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I. Introduction

Starting with Newtonian Dynamics, the Physics became gradually dominated by the idea that properties of matter extend indefinitely in Space. The Modern Physics still has remnants of that conception, in spite of sometimes explicit denial of the fact. When it comes to checking the truth of that assertion, it is easier to check directionally, for Man’s capability of displacement in Space is very limited. Inasmuch as the Space properties relate to the matter filling the Space, our Physics of Space is similar to Cosmology, and from Cosmology we can borrow one or two ideas regarding isotropy, the notion coming frontward whenever we discuss the matter in relation to Space. It turns out that, from cosmological point of view, the usual perception of isotropy, as invariance of properties of matter to action of rotations in Space, is not sufficient. In Cosmology, for the first time we face the idea that something is missing in the general picture about isotropy, and different reformulations of the conditions defining this general property for the needs of Cosmology show clearly what is this something: the scale of contemplation of the Universe (Sciama, 1961). Indeed, cosmologically the isotropy of the Universe is related to the scale we look at it: obviously even for the naked eye the intragalactic space is not isotropic with respect to distribution of matter. In regards to this, the primary drawback of the human condition is that most of the time we cannot talk about isotropy but at a scale where the Universe is, at least to a certain degree, a figment of our imagination, for instance at extragalactic or atomic level, to take only the two extreme examples. Thus, the judgments about the isotropy of Space are marked by all kinds of uncertainties. However, in spite of all uncertainties, we seem to have a possibility to rightfully declare that this way of taking the scientific freedom is the only reliable one, for the state of the case in Cosmology is just a natural one if we judge it from Geometrical point of view.

    In order to prove this, notice first of all that the Man cannot be stopped from inventing alternatives to the solution of problems in general: this is an essential manifestation of the Man. The freedom here, as everywhere where we speak of Man, is to choose from alternatives, for it only afterwards can be seen if the choice was right or not. This is the only irrefutable criterion of truth. Putting it in scientific terms, we first need to know all the alternatives we have in a certain situation and then set up a choice. In Physics or Mathematics setting up a choice means state a problem. We need to have all alternatives because if alternatives are missing there is also room for further choices and this process can continue indefinitely. The continuation may be good from a technological point of view, but in Fundamental Science it is certainly inadequate. Here one might find the excuse that sometimes we cannot recognize that alternatives are missing. However, the statement of a problem is liable to offer, more than anything else, a criterion of completeness of statement, inasmuch as this statement reflects the (subjective and objective) level of knowledge even to a greater extent than the solution itself. For, as always, the solution to a problem may be partial at best, if not a panacea, but in the statement of that problem our ignorance pops out bluntly: not known, not mentioned, not taken into account as a condition. And the correctness of a problem’s statement can be positively taken into consideration in cases where the problem is supported by a group of transformations, for in that case the alternatives are exhausted by this very property. This fact has been realized first in Geometry (Klein, 1891) and afterward in Physics and Cosmology. Regarding our subject, we cannot express the usefulness of groups of transformations better than by the words of Jaynes, referring to a fundamental principle of knowledge where this tool proved to be involved (Jaynes, 1973):

“…the principle of indifference may, in our view, be applied legitimately at the more abstract level of indifference between problems; because that is a matter that is definitely determined by the statement of a problem, independently of our intuition. Every circumstance left unspecified in the statement of a problem defines an invariance property, which the solution must have if there is to be any definite solution at all. The transformation group which expresses these invariances mathematically imposes definite restrictions on the form of the solution, and in many cases fully determines it.” (Italics ours).

Now it is clear that, if the isotropy can be conceived as based on properties of a group completely characterizing the Space, invariance with respect to this group refers to the circumstances left unspecified in the statement of the problem of isotropy. Initially this group was obviously that of rotations – the Universe appears the same in every direction, indifferent of distance – but the discoveries of the last century forced essential amendments to this perception. Indeed, it was discovered that the Universe does not look the same but starting from a scale where the nebulae are fundamental units. If our Galaxy is a nebula, the intragalactic space is not isotropic with respect to distribution of matter. The isotropy is not an invariant property with respect to scale, thus showing that the definition of isotropy as invariance to rotations only is not complete. Something must have been forgotten when the problem was stated. And indeed, this is the case.

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