[FOM] What would von Neumann say?

A. Mani a_mani_sc_gs at yahoo.co.in
Thu Jun 24 21:07:55 EDT 2010

Von Neumann wrote his book 'Mathematical Foundations of Quantum Mechanics' in 
1932. But his views changed substantially in subsequent years. He did not like 
the Hilbert Space formalism from the perspective of 'quantum logic' as he saw 
it as a direct extension of the formal apparatus of vectors in Euclidean 
spaces; Modularity does not hold in the Hilbert lattice in general. So the 
Hilbert lattice could not be interpreted as an event structure for the 
relative frequency interpretation of probability theory (non commutative). 
Further no dimension function (with discrete/continuous range) 'c' taking 
values in [0,1] and satisfying c(A) +c(B) = c(A\vee B) + c(A\wedge B) can 
exist on the Hilbert lattice.

His view therefore shifted towards the 'Von Neumann algebras' and related 
framework of continuous geometry. The dimension function on abstract 
continuous geometries was interpreted as a probability measure. Till 1936, he 
tried to get as close as is possible to the classical frequency
interpretation of probability and then abandoned the Von Mises theory because 
of the difficulties with the 'randomness' requirement/ equivalents thereof. 
After that he tried to arrive at a better theory.

The modularity postulate of quantum logic due to Birkhoff and Von Neumann in 
1936 was part of an attempt to capture both a probabilistic aspect and the 
generalization of classical logic. 

I do not think he attempted analogies or actual extensions of set theoretic
methods to the quantum domain. At least that is what most historical studies
say. (see Redei's many papers on the history of quantum logic and Von Neumann 
for more details of all of the above claims)


A. Mani


A. Mani

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