[MD] TOPOS

[Ron Kulp]

JOS,
Thanks again for the link! This ties in directly about what I'm saying about classical
generalised assumptions And why it does not serve us at the quantum level.  
 
"In regard to the three conditions listed above for a ‘realist’ interpretation, our
scheme has the following ingredients:
1. The concept of the ‘value of a physical quantity’ is meaningful, although this
‘value’ is associated with an object in the topos that may not be the real-number
object. With that caveat, the concept of a ‘property of the system’ is also meaningful.
2. Propositions about a system are representable by a Heyting algebra associated
with the topos. A Heyting algebra is a distributive lattice that differs from a
Boolean algebra only in so far as the law of excluded middle need not hold,
i.e., 
 ∨ ¬
  1. A Boolean algebra is a Heyting algebra with strict equality:

 ∨ ¬
 = 1.
3. There is a ‘state object’ in the topos. However, generally speaking, there will
not be enough ‘microstates’ to determine this. Nevertheless, truth values can be
assigned to propositions with the aid of a ‘truth object’. These truth values lie
in another Heyting algebra.
This new approach affords a way in which it becomes feasible to generalise quantum
theory without any fundamental reference to Hilbert spaces, path integrals, etc.; in
particular, there is no prima facie reason for introducing continuum quantities. As we
have emphasised, this is our main motivation for developing the topos approach. We
shall say more about this later.
From a conceptual perspective, a central feature of our scheme is the ‘neo-realist6’
structure reflected in the three statements above. This neo-realism is the conceptual
fruit of the mathematical fact that a physical theory expressed in a topos ‘looks’ very
much like classical physics.
This fundamental feature stems from (and, indeed, is defined by) the existence of
two special objects in the topos: the ‘state object’7, 
, mentioned above, and the
‘quantity-value object’,"


I love it...topos comes very close to my line of thinking on the subject.
-Ron