What if Natural Numbers Are Not Constant? by Jerzy Krol

Dear Yuri,

Your question seems to be well-posed only when you consider the title of my essay. However, the title does not suffice to validate the whole approach.

Besides, what you mean by fix value of 'constants of physics'? This is what you measure? Is this the same for you as the value of a specific number is? The constants of physics are elemnets of formal structure of numbers or rather they belong to physics?

Please, read the essay than we come back to the discussion.

Especially:

'there are constructions in mathematics which require different models of natural numbers. Some of such constructions are relevant to physics.'

Not all.

Jerzy

Dear Cristi,

Thanks a lot for your looking at my essay and for your interest. Sorry for some delay with answering - fortunately Torsten got the point of your questions - thanks Torsten for that!

Let me add that on one hand exotic R4 are ordinary smooth 4-manifolds, though curved, on the other they carry depth of topological and geometric oddities. These are: Casson handles and wild embeddings which might correspond to a state of QG (as Torsten beautifully explained in his essay); exotics also can be parts of string backgrounds which gives the fresh relation to existing theory of QG; or they refer to quasi-modular data which again refer to string theory and to the Seiberg-Witten theory data on them. This last can be a way to find a natural gravitational instanton assigned to exotics. Previously (1985), it was Witten who considered exotic spheres (though higher dimensional, i.e. 7, 11) as the most 'physical' examples of gravitational instantons. Currently, we are just working on this point with Torsten and trying to dig out gravitational instantons in exotic R4. That is funny, but before the contest I found and downloaded one of your singularity papers when we tried to make some progress in this exotic stuff.

Regarding the connection of exotic with foundations of mathematics there is something in it which we do not understand fully yet.

Thanks again, and now I will try to go through your essay.

Regards and best wishes,

Jerzy

Dear Dr Fisher,

thanks for your kind interest in my essay and for your words. I think that abstract tools as used in mathematics, are just tools which enable us to express, sometimes, more than in ordinary language. Like electron microscope. In the same time, mathematics allows for detecting its own variation and makes it precise. It does not mean that all this is physically valid. But, sometimes, it is the case.

I wish you good luck too,

Jerzy

Thanks Torsten for your questions and interest! I only briefly commented about these crucial issues in the essay.

Yes, indeed the inclusion of the non-isomorphic models of the numbers lead automatically to the construction of 'different', which, at this stage, are model-theoretic self-dual. This means, in particular, that, in their construction, one makes essential use of tools of model-theory. However, the model-theoretic self-duality allows for considering structures which 'survive' such constructions as 'classical' objects. Even the resulting structures would be finally expressible purely in terms of standard real numbers, they can not be standard smooth structures. This is Theorem 2. Now, the point is that that this formulation in terms of standard reals is possible only in dim 4, because, as you know well, there are only exotic [math]$\mathbb{R}^4$[/math] and there do not exist exotic [math]$\mathbb{R}^n, n\neq 4.$[/math] However, model-theoretic self-dual smooth structures might exist in different than 4 dimensions, though they can not servive as a classical objects. If they did, there would existed exotic [math]$\mathbb{R}^n$[/math] for general n.

It is conjectured, that the intrinsic 4-d reasons explaining this situation, are forcing constructions. Forcing is known from the history of set theory as the way of proving some independence results and as the changing the models of set theory (ZF). In our case, only in 4-d there are infinite trees associated with Casson handles and on such trees one can define non-trivial forcing. This is called forcing adding the Casson handle. This indicates on the origins for changing the models of set theory, and the reals in these models, exclusively in dim. 4 and in the context of exotic 4-geometry.

That is why, extending the GR equivalence principle over different models of reals includes the effects of exotic [math]$\mathbb{R}^4$[/math] into a theory.

Best wishes,

Jerzy

Dear Torsten,

... in my last post 2 words are missing: 'smooth structures', so the sentence should read now:

'Yes, indeed the inclusion of the non-isomorphic models of the numbers lead automatically to the construction of 'different' smooth structures, which, at this stage, are model-theoretic self-dual.'

sorry for this omission...

Dear Ben,

thanks a lot for your interest in my essay and exotic 4-smoothness. Your questions are excellent. I mean they indicate the essence of the approach.

Let me try to answer them as far as I can.

A.1. In fact the 'classical' non-standard models of reals used by me are those found by A. Robinson in 1960s (nonstandard analysis). They contain infinitesimals and infinite large numbers. However, the nonstandard models are related to the first-order language and properties, only. Topology on R, as a family of open subsets, requires 2-nd order language. There exists however different perspective, yielded by category theory especially toposes. E.g. in the smooth topos found by Moerdijk and Reyes, one has notion of differentiability, continuous topology, and reals in such topos are nonstandatrd reals by Robinson. Naturals as well. From that perspective one can approach more directly the fake differentiability on exotic R^4 (cf. J. Król, Exotic smoothness and non-commutative spaces. The model-theoretic approach, Found. Phys. 34, 843, 2004). However, logic and set theory are not any longer classical - they are intuitionistic.

It is fair to say that the relation of models constructions and exotic 4-smoothness is still rather conjectural from purly mathematical point of view, though work on completing this programm is under development and relation with models for spacetime in physics is promising.

A.2. There exists, more-or-less natural notion of order and topology on general families of small and large exotic R^4s. However in our, with Torsten, approach we propose to relate to the radial family of small exotic R^4s. In this case exotic R^4s are parametrized by the real radius and all are embedded as open subsets in standard R^4. The radius is, in the same time, the value of the GV invariant of codimension-1 foliations of certain 3-manifold, say S^3. This is, in fact our main technical construction allowing for the relation with QM. Geometry of foliations is non-commutative geometry of Connes.

A.3. Exotic R^4s can not be scaled smoothly, i.e. thgeir smoothness has to extend to infinity. Otherwise we would have immediately exotic S^4, but we do not (open 4-d smooth Poincare conj.).

A.4. Given the radial family, as above, of exotic R^4s we (with Torsten) showed that magnetic monopoles in 4-spacetime have similar action as open 4-regions with exotic smoothness in this spacetime. As the consequence the quantization of electric charge can be explained by the exotic 4-region in spacetime and not necessarily by the monopoles. There is also the connection of exotic R^4 with quantum (condensed) matter, heavy fermions etc. Interestingly, this kind of relation is obtained (the work in progress) from string theory and Seiberg-Witten YM theory via quasi-modular expressions. I think it is also fascinating thread (though, certainly, not in the main-stream). The exotic R^4s which were recognized as having the connection with Kondo state are those with integer GV invariant (the square-root of the radius in the radial family).

A.5. a) Even though the relation of exotic R^4s with model theory is still conjectural mathematically, there exists a considerable chance for its completion. All exotic R^4s have the leyer of 1-st order model theoretic properties, so the model theoretic-self-duality is the case for all of them. But, we do not have a proof or the construction at present. So, in principle 'yes' but, in fact, we do not know. On the other hand, maybe, there is something else in the model-theory constructions appearing here, which would be ineteresting and were not simply reduced to exotic R^4s.

b) In some natural understanding of the relation (Robinson's models) 'Yes'. But when turning to categories it is not so sure (I would have to give more precise statements and definitions).

c) I am not quite sure what you mean; if it is the commutativity of the diagram like:

N ---> R

| |

*N -->*R

then, it is indeed obviously commutative.

B.1. Exotic R^4s are ordinary, smooth, Euclidean 4-manifolds, though, necessary curved. The absolute Poincare invariance is not the case as usually on curved manifolds (the notion of particle is not absolute). However, exotic R^4s are connected with the non-perturbative regime of some QFTs (monopoles and others) in a curious way. This means that exotic 4-geometry is determined by effective matter (or couples to the effective matter) not just by energy-momentum tensor as in GR but in a different way. In fact this is our (with Torsten) current field of interest and we try to understand this better. We do not know precisely what is the fate of Poincare invariance with respect to the effective matter coupled, in the above sense, to exotic 4-geometry.

B.2. Exotic R^4 is just classical Riemannian 4-geometry (see 1 above), but it is related to foliatioons so to non-commutative geometry. When smoothness is standard, there is no foliation (and no wild embedding as appearing in Torsten's essay) and it remains only classical geometry. Exotic smoothness indeed crosses the classical-quantum border.

B.3. Certainly there should be relation to path integral - exotic smooth spaces should be present in QG path integral. Moreover, we made an attempt in calculation of path integral on exotic R^4 directly (arXive: ). Another impacts derives from string theory and quasi-modularity. By this one can find gravitational instantons associated with exotic R^4s such that the semi-classical calculation of path integral can be expanded on these instantons.

I hope that this helps and I was able to answer partly your very essential questions. I appreciate your interest and the questions.

Now, I turn to reading your essay. Please, give me a couple of days.

Regards,

Jerzy