We present here the results of analysis of many systems of different kinds, considered from one point of view. The goal of the study was to consider the qualities that are characteristic of all systems and are available for quantitative assessment. As the basis of our approach we can postulate that every system consists of parts.
The number of parts in various systems was studied; subsystems were treated both as parts of systems and as independent systems built of their own parts. First,the average number of parts was estimated for various systems. This average number [geometrical average, since multiplicative approach seems natural for making relative comparisons and studying hierarchical systems] was found to be close to number pi or its integer powers for many different systems.
The first impetus to the research was given by the discovery of unexpected (though not very distinct) size border between large taxonomic groups of sea bottom fauna at the body length of about 2 mm. Further research showed that oceanic bacteria, flora and fauna show clearly visible periodical structure of sizes, in which adjacent groups differ from each other in size (calculated as a cube root of volume) by a factor close to number pi [1].
After that, extensive material was gathered on animals and plants living not only in sea water, but on sea bottom, in fresh waters and on the land, confirming the existence of this structure and the value of the factor as equal to pi,and after the statistics on more than 100,000 species were presented (1969) for the book [2], there appeared much more material, confirming all results of this book.
The very first attempt to link the found order with taxonomic structure of organisms led to the important discovery that the geometrical average number of species in a genus, genera in a family etc. to classes and types is also close to number pi.
The diverse sources of studied data (body sizes and number of taxa) and presence of number pi in metrics of physical space stated three major goals for research: to check whether the discovered factor is really pi, with highest possible accuracy; to find out what other kinds of systems contain this constant; to give a theoretical explanation to this result if it is confirmed by further research.
First,the research was continued on taxonomic structures using for each taxon the latest and most reliable descriptions; no special classifications were made. The average decimal logarithm of the factor for all 40000 genera of world fauna with world surveys available was found to be 0.497 (lg(pi) = 0.49715). The order of the error is equivalent to the one expected from rounding of decimal logarithms to the second digit used in hand calculations. Then, other structures were studied. The analysis of all 111 plays of ten most famous Russian drama writers showed that the average factor, derived from numbers of acts in plays, scenes in acts, appearances in scenes, actors and remarks of each of them in appearances, was different from pi only in the third digit.
Similar studies of large numbers (hundreds) of other structures and classifications, different in size and nature, such as Universal Decimal Classification of library topics (a sample part),human skeleton, contours of letters and digits (as consisting of primitive strokes, in their simplest form), Russian syntax and morphology, different pieces of poetry and prose (including New Testament and Soviet Constitution), world soils, Japanese characters and others always revealed a factor approximately equal to pi. In some cases the factors were close to integer powers of pi, which could mean that some structural levels were ignored in systematization. It is interesting that the average deviation, calculated on a sample taxonomic material, was also found to be close to pi [3].
One can argue that the taxonomic structures we have studied are only semi-objective, while drama and library catalogues are completely artificial [though they, of course, must reflect structures of the things they are describing as well as mental patterns of their creators, shaped by the real world] and that we study no more than the structure of human thought. To make our point clear in this eternal philosophical discussion, we can agree that the subject of our studies is The World As We See It - the only thing that we have available.
It is interesting that the relation of projection between a circle and its diameter, which gives number pi in physical world, is exactly the relation between a subtaxon and a taxon, or, philosophically, between particular and general, the relation in which the extra dimension of specific features is "squashed" - and the factors between the numbers of points are also the same.
Hence, it would be reasonable to suppose that there can be common formalism describing basic structures of physics and semantics of our world - the formalism of Euclidean space.
Of course, world semantics is not a Euclidean space of any dimension, as well as our physical space is not; but if we notice the similarity, it is convenient to think they are - and as always the difference between ease of thought and truth is very vague.
If we try to find the exact value of pi experimentally by measuring circles in our physical space, we'll face two major obstacles: inaccuracy of our experiments, resulting from the imperfection of instruments, and the distortions of our space by various forces; knowing the value of pi in advance, we can estimate the quality of tools we use for measurement and make hypotheses about presence and strength of forces distorting our space.
When the force is removed, the space tends to uncurve and the factor of projection approaches number pi, which in our case means balancing the hierarchical tree of the system's internal structure so that the number of substructures on each level comes closer to pi; we can guess it from the fact that existing systems have come to such balance; we can also watch this relaxation process in development if we apply our statistics to different stages of system's evolution. It also seems possible to estimate the stage of evolution and the degree of perfection of the system by studying its structural factor.
The deviation of the real factor from number pi seems to correspond to the strength of the force distorting the space; if we learn to measure this strength, we can probably write an "energetic" equation linking curvature of the space with value of the force and find a "semantic potential" preserved in their transformations.
Number of parts in a whole seems to be the simplest thing one can count in structures. The next idea was to consider frequency distribution of parts. If we sort words in any book by their frequencies (Zipf distribution) or genera of fauna by their number of species etc., we will see that on logarithmic scale the graphs "place in list - number of occurrences" look like straight lines with tangents grouping around integer powers of other famous values, the golden mean (1.62) and golden wurf (1.31). And here, again, we can in different cases find the same relaxation process.
For instance, if we mark points representing number of people in three major races on the logarithmic scale, we will notice that they lie near a straight line with a tangent close to [minus] golden mean. If current relative life conditions of the three races are more natural and fair than they were in the past, then in the "relaxation" process the points must be approaching the line and the line itself must be slightly turning to accept the corresponding tangent. For this, the first (yellow) and the third (black) races must outpace the white race in growth, with black one growing faster (and/or longer) than yellow. Than, after balancing the line, they must accept equal growth rate. Needless to say, it is exactly the description of current demographic trends.
Of course, there can be many possible explanations to this and other "relaxation" processes, including that of coincidence. But, whether systems come to such balance in the course of evolution or they just maintain it from the moment they come to life, the existence of structural factors works against the destructive influence of entropy.
It seems that many ideas of esoteric fields and fluids as well as the
theory of morphogenetic fields suggested by Rupert Sheldrake are built on facts
based on action of the mentioned and other similar regularities, representing
basic structure of semantic space. We can also suggest methods for testing
other interesting features of these fields:
Non-physical character of the fields can be confirmed if we find that parts
of natural systems developing in physically isolated locations ( e.g., taxa of
land animals on different continents) show coordinated development.
Trans-temporal causation. The evidence for it can be collected if we prove
that structures of systems outlined for all history of their development obey
these or similar laws. Partly supportive to it is the fact that available
structural descriptions of non-static systems (processes), such as geochronological
table, reveal the same regular patterns.
We see the above mentioned and other discovered quantitative system regularities as the basis of a new approach to practically applicable system studies and hope that extensive exchange of ideas in this area will lead to better understanding of semantics of our world.
We would be glad to communicate with all individuals or groups interested in these ideas or carrying out similar research.
Please send your comments to Alexander Chislenko
sasha1@netcom.com