Column 1:
Discussion
(full PDF) by Jan van Rongen
("MrOoijer"), March 3 2012
The following cuts up the
PDF in passages.
(In Dutch, Jan van Rongen
has written this
text,
another accusation of bad
mathematics, and there his links to this PDF.)

Column 2:
Thomas Colignatus, March
6 2012
My work in mathematics is
not in axiomatics and not in pure math. Axiomatics is basically a formal
exercise but I work in the "intended interpretation". When mathematicians
complain that what I do does not fit in the current axioms then my answer
is that this may indicate that the axioms require change. This is how it
works.
There are many mathematicians
who do not understand this. Jan van Rongen appears to be one of those.
His pronouncement that my work is "pseudo mathematics" is actually slanderous.
It should be sufficient for him to conclude that I don't work in the standard
axiomatic framework. Instead, he goes over the top and implies that I would
not understand standard math, while I do. It is precisely because I don't
like the consequences of standard math that I look for alternatives. Van
Rongen suggests that I am a quack and that I mislead people, while I take
pains in explaining what I do and clarifying the limitations.
Again, this explanation:
I have university degrees in econometrics and teaching of math. This is
not an argument of authority but an introduction and a reply to Van Rongen's
introduction of himself. I use math in econometric applications. In teaching
math I concentrate on the didactics. As a student I found the logical paradoxes
fascinating, and the standard answers provided by mathematicians rather
strange, and was inspired to write "A Logic of
Exceptions". My book on the didactics of math "Conquest
of the Plane" discusses the derivative, and this caused me to look
at the notion of a limit again and at the definition of the real numbers.
This resulted in the paper "Contra
Cantor Pro Occam". It is OK when Jan van Rongen doesn't buy all this
but it would be nice to see whether some young mathematicians might fit
these into a new axiomatic framework.
I offered Van Rongen my comments
below and suggested that he corrected some points, but he declined and
put his memo on his website. 
Column 3:
Wrong representation (W)
or slanderous (S) 



ALOE and COTP have also been
reviewed by Richard Gill (2008,
2012).
The latter review caused these Reading
Notes. There are also reviews of the books by the EMS (2011,
2012).

The "review" of COTP by Spandaw
in Euclides [2012] is slanderous, and it is curious that Van Rongen does
not refer to my answer, both in general
and in detail.

"that the reals are countable"
may be too short as a summary. I provide a notion "bijection by abstraction"
such that N ~ R. If one uses "countable" to be only a potential infinity
then this is not quite the "countable" of actual infinity. The notion of
"bijection by abstraction" however allows us to regard these two notions
of infinity as two sides of the same coin.

ALOE explains that already ancient
writers looked at the option of a third value. Van Rongen's choice of words
suggests that I would seem to think that I present something wholly new
? My suggestion is only that there is a new implementation.

"emphasis on the errors" ? A
curious aim. The news is not relevant, or all news is error ?

Incomplete and tendentious 
1 


ALOE explicitly explains that
it does not target at a formal treatment. There is only a chapter for suggestions
for formalisation.

"pseudo mathematical mess":
Apparently Van Rongen is an expert in mathematical logic, has done his
best to turn the new findings into an axiomatic system and has proven that
this cannot be done ? But OK, this is only an introduction.

W & S 
2 


It is not full of half truths
and errors.

Reference to my critique on
Spandaw would help. There are also the reviews by Gill and EMS.

W & S 
3 


When contacting Van Rongen my
hope was that he would look at COTP and the slanderous "review" of Spandaw.
My discussion in CCPO is more experimental and trickier to discuss since
I have done hardly any work in axiomatics, formal math and number theory.
Van Rongen however seemed to agree with the slander by Spandaw and then
zoom in on CCPO.

My reason for CCPO is targetted
at finding a consistent story to tell in highschool without needing to
discuss the transfinites, see Neoclassical
mathematics for the Schools. I hope that it also deters math students
to spoil their lifes on the paradoxes of infinity (yet they are free to
do so, of course).

CCPO has been on my website
continuously since August 2007, updated once and while.

W & S 
4 


If Spandaw is referred to above,
then Gill might have been mentioned above too.

Gill's review is not "slightly
favourable" but a warm recommendation.

The differences for ALOE 2007
and 2011 are (a) typing errors, (b) now also N ~ R

"That can be the problem with
selfpublished material": The editions have new ISBNs, like with any publisher.

The same correct reasoning can
be found in [3].

W & S 
5 


This is page 129, see ALOE.

It is true that at that point
threevalued logic has not been introduced yet. But later it is explained
that twovalued logic also implies threevalued logic, when constructs
like the Russell set format are excluded (which is only a formal argument
to consider them nonsensical).

The translation of "and" into
that intersection is correct but it does not imply that "every subset"
is possible.

It is only that (2), (3) and
(4) allow infinite regress for values of y other than the set itself. Therefor
page 129 clarifies that the true form uses a switch between y = Z* and
y =/= Z* and that (2) is only a shorthand reminder.

4W 
6 


The switch is between y = Z*
and y =/= Z*. The first " y element of Z* " is incorrect and must
be " y =/= Z*".

This is on page 129, already
in 2007. What happened in March 2012 is that a reader of CCPO complained
that CCPO used the shorthand notation that allows infinite regress,
so that CCPO now includes the full form.

2W & S 
7 


This is only an inconsequential
gimmick in COTP and I am surprised that serious mathematicians like Spandaw
and Van Rongen spend so much attention on it.

The idea is to drive home that
the derivative of exp[x] is exp[x] itself again. Since COTP uses another
approach to the derivative we need another introduction and this might
work. If it doesn't, fine with me.

A common story to introduce
the fixed point notion is about a mountain walk, where you climb uphill
on Saturday 1214 hours, and go downhill on Sunday 1214 hours. At one
point in time you will also be at the same height. For this story the start
and finish are the same, making the problem indeed trivial. But, it is
a common story, and the graph allows us to tell it.

The reference to Spandaw is
incomplete. There is also my response to him. My discussion with Richard
Gill resulted in the Reading Notes, referred to above.

"high "educational" value" is
slanderous, since I have not claimed that it has a "high value". I have
only suggested that it is a nice way and that it might work.

W & S 
8 


The notation is from Mathematica,
a system for doing mathematics on the computer, distributed by Wolfram.com.

I exclude the origin so that
the endpoint still is included and an obvious fixed point, and use "at
least".

"that he does not mean the endpoint"
is a curious phrase. It is so slanderous.

"he gave no further explanation"
is false, see my reponse to Spandaw and the Reading Notes.

2W and S 
9 


As a student I have done exams
in Takayama "Mathematical economics" that uses the fixed point approach
to general equilibrium. I know the conditions of Brouwer's and Kakutani's
theorems, and have discussed these with Richard Gill for his review.

The inference "It is clear that
he does not understand Brouwer's theorem to the full" is slanderous. This
cannot be inferred from a didactic experiment.

W & S 
10 


Read COTP to see the new algebraic
approach to the derivative.

It simplifies the discussion
when you can make it intuitively acceptable that there is some f such that
f ' = f again. And this we call exp[x].

W 
11 


The series Ai shown is only
"abstraction" and not "bijection by abstraction". You have to distinguish
those two notions.

What he calls an "example" is
the key definition of the natural numbers. The creation of the natural
numbers requires abstraction. Unless you are able to hold an infinity of
numbers in you mind, all individually ...

2W 
12 


CCPO of March 6 is adapted a
bit. This is no book with an ISBN but work in progress.


13 


This is a selective representation.
A section below that, I explain that the "map" is not constructive but
an "bijection by abstraction", so that there is a difference with concepts
in the Cantorian school.

It is true that I understand
the Cantorian definition and that I don't like it. "Classical" is Aristotle,
and I propose a "neoclassical" approach that skips Cantor.

I have made that comment about
calculus to clarify that I have no particular interest in Cantor and number
theory or axiomatics in this realm of discussion, but this statement should
not be interpreted as if I don't accept standards of rigour for these domains.
My hope is that I can express my misgivings in a sufficiently clear manner
so that the specialists in these areas can design new axiomatic systems.

W & S 
14 


The version of CCPO in 2011
already discussed Cantor's original proof of 1874. I appreciate Van Rongen's
effort to help by working out his version. I don't see an essential difference
and it is curious why he didn't understand my earlier objection.

I had already formulated the
objection: that it assumes actual infinite R but then proceeds in potential
infinity. The point is that R is only constructed in the abstraction from
potential to actual infinity.

Van Rongen's blindness at least
had the result that I now included an Appendix where this is explained
in smaller steps, see the version of March 6. (If he had been open to the
argument, he could have done so himself, perhaps in a way that pure mathematicians
find more agreeable.)

It is a wrong representation
of my position that I would only "feel" that it cannot be right. There
is a well developed argument, with the notions of "abstraction" and "bijection
by abstraction".

W & S 
15 


This seems like a noble attitude,
but is selfserving. It is slanderous to present my work as pseudomathematics.

"presents himself as an expert
in the field" is unqualified. I am very careful in my explanation about
what I do (and this leads to complaints that these texts are too long).

One doesn't ignore a person
but gives counterarguments on content.

W & S 
16 


Richard Gill wrote favourable
reviews of ALOE and COTP not because of a "few" interesting ideas. Richard
Gill hasn't written a review on CCPO.

What Van Rongen calls "errors"
in ALOE and COTP in the above, are not "errors", so it should not surprise
that Gill can write favourable reviews.

Spandaw's "review" is slanderous
and it would have been better when Van Rongen would have corrected that
instead of providing a supposedly good sounding argument that having some
critique is more important than whether it is sound.

W & S 
17 


Thomas Cool / Thomas Colignatus
(b. 1954) graduated in econometrics in Groningen in 1982. If his professor
in logic had had an open mind then this would have been with an honours
degree. He graduated in 2008 as teacher of mathematics in Leiden. He advises
to a boycott of Holland
because of the censorship of science by the directorate of the Dutch Central
Planning Bureau.


18 


These publications of mine are
under the name Colignatus, that I use for my scientific work.

My response against the "book
review" by Spandaw can be mentioned.

Richard Gill has also a review
of COTP.

We might think that some limitation
is OK since it is just a note, but Van Rongen put the note on his website
and then one would tend to require that the sources aren't selective. PM.
In that Dutch note, he suggests that quacks are attracted to Cantor's diagonal.
This is a bit curious, as Cantor was attracted to it himself. (Van Rongen
could have mentioned Brouwer in this context too, another quack who rejected
Cantor.)

W & S 
19 

Note: I sometimes write
for another weblog. There
I presented my advice for a parliamentary
enquiry into the education in mathematics, relating to COTP. There
Van Rongen didn't discuss the arguments but went off on the other angle
of my analysis on CCPO. My suggestion was to discuss both via email rather
than via the weblog reaction windows. It is fine that he agreed to that
exchange but it is a pity that this opportunity didn't result into something
positive. People get nervous when there is a heckler. Aggressively blind
mathematicians might cause that people become more nervous about that suggestion
of a parliamentary enquiry. 

20 