[FOM] What would von Neumann say?

Tom Dunion tom.dunion at gmail.com
Tue Jun 22 14:26:44 EDT 2010

To motivate the question, a brief historical overview.

In the early decades of the 20th century two significant
intellectual edifices were axiomatized, by a number of
researchers, to great effect: Set Theory, and Quantum
Mechanics (QM).

Enter John von Neumann, with his contributions (in the
1920s) to establishing the cumulative hierarchy as
something which answers to and motivates the ZF axioms.
Subsequently he identified physical constructs in QM
with mathematical entities within a Hilbert space.

At first glance, it may appear that the driving motives
in these two communities were not at all similar.  In
the case of Set Theory, a large concern was to block
the paradoxes.  In QM, physicists were faced with
experimental results which classical methods and
assumptions could not explain.

Yet it seems there is a certain sort of intellectual
point of view in both fields, common among many
modern researchers.

Rather than try to define this view, I will simply
point to two common claims -- the first, that the CH
is vague and/or indeterminate (suggesting there is no
actual truth of the matter); the other, that there is
no such thing as simultaneously existing position and
momentum for particles (and not, as the Bohmian view
would claim, that there is merely a fundamental
limitation on the capacity to simultaneously measure
them with arbitrary precision).

I wonder, does anyone have a considered opinion about
either what was said *by* von Neumann about whether or
not his work in QM was similar in outlook to his work in set
theory; or what might be said *about* von Neumann's point
of view, such as whether his later QM work was perhaps
biased toward a "we cannot say more than this " position,
by his previous awareness of the limitations of Set Theory?
(By limitations, I mean such things as the irresolution of
the CH, and Goedel's incompleteness results.)

Tom Dunion

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