Dear Elias,

Thank you for your reply from Fri, 15 Feb 2002 08:11:04 -0000. You wrote:

> Regarding your question I would like to say the following:
> The geometric line if we choose two distinct points on it can be
> considered as a commutative ring with the two points denoted by
> 0 and 1.
> The line endowed with its commutative ring structure is denoted by
> R.
> Similarly the geometric plane can be identified with R^2.
> If somebody furthermore made the assumption that for any two
> points on the plane, either are equal, or they determine a unique
> line [like Euclid did] then he has silently assumed that R is not only
> a commutative ring but actually a field or else that the set A={x in
> R: x^2=0} contains 0 only.
> By dropping the last assumption we get the notion of square zero
> infinitesimals which live in R, being just a commutative ring, but
> not a field.
> This is equivalent to sacrifying the law of excluded middle. It is
> amazing that Protagoras was the first who questioned Euclid's
> hypothesis by bringing the argument that the intersection of the unit
> circle x^2+[y-1]^2=0 with the x-axis is exactly the set A, containing
> more than the single point 0.

It is not clear to me how do you model the infinitesimal, as in Thompson's lamp paradox (please see my preceding email printed below).

It seems to me that all real numbers should literally emerge out of some seemingly simple algorithm, (t_2 - t_0) - (t_1 - t_0) = 2 , resembling renorm recipes. We have not two but three animals here, don't we? If so, where is that mysterious sliding cutoff,  t_0 ?  Please see also my email to Joao Magueijo,

http://members.aon.at/chakalov/Magueijo.html

Best regards,

Dimi
http://members.aon.at/chakalov
http://members.aon.at/chakalov/dimi.html

----
Subject: quant-ph/0202057
Date: Tue, 12 Feb 2002 19:20:43 +0200
From: "Dimiter G. Chakalov" <dchakalov@surfeu.at>
To: e.zafiris@ic.ac.uk
CC: i.raptis@ic.ac.uk, k.savvidou@ic.ac.uk, amallios@cc.uoa.gr
 

Dear Dr. Zafiris,

I am very much interested in your ideas [Ref. 1], but my math is too week to understand them. May I ask you for help in clarifying the issue of mapping infinitesimals to real numbers.

We are inevitably forced to resort to real numbers, although we're fully aware, after Lucretius, Zeno, Leibnitz and Newton, that the infinitesimal is an entirely different entity. Consider, for example, the paradox of Thompson's lamp.

Imagine a lamp that is 'on' for 1 min, then 'off' for 0.5 min, then 'on' for 0.25 min, etc. Do we have a limit?

If you say 'yes', what is the state of the lamp at the instant of time at the "end" of this limit? We have not two but *three* puzzles here: the state of the lamp at the initial instant (t_1), the state of the same lamp after two min (t_2), and the nature of the cutoff  t_0. The latter can approach The Beginning but can never reach it.

Also, the "duration" of the two instants, t_1 and t_2, can only tend asymptotically toward the "singularity" (Planck length?).

Also very intriguing to me is that the cutoff  t_0, with which we measure any *finite* intervals, like  (t_2 - t_0) - (t_1 - t_0) = 2 , can approach asymptotically  t_1  but *not*  t_2. Hence it seems that some time-asymmetry is pre-build in this story about the infinitesimal.

Hence my questions: how do we move from the infinitesimal to real numbers, and back to the infinitesimal?

Also, can you suggest some 'generalized projection postulate', which would include the case of measurements in classical physics as a limiting case of a broader situation including quantum observables? In both cases we have the infinitesimal "filtered" through Boolean reference frames, don't we? Do you believe one could unravel this "filter" with Synthetic Differential Geometry or topos theory?

With kind regards,

Dimi Chakalov
http://members.aon.at/chakalov

----
[Ref. 1] Elias Zafiris. Topos Theoretical Reference Frames on the Category of Quantum Observables. Mon, 11 Feb 2002 12:22:47 GMT,
http://xxx.lanl.gov/abs/quant-ph/0202057

"Quantum theory stipulates that quantities admissible as measured results must be real numbers. The resort to real numbers has the advantage of making our empirical access secure, since real number representability consists our form of observation.
...

"Hence it is reasonable to assert that an observable picks a specific Boolean algebra, which can be considered as a Boolean subalgebra of the Quantum lattice of events.

"In essence an observable schematizes the Quantum event structure by correlating its Boolean subalgebras picked by measurements with the smallest Boolean algebra containing all the clopen sets of the real line. In the light of this Boolean observables play the role of coordinatizing objects in the attempt to probe the Quantum world. This is equivalent to the statement that a Boolean algebra in the lattice of Quantum events picked by an observable, serves as a reference frame, conceived in a precise topos-theoretical sense, relative to which the measurement result is being coordinatized, suggesting a contextualistic perspective on the structure of Quantum events. Philosophically speaking, we can assert that the quantum world is being perceived through Boolean reference frames, regulated by our measurement procedures, which interlock to form a coherent picture in a non-trivial way.
...

"The passage from a system of prelocalizations to a system of localizations for a quantum observable is achieved if certain compatibility conditions are satisfied on the overlap of the modeling Boolean charts covering the quantum observable under investigation."