PART 1
Oxford’s oral tradition, at least, suggests that E.E. Evans-Pritchard (Oxford’s “EP”) 1902-1973 ( “strangest” ethnographer who believed that social anthropology could be a part of natural sciences ) made the first historical attempt to introduce Quantum Mechanics for social anthropologists in the last century. Presently, Vice –Chancellor of Oxford University and experimental quantum physicist realized the second one at Humanities and Science panel discussion ( Randomness and Order , 10 February , www.torch.ox.ac.uk/humsciox ). In his presentation Jan Walmsley reduced the best scientific tradition of the 21^{st} century to the problem of non-classical or quantum probability, correspondingly , he used quantum card game and quantum notion superposition in order to explain real and dramatic challenge of quantum mechanics for humanities and “social sciences”. Definitely Jan used some sort of simplification, because there is a general ignorance in official and educational circles of what quantum mechanics is. In particularly, real drama is that all Western systems of education for humanities and so called “social sciences” are based on non-quantum ( more exactly – anti-quantum ) principles of classical materialism and classical physics of the 18^{th} century. Even Albert Einstein in 1930s considered “quantum revolution “ in physics as return of “wrong idealism “ in science. He famously asked quantum theorists :” the Moon exists when nobody is looking at it ?” And in 1935 Einstein, Podolsky and Rosen published scientific paper where they described seriously impossible experiment which must stop development of “wrong “ idealistic foundations of quantum mechanics.
We may describe today such “EPR-experiment” (in later David Bohm's terms ) as an impossible experiment where a particle with no spin ( with no magnetic circular movements described by “imaginary numbers “), while at rest, decays into two identical particles ( labelled 1 and 2), each with spin 1/2. Since momentum is conserved, the particles fly out in opposite directions. And since spin is conserved, the two spins must add up to zero. Therefore, in good agreement with “right “science, if
the spin of particle 1 is measured to be “up” along some specific direction, then the spin of particle 2 must be “down'” along some specific direction and there is no such thing as “observer dependent spin” at all. Following the results of such sort of EPR experiment, it became obvious that any observer dependent reality, predicted by quantum theorists in 1920 - 1930s, is simply a nonsense.
In the mid-1960s, however, mathematician and experimentalist John Bell ( CERN ) showed that it was actually quite possible to realize the EPR-experiment, when the two particles are emitted with definite spin directions, which are locally fixed at the decay. These directions, according to Bell,nevertheless, might be Unknown to the experimentalist. He then showed that if we measure the spin of particle 1 along one direction, and the spin of particle 2 along another direction, the results will be correlated. For instance, if we measure the spin of both particles along the same direction, particle 2 will always have the spin down when particle 1 has the spin up.
Thus, a nonsense, described by Einstein - Podolsky - Rosen in 1935 ( in which strong correlations are observed between presently no interacting particles, even if they are detected arbitrarily far away from each other ) became a part of experimental physics !
Since Bell's discovery, a number of physical tests have been performed successfully (by J Clauser and S Freedman (1972), A Aspect, J Dalibard, and G Roger (1982), and G . Weihs, Ch Simon, T Jennewein, H Weinfurter, and A Zeilinger (1998)] etc.etc
Moreover, in 21st century whole new non-nonsensical physics with their ideas of quantum computers, quantum telepathy, quantum internet , quantum teleportation, quantum cryptography, quantum credit cards, quantum markets, quantum elections, quantum space communications and even new quantum anthropology are emerging…
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John and Huon, very simply -
Schrodinger adopted partial differential equations ( PDE ) where m is mass, u is velocity and p - generalized impulse ( in N.Bohr sense ). I suspect that PDE could be very effective tool in future anthropology. Systematic argument about Schrodinger - Shirikigiroff/ Schrodinger - Levi-Strauss I 'd like to consider in Part 5.
Huon, I cannot find any " quadratic - looking equation " and any mathematical function in Levi-Strauss's formula. It is mathematical category logic, not mathematics.
Part 5
Next very important step. In 1925-1926 there were two quantum mechanics – Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics. They used different ways of thinking on the same object. Whereas Heisenberg suggested that idea of quantum noncommutativeness ( i.e. idea that 2 × 1 is not 1 × 2 ) is very important in new physics:
ίh×δ/δt = uH – Hu
where ί is imaginary number; h is Planck constant, δ some dynamical magnitude, H is Heisenberg’s matrix expressed non - commutativeness ( uH – Hu ), Schrodinger, nevertheless, believed that idea of wave function is fundamental as well :
▼²ψ + 2m/ һ² × ( E – V )×ψ = 0.
In 1926 Schrodinger had found remarkable “mathematical trick”( which have been proven by calculations later ) – he rewrites his equation as
ίh×δ/δt ×ψ = Hψ
where ψ is wave function and H is Heisenberg’s matrix ! Thus, quantum mechanics became united physical theory which is able to contain both great ideas within the same whole. In fact Schrodinger produced some intuitive revolution in human (un)consciousness and created a new sophisticated mathematical subculture. Because for without the wave function no laws of Nature could be formulated in the domain of quantum physics – we can also say that for without non- commutativeness no laws could be formulated also…
Remark 5.1 Schrodinger and Shirikigoroff.
God and Governments are not Constructors of the ethnic groups, and correspondingly, merely natural self-organization must be accepted as foundation for ethnogenesis. But what kind of mathematical equations can be used to describe such universal processes ? Ethnograper and statistician Shirokogoroff in 1922 – 1930s ( Vladivostok, Shanghais ) attemped to establish this type of equation in his project of ethnos theory. He had found that some Diffusion equations ( Fourier equation ? ) could be considered as the best candidates to realize it. Next step was made by Gumilev ( had ideas similar with EP) and Ermolaev in 1990s . They published paper where described analogy between quantum physics of lasers and ethnogenesis mathematics taking seriously. Today some mathematicians of Moscow Keldish Institute of applied mathematics, Synergy academic group and some computer scientists ( “Mathematical History” project ) try to understand this very strange finding. My own observations can also suggest that may be there exists unreasonable analogy between Diffusion equations ( quantum Fourier equations ) and …the Schrodinger equation indeed. If it is correct, it could be new fantastic math behind it.
Remark 5.2. Schrodinger and Levi-Strauss.
If I understand, most anthropologists and mathematicians may agree now that Levi-Strauss’s “algebraic formulae “ ( some anthropologists define it as “quadratic – looking equation” )
Fx(a):Fy(b) :: Fx(b) : Fa¯ 1(y)
are respectively either superfluous or problematic…Some have maintained that such sort of “algebra” does not add anything to the understanding of myths and kinship
( Cargal,1996 , Albert Doja, 2003,...+ P. Gow )… But in 2003 theoretical physicist Jack Morava made very interesting analysis of LS logic and he had found that :
“The formula is thus intrinsically unsymmetric: it is not required that the character b have an associated function b −1, nor that the function x have a sensible interpretation as a character. This suggests that the canonical formula can be paraphrased as the assertion: in a sufficiently large and coherent body of myths we can identify characters a, b and functions x, y, such that the mythical system defines a transformation which sends a to b, y to a−1, and b to y, while leaving x invariant.” ?
And in conclusion he said :
“When I first encountered Levi-Strauss’s formula, my reaction was bemusement and skepticism; I took the question seriously, in large part because I was concerned that might represent an aspect of some kind of anthropological cargo-cult, based on a fetishization of mathematical formalism. I am an outsider to the field, and can make judgements of Levi-Strauss’s arguments only on the basis of internal consistency in so far as I am competent to understand them); but I have to say that I am now convinced that the man knows his business”… ( ArXiv 2003 )
Jack used category theory ( branch of mathematical logic, not mathematics of Schrodinger type ) to make this conclusion. Some category theorists work in area of foundations of quantum mechanics and they called it as “ Quantum Structuralism “ . Thus, some reasons for re-interpretation of LS logic we have already and I hope “quantum mechanics for anthropologists “ can help it.
John, thank you. Some kind of refinements - I suppose that mathematics is Universal Language of Science, correspondingly, it could be difficult to define as "its own tribe's language". In Jack Morava's context we can imagine two different strategies in understanding of LS formalism: (1) “Traditional” strategy : to reject Levi-Strauss formalism, to make announcement that social and cultural anthropology IS NOT science, to make announcement ( for naïve Governments ) that we anthropologists live in parallel Universe and our Universe contains absolutely unsolved Big Data problems, to abolish any mathematical symbols and to use Geertz-like secret language ( “ of the fifty-seven matches for which I have exact and reliable data on the centre bet, the range is from fifteen ringgits to five hundred, with a mean at eighty – five and the distribution being rather noticeably trimodal…” Notes on the Balinese Cockfight , 426 ) for mathematical games in the field. And
(2) Non-traditional strategy which Jack Morava demonstrates ( see, also Part 5 ):
- 1. to take imperfect Levi - Strauss formalism seriously,
- 2. to improve it ,
- 3. to prove an existence of new refinements, simplifications ( Category theory, Grotendick's "childs pictures",etc )
- 4. to find analogies for it ( Topological algebra, quantum math, Manin's ethnomathematics, etc )
- 5. to prove it,
- 6. to generalize it ( may be there are new theorem, new math, new counterexample,…etc ),
- 7. to prove it,
- 8. to find anthropological, physical interpretation ;
- 9. to formulate hypothesis,
- 10. to test it in the field, numerical or laboratory research.
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