No good deed goes unpunished
About the effort to communicate in 20122015+ between TC and JB on the
derivative, COTP, meadow, sky, the work by Pierre van Hiele, the fraud by Hans
Freudenthal, arithmetic, and mathematics education and its research (ME and MER)
and its gap with research mathematics (RM) in general
Thomas Colignatus
(TC) (ME and MER) 20160506 & 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 20122014. Information has been anonimised where needed.

The
evaluation document
of 20160506 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.

(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.

(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").

(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.

(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 !

(s) W.r.t. mathematics education, I informed Bergstra
and his KNAW section about the misstate, 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
