Monday, July 11, 2011

And/Or Problem

Filed under: General,Geometry — Tags: — m759 @ 11:59 PM

"It was the simultaneous emergence
and mutual determination
of probability and logic
that von Neumann found intriguing
and not at all well understood."

Miklós Rédei


Update of 7 AM ET July 12, 2011—

Freeman Dyson on John von Neumann's
Sept. 2, 1954, address to the International
Congress of Mathematicians on
"Unsolved Problems in Mathematics"—

                                     …."The hall was packed with
mathematicians, all expecting to hear a brilliant
lecture worthy of such a historic occasion. The
lecture was a huge disappointment. Von Neumann
had probably agreed several years earlier to give
a lecture about unsolved problems and had then
forgotten about it. Being busy with many other
things, he had neglected to prepare the lecture.
Then, at the last moment, when he remembered
that he had to travel to Amsterdam and say something
about mathematics, he pulled an old lecture
from the 1930s out of a drawer and dusted it off.
The lecture was about rings of operators, a subject
that was new and fashionable in the 1930s. Nothing
about unsolved problems.
Nothing about the

Notices of the American Mathematical Society ,
February 2009, page 220

For a different account, see Giovanni Valente's
2009 PhD thesis from the University of Maryland,
Chapter 2, "John von Neumann's Mathematical
'Utopia' in Quantum Theory"—

"During his lecture von Neumann discussed operator theory and its con-
nections with quantum mechanics and noncommutative probability theory,
pinpointing a number of unsolved problems. In his view geometry was so tied
to logic that he ultimately outlined a logical interpretation of quantum prob-
abilities. The core idea of his program is that probability is invariant under
the symmetries of the logical structure of the theory. This is tantamount to
a formal calculus in which logic and probability arise simultaneously. The
problem that exercised von Neumann then was to construct a geometrical
characterization of the whole theory of logic, probability and quantum me-
chanics, which could be derived from a suitable set of axioms…. As he
himself finally admitted, he never managed to set down the sought-after
axiomatic formulation in a way that he felt satisfactory."

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