It appears that mathematicians are trained for abstraction but in class they are confronted with real life pupils and students. Their tradition is to resolve their cognitive dissonance by relying on tradition. Mathematics then becomes a tool for authority and stagnation. I rather see it as an exercise in freedom, where nothing can force you to accept a proof but the proof itself.
Education in mathematics
An econometrician dismissed from a Central Planning Bureau can become a teacher of mathematics. The didactics of mathematics became a new problem area and resulted in two books Elegance with Substance (EWS) (2009) and Conquest of the Plane (COTP) (2011c) and this memo Neoclassical mathematics for the schools (2011d).
There are now two open minded reviews that show the strength of mathematical thinking. Gamboa (2011) reviews COTP, admits to feelings of unease, but ends with a positive “enjoy”. Richard Gill (2012) reviews both EWS and COTP. It was special to me that I could find something about the notion of a derivative, as developed by Newton and Leibniz, but it is even more special that Gill spends a longer open minded discussion on it. He is no expert on the didactics of mathematics and he doesn’t discuss other improvements in EWS and COTP, but it is a quality review. He was trained in Cambridge, now professor of mathematical statistics in Leiden and member of the Royal Dutch Academy of Sciences. Earlier Gill (2008) reviewed my book on logic A Logic of Exceptions (ALOE) (1981, 2007, 2011a).
The CPB directorate not only blocked my analysis on unemployment but all my papers, also the one on Kenneth Arrow’s Impossibility Theorem. Nobel Prize winners Kenneth Arrow and Amartya Sen misstate their results, where their interpretation does not cover the mathematics. I had hoped on some support from mathematicians. On the contrary, it appeared that social choice theorists and mathematicians did not understand the issue themselves. I developed my analysis into the book Voting Theory for Democracy (2001, 2011b). In 2001 I had to write this report and note of protest. A recent event causes this short paper, Colignatus (2011e). Our modern democracies are not so democratic while improvement is blocked by its very scholars who are not behaving so scholarly.
We need to distinguish between mathematicians and engineers, where the latter will have more eye for reality. However, “financial engineering” may still lack the code of honour that bridge builders have, see Steinsaltz (2011), who favourably refers to Nicolas Bouleau. A good exception is also financial engineer Paul Embrechts who participated in Danielsson et al. (2001) in a warning on Basel II. On balance however, I maintain (also in Elegance with Substance) that part of the responsibility for the current crisis falls to “mathematicians” as well. Let them work hard towards improvement.
The world crisis has caused some economists to worry about the influence of mathematics on economic theory and practice, see this page by professor Geoffrey Hodgson et alii. According to my analysis they do not worry enough. I am strongly in favour of mathematics, like Leibniz would say “Let us sit down and look at the formulas” (no quote). Only through mathematics we can establish what would be an improvement in education, democracy, finance and economics. But the latter are also empirical sciences and it would be the fallacies of miscomposition and misplaced concreteness to mistake the one for the other.
Thomas Colignatus is the preferred name of Thomas Cool in science.
Colignatus (1981, 2007, 2011a), “A logic of exceptions” (ALOE), see http://thomascool.eu/Papers/ALOE/Index.html
Colignatus (2009), “Elegance with Substance”, http://thomascool.eu/Papers/Math/Index.html
Colignatus (2011b), “Voting Theory for Democracy”, 3rd edition, http://thomascool.eu/Papers/VTFD/Index.html
Colignatus (2011c), “Conquest of the Plane”, http://thomascool.eu/Papers/COTP/Index.html
Colignatus (2011d), “Neoclassical mathematics for the schools”, http://thomascool.eu/Papers/Math/2011-09-06-NeoclassicalMathematics.pdf
Colignatus (2011e), “”A short response to a supposed “review”, with reference to a longer discussion, also on Arrow’s Impossibility Theorem”, http://thomascool.eu/Papers/VTFD/2011-11-25-ShortResponse.pdf
Danielsson, J. et al. (2001), “An academic response to Basel II”, http://www.math.ethz.ch/~embrechts/ftp/Responsev3.pdf
Gamboa, J.M. (2011), “Book review. Conquest of the Plane, by Thomas Colignatus”, http://www.euro-math-soc.eu/node/2081
Gill, R.D. (2008), “Book review. A Logic of Exceptions: Using the Economics Pack Applications of Mathematica for Elementary Logic, by Thomas Colignatus”, NAW 5/9 nr. 3 sept., http://www.nieuwarchief.nl/serie5/pdf/naw5-2008-09-3-217.pdf
Gill, R.D. (2012), “Book reviews. (1) Elegance with Substance, (2) Conquest of the Plane”, by Thomas Colignatus”, http://www.math.leidenuniv.nl/~gill/reviewCOTP.html
Steinsaltz, D. (2011), “The Value of Nothing: A Review of The Quants”, http://www.ams.org/notices/201105/rtx110500699p.pdf