“Georg Cantor said: “The essence of mathematics lies in its freedom.” But I would like to augment this with the following: Where there is no mathematics there is no freedom. So mathematics is essential to our freedom, to the functioning of our democracy. (…) Our ignorance can be misused by the powers that be. And for us … as citizens in this Brave New World … we have to be more aware of mathematics, we have to know and appreciate its power – to do good but also to do ill.” (Edward Frenkel, Jan 19 2014)
We can only applaud this. In my Elegance with Substance (EWS)(2009):
“Mathematics is a great liberating force. No dictator forces you to accept the truth of the Pythagorean Theorem. You are free to check it for yourself. You may even object to its assumptions and invent non-Euclidean geometry. Mathematical reasoning is all about ideas and deductions and about how far your free mind will get you – which is amazingly far. But you have to be aware of reality if you say something about reality.” (EWS p9)
“Democracy is an important concept. The mathematics of voting is somewhat complex. It would be beneficial for society when its citizens understand more about the mathematics behind election results. Students in the USA have a Government class where such aspects can be indicated. Political Science as a subject has not reached highschool in general. Much can be said in favour of including the subject in economics, since the aggregation of preferences into a social welfare function is a topic of Political Economy. See page 59 and Colignatus (2007b) Voting theory for democracy (VTFD) for details and other references. Most economists will be unfamiliar with the topic and its mathematics though and thus it may well be practical to include it in the mathematics programme.” (EWS p48)
However, let us also look at key criticism:
- Mathematician Kenneth Arrow presented his “impossibility theorem” in his 1951 thesis. It holds, in his own words: “there is no social choice mechanism which satisfies a number of reasonable conditions” Palgrave (1988:125) and quoted in Voting Theory for Democracy (VTFD)(2014) 4th edition p240. Thus collective choice would require us to be unreasonable. Mathematician Arrow continued in economics and got the Nobel Prize in economics for this and other work.
- Mathematicians, political scientists and economists have tried since 1950 to debunk Arrow’s result, but did not find real solutions. These areas of science have become a force against democracy. Collective choice would require us to be unreasonable, and this would be scientifically proven.
- When I showed in 1990 that Arrow’s words do not fit his mathematics, and a bit later that his result was either inconsistent or incomplete, hell broke out. My paper was suppressed from discussion and publication. A mathematician who was supposed to review VTFD (3rd edition) started slandering. See the journal Voting Matters (April 2013). See my point however that there is a distinction between “voting” (counting ballots) and “deciding”. And see VTFD for the more involved presentation (starting with matricola).
- It has been impossible to find someone in Holland to discuss this issue rationally. Here is a report in English on a working group in social choice theory. Here is a page in Dutch. On a website for highschool students, Kennislink.nl, deluded mathematician Vincent van der Noort, who did not properly study the issue, claims that “democracy isn’t entirely fair“, thus encouraging highschool students to use their ellbows. The editors refuse to correct this falsehood and selective use of sources (or mystery, since Vincent doesn’t define fairness).
I suppose that professor Frenkel discusses democracy in general, without thinking specifically about Arrow’s “Theorem”. Perhaps he doesn’t know about it, and would be surprised that it would be “mathematically proven” that some degree of dictatorship would be necessary. However, to some extent we can agree with him. Good education in mathematics will do wonders for liberty and democracy. But, my point again: the definition of “good education in mathematics” is subtle. See these quotes from EWS too:
“With respect to logic and democracy, Colignatus (2007ab, 2008b), updated from 1981 / 1990, considers statements by mathematicians that have been accepted throughout academia and subsequently society on the basis of mathematical authority. It appears however that those statements mix up true mathematical results with interpretations about reality. When these interpretations are modelled mathematically, the statements reduce to falsehoods. Society has been awfully off-track on basic notions of logic, civic discourse and democracy. Even in 2007, mathematicians working on voting theory wrote a Letter to the governments of the EU member states advising the use of the Penrose Square Root Weights (PSRW) for the EU Council of Ministers. See Colignatus (2007c) on their statistical inadequacy and their misrepresentation of both morality and reality.
Over the millennia a tradition and culture of mathematics has grown that conditions mathematicians to, well, what mathematicians do. Which is not empirical analysis. Psychology will play a role too in the filtering out of those students who will later become mathematicians. After graduation, mathematicians either have a tenure track in (pure) mathematics or they are absorbed into other fields such as physics, economics of psychology. They tend to take along their basic training and then try to become empirical scientists.
The result is comparable to what happens when mathematicians become educators in mathematics. They succeed easily in replicating the conditioning and in the filtering out of new recruits who adapt to the treatment. For other pupils it is hard pounding.” (EWS p10)
PM. See where Georg Cantor went wrong: Contra Cantor Pro Occam (2012, 2013).