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.
Brexit - US.exit cusp ?
Current US statistics suggest
Trump : Arizona 41.6%;Colorado 40%; Florida 45%; Iowa 46%; Nevada 44%;;North Carolina 41%;Ohio 45%;Penn 40%;Virginia 41%; Wisconsin 39,3%.
Thus, let us imagine that new Black Swan is coming. Mathematically, we have a function, say f(x), has cusp at some imaginary point T(Trump) If the slope plot is discontinuous at T. If the magnitude of the function becomes very large on either side of T and arbitrarily close to T and it is undefined, we can say that US.exit function or f(x) is singular at T. As result- Black Swan is emerging... Statistics suggest nothing new,but Propp's structure may suggest that some value of f(x) can becomes infinite...?
Trump's singularities can be of many unexpected types
1.it could be function on both sides of the singularity,
2.function which changed with upon passing through singularity,
Thus, in our hunting for the Next Black Swan, we must await that we have the slope curve ( described process ) which can possess A CUSP ( a kink, or break ).
other voices about Brexit
And another voice on Brexit (attached), entirely discursive and probably just as speculative. Can it have any value if it doesn't have the maths?
Yes, it can.
On History of modern corporate capitalism in 19th century Britain.
When we look at that period mathematically - ahistorically - anthropologically, we find evidence that the key institutions of the modern economy were created less as a way to promote rational investments in conditions of uncertainty, and more to channel what the British elite regarded as irrational herding of investors into areas which are less productive... The decisions of the British policy makers were based on usually incomplete and frequently incorrect information. However they were inspired by a good intuitive understanding of some hidden mathematics of some behavioural economics. There was wide appreciation of the power of the overconfidence and other phenomena. These folks are important in understanding today's Britain...Hence, by application of mythology of "Lorentzians" and highly unlikely events we can find that there is missing some important elements in history of modern capitalism...etc.
Brexit as a Black Swan effect was made under time and other pressures and with incomplete and incorrect information, if even it was inspired by a good public illusions... Black Swan has own intuitive math which is unknown in our time, including effects of overconfidence and other unknown phenomena that must be named and be explicated in future.
A long time ago I had learnt that the numbers arent in our head and no on the things independently, they are a kind of result of what we do with things...what we do it is not only possible to mesure with numbers or math, we can´t reduce to it...it is nice to have a debat around the complexety, that means...maths, facts, class, history...etc...there is no magic...Brexit was defined as itself and then with it is having many meanings and effects..maybe it is not the centre of the present times, it is just a big bag of beliefs to discribe by anthropologists, too, much more and far from only maths, in case with it. The situation it is not for taking helicopters and see from the clouds what happens on the earth...we should mix with the people to know more..as Mallinowski tradition.
As is known B. Malinowski had PhD in mathematical physics and he applied function analysis in ethnography. Following Malinowski, it could be interesting to refine function theory ( may be to reinforce by Taleb like functional analysis ) in theoretical anthropology. Leach ( Cambridge mathematician by education) made some attempts to introduce so called "coding theory", redefine Malinowski's notion of function in his functional-structural analysis.
Oh yes.... but he didn't get sitting doing numbers on a chair...
Could it be that we could make more progress by devoting less attention to what our intellectual ancestors are purported to have done and spending more of our time thinking about what is now possible?
I think, in particular, of the analysis and modeling of large data sets made possible by advances in computing and doing anthropology in places where governments and journalists as well as scholars, both native and non-native, provide mountains of information that anthropologists confined to the bandwidth of human brains have no hope of scaling.
To me, these are concrete issues. I set out a number of years ago to examine the Japanese advertising world in which for more than a decade I earned my living. My model for doing ethnography is the Manchester school approach as elaborated by Victor Turner. Step 1 is to map the relevant social structure. Step 2 is to examine in detail social dramas, conflicts in which latent contradictions and tensions surface. Step 3 is to interpret the symbols in play in social dramas and their ritual resolutions.
Sounds straightforward if we imagine a small village of a few dozen people who conduct most of their lives in public. Consider an industry that employs tens of thousands of people whose job is never-ending innovation, operating in a period of rapid technological change that is constantly changing the media environment in which they work. I have sampled data from a major advertising contest annual that identifies over 27,000 role relationships connecting 7,019 creators to 3636 pieces of advertising. I read selectively from an industry trade press that publishes several hundred pages of new information each month. And, since I once worked for one of Japan's largest agencies, I can arrange interviews with the individuals most central in the networks I analyze. Can I get my head around all this by myself? No way.
That said, a network analysis program called Pajek enables me to explore my data quantitatively and to visualize the results, performing in minutes analyses utterly impossible when I was in graduate school in the 1960s. Had I the time and energy, I could use NetLogo, a free agent-based modeling program to see what is likely to happen given the highly simplified assumptions I make. Tools like these make it possible to frame research in ways unthinkable even a couple of decades ago. The serious question is how to relate the anthropologist's personal individual experience and what he has been told by a handful of key informants to the results that these tools produce.
Trump or Clinton ?
Malinowski probably was the first who applied Banach spaces ( functional analysis) in ethnography. It is very interesting math connected with divided spaces of the whole superspace expressed by Banach algebra. Following Malinowski, we can imagine some superspace of all choices and ask to find the best addition of two subspaces associated with Trump and Clinton voters tomorrow. Logically, it is possible to assume that two spaces created by Trump and Clinton voters can produce more than only One Space of a whole. It is a famous Banach paradox based on his fundamental functional analysis. Thus, if and only if Black Swan is not becoming, we can test how Banach - Malinowski abstract lattice may emerge in completely unexpected area of political ethnography of USA.
Let us continue our systematic work...
ANALYSIS OF OUR SUCCESSFUL PREDICTION.
Step 1 (27 June)
Brexit is defined as a very rare unpredictable "Black Swan" (BS) ( in Taleb's sense ) event which can produce series of unpredictable events of bounded rationality.
Step 2. (4 July)
Attempt to develop some kind of mathematical theory ( intuitive prediction is not scientific ) where Brexit is considered as highly unlikely ( for professional observers ) event that can totally alter the trajectory of the political system.
Step 3. (Aug 26)
In the search of BS manifestations. Brexit can have unexpected development in USA - assumption.
Step 4.(Sept 24)
Attempt of formalisation. Trump/ USexit function is introduced
Step 5.(Nov 2)
Refinements. Black Swan has own unknown exotic math?
Step 6.(Nov 7)
Logical structure of BS. Reference to B.Malinowski' functional analysis. US Black Swan may have logical structure of Banach paradox
Step 7.(Nov 9)
Observation of Black Swan. What appear to be small risks ( for professional observers, liberal social scientists , developers of liberal narrative and old paradigm politicians) accumulated to certain irreversible harm. Trump is obvious winner.