclides, the journal of the Dutch association of teachers of mathematics. Euclides is not a scientific journal, but one would hope for fair representation. However, Limpens finds it necessary to wonder whether I would be a wierdo. The Dutch word is “zonderling” and the English translation is “wierdo”, since the English word “eccentric” still has a somewhat favourable sound. Eccentrics may be respected in England but in Holland they are abhorred. The idea that Holland would be a tolerant country is a fairy tale. Limpens found it necessary to inform the readers that the cover of the book reminded him of Don Quixote. The editors of Euclides refused to publish my answer. I am rather convinced that Limpens’s misrepresentation, name-calling and slander about EWS made it easier for others and perhaps inspiration for some to throw other rocks. In the present case, I cannot write to the editors of NAW that Limpens gave a positive review, and in fact, as a scientist I must provide all information and report that Limpens took from the book that I would think that Newton and Leibniz would be dumb people. This, alas, is what the editors of Euclides saw fit to print, and of which I must assume that most of my colleague teachers of mathematics have read.

Recently, there was another case of name-calling and reference to Don Quixote, now with respect to the issue when pi or tau “should” be the standard. See my former weblog text on this.

Blind or double blind refereeing need not be wise

Anonymous refereeing as NAW does is an invitation to start misrepresenting. Editors should however also keep in mind that there should be decent treatment of a submitted paper. The latter is important for the citations and career perspectives of the author. Simply referring to other publication possibilities is awkward when a journal like NAW has a monopoly position in Holland. It are my collagues who might wonder: why wasn’t it published in NAW, if it would be so important ?

If I were paranoid then I would argue that the editors of NAW already decided that I would be a crackpot, and that they themselves quickly wrote such an anonymous excuse to block the paper, or asked some of their crackpot-intercepting buddies to write such an excuse. But I am not paranoid and only wonder why such a referee should remain anonymous.

Above I explained that reviewing the work by others is part of your academic job. Thus let it be known who is performing so badly. This is part of the Elo-rating like in chess, where scores are adapted from game results.

The editors of NAW might refer to the case of Limpens. His name is known and I can protest about his maltreatment. The editors of NAW might argue that they want to protect their referees from such protests, as if these referees would only honestly report about their findings when they would have such protection. I doubt whether this is all fair and square. When referees don’t misrepresent, don’t name-call and slander, and don’t neglect or burke, then there should be no problem with arguments pro and con. Then there will be a scientific discussion, as we should all hope for.

(For Dutch readers I can refer to “forum theory” by De Groot, that should be translated into English a.s.a.p.)

Earlier texts on the publishing process

I already wrote on some aspects before.

  • This text in 2014 suggested to replace arXiv with vixra and PressForward.
  • This is its sequel.
  • See here on Timothy Gowers and his boycott of Elsevier 2012. Interestingly, Gowers has now started a journal using the Scholastica platform. Gowers uses arXiv as the farming base, but obviously arXiv can be needlessly selective (in a fear of crackpots), and such a base is lacking for many disciplines, and thus it makes sense to propose that each institute of HE creates its WPA.

Boycott Elsevier, designed by Michael Eisen 2012

Trig Rerigged 2.0 (draft) proposes a new didactic approach to trigonometry. The proposal has the form of a booklet since it reprints some pages from Elegance with Substance (2009, 2015) and A child wants nice and no mean numbers (2015). The format might change in the future, like the earlier discussion of Trig Rerigged 1.0 of 2008 (now legacy) was absorbed in Conquest of the Plane (2011).

The reader might start with page 15 with the main idea, and page 16 with the main graphs. When these make sense, then restart at the beginning. Trig Rerigged 2.0 is targeted at researchers in mathematics education, teachers of mathematics and trainers of teachers. Well, science journalists might step in too. When you are none of these, then you are advised to be satisfied with the following.

Angular circle with circumference 1. Hook disk with area 1

A circle is defined as the collection of points at a given distance from a center. This distance is called radius. The circle is a concept of circumference. There is proportionality with the radius. With radius r we have circumference r Θ.

A disk is defined as the collection of points at a given distance or less from a center. The disk is a concept of area. Area depends upon the square of the radius. The general disk area is π r 2. Areas of concentric disks however are proportional again.

The unit circle has radius 1 and circumference Θ (“archi”) and disk area π (“pi”). Also Θ = 2π.

The angular circle has circumference 1. Angles can be measured as arcs on the angular circle, as percentages of 1.  The angular circle has radius ΘH.  It is common to use the algebraic symbols instead of their numerical values Θ ≈ 6.28… and H = -1 (“eta”).

The hook disk has area 1. Angles can be measured as sectors on the hook disk, as a percentage of 1.  The hook circle has radius √πH.

Main conclusions
  • It is immaterial whether angles are measured as arcs on the angular circle or as sectors on the hook disk. In both cases we have perunages or percentages of 1. The unit of measurement is actually the plane itself. Another formulation is the number of turns around a circle.
  • Both Θ and π are useful symbols to denote these relationships. They support a rich didactic environment, that allows students to grasp the notions that are closer to their understanding, and develop from there.
Graph of angular circle and hook disk

The following graph from page 16 gives the notions in a nutshell.

The angle α is the arc AB along the angular circle, or the sector OCD on the hook disk. When the sector is extended from the hook disk onto the unit circle, then this sector might be called a “Pi hook”, for its value is α π.

The arc EF is the angle in radians, with the value α Θ.

The point {X, Y} = {x, y} rH has the property that X2 + Y2 = 1. It is useful to use the separate symbols X and Y for this point, since it determines the length of arc from {1, 0}. The point on the unit radius (ur) circle can also be described as a function of the angle α, as {X, Y} = {Xur[α], Yur[α]} = {Cos[α Θ], Sin[α Θ]}.

graphPotential implementation

Since these are suggestions for improved didactics, there must be empirical testing to determine whether these are improvements indeed. It are the students who must show that it works.

Earlier I discussed the US Common Core. This new development on the didactics of trigonometry can be included. See Trig Rerigged 2.0 for more on the relationship to the US Common Core.

I am not qualified for primary education, but the above would seem to be helpful. For example, young pupils could colour sectors of the hook disk, and determine that hooks are additive. At a next stage, they may see the frailer circumference, and see that e.g. 25% of hook matches with 25% of angle. The pupils would be able to determine the radii of the hook disk and the angular circle, so that they grasp proportionality, and that area goes by the square of the radius, and the relationships to Θ and π.

There is an intermediate stage at which {X, Y} = {Xur[α], Yur[α]} = {Cos[α Θ], Sin[α Θ]} will be discussed, and their inverses. Parts might already be done in elementary school, but it would surely be done (repeated) in the early phase of secondary education. PM. The animation at wikipedia for the sine is fairly good, but one would want to be able to manipulate the position, and the choice of yellow for the vertical position is too light.

At the end of highschool, students should be able to deal with radians and sine and cosine. Those functions remain key because of the derivatives. However, the working horses will likely be Xur and Yur, for in trigonometry it is natural to work with turns.

Acknowledgement and word of protest

Above idea basically builds upon Trig Rerigged 1.0 from 2008. The issue here is didactics of trigonometry.

Michael Hartl published a tau manifesto in 2010, and MSC published a reply pi manifesto. The issue here is rather curious. Hartl explains his approach: “π is a confusing and unnatural choice for the circle constant.” This doesn’t concern didactics but concerns some notion of naturalness in some mathematical universe, as if criteria in mathematics itself would force a choice between either Θ or π. I don’t think that this is a relevant way to formulate the issue or discuss it. There is no need for an “archi manifesto” since the relevant issue has been stated in terms of didactics of mathematics in above books. Also, tau is an awkward choice of symbol, for it looks too much like the symbol r for the radius, especially in the handwriting of students.

Still, I read these manifestos and benefited from aspects of them, notably since this gave me the idea to define the hook disk as the disk with area 1, so that we can better see the underlying unity of the notion of angle or hook. Thus I acknowledge the contributions, but also must protest that it doesn’t help when these manifestos divert attention away from the proper question on didactics.

Earlier weblog texts on this issue have been here and here and this animation. On the use of H, see here.

The road from science and scientific discovery into political discussion is often via the channel of a particular party. Politicians of any party are less likely to discuss an idea when there is no party advocating it anyway.

In the USA, members of the Senate and House are elected via districts, which is District Representation. This likely caused the division between two main parties, Democrats and Republicans. The situation likely causes that there are a lot of Think Tanks that want to reach out across the division, to inform voters directly on their various own approaches. For Think Tanks it is important to find at least one representative who is willing to support their case. Bipartisan support is nice but not always necessary, as one can always wait for the next turn in the political cycle.

In Holland, there is Proportional Representation (PR). With 150 seats, it takes only 1 / 150 = 0.67% of the nation-wide vote to get a new party into Parliament. When an issue is important enough to start a Think Tank on it, then likely at least 0.67% of the voters would care about it nation-wide, and then it might be better to start its own party rather than a Think Tank. Political parties in Holland have their own “scientific bureau“, that can inform the rest of the world about their analyses.

This paper of mine compares DR and PR, with the example of the UK, and concludes that the Dutch system is most democratic. See also the short discussion of this in Mathematics Teaching 222 in the context of the UK referendum on PR in 2011.

Baudet starts a think tank rather than a party

Thierry Baudet (1983) started in 2015 a Think Tank “Forum voor Democratie” (FvD) (forum for democracy).

Unfortunately the FvD English page currently still gives a Dutch text on their mission. Let me translate. Their stated mission is to fight the deterioration of democracy and improve its quality e.g. by means of referenda and direct elections of mayors. They also want to move power from the EU back to Holland. They want a strict system of “green cards” for immigrants. They explain their perceived link of democracy to the latter by that “uncontrolled immigration threathens social peace” (my translation). (Like in Brexit, immigration pops up at unlogical spots, as if people stop thinking when the subject arises.)

It is remarkable that Baudet thinks that he cannot get 0.67% of the vote for such a noble cause as the defence of democracy. In Holland, the political party D66 also wants to improve democracy, but they are pro-EU and not anti-EU, and thus he cannot join up. However, as a Think Tank, Baudet would be forced to collaborate a lot with D66, because of the shared view on democracy.

Perhaps it might be easier to start a niche Think Tank rather than a political party though: for, a party requires capable representatives. It may also be a matter of temperament, as Baudet states that he has no affinity with politics itself and wants to remain “independent”. It is okay for other people to follow him but he will not follow others.

Baudet and his FvD helped initiating the 2016 Dutch referendum on the EU Treaty with the Ukraine, see my discussion here and here. Baudet is also prominent in the petition, discussed in the former weblog text. There I promised to look a bit closer at Baudet’s views, which I will do here.

A bit on Baudet’s background

Today’s society cannot do without education. It is always useful to look at what people got their diploma in. This is not intended for an ad hominem argument but helps to clarify their field of competence and way of thinking. The theme of the “Two Cultures” by C.P. Snow indicates that we must be alert on bridging gaps. (See e.g. here.) When people age and grow more experienced, they will tend to diversify from their diploma, but it is seldom that a person from the humanities acquires a taste for science and mathematics as well.

Baudet’s cv doesn’t state whether he did gymnasium A or B. Generally students with gymnasium B tend to specify this though. Also given his later studies in history and law there is a great likelihood that Baudet did A. We should not expect insight in science and mathematics.

He got a bachelor in history in 2006. At Vox Europ 2012 “The EU is an empire, and empires mean war“, the website claims that he would be a historian too, but generally this label would be reserved for masters, and Vox Europ better corrects the claim.

Observe that the general label “historian” is vague too. It is generally better when people study a particular field before they look into the history of that field. It is awkward to look at an issue in the past when you don’t know about the very field of study itself. Grand themes might be an exception since it is impossible to study everything, but check out this discussion on David Armitage.

Baudet’s 2012 thesis,The significance of borders. Why representative government and the rule of law requires nation states“, is a thesis in law, supervised by law professor Paul Cliteur and philosopher Roger Scruton. Thus it is not a thesis in history, though the thesis refers to historical events.

PM 1. The other members of the thesis commission are in law too, except for Alfred van Staden who is a political scientist and professor in international relations. Would he vouch for these aspects in this thesis ? PM 2. The meaning of a thesis is that it is one way of showing that you are qualified to do scientific reseach in that particular field. It doesn’t necessarily mean that you fully proved a particular argument. PM 3. An objective of a thesis is that the new doctor learns modesty about what can actually be proven. PM 4. Cliteur states on his website that he looks at issues of free speech, see also his lecture. I informed him about the censorship of science since 1990 by the directorate of CPB, and he doesn’t show an interest. Apparently Cliteur doesn’t see that it is a no-brainer to say that religious fundamentalism and terrorists who abuse religion present a problem to free speech. Those groups enjoy that he pays attention to them because they thrive on attention and it makes them more important than they are. In the mean time, Cliteur doesn’t defend the freedom of scientific thought right on his doorsteps, while it would be important for a free society that such defence is provided.

I am still looking for a review of Baudet’s thesis by an independent reader.

Potentially the mentioned short Vox Europ article has the same theme as the thesis. The scheme of that short article is that imperialism causes wars, that nationalism is opposite to imperialism, and (thus) that nationalism would support peace. Also Baudet classifies the EU as imperialist. Whether these definitions and statements are supported by scientists working in this field remains to be seen. I am more inclined to interprete developments in terms of political economy, and I haven’t read a key (convincing) statement by Baudet yet why his approach from law should generate key conclusions.

For example, Robert Mundell’s theory of the optimal currency area starts from economics and then provides some historical data that confirm the point. It is open for falsification from history. Baudet seems to turn this around, and starts with historical cases like Napoleon or the USSR and transfers insights to the present EU. This complicates the issue very much, since it suggests that we all must be historians of Napoleon and the USSR before we might discuss the EU. Instead, I prefer a background in political economy, and look at the EU and its future, while I am open for falsifications by historians who suggest parallels in their area of study.

For example, Deirdre McCloskey in her work as economic historian started out from economic theory and the philosophies of ethics and liberty before she discovered the key role of Holland around 1650 in the transformation from the Middle Ages towards the modern world economy. I think that McCloskey is a fine economist and historian, and her discovery of the key role of virtue ethics in this historical process is very convincing – i.e. the change of the social view of the merchant as a robber towards that of admiration and high social status, with the whole social infrastructure of bourgeois society supporting that change of perception. A good historian is always aware that one should not read modern ideas into the past. However, scientific laws are the same over time, and economic processes work the same too.

Incidently, Hubert Smeets, a journalist who has been reporting about Russia and the former USSR over many years, suggested in NRC-Handelsblad last weeks, that Baudet, Kelder & Wellens (from the inititative) would have compared the EU to the former USSR. This is a strong accusation, since the USSR was a totalitarian state. Wellens asked the NRC Ombudsman for a correction. The Ombudsman Sjoerd de Jong gave a fallacious reply. This is my deconstruction (in Dutch) of this affair. Conclusion: Smeets made a false accusation. Baudet’s comparison concerns imperialism which is a different issue, and what Baudet wrote by himself doesn’t have to be supported by Kelder & Wellens. The Dutch Ombudsmen do not work well, see my letter of 2013 to the international organisation of Ombudsmen.

Comparison with Hans van Mierlo and D66 who are pro-EU

In 1966, master of law and journalist Hans van Mierlo (1931-2010) founded the political party D66 (“Democrats ’66”). The “crown jewels” of D66 are: (1) a change from PR to DR, (2) direct elections of mayors and prime minister, and (3) referenda. Thus:

  • Baudet cannot join D66 or their scientific bureau (named after Van Mierlo who didn’t do science) since they are pro-EU and he is anti-EU. But he would be forced to collaborate with D66 a lot because of the shared views on the “crown jewels” (except perhaps DR ?).
  • Scientific analysis of democracy shows that these D66 “crown jewels” actually are less democratic. See my book “Voting theory for democracy“.
  • As far as I know, Hans van Mierlo never studied democracy and its electoral systems. Van Mierlo only was in love with the USA of JFK, and in Holland in the 1960s these ideas sounded new.
  • As far as I know, nobody else in D66 studies democracy. See how they disinform the UK.
  • As far as I know, Baudet never studied democracy and its electoral systems either. I am not aware of a clarification by him why D66 never succeeded w.r.t. its crown jewels. Apparently, Baudet only buys uncritically into the propaganda by D66 as if referenda and direct elections would be more democratic. Curiously, Baudet’s 2012 thesis,The significance of borders. Why representative government and the rule of law requires nation states“, discusses representative democracy and not “democracy” by plebiscite.
  • The Brexit referendum is rather disastrous from the scientific view on democracy, but it requires some study – see here – to cut through the dogma that a referendum is pure democracy by definition.
Legalistic / Popular Scientific
Pro EU and euro Van Mierlo, D66: crown jewels
Anti EU and euro Baudet, FvD: referenda, direct elections, vague on DR vs PR
Pragmatic on EU and euro Me, SvHG: anti-crown jewels

When Van Mierlo deceased in 2010, I honoured him with the pamphlet “Laat D66 zichzelf opheffen” (Let D66 abolish itself). About the dead nothing but good, and the pamphlet was intended as an antidote for his sectarian followers in D66 who might turn him into a saint and martyr of democracy. Observe that I signed this pamphlet under my personal and not scientific name, since it is a personal political opinion that a political party better abolishes itself.

Pamphlet 2010: Let D66 abolish itself

Pamphlet 2010: Let D66 abolish itself

PM. There is also the Dutch LibDem Party (LDP), founded in 2006 by Sammy van Tuyll. They are social liberal like D66, like my suggestion from 1993 of a Social Liberal Forum (SLF). Van Tuyll has a background in medicine, economics and law, and should be able to understand my economic analysis. It is not clear to me why he doesn’t study and discuss it. Van Tuyll and I met in 2007 and I explained about the censorship of science, and it didn’t ring a bell. I can only suppose that when Van Tuyll ever is elected into government then he will continue with the censorship of science by the Dutch government.

Meeting Baudet in 2010

I met Thierry Baudet at a book presentation in 2010, when he was co-editor with Michiel Visser of a collection of essays on conservatism. My comment at the book presentation was that a good starting point would be the natural conservatism in classical liberalism as formulated by J.S. Mill and J.M. Keynes. Of course my background is in economics. The book title suggests the conundrum that conservatism actually is progressive, but the content of the book did not clearly resolve this conundrum. Overall I thought that the book was useful, but did not feel that I should buy the second volume.

I gave Baudet a copy of the book by Hans Hulst & Auke Hulst in collaboration with me (1998) Werkloosheid en armoede, de oplossing die werkt” (W&A) (Unemployment and poverty, the solution that works). In response, Baudet gave me his business card, whence I sent him a note on the next day, April 13 2010, to confirm contact. The card and this link show that Baudet was already active in improving democracy.

Baudet's business card of 2010, referring to Dutch Parliament with 150 representatives

Baudet’s business card of 2010, referring to Dutch Parliament with 150 representatives

My presumption was that Baudet would read W&A, and that there would be a discussion proceeding from there. In some interviews Baudet is portrayed with stacks of books in the background so there is the suggestion that he might read books. However, while I read the book that he and Visser edited, I did not get a reply on W&A and neither on my suggestion to have a further discussion. One possibility is that he was too busy with his 2012 thesis (though W&A is relevant for that topic too). But after completion of the thesis, there still is no sign of interest.

There is my warning from January 2012 to various young Dutch intellectuals who might come across as “Young Turks“, including Baudet, that they should not forget about the need for a solid scientific approach to change of society. I knew that Baudet was a PhD student but not that he would present his thesis in June that year. Perhaps Baudet thought this warning superfluous since he was working on that thesis at that time. Perhaps it is okay to put on blinders for a thesis when finishing it. The very purpose of a thesis however is to teach you the scientific attitude that one should not neglect criticism.

In 2012 I highlighted the issue that now surfaces in the petition again, namely the link between the EU and euro crises to the censorship of science by the directorate of the CPB.

If Baudet and his FvD are so much interested in improving democracy, why are they not interested in my analysis of the failure of Trias Politica, and the need for an extension with a constitutional Economic Supreme Court ? Why doesn’t Baudet write a review of “De ontketende Kiezer” (2003) ? Why this island mentality and burking and elbowing out of views of others ?

Baudet doesn’t inform Kelder & Wellens at

Baudet in 2015 collaborated with master of law and journalist Jort Kelder and management accountant Arno Wellens on the petition that wants an enquiry by Parliament about the creation and future of the euro. See my discussion of in the former weblog entry.

Kelder & Wellens confirm to me that Baudet did not inform them about W&A and this warning of mine of 2012 to the “Young Turks”. If they want Parliament to provide “full information”, then I would hope that they themselves acknowledge that they had a glitch in their own information amongst themselves. They disinformed the 40,000+ people who signed their petition.

Because of Baudet’s neglect since 2010 of key information about economics and censorship of science, there now is this initiative that focuses only on the euro, while the relevant enquiry should be about unemployment, role CPB … and euro. The euro is only a symptom, and an addition to what went wrong already before.


Jort Kelder, Arno Wellens and Thierry Baudet, screenshot 2015-12-14

Council of Recommendation

The format of a Think Tank for Baudet’s FvD allows academics to join up in a council of recommendation, too, which some might find problematic if it were a political party.

Member of the FvD council of recommendation are professors in constitutional law Jos Teunissen and Twan Tak. They should understand my approach that there should be no taxation on minimum earnings. See the short text “Don’t tax sweat“.  Teunissen has this useful text “Vrijheid, gelijkheid en belastingen” (2010) on couples, but it is better to start with individuals, and then see DRGTPE p131-132 on couples. Constitutional lawyers should also understand the failure of the Trias Politica model of democracy and the need for an Economic Supreme Court (per nation).

Seeing the names of Teunissen and Tak causes the hope that they will be able to explain these things to the other members of the council, and that all agree that FvD can be abolished as it has been based upon a wrong analysis, neglect by Baudet and disinformation since 2010.

Here we find Baudet’s thesis advisors Paul Cliteur and Roger Scruton again. Obviously the thesis differs from the mission of FvD and it is a bit remarkable that the supervisors travel along, though the direction of travelling might also have been the other way around (from Euroskeptism towards thesis).

To my surprise I also see: Deirdre McCloskey ! After some search, though, we see that Baudet explains in his cv that he taught “between 2010 and 2011” at Arjo Klamer’s school “Academia Vitae” (though it filed for bankruptcy in February 2010), when Jos de Beus (1952-2013) got ill. McCloskey may have taught at this school too. Arjo Klamer was close to De Beus and gave an impressive presentation at the memorial meeting – see my comments on this. It is important to know that Jos de Beus did not understand Kenneth Arrow’s impossibility theorem for collective decision making. It is important to know that there is a line in economic theory from Jan Tinbergen to his PhD student Hans van den Doel to me, with a floundering branch to political theorist Jos de Beus, who collaborated with Van den Doel. Jos de Beus and I met when I presented Van den Doel with the Samuel van Houten Penning in 1994. We had occasional contact but to no effect.

As an economist, Arjo Klamer could help out by studying my work, but he doesn’t. Klamer however is also in the council of recommendation of FvD. For some reason, economists Klamer and McCloskey prefer Baudet’s non-economic approach in theory of law above my development in economic theory from Jan Tinbergen and Hans van den Doel. If only they studied my analysis and stated why they disagree, but now the world must wonder why they don’t look at it at all. And why would they not understand that they cannot see the full analysis yet, because of the censorship ? Ergo, that this censorship must be lifted ?

A member of the FvD council of recommendation is philosopher Ad Verbrugge. He is founding chairman of “Beter Onderwijs Nederland” (BON) (for “Better Education”). At the website of BON, some mathematicians are slandering about my work on mathematics education. Verbrugge doesn’t do anything about this. There is this letter of 2009 (my website has moved to I have rephrased some questions again this Summer for fellow math teacher Karin den Heijer, now board member of BON, see page 11 here.

The link to mathematics education is important. See my letter to the president of KNAW and directorate of CPB 2016, that explains that maltreatment of my work on mathematics education hinders other people to also see the value of my work in economics.

Member of this council of recommendation is Kees de Lange, emeritus professor in physics and former chair of an association on pensions NPB. De Lange might have looked at my suggestions on mathematics education, see my suggestion on what physicists might do. I am not impressed by De Lange’s understanding of economics. I am not aware of someone in the Dutch world of pensions who warned about the 2007+ crisis. In 2009 I contacted De Lange as chairman of NBP and informed him about the censorship of science since 1990 by the directorate of CPB. His reply was sympathetic to my feelings, as if that were a relevant issue, and that NBP did not look into economic analyses, and that my approach might only be discussed when shared by more economists (but they didn’t look at analyses anyway). I came away from this with the impression that De Lange was lost, both as a scientist and chairman of NBP. Later in 2010 De Lange helped found a political party 50Plus, he was elected in the Dutch Senate as member of a two-man fraction of OSF 2011-2015, but then continued independently.

PM. At this spot it is useful to mention that Baudet, Wellens and De Lange also perform in video channel “Cafe Weltschmerz“, created by (bachelor in business and marketing) journalist Willem Middelkoop (LinkedIn), who after the 2007+ crisis got rich by telling people to get into gold rather than have a parliamentarian enquiry into unemployment and censorship since 1990 by the directorate of CPB. One of Middelkoop’s books was published by Amsterdam University Press and by standard arrangement adopted by the University of Chicago Press, but it should have been accepted at neither place since there is no link to science. See my discussion of the gold bugs. One supposes that Middelkoop likes it when Baudet, Wellens and De Lange continue to create uncertainty amongst viewers, so that the market for gold as a “safe haven” remains strong. It is a pity, though, that this circus also draws in young people looking for answers, like psychiatrist Esther van Fenema (wiki) and mathematician Anna Grebenchtchikova (LinkedIn) and lawyer Hester Bais. They, with their higher education that should guard them, might be falling in the journalistic trap to look at symptoms rather than causes.

Member of the council of recommendation is Tom Zwart, professor of international and European law, since 2007 director of the Dutch School of Human Rights Research. Perhaps freedom of expression is also a human right of a scientist ? Or is the option to do science no human right ?

Member of the council of recommendation are other economists Edin Mujagic, Bruno de Haas and Daniel Lacalle. Let me invite them to study my work, starting with DRGTPE (before the crisis) and CSBH (after the crisis). Mujagic hasn’t responded yet, though my analysis dates from the fall of the Berlin Wall, that also affected his past. Lacalle is a hedgefund manager and could get very rich if he would start supporting my analysis (supporting the boycott of Holland, explaining to all that it is needed, and speculating on it).

Last but not least there is Theodore Dalrymple, who might be very happy to finally understand why the Dutch welfare state isn’t working as it is supposed to.

Thierry Baudet and Paul Scheffer

At “Cafe Weltschmerz” there is also this (tedious) interview of Paul Scheffer (1954, like me, Angela Merkel and Franςois Hollande) by Baudet on the Dutch referendum on the treaty of the EU with the Ukraine. Scheffer states that he would vote Yes for the treaty. Baudet participated in setting up the referendum, with the objective that people would vote No. It is fine that they can have this civilised talk, though it was so tedious that I quit watching after 10 minutes (though the referendum has already taken place).

Baudet was for one year a post-doc in 2013 with Paul Scheffer who has a chair in European studies in Tilburg. Originally, Scheffer first wrote a popular book on migration and the multicultural society, and then turned this into a thesis for Tilburg. The Leiden professor of social history Leo Lucassen stepped down from the promotion committee in protest that not enough had been done to make it a real thesis.

Scheffer did highschool HBS A, and graduated in political science in 1986. In his student years he joined the Dutch communist party, and later switched to the social democratic PvdA. He was at the Wiardi Beckman Stichting (WBS), the “scientific bureau” of PvdA in 1986-1992.

I was a member of PvdA in 1974-1991. When I was at CPB in 1982-1991 I developed my analysis on unemployment, with the conceptual breakthrough when the fall of the Berlin Wall in 1989 caused me to look at some fundamentals. My analysis was censored by the directorate. I sent a copy of my 1990 paper to Parliament, so that all parties were informed, and I was free to contact PvdA of which I was a member. I contacted WBS, and I assumed that fellow social democratic scientists would be interested in an analysis on unemployment. To my great surprise and dismay, they were not. See the letter reproduced in “De ontketende kiezer” (2003) p128. See my discussion “Soms loopt het zo” in “Trias Politica & Centraal Planbureau” (1994). My contact was with fellow econometrician Paul de Beer. I met Scheffer at a PvdA convention at that time so he was in the know. I met Scheffer again at the memorial service of Jos de Beus. I later discovered that Paul de Beer was an adherent of the idea of a basic income. See my discussion about the sectarian behaviour around basic income.

Director of WBS in 1989-2006 was Paul Kalma. I had had some contacts with earlier director Joop van den Berg (1981-1989), now fellow at the Dutch Montesquieu institute. The idea that there are drawbacks to the Trias Politica structure hasn’t arrived there yet.

When Holland succeeds in having this parliamentarian enquiry on unemployment and the role of the CPB … and the euro … then these events at WBS would be important to look into as well. As said at the beginning, the road from science and scientific discovery into political discussion is often via the channel of a particular party. Politicians of any party are less likely to discuss an idea when there is no party advocating it anyway. Thus it is very relevant to know why social democratic researchers at WBS were and still are not interested in a new approach to unemployment. I will be interested in hearing what has been happening as well. Obviously, Parliament will be hesitant to ask questions, since WBS is protected by the aura of science and by that parties will not easily look into dealings of other parties. But the notion of “scientific bureau” better be taken seriously, and scientists should be familiar with the idea of answering questions. Perhaps Thierry Baudet can already ask Paul Scheffer what his recollections are, and why Scheffer didn’t and still doesn’t do anything about the censorship when he heard about it.

The three Pauls (De Beer, Kalma, Scheffer), in 1991 at WBS (wikimedia commons and website De Beer)

The three Pauls: De Beer, Kalma, Scheffer, who were in 1990-1991 at WBS (wikimedia commons and website De Beer)