A Review by Karl H. Wolf, Emeritus Professor of Geology, Springwood, NSW Australia

Introduction
This is the 5th revised, enlarged edition of the successful book (‘earlier editions sold out in less than one year’, accompanied by several translations) as part of the major Springer monograph series entitled ‘Understanding Complex Systems’ and ‘Springer Series in Synergetics’ (see front page general orientation).
    This book is another important contribution to the study of complex phenomena increasingly encompassing all disciplines – because all natural and human relationships comprise systems, systems, systems: everything is complexly connected! Both Mainzer and Érdi describe the latest (up to 2006/7) advancements in our search for ‘understanding’ and ‘truth’. To paraphrase: ‘The theory of non-linear complex systems is now a proven problem-solving approach in the natural sciences … and also recognized that many if not all social, ecological, economical and political problems are essentially of a global non-linear complex nature. … no other holistic (in opposition to the traditional reductionist) perspective can hardly be achieved by any other methodology’.

Caveat
This review assumes familiarity with that of Péter Érdi’s ‘Complexity Explained’ on academici because numerous general aspects also apply to Mainzer’s book.  

Contents
The Preface of this 5th edition – with its all-too-brief chapter overview - is preceded by the earlier four Prefaces, thus allowing a glimpse of the book’s and the complexity-concept’s development.  A very good Introduction (with an excellent Interdisciplinary comparative summary table) precedes the next eight chapters. Chapter 2. Complex Systems (CS below): the Evolution of Matter. This is followed by Chapters 4 to 8 dealing with, respectively, CS: Evolutions of Mind-Brain, Computability, Artificial Life, Economics, and Human Cultures & Society. Chapter 9, then, offers an Epilogue on Future, Science and Ethics. The References are a good source for researchers. The Subject Index is rather incomplete, but ‘acceptable’. Mainzer’s numbering scheme of chapters and sub-sections is recommendable; however, Érdi’s text division into many shorter sections is especially welcome!  
    As to coverage of complexity research, both Mainzer and Érdi refer predominantly to the English-language literature. However, Érdi lists several German and numerous Hungarian publications, whereas Mainzer refers to many publications from both German and French sources; a few up to the years of 2006/07. In general, the two reference lists offer a good introduction into the CS theory and related topics. Many older publications are absent, e.g. W. Weaver, 1948. Science and complexity, Am. Sci., v. 36, 536-544; H.A. Simon, 1962. The architecture of complexity. Proc. Am. Philos. Soc., v. 162, 467-482.
    Throughout, Mainzer employs admirable comparative/contrastive reasoning as is indicated by the following often-repeated phrases: ‘philosophically speaking’, or ‘from the philosophical view point’, or ‘ … methodological, complex-systems, non-linear, micro-/macro-structural, neuropsychological, synergetic, theological … viewpoint’.

Scope of the ‘complexity theory’
The phrase  ‘complex theory ’ in isolation is almost meaningless unless we admit that any specific complex acts as a cognitive/intellectual ‘attractor’ for the integration of many newer concepts-cum theories (some may be natural laws now), such as chaos, order/disorder, reductionism, fuzzy logic, fractals, hierarchy, emergence, self-organization, cause-effect relationships, dissipative structuring, open/closed system characteristics, boundedness, degree of dynamism, bifurcation, periodicity, synchonicity, coincidence, connectedness, catastrophe, scale-controls, degree of freedom, randomness, entropy, accessibility, among a host of others – all required to investigate the huge complex system of reality.  Inasmuch as these concepts occur in hundreds of different combinations, there are equally hundreds of sub-complexes and subsub-complexities.  Thus, any definition of ‘complexity’ must automatically include a reference to the philosophy, methodology, techniques, … of these particular concepts in order to make ‘complex theory’ epistemologically, ontologically, heuristically, … meaningful.

Hierarchy of Knowledge Theory
Modern Theory of Knowledge (see François, 2004) constitutes a hierarchy. Simplistically, I like to consider Systems Analysis (and the near-synonymous Cybernetics, Network Theory, Holistics and Integration approaches) (all are different, in some ways, of course) at the top of a data/knowledge pyramid. Then follow beneath the pinnacle several levels of second, third, … order sub-EFPVs (Entities, Factors, Parameters, Variables) of Complexity, Chaos, Self-Organization, and many more. These latter ones constitute the lower ranks/levels of this knowledge pyramid-cum-pyramid – the division of ranks/levels depending on research needs and preferences.
Systems Analysis, Cybernetics, and other approaches have been described in many publications; see for example two Encyclopedias by the Union of International Association (1995) and Francois (2004).

Degree of completeness and readership addressed
Generally speaking, both Mainzer and Érdi cover well many aspects related to complexity theory, but differences, of course, are obvious – some examples given below.
    As to the readership addressed, let me reiterate the opinion by Rick Szostak in his book on ‘Classifying Science’ (see review in this academici Website in press) that the definition of ‘science’ ought to be much broadened because The Scientific Method(s) have been ‘modernized’ through the development of numerous new concepts (or hypothesis, theories, even natural laws) as exemplified by the complexity theory. As all these new approaches are applied in just about all knowledge domains, this book is addressing all disciplines’ readers.

Examples of application
    Both theoretical (conceptual) and applied (practical, pragmatic) applications of the Complexity Theory (indeed all other EFPV-members of the Systems Analysis philosophy or methodology) are too vast to even merely list them here. However, at least one exemplar: the present global economic/financial disaster (catastrophe, calamity, …), according to numerous experts (likewise numerous books available), could have been predicted (not so by Alan Greenspan et al.!) through the application of Network Theory. This theory, to reiterate, is a type of Systems Analysis – comprising Complexity Theory and other EFPVs.
    Now that the world has entered the economic ‘recession’ (concocted by financial experts?), experts again proposed that the Theories of Complexity, Chaos, etc., must be employed to understand ‘what went wrong’ and to cooperate globally to find a New Economic ‘Science’ (really?!) to prevent future occurrences of this shemozzle!  See Mainzer’s references to economy and in particular his section 7.5 (in the Chapter 7. Complex Systems and the Evolution of Economics) on Econophysics. To paraphrase: ‘Econophysics is an interdisciplinary scientific field that refers to economics research performed by physicists applying mathematical models … nonlinear dynamics, complex systems, and chaos theory.’ Over to you, experts/specialists! See also New Scientist (2008).
 
Comparison with Érdi’s book: some specifics
Although both authors’ offer a broadly similar coverage, in detail there are many differences, as described below.
(a) Concepts considered. Most, but not all, of the few dozen modern systems-contextual concepts referred to by Érdi are likewise dealt with in Mainzer’s book, some in a unique way (see fig. 2.28 of Super and Grand-Unification).
(b) Particular emphasis. Mainzer particularly emphasized that (i) one continually has to correlate micro- and macro-phenomena; (ii) many earlier linearly-considered systems represent in reality non-linear dynamics; (iii) of the many complexity-associated concepts he deals repeatedly and preferentially with synergy, order-parameters, reductionism/irreducibility, phase transitions, thermodynamic principles, numerous ‘quantum’-concepts, attractors, fractals/multi-fractals, cellular-automata, randomness, unpredictability, several additional ‘self-phenomena’ such as self-construction, time-series, power law, hierarchy, embodiedness, symmetry-breaking, emergence, entropy, expert systems and artificial intelligence, among others. He also emphasizes that (iv) the systems law of ‘the whole is more than the sum of its parts’ is too simple and needs to be supplemented (replaced?) by newly identified systems concepts, such as the ‘slaving principle’, for example.
 (c) Philosophical/cognitive tools considered. Of course, Mainzer too repeatedly refers to epistemology, ontology, heuristics, teleology, logic, paradigm shifts … opportunistically emphasizing that increasingly Philosophy of Science does have important pragmatic applications in ‘explaining’ the world’s complexity.
Here I like to add that this opinion is naturally contra to many (even ‘educated’ experts) that philosophy has no applied/practical value. According to Mainzer, this is false. As to pragmatic philosophies’ problem-solving abilities, see Baggini/Fosl (2003/07) for such cognitive/intellectual tools.
      (d) Knowledge domains considered. Both books consider the same domains but in detail their approaches vary considerably, as exemplified by the history of the complexity research, by the data related to life (biology, ecology, medicine, physiology, population, urban science), mind-brain (cognitive psychology, neurosciences, bionics, consciousness etc.) computability (algorithm, probability, quantum aspects, informatics, cellular automata etc.), mathematics (statistics, probability), artificial life, economics, econophysics(!), nanotechnology, and human cultures and society, linguistics (metaphors), earth sciences, cosmology/astronomy (black holes, galaxies), meteorology (climate, weather) ….Some specific phenomena referred to are: super-string M-theory, unification, wormholes, …. For historians: Mainzer referred to the philosophies and sciences of Aristotle, Plato, Einstein, Newton, Hamilton, Botzmann, Leibnitz, among others.
      (e) Less emphasized or ignored aspects. All research has to be selective, i.e. emphasizing certain aspects; this applies to all books. However, there are topics consistently(?) ignored to the intellectual detriment. For example: for years I have highlighted that philosophers ignore the differences between basic (pure, exact, fundamental) sciences in contrast to the derived (hybrid) sciences (and other disciplines). A comparative/contrastive study of the differences of their ‘complexities’ (and all other phenomena/concepts) is required. Books dealing with complexities in physics and economic systems do offer a glimpse of the differences of a basic science versus a derived/hybrid discipline – but the approach is so cognitively/intellectually and methodologically implicit and indirect that the crucial differences are hidden, not immediately obvious! The two books, of course, also missed that challenge.
    (f) Further differences of the two books. Mainzer ignores cybernetics, whereas Érdi deliberates it 22 times, providing some interesting information; the same applies to the catastrophe theory. Fuzzy logic could have been given higher priority in both books as there is a wealth of published data available (e.g. applied to engineering and industrial control systems, geology, etc.); linguistics, teleology, and other topics are also differentially treated.

Style of presentation
Considering that the author’s mother tongue is not English, the editing (by the author and/or by the publisher) is superb, i.e. the Edward Fry Readability Grade/Index is high! The syntax and diction are lucid and clear, i.e. there are no esoterically too-lengthy pseudo-sophistically twisted phraseologies. Yet, compare the layout of Mainzer’s with that of Érdi’s book, and you will understand why I prefer that by the latter – he used several imaginative more-eye-catching and comprehension-enhancing approaches!
    The degree of coverage ranges from the basics (historically and technically) to the more-advanced. The latter is, however, not esoteric as the data is always pragmatic (practical, applied). Mathematics and principles of physics, chemistry, biology, economics and other disciplines are of a good 1st year university science level. The liberal use of 223 figures (including 4 color ones) and some tables enhance the text discussions, although there is always an opportunity for more – especially for comparative/contrastive summary tables, flow charts, mind maps, and the likes. Both books are indeed highly recommended – a wealth of information there!
    Both books list the References near the end prior to the Indexes. Mainzer lists them chapter-by-chapter, whereas Érdi uses the alphabetic approach (which I prefer) – both utilizing a ‘beautifully efficient’ cross-referencing numbering scheme.
                            
References

● Baggini, J., and Fosl, P.S., 2003/2007. The Philosopher’s Toolkit: a Compendium of Philosophical Concepts and Methods. Blackwell Publishing, 221 pp.
● Francois, C. (2nd Edition), 2004. International Encyclopedia of Systems and Cybernetics, Volume 1 & 2. Saur Publishers, Munich and London, 741 pp. (See review in the International Journal of General Systems, Vol. 34, No. 3, June 2005, 321-4.)
● New Scientist, 2008.  The Folly of Growth: How to Stop the Economy Killing the Planet. New Scientist, 18. October, ’08, 40-54.
● Union of International Associations, 1995. Encyclopedia of World Problems and Human Potential, Volumes 1, 2 & 3. Saur Publishers, Munich and London, 3,161 pp. (See review in Journal of Documentation, Vol. 54, No. 4, September 1998, 520-523.)
● Szostak, R., 2004. Classifying Science: Phenomena, Data, Method, Practice. Springer Verlag, 286 pp. (See review in the International Journal of General Systems, Vol.35, No. 4, August 2006, 472-478; and in this academici Website, book-review section; in press.)