Subject: Abstract Differential Geometry (ADG) and 'an empty set'
Date: Mon, 16 May 2005 17:19:39 +0300
From: Dimi Chakalov <>
To: Anastasios Mallios <>
CC: "Prof. Elemer Elad Rosinger" <>,
     Chris <>,
     Petr Hajicek <>,
     Ioannis <>,
     Norbert Straumann <>,
     Karel Kuchar <>,
     Roland Omnes <>,
     Hans Primas <>
BCC: [snip]

Dear Professor Mallios,

Thank you very much for your kind reply from Sat, 14 May 2005 21:42:19
+0300 (EEST).

> It was nice to see that you are interested in ADG; recent outcomes can
> be found in physics arXives: physics/0405112 and physics/0405111 this
> latter being also to appear (further elaborated) in Note di Matematica
> (2004). There will be also a two volume account on the subject, that is
> to appear in Birkhauser/Boston some time the next few months.

Thank you. My math is too weak to understand the technical details of ADG, and I will be very grateful to you if you could explain its philosophy.

It is my understanding -- please do correct me if I'm wrong -- that you want to cure the pathologies in classical differential geometry (CDG) [Ref. 1], since the latter allows the manifold of spacetime to "pack an uncountable infinity of events into a finite spacetime volume" [Ref. 2].

You and Ioannis explained the "manifold reasons against the spacetime manifold" [Ref. 3], and suggested the following [Ref. 4]:

"(T)he central didactic point learned from ADG is that one should in a sense turn the tables around and instead of using algebras of C[inf]-functions to coordinatize (as it were, to measure!) space(time) when, as a matter of fact, these very algebras derive from the differential manifold space itself, one should rather commence with a structure algebra sheaf A suitable to one's physical problem and derive space(time) and possibly its (differential) geometric features from it. Algebra (ultimately, dynamics) comes first; while, space and its (differential) geometric properties second."

Here comes my request: if possible, please explain the ADG approach to the main virtue of Einstein's GR -- general covariance cast in classical differential geometry (CDG) -- presented in modern terms with the invariance of physical laws under the action of the group Diff(M) of active diffeomorphisms.

In CDG, the standard interpretation of Diff(M)-invariance is based on *classical determinism*. As explained by C. Rovelli [Ref. 5]: "Therefore
classical determinism forces us to interpret the invariance under Diff(M) as a gauge invariance: we must assume that diffeomorphic configurations are physically indistinguishable."

Specifically, I wonder what ADG says on this interpretation of "physically indistinguishable", and how it can circumvent the Hamiltonian constraint problem in canonical quantum gravity, which too is rooted on CDG.

I believe the interpretation of Diff(M)-invariance in CDG is obsolete: classical determinism is *not* the only available option. We enjoy a perfect determinism in our brains, which is not at all "classical", as I tried to argue at my web site.

Please excuse my blasphemous stand, and let me try to explain this (new?) determinism. Then I'll go back to the issue of Diff(M)-invariance, and will try to explain the implications from this third kind of determinism.

Back in 1953, Wolfgang Pauli suggested that, apart from deterministic and statistical laws, the concept of finality (Aristotle, Physics 194b33) should be considered as a *third* kind of natural laws, which consists in "correcting the fluctuations of chance by meaningful or functional coincidences." [Ref. 6]

Just recall that there are nearly 100 billion neurons and 60 trillion synapses in the human brain, which, if left to chance alone, would literally destroy the brain. Deterministic laws are not applicable either: there is no privileged autonomous system in the brain (the so-called homunculus), which gives commands to 'the rest of the brain'.

Hence the third kind of determinism, which originates from Aristotelian
'final cause' and was endorsed by Wolfgang Pauli and Carl Gustav Jung
[Ref. 7], reveals the ability of the human brain to GUIDE the fluctuations of chance by "meaningful or functional coincidences." [Ref. 6] This is a *physical* interaction between 'the brain as a whole' and its constituents.

Here we arrive at the well-known idea of the Holon. [Ref. 8] Its *physical* nature can be revealed by recalling that the human brain has the ability to 'act on itself': we think *about* the brain, *with* the brain,

Note, however, that the *physical effect* of the Holon is totally "dark",

Going back to the cornerstone of Einstein's GR, it seems to me that we should interpret Diff(M)-invariance in a very different manner, or else we may never be able to construct a theory of quantum gravity. Here again I 'step in the shoes' of Chris Isham, particularly his type-IV scheme, in which we start "with a view of quantum theory and spacetime physics that is radically different from that of conventional theories, and with the expectation that these standard ideas will emerge only in some limited domain."

Thus, bearing in mind that we have to correct the interpretation of "one of the main results of twentieth century's fundamental physics" -- the mapping of the spacetime manifold  M  into *itself* [Ref. 5] -- I was compelled to take an iconoclastic position by stressing that at the fundamental level (supposedly at the Planck scale) not only the classical differential geometry (CDG) will break down, but also the very "distinction" between the physical and mental realms. That is, the Cartesian cut Res cogitans/Res extensa is strictly valid only in the limited domain of CDG, while at the Planck scale the quantum-gravitational reality should exhibit the behavior of a *third* kind of reality: a brand new 'both physical and mental' reality.

To explain this third kind of reality, we need to apply the principle of complementarity; please see the metaphor of 'elephant's trunk' at

The inevitable consequence from this suggestion is that we should model the universe at Planck scale as a human brain-and-mind. Thus, we can suggest a very simple interpretation of Diff(M)-invariance (see below), which, I believe, can show us the right track to quantum gravity. We can use *psychological* notions, such as Platonic ideas, because the fundamental reality is 'elephant's trunk': a *third* kind of reality, which is 'both physical and mental'.

The proposed interpretation of Diff(M)-invariance is based on Platonic ideas,

Think of the Platonic idea of, say, 'corner'. There are infinitely many -- actual infinity -- Diff(M)-invariant observable explications of 'corner', because each and every Platonic idea is indissolubly linked to the realm of 'the unknown unknown'. As John Wheeler put it, "Time is Nature's way to keep everything from happening all at once",

All these *observable* explications of 'corner' are being kept in the Holon at the Planck scale. Please note that we cannot talk about the *physical* manifestation of 'corner per se', but only about its spectrum of possible/potential Diff(M)-invariant explications.

Clearly, all we can say about this third kind of reality is about what it is *not*: it isn't a "normal" Archimedean reality,

It can "absorb" brand new events from 'the unknown unknown' along the evolution of the Universe, which does look like 'creatio ex nihilo' and breaks the "normal" unitary dynamics. Also, if we run the cosmological time backwards along the deflation time, it can "absorb" and "dissolve" the created events, again in violation of our beloved unitary dynamics,

It can "stop" only at The Beginning, where there would be nothing but 'pure light and cognition' [John 1:1].

I'm glad that my Easter greetings have reached you. They had very specific connotation, which I believe goes straight to your Abstract Differential Geometry [Ref. 1]. (I sent my Easter greetings also to Chris, Ioannis, Ntina, Charis, and Fotini, but it seems to me that only you and Chris have received them.)

Please excuse my intrusion into your field of expertise. I am indeed very much interested in ADG, and will be more than happy to learn about your approach to Diff(M)-invariance, as explained by a former pupil of Chris Isham [Ref. 5].

I also very much hope that you will not feel offended by my efforts. (Ioannis, for example, brushed me away four years ago with "don't teach Mother Nature what to do!") I know about your work with Elemer, and all I'm trying to suggest is that perhaps you may wish to include an 'empty set' in ADG, to model the non-Archimedean nature of the Holon.

This 'empty set' -- it is "empty" because we can say nothing about it -- could be the source of *absolutely everything*,

The math could be at the tip of your fingers [Mat 7:7].

The critical comments from you and your colleagues will be greatly appreciated. I will keep them strictly private and confidential.

With deep respect, and very best wishes,

There is no conversation more boring than the one where everybody agrees.
Michel de Montaigne

And a fool waits for an answer.
Heinrich Heine


[Ref. 1] Anastasios Mallios, Quantum gravity and "singularities",
physics/0405111 v1. 25 pages. Note di Matematica, 2004 (in press).

"2.1: to perform differential geometry, no "space" is virtually required (in the usual sense of the standard theory (CDG), viz. a smooth manifold), provided that one is equipped with a basic differential, d, alias "dx", along with the appropriate "differential-geometric mechanism", that might be afforded thereby.

"2.6: the (differential-)geometric mechanism, in the formalism of ADG, does not depend on (emanate from) any "background space", in the sense that the latter term is, at least, understood in the classical theory (CDG)."

[Ref. 2] Anastasios Mallios and Ioannis Raptis, Smooth Singularities
Exposed: Chimeras of the Differential Spacetime Manifold, gr-qc/0411121
v1, pp. 10-15.

[Ref. 3] Ioannis Raptis, Presheaves, Sheaves and their Topoi in Quantum Gravity and Quantum Logic, gr-qc/0110064 v1, Sec. 2.

[Ref. 4] Anastasios Mallios and Ioannis Raptis, Finitary Cech-de Rham
Cohomology: much ado without smoothness, gr-qc/0110033 v10.
Int. J. Theor. Phys. 41 (2002) 1857-1902.

"(T)he central didactic point learned from ADG is that one should in a sense turn the tables around and instead of using algebras of C[inf]-functions to coordinatize (as it were, to measure!) space(time) when, as a matter of fact, these very algebras derive from the differential manifold space itself, one should rather commence with a structure algebra sheaf A suitable to one's physical problem and derive space(time) and possibly its (differential) geometric features from it. Algebra (ultimately, dynamics) comes first; while, space and its (differential) geometric properties second.

Footnote 57: (W)hat is of importance for ADG is more the algebraic structure of the 'objects' living on 'space(time)' -- which algebraic structure, in turn, is conveniently captured by the corresponding algebraic relations between the sections of the respective sheaves -- rather than the underlying geometric base space(time) itself."

[Ref. 5] Carlo Rovelli, The century of the incomplete revolution: searching for general relativistic quantum field theory, hep-th/9910131
v1, September 6, 2002. J. Math. Phys. 41 (2000) 3776-3800.

[The lesson of general relativity: diffeomorphism invariance]

p. 3: "At the classical (non quantum) level, this novel view of space and time is expressed by the use of physical theories which are still defined over a "spacetime" differential manifold M, but which are invariant under (active) diffeomorphisms [psi]: M --> M of the spacetime manifold  M  into itself. The maps [psi] form a group, denoted Diff(M).

"One of the main results of twentieth century's fundamental physics is that at the fundamental level the physical world is described by theories with property (1).

"Property (1) implies that spacetime localization is relational, for the following reason. If (XXX) is a solution of the equations of motion, then so is (XXX). But [phi] might be the identity for all coordinate times t before a given t_0 and differ from the identity for some t > t_0. The value of a field at a given point in M, or the position of a particle in M, change under the active diffeomorphism [x]. If they were observable, determinism would be lost, because equal initial data could evolve in physically distinguishable ways respecting the equations of motion.

"Therefore classical determinism forces us to interpret the invariance under Diff(M) as a gauge invariance: we must assume that diffeomorphic configurations are physically indistinguishable. That is

        [XXX]                   (2)

where [-] means physically indistinguishable. A (classical) physical gauge invariant state of the system s is not described by a field configuration (or by the location of the particles), but rather by the equivalence class of field configurations (and particle locations), related by diffeomorphisms.

"The quantities that have physical meaning, namely that can be predicted by the theory once the state is known, or whose measurement gives information on the state of the system, are diff-invariant quantities, that is, functions Q of the dynamical variables [phi] and X that satisfy

        [XXX]                   (3)

"These quantities are the "observables" of a general relativistic theory. They do not have a dependence on the coordinates  x , and they are not functions on M. Indeed, anything which depends on the coordinates  x  or is a functions on M is gauge-noninvariant, and therefore does not represent a physical quantity."

[Ref. 6] W. Pauli, in: Der Pauli-Jung-Dialog und seine Bedeutung für die
moderne Wissenschaft, ed. by H. Atmanspacher, H. Primas, and E. Wertenschlag, Berlin: Springer, 1995, pp. 317-330.

[Ref. 7] The Collected Works of C.G. Jung. Volume 18. The Symbolic Life. Princeton University Press, Princeton, 1975, Par 7.

"Body and mind are the two aspects of the living being, and that is all we know. Therefore I prefer to say that the two things happen together in a miraculous way, and we had better leave it at that, because we cannot think of them together. For my own use I have coined a term to illustrate this being together; I say there is a peculiar principle of synchronicity active in the world so that things happen together somehow and behave as if they were the same, and yet for us they are not. Perhaps we shall some day discover a new kind of mathematical method by which we can prove that it must be like that."

Quoted after: Hans Primas, Time-Entanglement Between Mind and Matter,
Mind and Matter, 1(1), 81-119 (2003).

[Ref. 8] I am grateful to Ioannis Raptis for pointing to me the notion
of Holon, as elaborated by Arthur Koestler.


Subject: The need for Non-Archimedean Scalars
Date: Thu, 02 Jun 2005 11:16:58 +0300
From: Dimi Chakalov <>
To: "Prof. Elemer Elad Rosinger" <>
CC: "Isham, Christopher J" <>

Dear Elemer,

Thank you, once more, for pointing to me your math.HO/0505336 v1, from 16 May 2005. Please correct me if I'm wrong. Perhaps Chris could correct my statements, too.

You said (Eq. 2.6, pp. 9-10) that it is possible to construct "a well defined infinitely large scalar, with which one can do without any restriction whatsoever all the usual operations in a field." I wonder if it is possible to define rigorously Lebesgue measure on such non-Archimedean entity,

My primitive understanding of non-Archimedean entities boils down to their ability to "create" and "absorb" 'elements of reality', and remain *unchanged*,

Hence I was curious if Anastasios Mallios could introduce an *empty set* from which brand new things could emerge along the cosmological time arrow,

I just don't like unitary "evolution"; it's boring and isn't evolution. Besides, it seems to me that we have a lot of non-Archimedean "dark" stuff to consider,

As for the presence of zero divisors (p. 23), I recall that, in biquaternion algebra, you have zero-divisors, i.e., non-zero elements which product is *zero*; please see Elio Conte's "Biquaternion Quantum Mechanics",

I wonder if you or Chris can use this zero product to model the 'empty set' above, hence define 'creatio ex nihilo' along the cosmological time arrow, as well as 'dissolving into nihilo' along the deflation time, down to [John 1:1].

I'm kindda biased, you know. Sorry.

Best regards,


Note 1: I'm biased, yes. But not without rational reasons, I believe. Let me elaborate on the crucial issue of non-Archimedean virtual/potential reality in GR, as outlined here.

We need new physics to address the cosmological constant puzzle: the cancellation of all but one part in 10122, as mentioned here. I speculated in November 2002 about some virtual pool of negative mass, which could serve as the "dark" energy engine of accelerated expansion of the real, positive mass pool of the Universe in the local mode of spacetime. Hence the question: What could be the total virtual pool of the Universe in the global mode of spacetime, such that (i) it could produce a local mode of spacetime consisting of two worlds with "inverted" spacetime basis, material and tachyonic, separated by a timeless luxonic "point", (ii) provide a dynamic adjustable "fine tuning" of the dark energy, and (iii) provide for an almost zero cosmological constant at the current epoch, by negative mass/positive mass cancellation process: all but one part in 10122. We cannot even address these issues, since we haven't incorporated the three kinds of mass in GR; see Yakov Terletsky here. Currently, we eliminate the "nasty" negative sign of mass by hand (Tom Roman), but can neither explain the obvious conflict between quantum physics and gravity (Carlos Barcelo & Matt Visser) nor the empirical fact that the gravitational waves from the dipole mode are completely canceled, as confirmed by LIGO so far. We just "design" positivity mass theorems to explain the empirical fact that there are no observable effects of the negative energy densities at the local mode of spacetime, despite the clear evidence that QFT should be full of such "negative" stuff; see the famous cat Macavity.

Can we extend the phase space of GR to include the non-Archimedean virtual reality? A hint from Steve Carlip's web site might be elucidating, I hope. Perhaps we need additional complex filed degrees of freedom, as I tried to speculate here.

Got a headache? I do. All this is not my cup of tea, definitely. All I can do is to speculate on the possible "entry point" in the current GR, at which these speculations might be somehow "inserted": Lorentz and Levi-Civita's conservation laws [Ref. 9].

It seems to me that both Levi-Civita and Einsten were right, since they implied two different cases, the global and local modes of spacetime. Note also that the "entropy" of the virtual reality in GR must be always zero [Ref. 9], as we could expect from a non-Archimedean virtual reality resembling the Platonic realm.

It is simply a third kind of reality, both physical and mental, as I tried to argue above. It is the reality of 'elephant's trunk'. This is not an original suggestion; just recall Spinoza, Leibnitz, and Pauli & Jung. All of them being systematically ignored by the established theoretical physics community.

I could elaborate more on the "entry point" in current GR, by quoting from the beautiful innovative research paper by Fang-Pei Chen [Ref. 9], but I'm afraid I am getting far too boring, and will stop here. I personally read everything from Chinese physicists, since I am convinced that they are incredibly good. Ex Oriente Lux.

D. Chakalov
June 3, 2005


[Ref. 9] Fang-Pei Chen, A New Interpretation about the Evolution of the Cosmos, gr-qc/0506007 v1,

[Sec 2.2. Lorentz and Levi-Civita’s conservation laws of energy-momentum tensor for gravitational system including matter fields and gravitational fields]

p. 3: About eighty years ago Einstein did not agree with these conservation laws; the only reason given by him is that these conservation laws "do not exclude the possibility that a material system disappears completely, leaving no trace of its existence" [8], because Einstein believed that the relation expressed by Eq. (5) should make a material system, being [XXXX] in the initial state, to [XXXX] --> 0 spontaneously.
[8]. Cattani, C. and De Maria, M. (1993). Conservation laws and gravitational waves in general relativity. In: The Attraction of Gravitation. Edited by Earman, J., Janssen, M. and Norton, J. D. (Birkhauser. Boston)

"Therefore in the complete disappearance process of this gravitational system its entropy should decrease to S =0 from S >0; this is contrary to the theorem of entropy increase; hence a gravitational system can not disappear completely and spontaneously.

"The energy density of matter field is always positive, so according to Eq. (5) the energy density of gravitational field should be always negative.

"From Eq. (5) we get [XXXX] immediately, this relation means that for an isolated gravitational system if the energy-momentum of matter field increases, then the energy-momentum of gravitational field should decrease, i.e. the energy-momentum of gravitational field might transform into the energy-momentum of matter field.

"This possibility might occur in reality, since the number of microscopic states both for matter field and gravitational field should all increase in this process so that the entropy of the system increases. It is worth to remember that in the above process the absolute value of gravitational field energy is increasing, thus the number of microscopic states for gravitational field should increase also. This possibility could be used as an important basis for establishing an alternative cosmology.

p. 4: "On the other hand we have shown in section 2.2 that the eneergy-momentum of gravitational field might transform into the energy-momentum of matter field; this energy-momentum transformation is equivalent to the creation of matter field’s energy-momentum (and the decrease of gravitational field’s energy-momentum).

"The creation of matter field’s energy-momentum is a useful concept. This concept had been introduced first in the steady state cosmology [6]; in order to reflect the creation of matter field’s energy-momentum, Hoyle had modified the Einstein equations by adding a correction tensor D_mv.

pp. 11-12: "(3). It uses the Lorentz and Levi-Civita’s conservation laws as one of its theoretical foundations. It means that the energy-momentum of matter field might create from gravitational field.

"How is the energy-momentum transformed from the gravitational field into the matter field? These problems relate with the quantum theory of gravitational field. On account of a complete and consistent quantum theory of gravitational field has not been constructed yet till now, so we can not reply fully these problems at once now.

p. 13: "It is reasonable to interpret [XXX] as the density of dark energy.

"The SBBC has a starting state called big bang and assumes that the total energy of matter fields (including the inflation field) has existed from the big bang; moreover, this theory does not study the origin of the matter field’s energy.

"The new theory of cosmology established in this paper has no big bang, it is without a beginning and without an end; the space expands continuously. The view of no beginning means that the state t=0, p_M=0 does not exist.

"The new theory uses the Lorentz and Levi-Civita’s conservation laws as one of its theoretical foundations.

p. 15: "1). Testing the Lorentz and Levi-Civita’s conservation laws

"Various concrete experiments and observations using the specific properties of gravitational waves to test the Lorentz and Levi-Civita’s conservation laws were enumerated in Ref. [12]. These conservation laws are the foundation of the new theory; their correctness means that the energy-momentum of the matter field might create from gravitational field.

"So that to confirm these conservation laws is to confirm indirectly the new theory of cosmology and to disprove SBBC, since SBBC does not permit the creation of matter field’s energy-momentum.

p. 16: "As it has been explained in the introduction section of this paper, the new evidences of observations have brought out some crucial weaknesses of SBBC. It is necessary to introduce new concepts and new laws; the main objective of this paper is to show such necessity and to derive a new alternative theory of cosmology.

"The current work is only preliminary and it is hoped that this work may generate further interests and studies in establishing a better alternative theory of cosmology."


Note 2: Let the reader be reminded that I'm psychologist, and don't need quantum gravity to harness PHI. Why would a fish need a bicycle?

More at EPS13 in Bern, during the poster session on Tuesday, July 12, 2005, from 15:20:00 to 16:40:00 only. I will demonstrate an effect of the Holon of the human brain, which was predicted by the theory proposed at this web site. It's not "dark energy". No. Forget about "gravitational waves" either.

Interested? Bring your digital camcorder.

D. Chakalov

The Roman Rule: The one who says it cannot be done should never interrupt the one who is doing it.