Subject: How does one observe no bananas? Or empty space?
Date: Fri, 15 Feb 2002 20:25:33 +0200
From: "Dimiter G. Chakalov" <>
To: "John D. Fearns" <>
CC: Ioannis Raptis <>,
     Elias Zafiris <>,
     Jeremy Butterfield <>
BCC: [snip]

Dear Dr. Fearns,

I've been reading you beautiful article [Ref. 1] for over three hours now, trying to understand precisely how a mathematical universe can hide infinitesimals, which is the main issue in my speculations about the structure of spacetime,

May I ask two questions. You wrote that unnormalisable vectors (Sec. 4.1) may represent "physical states with the interesting property of not being able to be logically duplicated (or, more generally, considered to be part of a larger quantum system)."

In Sec. 3.1, you stressed that "We continue in the assumption that all quantum state-spaces referred to from now on are finite-dimensional", but in Sec. 4.2 you examined the case of infinite-dimensional representations.

Are the unnormalisable vectors somehow different in the cases of finite vs. infinite-dimensional state-spaces? A change in the behavior of probability maybe?

Also, have you thought about geometrical formulation of QM [Refs. 2 and 3]? If I understand your work correctly, the delinearization program [Ref. 2] would be the first target for applying your ideas.

I have actually tons of questions, but first will study your article with the help of some friends of mine from Sofia U. Dept. of Mathematics.

As to the question from the subject line and your observation that "the system itself neither exists nor does not exist, but instead lies somewhere inbetween" [Ref. 1], please see how the brain works at

You don't see the holistic entity "behind"  |dead cat>  and  |live cat>  (Ioannis calls it 'Holon'), simply because It can not be represented with any number *whatsoever*. It does lie "somewhere inbetween" and is very much alive and kicking thanks to our Intuitionistic Logic (IL), I believe. But then comes the issue of its Heyting algebra of observables (?), which I can't grasp at all, and probably never will.

With best regards,

Dimi Chakalov

[Ref. 1] John D. Fearns. A Physical Quantum Model in a Smooth Topos. Thu, 14 Feb 2002 17:20:00 GMT,

"Thirdly, the new viewpoint requires a greater repertoire of concepts, not all of which immediately sound physically plausible (how does one observe no bananas? Or empty space?).

"In the unlikely event that the definition above is not accepted, then to what extent the 'almost zero' vectors represent physical states is an unexplored area. Its study may be useful when considering the states of open quantum systems (systems with states that perhaps, even logically, should not be duplicated by tensor production).
It may force upon us the strange and exciting position of being able to talk about the entanglement properties of a system while having to accept that the system itself neither exists nor does not exist, but instead lies somewhere inbetween.

"We noted on our foray into quantum mechanics such observations as the fact that complex Hilbert spaces and finite-dimensional complex inner-product spaces need not be the same thing when one frees oneself from classical logic, and that the extension of simple quantum mechanics from finite to infinite dimensional systems seems to require either a change in the behaviour of probability, or a change in the collection of allowed physical states.

"Parallels might have been drawn far earlier between, say, the method by which the physical universe always hides virtual particles from the observer, and the method by which a mathematical universe can hide infinitesimals from the man who wishes to name one."

[Ref. 2] R. Cirelli, M. Gatti, A. Manià. The Pure State Space of Quantum Mechanics as Hermitian Symmetric Space.

"The true aim of the delinearization program is to free the mathematical foundations of QM from any reference to linear structure and to linear operators."

[Ref. 3] A. Ashtekar, T.A. Schilling. Geometrical Formulation of Quantum Mechanics.

"The geometric formulation shows that the linear structure which is at the forefront in text-book treatments of quantum mechanics is, primarily, only a technical convenience and the essential ingredients -- the manifold of states, the symplectic structure and the Riemannian metric -- do not share this linearity."