I have written this blog about the Foundations of Medicine because I believe that most of practioners of medicine dont know what the foundations are about.

They act as medical engineers trying to repair what they have broken and stumbling upon inventions they did not dream about before.

I also believe that the current foundation of Medicine, BioScience is not curing diseases but is generating a lot of new diseases that never are cured. They become Chronic always needing new medicine,

Medicine is the science and practice of establishing the diagnosis, prognosis, treatment, and prevention of disease whwere a disease is a particular abnormal condition that negatively affects the structure or function of part or all of an organism,.

Medicine is about the Quality of Life.

When you look at the facts you will see that current  Medical Science is realizing the vision of  the Rockefeller Foundation to take over control of Evolution to create a Super Human.

Many people are totally not aware of this and believe we are part of a spectacular rise of science that will not only Cure all diseases but will also make it possible to choose for an Optimized Body, an Optimized Brain connected to a Super Computer and Immortality.

We will be Gods with the Gods.

The History of Western Medicine starts with the creation of the Hospital by the Church of Rome.  At that time taking care of the sick was part of the Seven Acts of Mercy.

When the Church lost its wealth, the responsibility for the poor sick in  the hospitals was given  to the Merchants of the big Cities who started to Make Money out of Medicine.From that time on Medicine was Big Business.

The hospital at that time looked like a church.

Research in medicine started when it was allowed to Open the Body and examine all the parts.

Chinese medicine became holistic because it was not only Forbidden to open the body but also to touch the body only at the Pulse.

The practical treatment of disease was given to the (Guild of) the Barbers who even made use of acupuncture. Later the Apothecary took over a part of the job because of their knowledge of herbs and medicine preparation.

In the 18th century medicine started to treat Mental Health and created a Split between Brain and Body. Currently we know that mental problems can be caused by bodily problems and that the Imagination is capable to imagine an illness.

Because of Quantum Mechanics we now know that our material reality is a product of our mental imagination. This is called the Placebo-effect.

Western Medicine took a big step when new tools where invented to support diagnosis. Many doctors started to specialize in a  pavillion part of a new type of hospital. Later the many pavillions where joined into a big building that had to move up to use the available space. When Medicine went up it became an abstract theory without a relation to life.

We are now moving to Smart Medicine. In smart Medicine data about the body  is sensed with specialized sensors. The collective data is analyzed  with advanced statistical methods to find general patterns.

The big problem is that you See is what you Get. We measure what we Want to see so the sensors imply a certain disease and a disease is a measurable Distortion of the pattern we suppose is taking place.

Medicine is Creating its own diseases and pharmacy is treating the diseases we measure. This does not mean that the cured patient is balanced and in a good humores,

The new Hospital wil become a Call Centre and the Patient is treated at Home by a Programmed (controlled by protocols) Human.

The personal data is correlated with the collective patterns to find personal patterns and personalized solutions.

Until now these solutions are of a chemical nature (“pills”). In the future the new “pills” will be “Waves“.

To interpret the collective patterns we need a Shared Foundation.

This foundation is now mostly based on ChEmical theories with life as an interaction (“reaction-chain”)  between molecules as parts with the computer-program DNA in the center.

A body is NOT a complex Chemical Factory. the Mind is NOT a Computer (“Operating System“). The Brain (‘Up”) does not controls the Body (“Down”) just like the Management is always Up in de Building that Controls the Company. .

Humans are the result of big Cooperations of a huge amount of parts that consist of big cooperations that cooperate. The parts are coordinated by their shared E/M-fields that create shared fields.

In the future Quantum Mechanics will be the new foundation. The  Standing Wave pattern of the Body, the Biofield, will show the instabilities of the body .

This field will be stabilized (“healed“) with complementary wave patterns. Medicine will be based on =resonance.

Quantum Mechanics also explains Homeopathie.

In the future we will see a move from (bio)Bio-Chemistry to (Bio)Physics.

This  blog contains parts of two documents of the website of Marco Bischof.  Marco is one of the leading scholars in the field of  holistic medicine in the German speaking world. Both documents written by Marco are about Biophysics.

The most important Part (2) of this blog   is about the (Bio)Field.

It is possible to read Part 2 without reading Part 1.

Part 3 is about a possible mathematical foundation of Medicine called Global Scaling. Clobal Scaling is based on waves, fractals and continued fractions that explain harmonics.

Part 1 About the History of Biophysics.

From 1950 to 1970 biophysics developed faster than in the 100 years before. While in 1944, when the first volume of Otto Glasser’s „Medical Physics“was published, in the USA the number of workers in the field counted not more than 200, in 1950, when volume II appeared, about 200 institutes or departments were devoted to biophysics. At the same time the scope of the field also greatly expanded. Glasser’s handbook still remains one of the most complete treatments of the full range of biophysical investigations;

However, in the same period of the 1950’s molecular biology experienced its breakthrough with Watson and Crick’s 1953 model of DNA. The first phase of its development (1930-1950) which included the acceptance of the existence of macromolecules, the use of x-ray diffraction analysis (started by Bernal in 1934) for determining molecular structures and the development of extraction methods that avoided degrading of molecules, was characterized by a relatively broad definition of the subject.

The term „molecular biology“ was first used by Warren Weaver in 1938 to describe a grant programme by the Rockefeller Foundation promoting „experimental“ or „physiochemical“ biology. He claimed this was a new branch of science in which modern tools were „reaching deeper and deeper into the living organism“. The program was supported by such people as W.T.Astbury, who was the first to describe himself as a molecular biologist,

Schroedinger’s little book „What is life ?„ (1944), in which he took up Delbrueck’s idea that the gene could be a molecular structure (Schroedinger called it an „aperiodic crystal“), played an important role in the beginnings of molecular biology. The book inspired many physicists after the war to go into biology.

Delbrueck’s idea originally came from Niels Bohr: it concerned the question if not maybe certain aspects of the life of a cell, such as self-replication, could only be explained by a new kind of natural law still to be discovered,

THis natural law was transformed in the principle of autocatalysis detected by Stuart Kauffman. In autocatalysis structures are recreated to become self-reproducing structures (autopiesis).

in a similar way as classical physics had been replaced by the quantum physical worldview. The inspired search for such new natural laws was one of the main motivations of many of the newly converted molecular biologists Gunter Stent, Maurice Wilkins, and Francis Crick all have confessed to having been greatly influenced by Schroedinger’s book. Most of them participated in the 1st phage meeting at Woods Hole where a proper school of molecular biology was created in 1945.

However, this romantic, even somewhat vitalistic program of the very beginnings of molecular biology was soon turned almost into its opposite, the still ongoing, gigantic enterprise of analyzing systematically a broad range of molecular details. Maybe it was inevitable.

When in 1937 the eminent biologist Barbara McClintock met Max Delbrueck, maybe the single most influential figure of the founders of molecular biology, she considered his approach to be fundamentally flawed. „He has never developed a feeling for the organism“, she said.

The second phase of molecular biology (1950 onwards) clearly is characterized by a narrowing down of the subject to the investigation of biological „specificity“ (molecular „information“), involving the study of proteins and nucleic acids in view to understand genetical structures and mechanisms.

This trend was not without influence on the development of biophysics proper, as is reflected in the Study Program in Biophysical Science, Boulder, Colorado, 20 July-16 August 1958, organized by the Biophysics and Biophysical Chemistry Study Section of the U.S.National Institutes of Health. Its objective was to „encourage the further blending of concepts and methods of physical science with those of life science in the investigation of biological problems“, in other words, to further physicalize biology. Its approach was that of molecular biology.

Basically, molecular biology and mainstream biophysics as they have developed since the 1950’s because of their extreme emphasis on molecular and physicochemical aspects of physiology have to be considered as being in the tradition of physiological chemistry rather than physiological physics. They are based rather on the kind of work exemplified by Höber’s „Physical Chemistry of Cells and Tissues“ (1945)[14], and later, Setlow and Pollard’s „Molecular Biophysics“ (1962)[15], and represent a late triumph of the reductionist program of the Berlin school of 1847.

However, this approach has long remained foreign to the practice of physiological science. As Cranefield wrote: „there is ample indication that during the last few decades physiologists in all countries have not considered the elucidation of molecular mechanisms, the dissolving of physiology into biophysics and biochemistry as their immediate goal. (…) As for the theoretical bias of modern physiology, it is still common to define physiology as the physics and chemistry of life and to assume that the mechanistic stand is correct.

Nevertheless, a position sometimes called organicism holds a very wide popularity today: the position that is identified with the frequent use of such concepts as homeostasis, integration, adaptation, organization, and whole organism.

These biological concepts are essentially neutral but to some they carry a flavor of vitalism“. This alternative current of physiological and biological thinking which can refer to Claude Bernard who rejected German reductionism, represented by such names as A.G.Gurwitsch, E.S.Russell, E.Rignano, P.Weiss, J.Needham, D’Arcy Thompson, J.S.Haldane, J.Barcroft, W.H.Cannon, C.H.Waddington, A.Meyer-Abich and L. von Bertalanffy, has only quite recently fallen into disregard.

Correspondingly, there is also in biophysics another tendency which has developed since the 1920’s and has not always been eclipsed by the dominating school to the degree it seems to be today.

In Germany, it has been represented by the work of Walter Beier and his school at the Institute of Biophysics of the University of Jena (German Democratic Republic). Applying the findings of the school of the Austro-Canadian biologist Ludwig von Bertalanffy to biophysics, Beier has developed a systemic, comprehensive approach which always aims at the wholeness of the system considered. In this approach a system is not viewed as a conglomerate of parts and it does not investigate isolated phenomena, but rather the wide field of interactions. Biological processes on each systemic level, i.e., molecular, cellular, tissue, organ, and organismic, are considered in the context of the higher level in which they are embedded. Besides systems theory, Beier’s approach is based on the irreversible (non-equilibrium) thermodynamics of open systems (while earlier biophysics uses equilibrium thermodynamics), information theory, cybernetics, and uses tools such as factorial analysis. He also follows Rashevsky who introduced relational and similarity considerations and optimization principles into biophysics.

THE SITUATION TODAY As we have seen, today in biophysics such a holistic approach is not widespread. The textbooks rather reflect the fragmentation of contemporary science with its excessive analysis, lack of synthesis and of a viable theory of life.

Biology has lost the wholeness of life, if not life itself, from its view. In spite of the programmatic postulate which has accompanied it since 1847, biophysics still has not developed into a theoretical biology. Beier is one of the few biophysicist aware of the central importance of theoretical biophysics as a complex, interdisciplinary branch of science, and has also shown a practicable way to develop theoretical models. His approach, developed in the 1960’s and 1970’s, can now be used as a starting point for the development of a modern, holistic, and organicist (non-reductionist) biophysics which should place much emphasis on investigating the holistic functions of organisms, such as self-regulation in growth and healing, morphogenesis etc.

Besides the conceptual tools that Beier has brought into play, such a new approach in biophysics must call upon many new developments in science, such as nonlinear electrodynamics, deterministic chaos, cavity quantum electrodynamics, and the physics of the vacuum.

Full use of the consequences of quantum physics, such as the Heisenberg indeterminacy, the holistic principle inherent in the quantum picture, and non-locality, has not yet been made.

Dürr also points to the fact that in molecular biology and chemistry the phase relationships of the electrons are not taken into account (only the intensities) and that the phase structure of the wave formed by the superposition of the partial waves of the millions of electrons in the DNA double helix molecule, may contain, in holographically coded form, additional important information.

A most fundamental element of such a new approach in biophysics will probably be a shift from the molecular to the field aspect of living matter, as proposed by Welch and by Popp, among others, which seems to be the only viable way to synthetize the myriads of molecular details science has amassed, into a holistic model of the organism.

One of the most important goals of applied biophysics in the near future must be the development of non-invasive methods to assess the functional state of whole and intact organisms, and that of the organism’s subsystems in the context of the whole organism.

This goal may be realized by measuring the electromagnetic field produced by the living organism; any interpretation that leads to useful results has to be based on a model of the organism making regulation processes accessible at the field level.

BIOPHYSICS: THE PHYSICS OF BIOLOGY OR THE „BIOLOGY OF PHYSICS“ ?

Any discussion of the future of biophysics inevitably leads to the fundamental question of the relationship between physics and biology. It seems clear that, besides drawing upon new developments in physics, a new biophysics also must become more biological again.

Any program of reducing biology to physics and chemistry has to consider that the actual physical laws have been obtained from the investigation of non-living phenomena, easier to investigate by the physical approach because they are less complex.

Maybe the reason for the failure of biophysics to develop a theory of life is just this reductionistic approach postulating that life processes should be explained by the already known physicochemical laws of non-living nature.

Not a few scientists of past decades, most notably J.S.Haldane and Adolf Meyer-Abich, as well as contemporary ones (e.g., Dale A.Miller[24], Gunter Albrecht-Buehler) have postulated that the principles governing life are more fundamental than those of non-living matter, and therefore biology should replace physics as the paradigmatic „leitwissenschaft“ (leading science).

Without subscribing to any vitalistic notions,takes a similar stance and shows a way which may lead beyond the ages-old controversy. He postulates that physics has to change and enlarge in order to describe biological phenomena; he rejects a reduction of biology to contemporary physics, but does not exclude one to a future, enlarged physics. According to Beier, to achieve a mathematization of biology, the biophysicist should not simply copy physics. „

Biophysics has to contribute to the establishment of purely biological postulates and principles without reducing these to physics. We can consider this method as an enlargement of physics. Physics in a sense ‘incorporates’ biology, via biophysics. Such a procedure can be taken by the materialist as well as by the vitalist from their respective standpoints to be a justification of their views. This again shows clearly the uselessness of a mechanist-vitalist controversy“. Rashevsky wrote that according to the Goedel theorem there must be physical phenomena which cannot be deducted from the laws of physics, even when they do not contradict those laws, and it cannot be decided if biology does not belong to this class of phenomena. Rashevsky and his school based their work on the presumption that biological phenomena are governed by such non-deductable principles.

To identify and formulate these more fundamental laws of life of which the known physical laws may turn out to be special cases, in the same way as the laws of classical Newtonian physics turned out to be special, restricted cases of the laws of quantum physics, should be the most noble goal of a new biophysics. Biophysics is predestined to take on this task. These laws have to be formulated in a mathematical way, but not necessarily metrically, as Rashevsky has shown.

PART 2: FIELD CONCEPTS AND THE EMERGENCE OF A HOLISTIC BIOPHYSICS

Due to recent advances in several disciplines, the basic features of a holistic biophysics are now emerging. It is proposed that the postulates for such a field must include that it will be based on the intrinsic holism of quantum theory and the properties of macroscopic quantum effects, that it should include the principles of nonlocality, nonseparability, and interconnectedness, that it will be based on a field picture of reality and the organism, and finally must include consciousness.

This blog attempts to show why field models are appropriate tools for the holistic modeling of the organism, proposes a hierarchy of regulation systems based on fields, gives a review of field models proposed in biology, biophysics, consciousness research and social science, and discusses the possible role of fields in bridging the mind-body gap.

Finally, a discussion of the perspectives that may be opened up for biophysics by some recently proposed extensions of electromagnetic theory leads the author to suggest a role for the physical vacuum in the organism.

Introduction

“Can Physics deliver another biological revolution ?” asked an editorial in the journal Nature of January 1999¹. However, the so-called “new physics-biology agenda” which several U.S. universities, foundations and government agencies intend to fund heavily, is actually a very old agenda going back to the 1930’s, namely that of molecular biology, which now in a gigantic effort of total genetic mapping, called by some “the New Manhattan Project of Biology”, strives to bring the molecular biology project of total technological control over life to a final fulfillment.

Is this the kind of “new biophysics” we want to be supported with all that money and effort ? I am certainly not alone in asking this question, and with the growing dissatisfaction about the reductionist, molecular-genetic approach to biology the number of those who develop and support an entirely different kind of biophysics is increasing.

A look into the history of biology and biophysics shows that there always have been alternative traditions to the molecular-reductionistic approach now dominant in these fields.

In biology, during some periods, e.g., in the first half of this century – holistic or organismic approaches have been far from marginal, in some disciplines, such as developmental biology, even constituting the dominant trend. Biophysics itself, although a relatively young field of science, has already experienced several changes in emphasis in the course of its history. The German founders of biophysics at the beginning of the 19^th century, people like H.Helmholtz, E.Du Bois-Reymond, E.Brücke, C.Ludwig, E.H.Weber and G.T.Fechner, usually are depicted by historians of science as staunch reductionists and opponents of vitalistic and idealist ideas in biology. However,  in reality they were not in such a sharp opposition to the romantic Naturphilosophie spirit of the time, and nearer to the antireductionistic approach of Claude Bernard, than is generally assumed. Likewise, in the 1930s when molecular biology started its rise that has made it the dominant approach in biology and biophysics, there were a number of other viable biological programs, some of them holistic, that could have been chosen for promotion by the Rockefeller Foundation.

The reason why the officers and scientific advisers of the Foundation chose to promote the molecular vision of life was that they sought to develop, under the influence of Jacques Loeb’s engineering standpoint aimed at controlling life, a mechanistic biology as a central element of a new science of man whose final goal was social engineering.

Originally, the Rockefeller Foundation had intended to promote its goal of social control by eugenics, supporting several research projects concerned with eugenic selection. When eugenics lost its scientific validity amd social acceptance, the molecular biology program of creating a new biology grounded in the physical sciences and able to rigorously explain and eventually control the fundamental mechanisms governing human behaviour, promised a surer, if slower, way toward social control by eugenic selection based on the more readily accepted principles of genetics and protein science. Even within the scientific developments that stood at the origin of molecular biology, before it definitely took the reductionist turn leading to the present situation, at least one also forms an important element in the foundation of a modern holistic approach in biophysics: the quantum philosophy of Bohr, Schroedinger, Jordan and Delbrück.

Biophysics and the Quantum Revolution

The philosophical contribution to the rise of molecular biology originated in Niels Bohr’s speech „On Light and Life“ at the International Congress on Light Therapy at Copenhagen, August 15, 1932 ²³ where he postulated that a new physics was required for interpreting life; life was not reducible to atomic physics. A few of the key figures in molecular biology, mainly M.Delbrück and E.Schrödinger, were strongly influenced by Bohr’s views ²⁴, which Schrödinger took up in his book „What is Life ?“ ²⁵. The book inspired many young physicists traumatized by the wartime use of physical expertise, to go into biology and had a strong influence on the development of molecular biology ²⁶.

However, the book certainly was more influential by interpreting the genetic viewpoint of H.J.Muller, T.H.Morgan and M.Delbrück in physicist’s terms and by backing it with the prestige of physics than by convincing biophysicists of the Bohr-Schrödinger hope of discovering new physical laws through biophysical investigation of biological phenomena²⁷.

These ideas – which have to be seen in the context of the rather strong holistic tendency of the biology of the time – did not exert any lasting influence on molecular biology. On the other hand, they have been, and still are, a seminal influence for the later emergence of a holistic biophysics.

Today, molecular biology proposes itself as as the manageable project of refashioning life and redirecting the course of evolution that the Rockefeller Foundation and some of its early pioneers like H.J.Muller and W.Weaver had envisioned.

Because of the enormous technological and social power promised by molecular biology, even the increasing awareness of the bad science on which it is based in many respects, such as the many weak points in genetic and evolution theory²⁸, does not prevent it to carry us into such immature and dangerous projects as genetic engineering biotechnology in agriculture and medicine, and the “Human Genome Project”, whose deeper nature is revealed by the military epithet of a “Manhattan Project of the life sciences”.

The noted quantum chemist, Hans Primas, agrees with this fundamental criticism^29,30. He writes that molecular biology, as it exists today, is in fact engineering, not science. It is pragmatic, instrumental knowledge which aims at the power over nature, but not at understanding. It does not constitute a scientific theory of life able to give us orientation to live rationally with nature, but only provides technological control over life. Contemporary molecular biology has become a scientific technology which has lost contact with the epistemological sciences. Aspects of life that cannot be treated or understood from the molecular viewpoint, such as morphology, are glossed over. The assessment Robert Rosen made in 1967 ³¹ is actually still valid: „…it must be pointed out that the older problems [he refers to the questions that preoccupied an older generation of biologists] have merely been displaced and not solved by the recent developments at the molecular level. These problems involve the very core of biological organization and development: homeostasis, ontogenesis, phylogenesis“ .

For Prim, the primary shortcoming of molecular biology is that the holistic character of the physical world now recognized in quantum theory is either not acknowledged by the bioengineers or rejected as irrelevant. He emphasizes that molecular biology, though well grounded in empirical knowledge, has no foundation whatsoever in the principles of quantum theory, contrary to a widely held belief to the opposite. It uses the methods and technologies of quantum mechanics, but its way of thinking is still committed to the classical physics of the 19^th century and has not taken notice of the fundamental insights of quantum mechanics on the structure of the material world.

According to Primas, on whose statements the following is mainly based the atomistic-molecular view of matter and the reductionist-mechanist philosophy have no more any scientific foundation, according to the actual understanding of quantum theory. The description of reality by isolated, context-independent, elementary systems such as quarks, electrons, atoms, or molecules is only permissible under certain specific experimental conditions, and these entities cannot in any way be considered as „fundamental building stones“ of reality. Besides the molecular one, there are other, fundamentally different descriptions, complementary to the molecular one, which are quantum-theoretically equivalent and equally well founded. Quantum theory is much richer in possiblities than is admitted in the worldview of molecular biology.

In Primas‘ view, the feature of quantum theory that is most significant for biology is its intrinsic fundamental holism. For quantum mechanics, the scientific theory most widely recognized as fundamental and best confirmed by experiment, material reality forms and unbroken whole that has no parts. These holistic properties of reality are mathematically precisely defined by the Einstein-Podolsky-Rosen (EPR) correlations which are experimentally well defined.

Primas postulates that, by virtue of this, quantum mechanics constitutes the first and up to now only logically consistent, universally valid, mathematically formulated holistic theory In quantum mechanics, it is never possible to describe the whole by the description of parts and their interrelations.

With this view of quantum mechanics Primas follows Bohr and the school of Heisenbergwhile quark physics as founded by M.Gell-Mann continues to cultivate democritean atomism with their clinging to the concept of elementary particles bootstrap theory“ of G.Chew, D.Bohm’s „Causal Quantum Theory“ or „Holographic. Similar holistic views of quantum theory are the „ Theory of Reality“ , and others advocated by H.Stap, A.Goswami , Kafatos & Nadeau, Friedman, D.Peat, F.Capra, H.Atmanspacher, and many others.

This holistic view of quantum theory, although the phenomena on which it is based are not yet completely understood theoretically, cannot be rejected anymore because the strange EPR quantum correlations of non-interacting and spatially separated systems have been amply demonstrated in many experiments

Therefore the world-view of classical physics, atomism and mechanistic reductionism definitely cannot anymore be the basis of our worldview, and of biophysics. Quantum mechanics has established the primacy of the unseparable whole.

Another important epistemological consequence of quantum mechanics, complementarity, is also connected to its holism. As Primas writes, there is no single description, such as the molecular-reductionistic one, which can alone represent the whole reality of the subject of a scientific investigation, or is better or „truer“ than any other. Nature is extremely diverse and stratified; each description comprehends only a minute partial aspect of its unfathomable multiplicity.

Any scientific description of a natural phenomenon is only possible if we renounce the description of its complementary aspects.

Quantum theory can only be applied if we abstract from certain aspects and thereby break the holistic symmetry. However, the kind of abstraction we use is not prescribed by the first principles of the theory, such that quantum mechanics allows, and even requires, many different, but equivalent, complementary descriptions of nature. As an important postulate for future science, Primas therefore emphasizes that we will have to learn to work simultaneously with several complementary descriptions of nature .

In this perspective, the molecular view is legitimate and important and should not be abandoned; molecular biologists can be rightly proud of their successes. It should be cultivated, but not at the expense of other viewpoints. It is its extreme one-eyedness that must be criticised. However, as Primas points out, „biology is more than molecular biology.

He postulates that science must now redirect ist attention to the wholeness of nature, and therefore will have to ask radically new questions. It has to develop a concept of reality which does not exclude any part of it. Those properties which belong to organisms only as wholes must remain within the scope of science. Therefore, it will be necessary to consider the phenomena as well from „bottom-up“, as in mechanistic understanding, as from „top-down“, as in vitalistic and holistic understanding. According to Primas, the notion that the latter is not legitimate or secondary is a prejudice that must be overcome.

From the viewpoint of the quantum-theoretical worldview, both are completely equivalent, but lead to fundamentally different research agendas and insights. Even the criteria according to which the scientist decides what is scientifically defendable and interesting, are completely different from these two viewpoints. Also, according to quantum theory functional and teleological explanations are completely legitimate and equivalent to causal ones; even the primacy of causality has no foundation in the first principles of physics. Primas points out that it is not possible to distinguish between causal and final processes by purely mechanical means and that such a distiction only makes sense for irreversible processes. As to the existence of the hypothetical vitalistic forces, modern physics is well able to integrate new forces into its system.

Thus, matter has become „dematerialized“ by modern quantum theory, and this property of „thinglessness“ in the quantum worldview is closely connected to the property of „interconnectedness“.

The emphasis is no longer on isolated objects, but on relations, exchanges, interdepences, on processes, fields, and wholes.

Quantum theory is a nonlocal theory. It is important to see that we cannot retain the classical world of objects and only add the interconnectedness as a supplementary property of these objects.         They are two of the complimentary descriptions or aspects of reality which Primas has alluded to and cannot be used simultaneously; thus they rather should be considered as different diomensions of reality. The holistic interpretation of quantum theory in fact may also be taken as implying a multidimensional structure of reality.

In this view, there are, besides the world of objects, one or several more fundamental levels of reality where interconnectedness rather than separatedness dominates. Fields certainly belong into this category; however, apart from electromagnetic and other physical fields which are still among the phenomena considered as belonging to the four fundamental forces of the observable world, we must assume the existence of additional field-like levels of reality not directly observable at present, which may be beyond space-time and represent the realm of potentiality, or of the „noumena“, the realm behind the phenomena according to Newton, in contrast to the actuality of the observable.

The Schroedinger wave function of quantum theory actually describes this hidden domain of potentiality, of the non-observable, unmanifested, pre-physical world of non-local correlations and superluminal, instantaneous connections, rather than the world of observable phenomena ⁴⁸. Only with the act of measurement this infinity of potentialities, described in the Schroedinger equation as a superposition of all possible quantum states, is „collapsed“ into one single actuality. Connected to the concept of potentiality is the concept of „entanglement“ which describes the characteristic of interconnectedness. In the absence of any interaction (such as a measurement), two systems are in an entangled state in which neither system by itself can be said to be in a „pure state“, i.e., can be fully specified without reference to the other.

This hidden domain can be considered as a fundamental dimension of reality, a domain of dynamical connectivity, from which the patterns of the physical world arise. According to some authors, this realm of pre-physicality is not only the basis of the physical world and of matter, but also seems to be connected to, consciousness, which some see as the fundamental field underlying i. In physics, it is treated by the various models of the physical vacuum. Its possible relevance to biophysics as a basis for a true quantum biology ^62,63 seems obvious to us, as we will explain later. Therefore we postulate the development of a „vacuum biophysics“ (see last chapter) .

The „hidden domain“ of connectivity has characteristics completely different from those of the classical, macroscopic world of separated objects. For a long time, the quantum description that reveals the properties of phenomena belonging to this domain, or arising from it, was taken to apply only to the microscopic world of atoms and molecules, while the world of macroscopic phenomena of our experience was considered to be purely classical and not to manifest quantum properties. However, today we know that this is not true, and that there are many macroscopic quantum manifestations, although our knowledge about them is still limited^64-68. Biological systems obviously possess the characteristics of macroscopic quantum systems.

The Redefinition of Biophysics

From what we have just heard, we can conclude that traditional biophysics, based up to now on classical physics and equilibrium thermodynamics, needs a redefinition in terms of the revolution brought by the last few decades of quantum-mechanical experiments and interpretations. As there is still no agreement on the definition of the field, we are free to attempt such a redefinition. In the following I will try to determine what the postulates for such a new biophysics could be, as it actually is already in development since about ten or fifteen years.

First of all, it indeed will entail a revolution based on physical concepts – however, not of the kind alluded to in the Nature editorial. While the physical view will be fundamental, it will not be that of classical physics, and the goal will not be the reduction of biology to physics but an understanding of the physics of the living, and physics must not replace, but support profound biological understanding. Thus, the new biophysics should be more than just an empirically based bioengineering technology; it will need epistemological and philosophical foundations. Its goal should be to develop an adequate theory of life, and it should balance the mastery of life with the understanding of life.

As a complement to the onesidedness of the molecular approach, the new biophysics will focus on holistic aspects of organisms, and will attempt to provide a vision able to synthesize the wealth of molecular details accumulated by molecular biologists. Its basis must be the insight into the fundamental interconnectedness within the organism as well as between organisms, and that of the organism with the environment.

Therefore it must be inter- or transdisciplinary and truly integrate biological, biochemical and medical expertise into its physical models, but also connect to knowledge from fields such as geophysics, biometeorology, heliobiology etc.

It will be based on quantum theory, and not classical mechanics – therefore it may also be called „quantum biology“ –, and also, instead of equilibrium thermodynamics, it must refer to non-equilibrium thermodynamics. Organisms clearly are open systems far from equilibrium. Other central concepts of the new biophysics, related to the latter, will be coherence and macroscopic quantum states.

I postulate that field thinking and field models will have to be one of the central elements of the new biophysics, as a complement to the molecular view, as a means to sythesize the wealth of its details, and to adequately model thinglessness, interconnectedness and non-locality – therefore bioelectromagnetics will play a central role in the new biophysics. However, recent experiments have shown that the existence of hitherto unknown, non-electromagnetic fields in and between organisms cannot be excluded. Of course, the field aspect of the organism has to be seen in close connection, and constant interaction, with the solid aspect. Attention should also be paid to the field aspect of biochemical processes, for example in collective processes, reactivity and molecular recognition.

I suggest that the existence of a pre-physical, unobservable domain of potentiality in quantum theory, which forms the basis of the fundamental interconnectedness and wholeness of reality and from which arise the patterns of the material world, may provide a new model for understanding the holistic features of organisms, such as morphogenesis and regeneration, and thus provide a foundation for holistic biophysics – therefore I propose that the usefulness of the theories of the physical vacuum for understanding the phenomena of life is investigated – one important aspect of their usefulness may be as a link between the domain of biology and consciousness.

I postulate that the new biophysics needs to extend its interdisciplinarity even beyond natural science. Consciousness cannot be excluded anymore from biophysics, although the difficulties of such an extension should not be underestimated. There is now enough evidence showing that consciousness is a causal factor in biology and not just an inconsequential epiphenomenon. Starting from the analysis of the phenomenology and the experimental evidence for mind-body interaction, field models and vacuum theories may provide the necessary tools for bridging the mind-body gap. However, it is also necessary to acknowledge the limits of the scientific approach, and value the goal of understanding highly enough to include non-observables into our models, if this supports understanding.

Historical Antecedents of Holistic Biophysics

Not only theoretical considerations as those of Primas, but also the history of biology and of biophysics show clearly that the reductionistic concept of molecular biology is not the only possible concept for biological science; there even have been several periods in which holistic approaches have been dominating biology, or at least parts of it⁶⁹. Even in biophysics proper, there have always been holistic schools of thought, although they usually are not mentioned in contemporary reviews of the field, or not thought of belonging to biophysics.

An early instance of holistic thought in biology is the school of rational or transcendental morphologists of the late 18^th and early 19^th centuries, such as G.de Cuvier, E.Geoffrey St-Hilaire, and R.Owen, who deduced, from their detailed studies of organismic morphology, the existence of principles or laws of form operating in the biological realm. Similarly, the holistic biology and medicine of the Romantic period, including J.W.Goethe, A. v. Humboldt, C.G.Carus, K.E. von Baer and J. Müller, regarded the ideal form of a species as a transcendental guiding principle in morphogenesis. A strong tradition of antireductionistic physiology starting with F.Magendie was mainly created by Claude Bernard. Bernard’s concept of “internal environment” (“milieu interne”) developed in 1857, stood at the beginning of the research in physiological regulation, to which E.Starling, L.J.Henderson, J.Barcroft, and W.B. Cannon (“homeostasis) have been contributing; other holistic physiologists were J.S.Haldane, H. Selye, R. Dubos and A. Carrel.

In the period of 1900-1950, the main thrust of developmental biology, especially in the work of H.Driesch, H.Spemann, P.A.Weiss, A.Gurwitsch, R.Harrison, C.M.Child, J.v.Uexküll, J.Needham, E.S.Russell, J.H.Woodger, A.Meyer-Abich, and A.Müller, has been holistic and centered around the concept if biological fields ^69,73-74.

Very much fallen into oblivion has today the important work done during this century in the German tradition of holistic physiology, pathology and medicine, represented by names like G.v.Bergmann, H.Schade, F.Kraus, F.Buttersack, A.Bier, Th.Brugsch, B.Aschner, L.R.Grote, F.Hoff, R.Rössle, G.Ricker, H.Eppinger, W.Petersen, H.Pischinger, O.Bergsmann. Holistic schools of medical thought have also been active in Britain, the USA, and France, mainly in the interwar period⁷⁵.

Important contributions to holistic science came from Gestalt theory and other holistic theories in psychology, neurobiology, and psychiatry, as exemplified by W.James, Chr.v. Ehrenfels, M.Wertheimer, W.Köhler, K.Goldstein, K.Lewin, C.v.Monakow, C.S.Sherrington, K.Lashley, G.Murphy, V.v. Weizsäcker, A.Maslow, K.Pribram ^74-78.

The work of the Austrian-Canadian biologist L.v.Bertalanffy is particularly significant for the emerging holistic biophysics; Bertalanffy’s “organismic” conception of life, which also was the starting point of his “General System Theory”, actually constitutes a first attempt to create a holistic picture of the organism based on physical laws and detailed biological knowledge, and ist still a valuable guide for our contemporary efforts ^79-84. The German school of Beier ^85-86 has continued this tradition in biophysics.

An early tradition of holistic biophysics developed from Claude Bernard’s school in France, represented by A.d’Arsonval, W.Kopaczewski, F. Vlès, and G.Lakhovsky.

Russia has its own tradition of holistic biology, biophysics and bioelectromagnetics with the work of V.I.Vernadsky, A.L.Chizhevsky, A.G.Gurwitsch, E.Bauer, and A.S.Presman, among others.

An important contribution to holistic thinking comes from the tradition of mathematical biophysics with its concepts of optimality, and relational and similarity principles, represented by D’Arcy W.Thompson, N.Rashevsky, R.Rosen, and R.Thom.

Holistic concepts in philosophy of relevance to biology and biophysics have been developed by J.Smuts, A.N.Whitehead, O.Spann, A.Meyer-Abich, H.Conrad-Martius, H.Plessner, H.Friedmann, among others.

Last but not least, the contributions of psychosomatic medicine, such as those by V.v.Weizsäcker, F.Alexander, Flanders Dunbar, G.Groddeck, and G.R.Heyer, must be mentioned.

In the time of 1900 to 1950, holistic approaches in biology, usually connected to field concepts (see next chapter), far from being marginal, have formed one of the dominant schools of thought, at least in developmental biolog. For various reasons, the plausibility of this approach has strongly decreased from the 1950‘s^. The main reason was the rise of genetics as an alternative program to explain development.

Much of this work is anything but obsolete, but it is now necessary to actualize the fundamental observations and concepts developed by these pioneers of holistic science with the up-to-date knowledge in the respective fields.

Field Concepts as a Necessary Base of Holistic Biophysics

Modern physical field theories, such as those developed by Faraday and Maxwell, have their origin in metaphysical concepts arising from the participatory consciousness of archaic man. Field and particle theories arise from complementary modes of human self- and world-experience rooted in bodily awareness, with the field perception of reality probably being much older than particle theories, which could only arise when the participatory world-view slowly gave way to individual consciousness and the perception of an objective world, by way of an increasing differentiation between inside and outside, I and the world, the subjetive and the objective, and matter and consciousness. This objectivation of the world has reached its climax in the worldview of the classical physics of the late 19^th century, with the assumption that all there is to reality is solid particles assembled in various degrees of complexity.

The electromagnetic field theories of Faraday and Maxwell proposed in the second half of the last century have inspired biologists already from 1900 to the development of the first field theories, but it was only with Einstein’s General Relativity, quantum mechanics and quantum field theory, that physics overcame the mechanical world picture and conceived reality as a field phenomenon.

Biology still bases its picture of the organism on the building-stone view of classical physics which physics itself has already given up since a couple of decades. One of the first and most fundamental steps in the development of a holistic biology must therefore be not only to complement the classical view with the field aspect, but even to build its model of the organism on the field picture.

As I have shown elsewhere, the concept of the field has alraedy occupied a central place in the school of “organismic” or “holistic” biologists in the first half of this century⁶⁹.

Leading biologists such as H.Spemann, R.Harrison, P.A.Weiss, J.Needham, and C.H.Waddington, A.G.Gurwitsch used the hypothesis of a biological, or morphogenetic, field, introduced by Weiss and Gurwitsch in the early 1920’s, as a tool for understanding the phenomena of development, regeneration and morphogenesis and to make predictions for experimental testing. Although their field concepts referred to the model of physical, especially electromagnetic fields, the organicists generally considered the biological field as a purely heuristic concept and left the exact nature of the fields open. The time (and electromagnetic science) was not yet ripe for the notion of real electrical, electromagnetic or otherwise physical, fields of long-range force.

ELECTROMAGNETIC FIELD THEORIES IN BIOLOGY

At this time, there was not enough experimental evidence for the existence of bioelectromagnetic fields nor for the biological effects of EM fields. Thus, the various electromagnetic (EM) field concepts proposed since the first decades of the 20^th for biology, as those by Keller, Burr, Burr & Northrop, Crile, Lund, and Lakhovsky⁸⁹, were premature. The breakthrough and beginning of modern EM field theories in biology came only in 1970, with A.M.Presman’s report of the pioneering work of Soviet bioelectromagnetics researchers, which also contained a first outline of a holistic EM field theory of the organism and his relationships to the environment⁹⁰. Since then, there is ample evidence for bioeffects of EM fields and endogenous EM fields. It is now established that organisms react sensitively to the impact of electromagnetic fields, including very weak ones; effects of various types of endogenous physical fierlds on cellular organization and morphogenesis are very likely. We also know that several kinds of electromagnetic fields, including microwaves and optical frequencies (biophotons), are emitted from living beings. There is also evidence that weak endogenous electrical currents are involved in regeneration and growth of new tissue; the role of ionic currents in morphogenesis and development has also been demonstrated. Communication by electromagnetic fields is established for fishes and insects, which suggests this may be a more general phenomenon.

A number of further developments have contributed to the full development of modern biological field theories. They include the work of A.Szent-Györgyi on the excitation-deexcitation dynamics in biochemistry and the role of charge transfer in the organism^91-93; the work on open, non-equilibrium systems and the coordinated collective behaviour of particles culminating in Prigogine’s theory of “dissipative systems”; H.Fröhlich’s work on coherence of the EM fields coupled to these particles and connecting them ^94-97; the Dicke theory of the coherent emission of coupled multiatomic emitters; the theory of quantum coherence developed by R.J.Glauber ^99-101; and several recent developments in quantum optics, such as the work on non-classical light and Cavity Quantum Electrodynamics.

MODERN BIOELECTROMAGNETIC FIELD THEORIES

These achievements became important elements in the “biophoton theory” developed by Popp and his group. Like most modern field-theoretical proposals, it tries to reconcile particle and field approaches. Based on the evidence for the coherent emission of ultraweak luminescence by organisms, it conceptualizes organisms as biological lasers of optically coupled emitters and absorbers operating at the laser threshold. The solid part of the organism is coupled with a highly coherent, holographical biophoton field which is proposed to be the basis of communication on all levels of organization; the components of the organism are seen to be connected in such a way by phase relations of the field, that they are instantly informed about each other at all times. The biophoton field is also postulated to be the basis of memory and the regulation of biochemical and morphogenetic processes.

A related approach, “bioplasma theory, was developed from early suggestions by Szent-Györgyi, who pointed out that biomolecules in the organism are predominantly present in the excited state, and that the energetics of living systems are based on excitation-deexcitation dynamics which are also the basis of chemical bonding. Biological plasma is described as a „cold“ plasma of highly structured collective excitations produced by the dielectric polarization of biological semiconductors, which functions as a single unit. The collective excitations of the molecules propagate in the form of excitons.The complex aggregates and configurations formed by the plasma particles serve as an energy network in the organism. External and intrinsic radiation is stored in the bioplasma in the form of trapped cavity oscillations which form the biological field; it has a complex broadband holographic wave structure of great stability. The biological effects of external radiation are ascribed to resonance properties of the whole system, and not to any of its parts. Like biophoton theory, the bioplasma concept implies non-equilibrium and electronic population-inversion, and therefore laser-like processes, as postulated by Inyushin in the early 1970‘s.

Pribram’s „holographic theory of perception and memor, first proposed in 1971, has been an important contribution to modern biological field theory. It proposes that information from the sensory input is enfolded by Fourier-like transformations and stored in the brain in the form of holographic interference patterns, i.e., coherent EM fields. For reading it out in remembering, it is unfolded again by inverse Fourier transformation. In 1975 Pribram synthesized his model with the more general „holographic theory of reality“ proposed by Bohm in 197. It suggests that the organization of reality itself may be holographic, the world of objects we perceive (the „explicate order“) being a second-order manifestation of the more fundamental „implicate order“ or „holomovement“ forming the basis of the world’s fundamental unbroken wholeness.

The „holographic concept of reality” proposed by Miller et al is a useful attempt to sketch the outlines of a possible synthesis of field models emphasizing the particle aspect, like bioplasma theory, and those who focus on the connecting and/or underlying fields, like biophoton theory and the holographic theories. At the same time, it tries to elucidate the significance of biolectric phenomena and physico-chemical parameters like the acid-base and electrolytic balances and redox potentials within the bioelectromagnetic fields.

Today, the field view of the organism and its interactions is finding increasing acceptance in biology, biophysics and medicine ¹¹⁹.

Biophysics and Consciousness: On the Possible Role of Fields in Bridging the Mind-Body Gap

The German founders of biophysics in the early 19^th century recognized consciousness as the „ultimate problem of biology“, but science at that time was not prepared to include consciousness into biophysics, although the necessity was not denied¹⁹. Today, the situation is different. Since around the turn of the century, Freud and his followers have made the problem of consciousness a central topic of a broad research effort whose results widely influenced Western society, but were hardly taken seriously by the natural sciences.

The last few years have changed this: consciousness has ceased to be a „non-subject“ and is now definitely on the scientific agenda^127-128. An increassing number of authors are emphasizing the necessity of introducing consciousness into the scientific worldview, and some believe it should even become its very foundation ^{41,44-46,49,57,131-135}. As to the consequences of a „biology without consciousness“, Efron has pointed out the disastrous epistemological confusion the exclusion of consciousness from biology has caused.

Although the highly animated discussion in consciousness research is characterized by widely divergent standpoints, the decades-long dispute about the inclusion of the observer and the possible role of consciousness in quantum mechanicshas certainly been a significant influence. As can be seen from the new discipline of „Quantum Neurodynamics“ , Pribram’s and Bohm’s holographic theories, together with Eccles‘ suggestion that fields analogous to the probability fields of quantum theory could be responsible for the coupling between consciousness processes and neural events¹⁴¹, have also been of considerable influence. Thus, an important segment of the most recent efforts in consciousness research is based on the hypothesis that consciousness may have a field-like nature, and/ or that fields may play a mediating role between consciousness and the biological organism. This hypothesis has already a considerable history and is more widely held than commonly is known.

It was probably W.James who first introduced the concept of a field of consciousness into modern psychology in 1890. C.G.Jung’s “Collective unconscious”, first proposed in 1917, is conceived as a deeper, fundamental field-like unitary psychophysical reality (“unus mundus”) occasionally producing “synchronicity” effects. Gestalt Theory, initated by Chr.v.Ehrenfels and W.Köhler, postulates an isomorphism between psychological and psychophysiological processes, mediated by fields analogous to Maxwell’s electromagnetic fields and not bound to the nervous substrate, whose geometrical structure mirrors that of the perceived stimuli. In the 1930’s K.Lewin proposed in his field theory of social psychology that social interactions are best understood by a field model. In the 1940’s G.Murphy developed a field concept of the organism, of personality, and of communication and suggested the existence of an interpersonal field which he thought to be part of a wider universal field. Murphy also explained Psi effects in groups as a loosening of the usual interpersonal barriers and opening up to the interpersonal field. Paul Schilder demonstrated the existence of a “body-image”, a 3-dimensional picture of the “perceived body”, different from the body of anatomy and physiology, a kind of a constantly reorganized field that is constructed from the visual, tactile, kinesthetic, postural etc. experiences of an individual’s lifetime. In 1964, Aron Gurwitsch proposed a field theory of experiential organization in the tradition of Gestalt theory, Lewin and Husserl’s phenomenology. The Russian mathematician and philosopher, V.V.Nalimov, has recently proposed a theory of the “semantic vacuum”, according to which there is a deeper, unobservable process from which ordinary, reflexive consciousness emerges which he calls the semantic vacuum, in deliberate analogy to the concept of the physical vacuum.

A school of thought that has found wide interest and has led to a new branch of neuropsychology, has its origin in the work of Pribram and Eccles. In the early 70’s K.H.Pribram has proposed that coherent holographic fields mediate between consciousness and neurological processes; J.Eccles has postulated in 1977 that consciousness has an existence independent of the brain, and that the self interacts with the body and the material world using the brain as an instrument. H.Margenau in 1984 suggested mind may be a unique type of non-material field, analogous to quantum probability fields; this suggestion was taken up in 1986 by J.Eccles who proposed that this field may modify the probability of emission of neurotransmitters at the dendritic synapse. This finally led to the formation of the new field of Quantum Neurodynamics, based on the hypothesis that brain processes are to be understood on the basis of quantum field theory and are based on quantum fields, or potentials. Long-time memory is conceived as a structured complex of vacuum states; remembering as the emission of coherent biophoton signals from the vacuum state. The coupling of neurophysiology with the “quantum sea” of the vacuum is assumed to be the basis of brain processes.

Quantum neurodynamics illustrates the many recent efforts to find approaches bridging consciousness as an entity which is not directly measurable, and the solid material aspect of the organism, with the hypothesis of a mediating field domain. Similar efforts have been made in the last few decades in many areas, not least in connection with the scientific investigation of Eastern medical systems, such as acupuncture, of contactless healing and various other phenomena.

Observations and Experimental Studies of Interpersonal Fields of Unknown Nature

The existence of electromagnetic fields emitted by living organisms, including humans, is now well established, even if there is not much established knowledge about their biological functions. On the other hand, man has a long history of subjective experience of field-like interpersonal connections which usually are relegated to the realm of imagination by the scientifically minded. More recently, however, a number of scientific experiments have to some extent given evidence for the physical reality of these field observations.

As to the observations, an important example are the studies of nonverbal behaviour that have shown a synchrony of the body motion of speakers and listeners with the speech pattern, which probably serves to establish empathic resonance. A related phenomenon is the well established phenomenon of “emotional contagion”. Psychiatrists and psychotherapists have been familiar since decades with the “praecox feeling”, the field-like aura displayed by their patients announcing impending psychosis or schizophrenic episodes, and have been well aware of the contagious nature of these states. The phenomenon of “transference” between therapist and patient is equally well known and has led a number of authors to the hypothesis of an “interpersonal field”. In “mutual hypnosis” two persons create a common psychic field which in the deeper stages can turn into a shared hallucinatory or dreamlike reality. Families may, according to some psychotherapists, possess a common unconscious and shared emotional field.

A number of recent experimental studies give evidence that such interpersonal field effects may have some physical basis. Studies of empathically bonded pairs by J.Grinberg-Zylberbaum have shown interhemispheric and interpersonal EEG coherence and the appearance of transferred (evoked) potentials in the unstimulated partner after separation by a Faraday cage. The “field-REG experiments” done by the Princeton Engineering Anomalies (PEAR) Laboratory demonstrate anomalous influence of group events with a “high degree of subjective resonance between participants”on the random output of portable random events generators (REG) that suggest the presence of a field within such groups. Experiments on distant mental influence on living systems (DMILS) show that persons are able to exert direct mental influences upon various distant biological systems shielded from all conventional informational and energetic influences.

While these experiments suggest the possible non-electromagnetic nature of the studied fields, the measurements of the “Copper-Wall Project” performed by E.Green demonstrate that in healing sessions, exceptional subjects, such as healers and sensitives, are able to generate anomalous voltage surges in electrical body potential which are transmitted to and measured by electromaters attached to the four highly polished copper walls surrounding them in some distance.

Biological Fields in the Light of Some Proposed Extensions of Electromagnetic Theory

In some of these experiments showing the existence of interpersonal field phenomena (Grinberg-Zylberbaum, DMILS) electromagnetic fields have been excluded; therefore we must use the hypothesis that some kind of probably unknown, non-electromagnetic field(s) may be involved. Recently, a number of authors have suggested that electromagnetic potentials (vector and scalar potentials) may play a role in living system, and a series of preliminary experiments (which still have to be reproduced independently) seems to show biological effects of vector potentials different from those of ordinary electromagnetic fields ^147-153. It has been proposed that there may be a whole class of non-electromagnetic fields underlying electromagnetic phenomena, which have been called „subtle energies“ by some authors, following a suggestion by Einstein.

In fact, while the potentials have long been considered mere mathematical conveniences without physical reality, the reevaluation of their significance made possible by the groundbreaking paper by Aharonov & Bohm , is now opening up a new field of electromagnetic research which we suspect may turn out to be highly significant for bioelectromagnetics and biophysics in general. A number of recent attempts to formulate extensions of electromagnetic theory point to the existence of an additional, hitherto unsuspected dimension of electromagnetism, which seems to be able to interact with the very structural fabric of space and time ^155,156. Aharonov & Bohm  have shown that in certain cases the potentials act as real physical fields and must even be consdered more fundamental than the electric and magnetic forces; in the experiment they proposed the potentials exert an physical effect on charged particles in a field-free volume but not in the way force-fields do – they only influence the phase, and thus are fields of information.

However, the reason for the now well proven Aharonov-Bohm (AB) effect has only become evident in the wake of its analysis and generalization by Wu & Yang. Barrett  has given evidence that in the AB experiment the electromagnetic field, normally of U(1) symmetry, is „conditioned“ into SU(2) form (in other cases even higher symmetries can be obtained) by the geometrical constraints of the experiment, which adds a degree of freedom to the field allowing an interaction with the space-time metric (neutrino network) and its topological structure, and endows it with a gravitation-like essence and form. Barrett also has shown that the AB effect is only one of a whole class of effects where this is the case.

According to Tiller potentials have the important function of mediating between electromagnetic fields and the macroscopic quantum states of solid matter on the one hand, and the physical vacuum on the other hand, because of their property of controlling the phase of electromagnetic fields. He suggests that the „subtle energy“ fields of the vacuum domain, belonging to a higher dimension beyond space-time, organize the structure of space-time, which in turn, by the intermediate of the potentials, generates the corresponding electromagnetic fields. These finally give rise to the observed processes in space and time.

This hypothesis is supported by the work of Barrett on the conditioning of the electromagnetic field. In this process, the phase-controlling property of potentials is central. This is highly significant for biophysics, not only because of the coherence of biolectromagnetic fields; its importance can also be illustrated by the fact that the living organism with its many rhythmical processes basically is a complex system of oscillating fields coupled nonlinearly by their phase-relations.

Apart from potentials, a number of further non-electromagnetic fields have been forwarded in the various proposals for extensions of the Maxwell theory, as possible elements of an intermediate „subtle realm“ between particles and force fields and the vacuum, or as elements or aspects of the vacuum itself, for instance, „torsion fields“and the B(3) „ghost field“ of longitudinal magnetic polarization.

The Concept of Vacuum Biophysics

We are convinced that it is one of the central tasks of biology and biophysics, as it is of physics itself, to investigate the process of becoming and of manifestation, the arising of actuality from potentiality. It is clear that this is not yet completely realized in quantum physics, although the recent discussions about the interpretation of quantum theory and the alternatives to the Copenhagen interpretation have shown that it is groping in this direction. The same tendency can be found in the recent attempts at developing unfied theories of all physical interactions.

In the various unification programs the concept of the physical vacuum occupies a central place. „The vacuum is fast emerging as the central structure of modern physics“ (Saunders & Brown. We postulate that it also merits such a central place in biophysics ^62,63,147. It has in fact already been used in a number of recent models, e.g. by Conrad, Grandpierre, Laughlin, Laszlo, Jibu & Yasue, Shacklett, and Tiller. We would not be surprised if it would turn out to be the very foundation a holistic quantum biophysics needs. The holistic quantum logic of biological processes and structures may not be sufficiently understood without the explicit inclusion of the vacuum concept into biophysics. Biophysics should be able to explain how, in the generation and development of organisms, pre-physical potentialities are transformed into physical realities. For practical reasons, it should also be interested in improving our knowledge on the more subtle, early levels of biological manifestation, where we may have access, for instance, to the formation of preconditions for illness. The assumption of a pre-physical dimension of potentiality is a prerequisite for the full understanding of life. To quote the Heisenberg pupil Hans-Peter Dürr: „Living systems prove that actuality (factuality) is not all there is, but potentiality is also important. Like all macroscopic quantum systems, they are emergences of potentialities into factuality“ ¹⁷⁸.

We postulate that the concept of the vacuum is the appropriate framework to model the fundamental quantum-mechanical domain of potentiality. The vacuum is the „ground of being“ from which the information for the structured development and regeneration of inorganic as well as living forms arises. All the features of the unbroken wholeness of reality implicit in quantum theory – non-separability, non-locality, fundamental connectedness – which are so fundamental for biological understanding, are an expression of the properties of the vacuum. The vacuum is the origin of microscopic and macroscopic coherence, an essential feature of living organisms. And, finally, the understanding of the vacuum may provide the crucial insights on the role of consciousness in physical reality, and in the various stages in which the creativity of the ground of existence unfolds on its way from pure potentiality and information to physical manifestation.

The concept of of the quantum vacuum may provide an important tool in developing both, a holistic terminology, and a holistic methodology for the integration of biology and physics necessary for the emergence of a holistic biophysics, or quantum biophysics. Especially significant may be its usefulness as a suitable framework for the treatment of organisms as macroscopic quantum systems (cf. the significance of vacuum degeneracy).

However, it must be clarified that we are not only talking about the electromagnetic vacuum of zero-point fluctuations, but of a more inclusive and fundamental unified vacuum of all four interactions.

SUPERFLUID VACUUM MODEL OF THE ORGANISM

In biophoton research first considerations on the possible role of the vacuum have been made in the 1980‘s in connection with the stability and optical properties of DNA and the optimal signal/noise ratio in the information transfer by biophotons. The central role that the vacuum plays in the Dicke theory and in Cavity Quantum Electrodynamics is well known. In 1985, Popp has suggested that biophoton emission as measured may arise from a non-measurable, virtual, delocalized, highly coherent field within the tissue, denoted by him as the realm of „potential information“ in the organism; he conjectured it may be a kind of vacuum state ^179-180.

More recently, Zeiger has developed a „superfluid vacuum model“ for understanding the biophoton emission of seeds and its connection to their viability. According to this model, seed vitality and biophoton emission are two parallel expressions of the same underlying reality: the superfluid Bose-condensate of photons. He proposes that the radiation field coupled to biological systems has to be understood on the basis of a twofold ground state. It consists, on the one hand, of a non-perturbative, collective-coherent state responsible for stability, internal communication and photon storage, which endowes the organism with a quiet background field connecting all its components by long-range phase relations with each other and with the environment. The second ground state, a perturbative, fluctuating-coherent state consisting of the excitations of the collective-coherent state, is responsible for flexibility, adaptation and external communication, and from it the observed biophotons are emitted. The two states are separated by an energy gap which controls the behaviour of the system and is a basic measure for the overall state of the organism. It is a parameter that promises to become an important new tool in biophysics proving additional information on the living system. Zeiger’s model may be a significant step in the biophysical modelling of the process of the emergence of „becoming“ from the potentiality of the „ground of being“.

Thus, if we speak of the electromagnetic field, or biophoton field, of cells, tissues or of the whole organism, as opposed to the biophoton emission measured, we may actually be dealing not with an electromagnetic field in the usual sense, but with a virtual field, or vacuum state. This has actually been proposed by Bearden^181-183 and is partially supported by the work of Barrett and others, already mentioned, on one or more deeper level(s) of electromagnetism.

If we define vacuum physics as that branch of physics concerned with the fundamental pre-physical level of potentiality from which matter and fields arise and which contains the information for their dynamic structuring processes, I suggest that the corresponding field of biophysics concerned with the investigation of the biological role of the physical vacuum and of the mediating role of potentials and other nonelectromagnetic fields between the vacuum on the one hand and force fields and solid matter on the other hand, should be called „vacuum biophysics“. It may become an important theoretical element and research subject of the new biophysics.

Under the above assumptions a hierarchy of levels of biological function, or regulation systems, based on fields, may be envisaged as a working hypothesis, where we have, between the solid body on the one hand, and consciousness on the other hand, the intermediate levels of holitic regulation systems and physiological-biochemical regulation, bioenergetic (EM) fields, and finally bioinformation fields.

Conclusion

Can physics deliver another biological revolution ? There is good evidence it indeed can and will, but it is not the further perfection of the reductionist program of molecular-biological control of life devised in the 1930’s that will produce the much needed revolution, but rather the renewal of the alternative tradition of holistic understanding in biophysics whose features are now beginning to emerge more clearly. It will draw on the still valid findings of the various holistic approaches in physics and the biomedical sciences of the first half of the last century which it will actualize by linking them to the many recent advances in physics and molecular biology, and it will incorporate the interaction of consciousness with biological matter as a key element in understanding the phenomena of life. The new holistic biophysics will be based on a field model of the organism and its interactions with other living systems and the environment; besides electromagnetic fields it will also consider the possible role of novel, non-electromagnetic fields and of the physical vacuum in biological processes.

About Mathematical Biology (Robert Rosen)

Part 3: ABOUT GLOBAL SCALING THEORY

Scaling means logarithmic scale-invariance. Scaling is a basic quality of fractal structures and processes. The Global Scaling Theory explains why structures and processes of nature are fractal and the cause of logarithmic scale-invariance.

Scaling in Physics

 In 1967 / 68 Richard P. Feyman and James D. Bjorken discovered the phenomenon of logarithmic scale-invariance (scaling) in high energy physics, concrete in hadron collisions.

 Simon E. Shnoll found scaling in the distributions of macroskopic fluctuations of nuclear decay rates. Since 1967 his team discovers fractal scaling in the fluctuation distributions of different physical and chemical processes, as well as in the distributions of macroskopic fluctuations of thermic noise processes.

 In 1982 – 84, Hartmut Müller discovered scaling further in the distributions of elementary particles, nuclei and atoms dependent on their masses, and in the distributions of asteroids, moons, planets and stars dependent on their orbital properties, sizes and masses

 Scaling in Seismicity

Within the fifties Beno Gutenberg and Charles Richter have shown, that exists a logarithmic invariant (scaling) relationship between the energy (magnitude) and the total number of earthquakes in any given region and time period.

Scaling in Biology

In 1981, Leonid L. Chislenko published his work on logarithmic scale-invariance in the distribution of biological species, dependent on body size and weight of the organisms. By introducing a logarithmic scale for biologically significant parameters, such as mean body weight and size, Chislenko was able to prove that sections of increased specie representation repeat themselves in equal intervals (ca. 0.5 units of the log based 10 scale). See lso the Law of Kleiber

In 1984, Knut Schmidt-Nielsen was able to prove logarithmic scale-invariance in the construction of the organisms and in metabolic processes.

In 1981, Alexey Zhirmunsky and Viktor Kuzmin discovered process-independent logarithmic scale invariance in the development stages in embryo-, morpho- and ontogenesis and in geological history.

Scaling in Neurophysiology

We live in a logarithmic world. All of our senses perceive the logarithm of a signal, not the linear intensity of the signal itself. That is why we measure sound volume in decibels, and consequently in logarithmic units.

Sounds whose frequencies differentiate themselves by double, quadruple or eight-times, we perceive as a, a’ or a’’, the same sound. This property of our sense of hearing makes it possible for us to differentiate harmony from disharmony. The harmonic sound sequence 1/2 (Octave), 2/3 (Fifth), 3/4 (Fourth), 4/5 (Major Third), and so forth is logarithmic, hyperbolic scale-invariant.

Our sense of touch is also calibrated logarithmically. Assuming that one holds in the left hand 100 grams and in the right hand 200 grams; if one then adds 10 grams to the left hand, then 20 grams must be added to the right hand in order to sense the same weight increase. This fact is known in Sensing Physiology as the Weber-Fechner Law (Ernst Heinrich Weber, 1795 – 1878, Gustav Theodor Fechner, 1801 – 1887): The strength of a sensory impression is proportional to the logarithm of strength of the stimulus.

The Weber-Fechner law also touches on our senses of smell and sight. The retina records only the logarithm, not the number of impinging photons. That is why we can see not only in sunlight but also at night. Whereas, the number of impinging photons varies by billionths, the logarithm varies only by twentieths. (ln 1000,000,000 ≈ 20.72)

Our vision is logarithmically calibrated not only in regards our perception of the intensity of light, but also relative to the lights wave length which we perceive as colors.

Our ability to judge lineal distances is based on the possibility of comparison of sizes and the determination of relative measurement scales. The linear perspective assumes a constant size proportion that is defined by size enlargement or reduction factor. This factor is multiplied several times with itself in the perspective. From this an exponential function is defined which argument is a logarithm.

The function of our sense organs is concerned with acoustic or electromagnetic wave processes. The logarithmic scale-invariant perception of the world is a consequence of the logarithmic scale- invariant, construction of the world.

Scaling in Mathematics 

All natural numbers 1, 2, 3, 4, 5 … can be constructed from prime numbers. Prime numbers are natural numbers that are only divisible by the number 1 and themselves without leaving a remainder; accordingly, the numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 … are quasi elementary parts of the real number continuum. The distribution of the prime numbers among the natural numbers is irregular to such an extent that formula for this distribution cannot be defined. Of course, prime numbers become found more seldom the further one moves along the number line. Already in 1795, Carl Friedrich Gauß noticed this. He discovered that the set p₁(n) of prime numbers up to the number n could be calculated approximate according to the formula p₁(n) = n / ln

1. The larger the value of n, the more precisely is this law fulfilled; and means that the distribution of the set of prime numbers among the natural numbers is scale-invariant

Non-prime numbers can clearly be represented as products of prime numbers. One could also say that non-prime numbers are prime-number clusters. This means that non-prime numbers are composed from several prime numbers. In this interpretation, one can derive the prime-factor density distribution on the number line.

The logarithmic scale-invariance of the distribution of the prime numbers is a fundamental property of the continuum of numbers. Moreover, this is the only non-trivial statement that holds true for all prime numbers.

Logarithmic scale-invariance of the prime-factor density distribution means that one can speak about a standing density-wave in the number continuum. The prime factors, 2 and 3 produce the base-oscillation, and early prime factors produce the spectrum of overtones.

From the Model to the Theory

Oscillations are the most energetically efficient kind of movement. For this reason, all matter, not only each atom, but also the planetary system and our galaxy, oscillate and light is an unfolding oscillation and, naturally, the cells and organs of our bodies also oscillate

Based on their energy efficiency, oscillatory processes determine the organization of matter at all levels – from atoms to galaxies

In his most meaningful work, “World Harmonic”, Johannes Kepler established the bases for harmonic research. Building on the ancient musical ‘World Harmony’ of the Pythagoreans, Kepler developed a cosmology of harmonics.

Global Scaling research continues this tradition.

The Melody of Creation

Scaling arises very simply – as a consequence of natural oscillation processes. Natural oscillations are oscillations of matter that already exist at very low energy levels. Therefore they lose few energy, and likewise fulfill the conservation law of energy.

The energy of an oscillation is dependent on its amplitude as well as on its frequency (events per time unit)

Consequently, the following is valid for natural oscillations: The higher the frequency, the less the amplitude. For natural oscillations, the product of frequency and wave length as well as the product of frequency and amplitude are conserved. They limit the speed of propagation of oscillations in mediums, or the speed of deflection.

A standing wave in a homogeneous space arises only if in the direction of the wave penetration the space is finite and if the half wave length is equal to an integer part of the medium size.

As a consequence we can find for any low enough resonant oscillation mode frequency f₀ a higher mode frequency f₁ with an integer relationship n = f₁ / f₀. The frequencies of such resonant oscillation modes generate exponential series

Therefore, the complete resonant oscillation frequency spectrum can be represented as a set of logarithmic fractal spectra (1) with natural n = 1, 2, 3, … In this representation the generation of the complete resonant oscillation frequency spectrum can be understood as an arithmetical task, what can be reduced to the fundamental theorem of arithmetic, that every natural number greater than 1 can be written as a unique product of prime numbers.

In our example, the oscillation period of the 1. mode is three times longer that the period of the 2. mode, nine times longer than the period of the 3. mode and 27 times longer that the period of the 4. mode. From this follows the logarithmic, fractal, construction of the (repeating itself in all scales) oscillation representation. In this connection, one speaks of scale-invariance (Scaling). In nature Scaling is distributed widely – from the elementary particles to the galaxies. It is in this connection that one speaks of Global Scaling.

Natural oscillations of matter produce logarithmic, fractal spectrums of the frequencies, wave- lengths, amplitudes and a logarithmic fractal network of oscillating nodes in space.

In physical mediums, base-tones, upper- or undertones are produced simultaneously, and in this there arise consonances and dissonances. Not only our hearing can distinguish consonance from dissonance; this capability extends to all matter, and has to do with the energy expenditure necessary to produce an overtone.

A musical ‘Fifth’ arises most easily (the least expense of energy per oscillation period), because merely a frequency doubling and trebling is necessary in order to produce an overtone in interval of 3/2 of the base frequency. Somewhat more energy is necessary to produce a musical ‘Fourth’, 4/3, while this additionally requires a quadrupling of the base frequency; and even more energy is necessary for the production of the larger musical ‘third’, 6/5, of the same amplitude, and so forth.

The musical intervals accordingly play an energetic, key-role in the spectrum of the natural- oscillation-modes. In fact, this spectrum is constructed like the spectrum of a melody.

Natural oscillations of matter are probably the most important structure-forming factor in the Universe. For this reason one finds fractal proportions everywhere in nature. The logarithmic, fractal distribution of matter in the universe is a consequence of natural oscillation processes in cosmic, space and time measurement scales. In this connection, one speaks of the “Melody of Creation”.

Leonhard Euler: About Continued Fractions and FRactal Geometry

In the works “About Continued Fractions” (1737) and “About the Oscillations of a String” (1748), Leonhard Euler formulated problems whose solutions would keep the field of mathematics busy for 200 years.9

Euler investigated natural oscillations of an elastic, mass-less string of pearls. In connection with this task, d’Alembert developed his method of integration for a system of linear differential equations. Daniel Bernoulli asserted his known statement that the solution of the problem of the free-oscillating string can be represented as trigonometric sequence, something which started a discussion between Euler, d’Alembert and Bernoulli lasting a decade. Later, Lagrange showed more correctly how one arrives at the solution of the problem of oscillations of a string-of-pearls and the solution of the problem of a homogenous string. The problem-solution was first completely solved by Fourier in 1822

Almost insurmountable problems arose in the meantime with pearls of various mass and irregular distribution. This task led to functions with gaps (or voids). After a letter from Charles Hermite, May 20, 1893, who additionally added, “Reject in nervous horror the lamentable annoyance of the functions without derivative”; T. Stieltjes investigated functions with discontinuities and found an integration method for these functions which led to continued fractions.

Meanwhile, Euler already recognized that complex, oscillating systems can contain such solutions (integral) that themselves are not overall differentiateable, and left behind to the future mathematical world an analytical “monster” – that is, the non-analytic functions (this term was chosen by Euler himself).

Non-analytic functions provided ample and profuse study up to the 20^th century, after the identity crises in mathematics, appeared to be conquered.

The crises began, and lasted until about 1925, when Emil Heinrich Bois Reymond, in 1875, reported for the first time about a Weierstrass constructed continuous, but non-differentiable function. The main players were Cantor, Peano, Lebesgue and Hausdorff; and as a result, a new branch of mathematics was born – Fractal Geometry.

About Fractals

Fractal comes from the Latin ‘fractus’ and means “broken-up-in-pieces” and “irregular.” Fractals are consequently distinct, fragmentary, tricky mathematical objects. Mathematics in the 19^th Century considered these objects as exceptions and from there, tried to derive fractal-objects from regular, continuous and smooth structures.

The theory of fractal groups made possible in-depth investigations in “non-analytic”, manifold, granular or fragmentary forms. It immediately it becomes apparent that fractal structures are by no means so infrequently encountered in the world. More fractal objects are discovered in nature than ever suspected. Moreover, it suddenly seemed as if the entire natural universe were fractal.

Especially, works of Mandelbrot finally advanced the geometry to the position where fractal objects could be mathematically correctly described: fragmented crystal-lattices, Brownian motion of gas molecules, complex, giant polymer-molecules, irregular star-clusters, cirrus-clouds, the Saturn rings, the distribution of moon-craters, turbulences in fluids, bizarre coast-lines, snake-like river- streams, faults in mountain-chains, development branches of the most varied kinds of plants, surface areas of islands and lakes, mineral-formations, geological sediments, the distribution in space of raw-material-occurrences, and, and, and……

A deciding factor in the accurate treatment of fractal objects was the introduction of real and also irrational dimensions, in contrast with the whole-number dimensions of Euclidean Geometry.

Let’s consider an example: In Euclidean Geometry, a disappearing, small grain-of-sand has dimension 0. A line – dimension 1. But which dimension does sequence of grains-of-sand arranged one after the other?

The Euclidean point of view only knows boundary conditions: Either one proceeds on a wide path – until one cannot recognize anymore grains-of-sand and then assigns to this object the dimension 1, or one recognizes the grains-of-sand as objects of dimension 0. and while it is usually known that 0=0=…+0 = 0, the grains-of-sand, are likewise assigned dimension 0. That through this the essential is lost is obvious.

The first step of an in-depth analysis of this situation was undertaken by Cantor in his letter of June 20th, 1877 to Dedekind, the next was followed by Peano in 1890.

The mathematicians recognized that an accurate understanding of fractal structures could not be reached when one defines ‘dimensions’ as a number of coordinates. Therefore, in 1919 Hausdorff defined a new conception of dimension. The fractal (broken) dimension D completed the topological (whole-number) dimension through logarithmic values.

The fractal dimension of a grain-of-sand sequence of N grains-of-sand of the relative (In comparison to the total length of the sequence) size 1/k, where D = log(N) / log(k). Assuming a sequence of 100 grains-of-sand is 100 mm in length and the size of a grain-of- sand 1 mm. Then D = log (100) / log (100) = 1. However, if the sequence only consists of 50 grains-of-sand, then D = log (50) / log (100) = 0.849485. The fractal dimension D is, accordingly, a measure for the fragmentary nature of an object. The larger the gaps, the further D is from integer number values.

The application of the Hausdorff-Dimension in geometry now makes it possible to deal with not only completely irregular real mathematical objects, but at the same time provides the formula for the creation of home-made fractal creations. The creation of various Mandelbrot- und Julia-groups using the computer gave rise to a popular mathematical sport. The Mandelbrot-group is still today the object of non-resolved theoretical investigations. However, it is important that these through mathematics that their connections become visible, and are being investigated in the most varied of specialist fields.

Nevertheless, the fractal ‘Grain-of-sand’ sequence strongly reminds of Euler’s ‘String-of-Pearls.’ Both of these objects are fractal.

In 1950, the Leningrad mathematicians, F. R. Gantmacher and M.Krein regarded the line-of-deflection of an oscillating string of pearls as a broken line. This initial step even enabled them a fractal view of the problem, without which they were unaware (Mandelbrots Classic “Fractal Objects”, appeared in 1975 and was his first works of which 50 were in the field of Linguistics). They first brought the Fractal visibly into the situation, and to completely solve (and also for the most general case) the 200 year old Euler Problem of the oscillating “Sting-of-Pearls” for variable masses and irregular

In their work, “Oscillation-matrices, Oscillation-nuclei, and Small Oscillations of Mechanical Systems” (Leningrad, 1950, Berlin, 1960), Gantmacher and Krein show that Stieltjes-Continued- Fractions are solutions of the Euler-Lagrange equations for natural oscillating systems. These continued fractions produce fractal spectrums.

In the same year, the comprehensive work of Oskar Perron appeared “The Theory of Continued- Fractions.” The theme was also worked on by N. I. Achieser in his work, “The Classical Moment Problem and Some Associated Questions About the Analysis (Moscow 1961). Terskich generalized the (with regard to the contents) Continued-fraction method on the analyses of Fundamental-oscillating, branching chain-systems (Terskich, V. P., The Continued-fraction Method, Leningrad, 1955). Khintchine resolved the meaning of Continued-fractions in arithmetic and algebra (Khintchine, A. J., Continued fractions. University of Chicago Press, Chicago 1964).

Additional works by Thiele, Markov, Khintchine, Murphy, O’Donohoe, Chovansky, Wall, Bodnar, Kučminskaja, and Skorobogat’ko, etc. helped lead to the final break-through for the continued- fraction method, and in 1981 enabled the development of efficient algorithms for the addition and multiplication of continued-fractions.

Each real number – and also each measurable value – can be clearly represented as a ‘normed’ continued fraction (all partial denominators are 1). Finite, normed continued fractions converge to rational numbers. Infinite continued fractions converge to irrational numbers.

The simplest continued fraction generates the ‘Golden Number’ proportion. All of its elements are Supposedly this is why the ‘Golden Mean’ is so wide-spread in nature. The sequence {1, 1/2, 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, …} produce the sequence of Fibonacci-numbers.

A quite special structure is exhibited by the continued fraction of the Euler-number e= 71828 … This continued fraction contains the sequence of all natural numbers and the sequence of all musical intervals.

The convergent Continued-fraction for e is composed of the reciprocals of the musical intervals (Prime 1/1, Octave 1/2, Fifth 2/3, Fourth 3/4, Major Third 4/5, Minor Third 5/6, ….).

The number 1 – and each other integer number – also can be represented as a continued fraction.

In its continued-fraction representation, every number is an oscillation-attractor. Khinchine could prove that convergent continued-fractions yield the best approximations of the irrational numbers because they themselves converge the most quickly on the eigenvalue of the continued-fraction.