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.)

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.

Wrong representation (W) or slanderous (S)

1. 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).
2. 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.
3. "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.
4. 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.
5. "emphasis on the errors" ? A curious aim. The news is not relevant, or all news is error ?

1. ALOE explicitly explains that it does not target at a formal treatment. There is only a chapter for suggestions for formalisation.
2. "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.

1. It is not full of half truths and errors.
2. Reference to my critique on Spandaw would help. There are also the reviews by Gill and EMS.

1. 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.
2. 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).
3. CCPO has been on my website continuously since August 2007, updated once and while.

1. If Spandaw is referred to above, then Gill might have been mentioned above too.
2. Gill's review is not "slightly favourable" but a warm recommendation. 
3. The differences for ALOE 2007 and 2011 are (a) typing errors, (b) now also N ~ R
4. "That can be the problem with selfpublished material": The editions have new ISBNs, like with any publisher.
5. The same correct reasoning can be found in [3].

1. This is page 129, see ALOE.
2. It is true that at that point three-valued logic has not been introduced yet. But later it is explained that two-valued logic also implies three-valued logic, when constructs like the Russell set format are excluded (which is only a formal argument to consider them nonsensical).
3. The translation of "and" into that intersection is correct but it does not imply that "every subset" is possible. 
4. 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.

1. The switch is between y = Z* and y  =/= Z*. The first " y element of Z* " is incorrect and must be " y =/= Z*".
2. 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.

1. 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. 
2. 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. 
3. A common story to introduce the fixed point notion is about a mountain walk, where you climb uphill on Saturday 12-14 hours, and go downhill on Sunday 12-14 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.
4. 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.
5. "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.

1. The notation is from Mathematica, a system for doing mathematics on the computer, distributed by Wolfram.com.
2. I exclude the origin so that the endpoint still is included and an obvious fixed point, and use "at least".
3. "that he does not mean the endpoint" is a curious phrase. It is so slanderous. 
4. "he gave no further explanation" is false, see my reponse to Spandaw and the Reading Notes.

1. 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. 
2. 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.

1. Read COTP to see the new algebraic approach to the derivative.
2. 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].

1. The series Ai shown is only "abstraction" and not "bijection by abstraction". You have to distinguish those two notions.
2. 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 ...

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

1. 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.
2. 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.
3. 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.

1. 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.
2. 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.
3. 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.)
4. 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".

1. This seems like a noble attitude, but is self-serving. It is slanderous to present my work as pseudo-mathematics. 
2. "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).
3. One doesn't ignore a person but gives counter-arguments on content.

1. 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.
2. 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.
3. 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.

1. 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.

1. These publications of mine are under the name Colignatus, that I use for my scientific work.
2. My response against the "book review" by Spandaw can be mentioned.
3. Richard Gill has also a review of COTP.
4. 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.)