Thomas Colignatus (TC) (ME and MER)
2016-05-06 & 09

My books ALOE and EWS and COTP and FMNAI apply mathematics. I am an econometrician and teacher of mathematics. I don't aspire to be a mathematician. These books aren't written for mathematicians but for the area of application, with students in mind. See my explanation that I do applied mathematics.

It is obvious that one can say about my books "they are not mathematics".

When a mathematician says "they are not mathematics" then this becomes a tricky situation however.

Beware of the potential abuse of language and the authority of mathematics.

• A mathematician better aspires at accuray, which is their profession, and says "these books haven't been written by a mathematician for mathematicians".

• When people hear a mathematician stating that something "isn't mathematics", then they might think that it is rubbish. Perhaps such mathematician intends people to think so indeed ? But then there is an abuse of language and authority. How can a mathematician judge about a book that doesn't fit his or her profession ? For a judgement, the mathematician would need to study the relevant field. This also holds for the empirical science of "mathematics education research" (MER). One would give an argument on content. Stating ex cathedra "it isn't mathematics" is an abuse of language and authority.

Jan Bergstra (JB) (UvA, homepage, research mathematician (RM), no mathematics education researcher (MER), secretary of the math section at KNAW) states about this paper on a "sky": "It isn't mathematics." He abuses both language and his position of authority.

This paper was an effort on my part to reduce potential misunderstandings by mathematicians (e.g. Bergstra) w.r.t. the algebraic approach to the derivative in ALOE (2007), EWS (2009) and COTP (2011).

My idea is to bridge the gap between my field of ME and MER and the field of RM. When the latter would understand more about the algebraic approach, they might write their mathematical papers for each other, and this might cause support by them for this approach. However, Bergstra states that he cannot use that bridge since "it isn't mathematics". He states even that he doesn't "understand" it. For him, bridges to RM must already be RM. He insists on doing a particular substitution while from the definition it is clear that this cannot be done. (A similar confusion can be seen in complex numbers, see here.)

My original contact with Bergstra was on the algebraic approach to the derivative. One possibility was that it might be related to his own approach on resolving "division by zero", namely a "meadow" in group theory. It later appeared that he used the phrase for error handling and not for finding a solution for "division by zero". Bergstra's approach doesn't assume simplification (or computer algebra). His approach might be relevant e.g. for statistics on databases (or when such simplification causes error).

I came in contact with Bergstra via mutual friends. Holland is a small country. I have no doubt that Bergstra started out on good intentions. For both of us it will hold: no good deed goes unpunished.

• It was strange to observe that Bergstra apparently did not understand the algebraic approach. It was strange to see that he accepted "sloppiness" in the current highschool treatment of derivatives but did not accept my treatment that is clearer for highschool.

• To my question "how would you formulate it then ?" his response was "I can only formulate it when I understand it". A nice phrase is also: "What one cannot understand (your own drafts), it is probably false." He should know however: (a) the derivative, (b) the treatment in highschool, (c) criticism w.r.t. the latter, and from (a) - (c) he should see how the algebraic approach improves on (b) and (c).

• When I ask whether he could refer to someone else with a background in both mathematics and the empirics of MER, like a trainer of mathematics teachers, he answers that he cannot do so, stating that "everyone who he knows" would insist that "he himself would resolve all unclarities first". Such a person however would understand that MER is not the field of Bergstra, and Bergstra's claim is false unless he really makes himself believe that he doesn't know such people. Bergstra is secretary of the math section at KNAW, and has the option for an international search for such a researcher. Could he really not invite some trainers of mathematics teachers to look into this ? The rejection is a fallacy and disingenuous.

• Bergstra should not have abused language and authority. He should have understood my protest on this. In my experience he has become a "hostile witness" and has breached integrity of science.

• My best response was to answer to each detail of misunderstanding and (deliberate) misrepresentation. The collection of these points resulted in above paper on the Sky. One might say that the paper is a positive result of this exchange. However, the paper is to a large extent a restating of the obvious. The curious phenomenon is that it took me quite some effort and that I still think that it is rather obvious, and that Bergstra's attitude is that it would not be obvious for mathematicians (and perhaps only because they make it their profession to question everything because they don't have to teach in the schools).

Subsequently, in the context of the KNAW conference of 2014 on arithmetic (theory and (elementary) school education and assessment of arithmetic ("rekentoets")), it dawned on me only then (though apparently it was on his website) that Bergstra was also secretary of the KNAW math section, and organiser of the event. There is this collective breach in integrity of science.

A letter to NWO / NRO / PROO of April 2016 makes it necessary for me to document this TC - JB exchange. For researchers in (mathematics) education and historians and philosophers of science, it appears useful to document my contact with Bergstra in 2012-2014. Information has been anonimised where needed.

• The evaluation document of 2016-05-06 is an effort to evaluate the situation, and it summarizes the events from 2012 to April 2016 (though still incomplete).

• The email collection of 2012 w.r.t. COTP and an effort of me to understand Bergstra's work on the "meadow" (see group theory and field - Dutch "lichaam").

• The email collection of 2014 w.r.t. the papers on sky and Van Hiele and Tall (in 2015 more extended on Freudenthal).

• The email collection of 2014 w.r.t. KNAW conference on arithmetic and on KNAW / LOWI. This involves more people. Obviously one is grateful for the effort shown by more people, but it is a breach of integrity when criticism isn't responded to. To have an appearance of talk but turn a deaf ear, doesn't work in science. The KNAW 2009 report showed that "realistic mathematics education" (RME) hadn't been based upon proper research. A scientist concludes: it is an ideology, comparable to astrology or homeopathy. For that reason I think that it is necessary to debunk the claimed authority of the "Freudenthal Institute", and call it by a proper name, like "Freudenthal Head in the Clouds Realistic Mathematics Institute" (FHCRMI). Bergstra objects to this and calls it disrespectful. He allows that Spandaw misrepresents and slanders my work and person, but he objects to protesting against and debunking of FHCRMI. Let me warn that Bergstra is no empirical scientist, and that he hasn't studied mathematics education and its research. His argument about "respect" is confused. Of course one treats people with respect, but mentioning and clarifying that ideology is no science cannot be classified as disrespect.

Let me label points and sentences.

1. (a) Bergstra isn't aware or doesn't seem to care when a halftruth "this isn't mathematics" causes people to neglect my work, or perhaps he even aspires that objective (because "it isn't mathematics"). (b) Bergstra will also say that he only gives his own modest opinion, and he only happens to be professor and secretary of the math section at KNAW. (c) Such abuse of language and authority is a breach in integrity of science.

2. (d) Bergstra in his position should acknowledge that my books apply mathematics, he should acknowledge that there is criticism on present views, welcome discussion on this, insist on a discussion on content, resist abuse and ad hominems by Spandaw and others, and, if he takes a position himself, then give a public statement in sufficient detail so that possible critique is clear, instead of burking as he does. (e) That is, Bergstra has gone at length in an email discussion on above paper on "meadow" and "sky", which one may appreciate, but I am not aware of a public statement that I can refer to. (f) Potentially, Bergstra does not want to draw attention to the books ("because they aren't mathematics").

3. (g) Bergstra also employs the phrase "I do not understand what you mean (or your formula)", which is of course a strong indication that what is said isn't mathematics, as this would be clear. (h) Apart from common instances in communication when this is fair use, Bergstra however also applies that phrase when it is clear that there is obstruction on his part. (i) Common communication has moments such that one might say "if you formulate it such, then I agree" (and then don't give other content). (j) He takes the attitude as if it would be my problem to inform or convince him, or as if he can indicate that he as a mathematician takes his responsibility to clarify that something isn't mathematics. (k) Potentially this is only the attitude of a professor w.r.t. a Ph.D. student who must be able to overcome critique by other mathematicians. (l) However, my work isn't in this position.

4. (m) W.r.t. the algebraic approach to the derivative, my question was whether he could help with acceptance amongst RM. (n) The current development is proper for education in highschool, but it remains useful to see what RM think. (o) They cannot say that "this isn't mathematics" since the algebraic approach gives the same results for highschool functions as Weierstrasz "that is mathematics". (p) (This is what COTP proves.) (q) If Bergstra objects, let he give a public statement with adequate detail. (r) The approach is available since 2007 !

5. (s) W.r.t. mathematics education, I informed Bergstra and his KNAW section about the mis-state, and it is a breach of scientific integrity that they didn't do anything about it. (t) Their official mission is of course to support RM and research in mathematics, but they should be aware that they may be in a better position than me to inform the proper authorities (like they also organised the conference on arithmetic education).

Advised reading for background information

• What a mathematician might wish to know about my work
• After 45 years of unemployment: If Holland had been just a bit nicer and more competent