#### 9.01. Cosmology and calculating models

Cosmology, the science of the entire physical Universe, from the beginning of its contemporary development, that is from the time when Albert Einstein wrote down a single set of mathematical equations and suggested that it represent the entire material world, has employed models for presenting its results. Models, in the sense used in cosmology, are strict mathematical conclusions drawn from simplified but distinctly formulated assumptions. Other branches of astronomy and physics also used models. There were models of atoms, models of stellar interiors, models of the Earth's core....Models were first used for understanding structures and processes not accessible to direct observation. The atoms were too small to be observed; the stellar and terrestrial interiors were screened by the outer shells of those bodies. Similarly, the regions of the Universe situated beyond the cosmological horizon were unobservable and thus needed to be represented through models.

However, the difference is that in other sciences those processes, though unobservable directly, have some impact on the phenomena accessible to human experience and so can be indirectly compared with the observations. Thus, models can be somehow checked, compared with reality in the same way as any other hypotheses. So it is the case with, for example, quantum mechanical models. Even the quark models of elementary particles yield some observable consequences. Yet the matter looked otherwise, from the very beginning, for cosmological models. When the cosmological horizon appeared in the first model calculation of the Universe in the 20th century, it was obvious that all occurrences beyond the horizon can have no influence on the observable part of the Universe; they never cause any observable phenomenon. Thus, one has a limited possibility of comparing the properties of calculated cosmological models with reality; it can be done only locally, in an exceedingly small part of the Universe. All the global features of models must, however, remain forever unchecked in any direct way. Thus, in other branches of human knowledge, models served as supplementary tools for investigating reality. In cosmology, they were rightly considered the only method possible. Constructing models became something characteristic of cosmology. Those first models appeared in times when numerical calculations had to be performed with the help of arithmometers, mechanical calculating machines, when the obvious tools of theorists were the analytical, mathematical deduction and transformations of formulae. This changed diametrically with the invention of computers. Making models became easier. At present, thinking in terms of models is standard in almost all branches of the sciences and even in the humanities. Sometimes constructing models is considered theoretical work. Some people do not discriminate between the basic significance of theoretical considerations and the secondary significance of numerical calculations based on them. Models are also provided in disciplines where reality can be observed without difficulty. In such situations, models serve as a simplified way of depicting reality. Such simplified models, simplified pictures of reality, can be used for easily predicting events in any possible domain of interest. In this respect, cosmology with its numerous models of the Universe does not seem to be so different from other sciences. There is little understanding of the difference between cosmological models and other types of models widely used today.

#### 9.02. Cosmology without models

Question (1) may be put thus: is constructing models the only possible way of getting knowledge of the Universe as a whole? However, before we start to discuss this question, a more precise notion of the scientific model has to be introduced. In principle, any case of simplification in science may be called a model. One could say that even in the simple situation when we explain the motion of a thrown stone by using notions of free fall and inertia, we use a model. And, in fact, the transition between applying such straightforward concepts like inclined plane and very sophisticated ones like steady state Universe is a continuous one. Nevertheless, some line of division, even when not a sharp one, can be drawn here.

Even in our daily life, we distinguish between 'concept', which the Germans call 'Begriff' and 'mental picture,' or 'representation' (German: 'Vorstellung'). One could say that it corresponds more or less to the difference between 'concept' and 'conception.' To communicate our thoughts to other people, or even to comprehend our own thoughts better, we have to use some 'notions', some 'concepts,' some logical units which are either so simple and need no definition (we call them elementary notions), or which can be defined with the help of other notions. Even though such concepts can be used for emotional descriptions, they remain inherently objective. When a human being wants to comprehend some sensual perceptions, to describe them in an impersonal way, he uses concepts. We make efforts to put our feelings aside when creating concepts. However, our life, even our scientific life, would be completely impossible if we were to restrain from any feelings and emotions. Therefore an important ingredient of our activity is the power of imagination. When we personally experience a single notion (concept) or a complex one, when we supplement them with our feelings, our personal propensities, we create mental pictures, conceptions. When such a conception gives rise to a more or less compact picture of some physical reality, it can be called a "model" for this reality. Cosmological models are special examples.

A theory, as regarded in this book, in opposition to models, is a logical structure consisting solely of concepts in the given sense. As (1) stated above, there is no sharply limiting line between theories and models. Nevertheless, there is an obvious difference between such constructions from various epochs like the Theory of Four Elements, the Theory of Flogiston, the Theory of Evolution, the Theory of Sets or General Relativity on the one hand, and, on the other hand, such model descriptions as the Cosmological Model of Eudoxos, the Model of Jupiter's interior, or models of economical growth.

So understood, modeling is of long standing in cosmology. Mathematically calculated cosmological models were used even in antiquity. Geometrically conceived "systems of worlds" were constructed in times when constructions of such a type were completely unknown in other disciplines. However, this tradition of modeling in cosmology is not the only one. If we choose to take a closer look at antiquity we find not only Eudoxos with his model, but also, for example, Epicurus with his general, philosophical considerations of the Universe. At that time, philosophical argumentation was a received style of working in the domain that is now called science.

In more recent times, a classic example of the non-model kind of cosmological research was the cosmological paradoxes. Let us analyze the Photometric Paradox, called also Olbers' Paradox. There is a combination of assumptions. The mathematical assumption: (i) our space is Euclidean. The physical assumptions: (ii) the surface brightness of a body is independent from its distance from the observer, (iii) the principles of propagation of light are valid in space up to infinity. The astronomical assumption: (iv) there is no obscuring matter in astronomical space. And, the philosophical assumptions: (v) the Universe is homogeneous and isotropic, (vi) the Universe is infinite, (vii) there is no general evolution of and in the Universe. In fact, some other assumptions were involved here too, but they seemed at that time and to date, so obvious that as long as nobody introduces any assumptions contradicting them, they are not worthy of mention. Such hidden assumptions include, for example: the validity of ordinary arithmetic, the linear character of time, the validity of our logic in all times, etc., etc. An elementary calculation made under those assumptions shows that the line of sight, in every direction, within a certain distance must intersect the surface of some star. Thus, one can draw a conclusion that in every direction the celestial sphere should shine with the surface brightness equal to that of an average star. The "paradox" carne about because all the assumptions were considered at that time as the most obvious ones, and still the conclusion did not agree with the elementary observation that the night sky is quite dark.

No model is constructed here in the sense given above. Out of some general assumption, a simple but general conclusion is drawn. And the fact that this conclusion did not agree with observations showed that at least one of the assumptions was wrong. Thus, out of this argument we can learn something about the entire Universe: one or more among the accepted properties are not valid over all of it. This is certainly a negative kind of knowledge, but by eliminating some possibilities we do approach the true idea of the Universe. Most of today's cosmologists are convinced that out of the 7 main assumptions, only two (v - homogeneity, and vi - infinity of space) remain (approximately) valid. All the other ones are wrong. In fact, this last statement is founded on the knowledge of only the observable part of the Universe. The Universe at large may be, for example, Euclidean, and perhaps there is no general evolution (e.g. model of Jaakkola). Thus we cannot be sure which ones among the assumed properties are actually wrong. However, we can be absolutely sure that some of them (at least one) are unfulfilled in the Universe at large. The cosmological paradoxes are usually underestimated. In fact, they demonstrate clearly the power of thinking, which is able to make up for an absence of other types of perception and penetrate otherwise inaccessible regions.

We have arrived here at an answer to Question (1). This answer is negative, although few examples of non-model cosmological results can be cited to date.

#### 9.03. Cosmological principles and the evolution of world views

The premises of cosmological paradoxes are of the same type as general assumptions formulated as cosmological principles. The very foundation of the Generalized Copernican Principle, the assumption of homogeneity and isotropy, can easily be found among the assumptions used for formulating the photometric paradox. Question (2) can be stated thus: are assumptions in the form of cosmological principles necessary for cosmological research at all? (cf.: Klotz 1979) To address that question, let us once more make a short review of the basic cosmological principles. The first known cosmological principle was conceived by the ancient Indian culture, where spiritual existence was considered the only important mode of being. This principle is so sublime in content that it is still impossible, even applying modern mathematics, to construct any specific model based on this principle. The cosmological principle based on the half-materialistic world view of the ancient Greeks is also half-materialistic and looks, from today's standpoint, like a mockery of reality. It is neither truly spiritual, nor strictly materialistic, trying to reconcile two, in fact contradictory, tenets. Copernicus furthered this process of considering the sensual world an agglomeration of material bodies. His cosmological principle, in contrast to their predecessors, regards also planets (i.e. a certain kind of celestial bodies) as physical bodies. A generalization of his principle leads us to consider all that we can perceive with our senses - all the Universe - a physical entity. In the course of this process, the Universe becomes, in the minds of scientists, more and more homogeneous.

The generalization of Copernican views seems to be self-evident for cosmologists. Sometimes they even do not take notice of what they assume (Einstein). However, here an important fact is involved. There was increasing awareness in the first half of the 20th century that there is something like a cosmological principle and that it is not given from above but rather assumed by cosmologists by choice. The period of purposefully creating cosmological principles began. One can find attempts to weaken the Copernican Cosmological Principle (the Softened Copernican Principle of Zieba, see 4.22). However, these attempts are not very typical. Better known and more characteristic of this epoch is the attempt to make the Copernican Principle stronger and narrower, the Perfect Principle. From a certain point of view, the birth of the Perfect Principle can be regarded as a final move towards a thoroughly materialistic view of the Universe, as it is indeed viewed by some conscious and dedicated materialists. For example, Jaakkola (1989) considers the Perfect Cosmological Principle as the only one satisfying the requirements of strict scientific thinking free from any metaphysical ideas. Therefore, he proposes to call it simply "the Cosmological Principle" without any other adjective. Can one proceed further in this direction? In an abstract way, yes; in any relation to reality, no (cf.: 5.10).

In the middle of the 20th century there began a peculiar process. Some considered it a new trend towards a spiritual comprehension of the world, some - just the opposite - as a tendency to reduce everything (including life) to a sub-physical reality, and still others considered it a complex mixture of both. Whatever it is and however it will be called by posterity, this trend marked a distinct departure from the classical form of materialistic science of the 19th century. Carter (1984) maintains that the Copernican standpoint (that the Earth is an average celestial body) was merely an unfounded reaction to the ancient Greeks' endowing a privileged position to the Earth as the center of the Universe. It seems that there exists a general trend today at least to bring together, if not to reconcile, many old and new convictions and principles, not only the cosmological ones. What are the main symptoms? The strict causal relations in physics are replaced with probabilistic causality for microcosmic phenomena and in certain sense also for macrocosmic ones (cf.: Neyman and Scott 1959). Some practices of folk medicine, rejected before as sheer superstitions, are generally accepted and practiced by certified physicians (acupuncture, acupressure etc.). The same pertains to some systems of esoteric medicine (homeopathy). Some astronomers do believe that the entire Universe is a living being (cf.: Hoyle 1988). Some beliefs of astrology are tested in scientific ways and accepted as scientific reality (cf.: Culver and Philip. 1977). Telepathy, the ability of foreseeing the future, telekinesis, and similar phenomena are being examined in official scientific institutes. Certainly this is a period of some scientific turmoil. It is too difficult to evaluate it at present and predict in which direction it will eventually go. In any case, a new cosmological principle - the Anthropic Principle - arose just in this period. Is this a spiritual principle which brings us closer to the idea of purposeful creation of the world, or is it, rather, a mechanistic perspective of seeing the Universe, which make it possible to claim that automata (intelligent and feeling) will replace humanity? The matter here is much confused. This particular cosmological principle is a true child of its scientific time.

#### 9.04. Cosmology and the Gaia Hypothesis

It seems to be not incidental that more or less simultaneously with the Anthropic Principle, another scientific conjecture which also attempts to unite physical existence with elements of life and consciousness appeared. It is the Gaia hypothesis (Lovelock 1979). The hypothesis (which in its original formulation and in the opinion of other scientists, considers the Earth as a more or less "self-aware" organism capable of exploiting other beings for its own good and capable of defending itself but also endowed with some kind of "feelings" toward earthly mankind) belongs - from the formal standpoint - to geophysics but is closely related to cosmology as well (cf.: Follgett 1988). If the Earth is an "organic" being, then what about the other planets? What about all the celestial bodies, what about the entire Universe? One could even think that this hypothesis says more than the Anthropic Principle in its three versions. The Anthropic Principle states that the Universe is capable of producing and maintaining intelligent beings.

If we extend the Gaia Hypothesis to all celestial bodies and their agglomerations, then we obtain a Universe which is not only able to produce and maintain intelligent beings but is also intelligent in itself.

This opinion is close to the old Indian views that the Universe is a Body of some spiritual being (possibly even of the Highest Spiritual Being). However, the basic difference is that the Indians considered only spirit of importance, whereas in the Anthropic Principle and the Gaia Hypothesis such properties as life, intelligence, and consciousness are all closely related to or even (in some interpretations) deduced from material existence. It may be said that in the old Indian times the spiritual aspect of existence was overestimated, and only the later evolution of scientific views led to ascribing more significance to the material aspect of life. Ideas of that kind reached their climax in the Perfect Cosmological Principle. And now it tends towards equilibrium of some sort: acceptance of the physical and the spiritual aspects of the Universe as equally valuable. The spiritual side is sometimes understood as autonomic or even as primary - that which brought the sensory world to existence. Sometimes it is considered secondary, something descending from physical reality. Both these versions do somehow coexist in science. In this new approach to the Universe and to its spiritual-physical existence the modern tendency to attain a certain kind of balance between spiritual and materialistic world views manifests itself.

One can say that the Gaia Hypothesis supports the Anthropic Principle. In fact, the situation is not so simple. The Anthropic Principle provides the argument that the size of a physical and intelligent being striving to knowledge must be approximately the size of a man. If we, following the Gaia Hypothesis, allow the physical intelligent beings to be as large as the earthly globe (if not the stars), then some important arguments usually used in support of the Anthropic Principle have to be rejected. Both the hypothesis and the principle arose out of the same philosophical attitude. However, they do possess some features which are mutually contradictory.

#### 9.05. Future cosmological principles

Can we expect some further cosmological principles? What kinds will appear in the future? That strictly depends on the evolution of scientific world views. So far, we can see an enormous proliferation of models in all the fields of science. This is an obvious consequence of the proliferation of computers. And understandable, too. When there are new tools available it is profitable to see what can be obtained using them. As long as this development proceeds, mathematical modeling will not ebb. Such models must be calculated from well-defined assumptions. For cosmology it means that cosmological principles (old or new) will remain fundamental for cosmological research as long as constructing mathematical models dominates scientific research.

This demand for strictly and consciously formulated assumptions for the purpose of ca1culating models brought about the formulation of the "minor" cosmological principles discussed in the Chapter 7.

However, no scientific style of research has lasted forever. Today we can see the development of some quite novel approaches to scientific investigations, methods which are likely to replace the old ones. Almost all of present day science has developed in the following manner: one makes a hypothesis and then tests whether or not observations and experiments confirm it. Fritz Zwicky (1957, 1959) developed another approach to reality. He proposed not to form individual hypotheses but to take into account all plausible hypotheses and then exc1ude those that have not withstood verification. One thinks here in a rather deductional way, and the research is done using as few assumptions as possible. Zwicky's method has been developed within the general stream of Goetheanism (see. Appendix), which in the last decades has spread among some scientific circ1es. Is this a trend to deliver cosmology from accepting a priori philosophical assumptions, from using cosmological principles?

After the epoch of direct, mental seeing of the truth (Ancient India), after the epoch of coming upon intellectual truths through revelation (Egypt, Chaldea, Babylon), after the epoch of deriving all truths from logical thinking (Greece, Rome, up to the Middle Ages) comes an epoch which tries to found all logical thinking on physical experiment. In its contemporary stage it attempts to calculate models of everything or states hypotheses which have to be checked afterwards by comparing them to reality. In all these epochs the power of thinking was used - but differently in each one.

Should we be so self-satisfied and arrogant as to think that our approach to thinking as a scientific tool is the superior and concluding one? Are we to believe that human thinking cannot be used for scientific purposes in a still better way?

Fritz Zwicky used to say that the most important scientific instrument is the scientist himself and that this instrument must be well set, well adjusted. Do we really think that the scientific attitude most common today is the best possible?

#### 9.06. Is any simplification a shortcoming?

The above considerations were stated with the implicit assumption that the cosmological principles' simplification of the structure of the Universe is their weak point. We want to obtain a true picture of the Universe and so look for a possibility to bypass cosmological principles. This is not the only stance possible. Stanislaw Zieba (1991) maintains that cognition of the actual structure of the Universe cannot be regarded as the true aim of cosmology. The Universe is too complicated, even in its most general outline, to be comprehensible for us. We simplify its picture using cosmological principles because we want to do so.

The principal aim of science is to provide simplified pictures of reality: Even the logic we use as the basis of every scientific argumentation is - according to Zieba - not the way we actually think, not the way we distinguish truth from falsehood, but just a model of our thinking, a simplified model of human cognition of reality. Woe betides him who trusts the received laws of logic as absolute ones. The same situation exists in other disciplines of knowledge. And so it must be in cosmology. The physical Universe is ex definitione the largest object of sensory investigations, and it is enormously complicated, even in the regions accessible by observation. When we want to describe the distribution of extragalactic objects in neighboring regions of our Galaxy, we do not give spatial coordinates of all extragalactic objects, even if they are determined. Such a true picture of the distribution of objects is just incomprehensible and thus useless for us. Also, in cosmology we have to look for the most basic features of the structure of the Universe in order to understand it. In this sense, cosmological principles will be always needed and useful. There can even be a number of cosmological principles, not necessarily consistent with each other, each one revealing a different feature of the Universe. The problem consists in remaining aware of the extent to which each of the principles accepted approximates reality.

One cannot object to this idea, but a remark should be made that in Zieba's view the cosmological principle is regarded more as a result than as a tool of cosmological research.

#### 9.07. Models without principles

We have .arrived at the conclusion that it is possible to use cosmological principles without constructing models and also that one can practice cosmology without involving cosmological principles. The question still remains if it is possible to create models without resorting to cosmological principles. Ellis wrote that "...we are unable to obtain a model of the universe without some specifically cosmological assumptions which are completely unverifiable," - but what sort of unverifiable assumptions did he have in mind?

This question can be understood in various ways. If we ask whether one has to consciously accept some philosophical assumption as a cosmological principle, then obviously the answer is negative. Neither Eudoxos nor Hipparchus was aware of the Ancient Greek Cosmological Principle as such. Einstein, when introducing the assumption of homogeneity of the Universe, considered this just a simplifying mathematical assumption and did not expect that in the future it would be ranked as a cosmological principle. A cosmological principle has to be used first - explicitly or implicitly - by a certain number of scientists and only later acknowledged as such. Even the deliberately created Perfect Principle, which was so named right after its conception, does not breach this rule. However, it may be asked whether one can construct a model involving no philosophical assumptions, only mathematical and physical ones. Of course we do not like to ask silly questions. We are aware that philosophical assumptions are involved in all mathematics and physics and many of them are unverifiable. Thus perhaps the question should be so formulated: is it possible to create a model of a Universe without introducing any assumptions not commonly used in mathematics and physics? This was done several times. It was the case with the model called Schwarzschild's solution in general relativity (cf.: 2.12). This solution was found by Schwarzschild in 1917, when Einstein constructed his first model. It describes the case of all mass concentrated in one point in empty curved space. This was considered at first by all the physicists and by Schwarzschild himself as a solution useful only for celestial mechanics and not relevant to cosmology. Only many years later was it realized that this is a kind of Universe model, remarkable from a methodological point of view. Schwarzschild's solution found an application in the theory of black holes, which are sometimes considered as little universes. Thus it obtained the status of a cosmological model, even if not corresponding to the actual Universe.

Apparently the model of Gödel (1949) was also constructed without any particular cosmological principle. However, in fact, it fulfills an assumption very similar to that of the Zone-Model of Zieba. The mean density of mass in both models is constant all over space. There is a preferred direction in any space point. It is the rotation axis direction for Gödel; it is the direction perpendicular to local density layers in the Zone-Model. The anisotropy distinguishes it from the Generalized Copernican Principle. The model of Gödel also fulfills the Softened Copernican Principle (4.22).

Ellis and his collaborators (1978) proposed the so-called Static Spherically Symmetrical (SSS) Universe Model with our Galaxy in the center of the Universe. Instead of any cosmological principle, two features were assumed, static character and spherical symmetry. One can say that this was a particular philosophical assumption, not a cosmological principle. However, if there were a number of models based on that assumption of spherical symmetry, it would certainly deserve to be ranked among the cosmological principles. In 2.13, I called it the Generalized Ancient Greek Principle.

#### 9.08. What is a cosmological principle?

Up to now, we have considered more than thirty cosmological principles, most of them known only to a small number of specialists. As we have seen, six of them exerted a considerable effect on cosmology. Out of those six, three are often used and discussed in contemporary cosmology. We reviewed specifications of a number of cosmological principles and found that most of them are useful in extrapolating the properties of the observable parts of the Universe to the unobservable ones. We saw also that not all statements, not all hypotheses, proclaiming something about those unobservable parts, can count as cosmological principles. However, we did not come to any general definition;

we still do not know what is a cosmological principle and what is not.

Exact definitions are usually not easy. We had trouble defining what a theory is and what a model is. The same is true for most other areas. Can we strictly define what is (and what is not) a philosophical idea, what is (and what is not) a law of nature, what is (and what is not) science? We have some ideas about them but there are no strict definitions which would be acceptable to everybody.

It cannot be strictly delimited what is a cosmological principle and which, even remarkable and general, statements about the Universe are to be considered as cosmological hypotheses but not as cosmological principles. However, allowing myself to present my rather lenient demands in this area, I propose that a cosmological principle is a statement fulfilling the three following conditions:

a. it states something about important features and properties of the Universe as a whole;

b. it indicates how to extrapolate our local scientific results to the entire Universe, or it explains why such an extrapolation is not possible;

c. it is embedded in some general attitude concerning human knowledge.

Of course any group of cosmologists, or even an individual cosmologist, can have their own judgment regarding what is an "important" and what is an "unimportant" property of the Universe. And naturally everyone will have his own general attitude concerning human knowledge.