An attempt to show how applications of pure mathematics can work in social sciences.
Great French mathematician Andre Weil ( 1906 - 1998 ) is well known in history of social anthropology. In fact, he was co-author of Levi - Strauss 's The elementary structures of kinship and his a very short algebraic introduction in Levi-Strauss solution is contained in LS book on kinship. In 1940s Andre Weil was ready for such sort of collaboration with anthropologist. In 1920 he taught himself Sanskrit and had deep interests in Greek, Latin, correspondingly - Hinduism also. His number theoretical structuralism was connected with the mathematical separation of the infinite descent argument into 2 types in newly created Algebraic Geometry. However, after publication of LS book their cooperation became less intensive and Levi-Strauss changed his research direction to Trubetskoy - Jacobson's linguistics. Today, nevertheless, it could be interesting to introduce another unknown Weil's mathematical findings and theorems in theoretical anthropology.
Another masterpiece of Weil is his theorem on representation of irrational numbers which has remarkable applications in mathematical geography, comparative demography and comparative company assessments in international business. I try to apply Weil's theorem for Brexit prediction.
Theorem. Let the first digits of number 2^n, where n =0;1;2;....Then statistics of their distribution is Weilian or 1;2;4;8;1;3;6;1;2...
Demonstration. Number 1 has a statistical weight 29%, 2 has 21%, 3 has 11%, 4 has 10%, 5 has 6%, 6 has 6%, 7 has 8%, 8 has 3% and 9 has 6%.
In 1980s Konstantinov and Arnold in 1997 showed that distribution of areas and populations in current national boundaries is Weilian .For example, Weil's index or relation of area : population for France is 5/6; UK 2/6; Italy 3/6; Ukraine 6/4; Georgia 6/4. It is very remarkable that our highly abstract theorem is able to catch essential characteristics of reality. Indeed, two currently disintegrated nations - Ukraine and Georgia have the same Weil Index!
Prediction. In comparison with Weil index for disintegrated nations, Italy and UK have some kind of reserve of stability - UK 2/6 and Italy 3/6. Thus, it could be difficult to represent UK in some deepest fundamental spatial sense as disintegrated nation. Hence, any kind of referendum of Brexit type cannot change status quo and it is quite logical that on 23.06.2016 majority of UK population must vote STAY in the EU community.
Sorry am being a bit slow, what does the 2 and the 6 each mean in UK = 2/6 ?
Very interesting results - 51,9% ( Leave) vs 48.1% (Stay). 33 millions voters. Weil's theorem works as approximation without any mathematical model. This means incorrect prediction of Weil's theorem suggests only one thing - refinement of theorem is necessary.
Abraham Heinemann said:
Sorry am being a bit slow, what does the 2 and the 6 each mean in UK = 2/6 ?
It is very simply - 243.610 (= 21%) area in km^2 / 65.542.579 (=6%) population of UK, hence the first digits are 2 and 6. Theorem is used for very quick approximations.
BLACK SWAN EFFECT.
Usually Weil's theorem is working well. However, even for Government professionals Brexit is defined as "riskier than you think" event... I suppose Brexit is indeed very rare unpredictable Black Swan event which can produce series of unpredictable rare events of so called " bounded rationality " in Scotland, European Union and may be in USA? Black Swan has no any established linear theory, and I suppose it cannot have such theory because modern economics is classical theory used linear classical equations and based on pre-quantum economic determinism of the 19th century.... Thus, probably here pure anthropology of bounded rationality meets pure mathematics ?
Groan. Does that mean Italy will be next to leave?
Maybe it's time to get to the bookies.
I await the next Black Swan not in Italy but in financial markets. There is sufficient reason to expect that Brexit will set off a limited scale financial crisis... Joint Anthropology of Bounded Rationality (BR) can find new arguments under uncertainty.
Black Swan is a mathematical event which Weil theorem cannot predict. It is highly unlikely event that can totally alter the trajectory of the political system (in the case of Brexit ). Black Swan was introduced by Nassim Taleb in order to describe how what appear to be small risks accumulate inevitably to certain irreversible harm in the terms of applied statistics. Taleb made references to Soviet school of probability theory of 1930s ( Gnedenko,et el ) as well as Ancient sceptical philosophers...
Speaking generally, however, Black Swan statistical theory can also be applied to MYTHOLOGY as a whole where in comparison with Kenna' networking approach, every myth mathematically represents a story of highly unlikely events that can totally alter natural way of life of the system, indeed.
Let us formulate more exactly mathematical foundations of Black Swan conception. Gauss introduced his law of errors or "normal distribution" which became well established principle in science. Its universality stems from the central limit theorem ( the normal distribution arises from averaging a sufficiently large number of random variables regardless of which distribution rules their fluctuations on an individual basis). While the normal or ergodic type of fluctuations describes much of the physical world in the last century, it can fail dramatically for highly complex systems today.A different class of fluctuations for which the central limit theorem does not hold, provides a new paradigm associated with Taleb's Black Swans (BS). The main property of such BS fluctuations is that they are scalefree.
When it is combined with impact of the outliers, scalefree fluctuations give rise a new notions such as kurtosis - risk, extreme risk, supernatural fluctuations and events , Taleb's Black Swans etc - highly unlikely events that can dramatically change trajectory of the system. Such paradigm can describe every myth, action or policy with suspected risk of causing severe harm to the public domain and in the absence of scientifically( computationally) near - certainty.....
Let us continue our educational online experiment. Next step -
BREXIT AS AN EMERGING MYTH in vivo
The reader will also easily find that in the context of BS approach, Brexit could be considered as an emerging myth in vivo.There are three mathematical definitions of myth in modern anthropology :
1. Levi - Strauss & Morava definition, where some myths we can identify characters a,b and functions x,y such that the mythical system defines a transformation in which a goes to b, y to a^-1 and b to y, while leaving x invariant.(2003).
2.Kenna - Mac Carran's definition where mythical system is a comparable graph in which vertices represent individuals and undirected commutative edges represent interactions between them ( 2013 ).
3.Post - Taleb's definition where myth is a Lorentzian or fat - tail type of distribution for fluctuations when the central limit theorem does not hold.( Please see previous post.)
Thus, Brexit is defined as Lorentzian, hence we can make some very careful predictions and simulations.
Current post - Brexit development : NATO Summit by 9 July 2016 in Warsaw defines Brexit as a problem of international security and T. May suggests that Brexit can work for UK.
UNEXPECTED MORPHOLOGY OF BREXIT.
Following structuralist formula of V.Ya. Propp 's Morphology of a Fairy Tale ( 1928 ) - a disaster happens.The hero is appealed for help.He sets off searching .On his way he meets someone who tests him and awards him with a magic device.Hero finds his object.The hero returns. Mathematically, it is a kind of lemma for complex problem (black swan) solution in some imaginary numbers by some ideal primitive ( Turing like computer )...
Brexit as rapidly evolving fairy tale is faced with new remarkable refinements - on Wednesday British Brexit hero Nigel Farage and USexit future hero Donald Trump had found a new details for Propp's structuralism. Brexit can have unexpected development ... in USA....?
While they were speaking about the oposition the event remembered me https://www.youtube.com/watch?v=zrzMhU_4m-g
It is sad...we are in a chapter of a tale very dark!
and as Propp tale structure analyses the structure of the many monsters head ....
ASTONISHING MATHEMATICAL ANALOGIES
Lorentzian ( Brexit - myth as mathematical imaginary object in my last definition ) is the Cauchy distribution f ( x ; x°, z ) or some canonical example of a "pathalogical" distribution where both its means and expectations are undefined. Lorentzian in mathematics was studied by mathematicians of the 17 th century and was known under the title
WITCH OF MARIA AGNESI
or strange asymptotic curve having remarkable characteristics of Propp structures !...