Vinod Wadhawan’s new book ‘Revisiting the Scientific Method’

‘REVISITING THE SCIENTIFIC METHOD: The Need to Make Science More Inclusive in Scope’ 

by Vinod Kumar Wadhawan 

 

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This book is a follow-up to the previous book by the author (‘The 8-Fold Way of the Scientific Method’). The previous book introduced the reader to the way science is done: Strict adherence to objectivity, rationality, and transparency in handling information about a natural phenomenon we want to understand. The present book goes a step further and takes a critical look at the Scientific Method to see what can be done to make science more inclusive in scope; for example by giving due importance to subjective or experiential information also, and not only empirical information. Together the two books provide a fairly comprehensive account of the nature of scientific research, and can serve as course material for the training of an aspiring scientist. The theme of consciousness runs throughout the present book, because it is the most important example of a nonphysical phenomenon or entity that current science, by and large, tends to stay away from. Ways are suggested for dealing with this problem by relaxing in a carefully guarded manner some of the eight tenets of the present Scientific Method. Big Data is a rather recent development in the history of science. Its availability is going to have far-reaching consequences for the way science is going to be done now. The book discusses its promises and pitfalls. Present-day science, which is mostly reductionistic in approach, is not adequate enough for dealing with complex systems. Here again, Big Data may be of big help because pattern formation is an important characteristic of many complex systems, and Big Data is very good (rather too good!) at discerning patterns or correlations. Discussion of Karl Popper’s falsifiability criterion occupies substantial space in this short book. This is because this criterion is the main reason why the present Scientific Method labels many questions about Nature as unscientific or nonscientific, thus limiting the scope of scientific enquiry. In a more inclusive approach one would also give due importance to the philosophical rival of falsificationism, namely verificationism. Another way of making science more inclusive is to use a diluted version of falsificationism, formulated by Imre Lakatos. There is also a discussion of the work of some other philosophers of science, notably Nicholas Maxwell and Thomas Kuhn. Popper’s philosophy for doing science has proved to be very successful, but it is desirable that the student of science be also aware of other models of how science can progress. 

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CONTENTS 

 Preface vii 

 

1. The Scientific Method 1 

1.1 Asking the right question 2 

1.2 Objective observation of the world 4 

1.3 Coming up with hypotheses for understanding the data 6 

1.4 Reproducible verification of predictions of hypotheses 12 

1.5 A theory for explaining the hypotheses 15 

1.6 Use of unambiguous language and logic 18 

1.7 Choice of the smallest necessary set of axioms 21 

1.8 The falsification requirement 25 

 2. Complex Systems, Complexity Transitions 27 

2.1 Complex systems 27 

2.2 Towards a formal definition of a complex system 30 

2.3 Complexity transitions 34 

2.4 Emergence 36 

2.5 Reductionistic science is inadequate for dealing with complex systems 38 

 3. Big Data and the Future of the Scientific Method 41 

 4. Life, Intelligence, Consciousness 53 

4.1 Life 53 

4.2 Intelligence 60 

4.3 Consciousness 62 

4.4 One consciousness or many? 73 

 5. Artificial Life, Intelligence, Consciousness 77 

5.1 Artificial life 77 

5.2 Artificial intelligence 80 

5.3 ChatGPT 81 

5.4 Artificial consciousness, or machine consciousness 87 

5.5 Can a robot acquire consciousness? 90 

5.6 Have some of our machines already become sentient? 91 

5.7 Apocalyptic AI and transhumanism 94 

 6. Towards a More Inclusive Scientific Method 99 

6.1 Asking the right question 99 

6.2 Observation of the world 100 

6.3 Coming up with hypotheses for understanding the data 102 

6.4 Testing of predictions of hypotheses 104 

6.5 A theory for explaining the hypotheses 105 

6.6 The language and logic of science 106 

6.7 Choice of axioms 107 

6.8 How justified is the falsifiability requirement? 108 

 7. Concluding Remarks 115 

 Bibliography 123

 Index 135 

 == 

Preface 

 Nature is all there is. All phenomena are natural phenomena, and the Scientific Method is the method of choice for investigating them. The study of all natural phenomena should come within the purview of science. But the strictness of the Scientific Method presently used makes it inapplicable for investigating certain nonphysical phenomena in their entirety, a good example being that of consciousness. In a general sense, consciousness is a state of awareness of one's thoughts, feelings, sensations, and surroundings. It is the subjective experience of perception, cognition, and emotions that is sometimes described as the ‘sense of self’. Consciousness is a fundamental aspect of human experience and an essential characteristic of the human mindbody, and of many other life forms. An important requirement of the Scientific Method is that in any scientific discourse, every word used must convey the same meaning to all concerned. Words like ‘consciousness’ create problems on that score. Although the neural correlates of consciousness help us in taking an empirical view of the situation, the problem remains that we do not know how physical processes in the brain give rise to subjective experiences. How to remedy the situation so that we can have meaningful scientific dialogue on such natural phenomena also? In my previous book (The 8-Fold Way of the Scientific Method) I gave a fairly comprehensive discussion of what the Scientific Method is all about, and how science is done by applying it. The achievements of the method have been spectacular, and we humans can be truly proud that we invented it. But, as pointed out near the end of that book, there is a need to relax its dictums (in a carefully guarded way) so that the scope of scientific investigations can become more inclusive. I suggested some ways there, and discuss the matter more comprehensively in the present book. There is a related issue, namely that of understanding complex systems. The human mindbody is perhaps the most complex system of them all. The present Scientific Method is suitable mainly for doing reductionistic science. A characteristic feature of a complex system is that it must be investigated as a whole, and not by reducing it into parts and assuming that if we understand the parts, we understand the whole. For complex systems, the whole is more than the sum of the parts, in a mutually interactive manner. An extended Scientific Method that is effective for studying complex systems will automatically become more inclusive in scope. Big Data is an inalienable feature of life in modern times. It is already influencing science and many other activities like finance, business, advertising, entertainment, government, warfare, etc. There is a viewpoint that Big Data can make it possible to do science without having to postulate hypotheses/models/theories beforehand. This may well be too optimistic, but there is no denying the fact that the availability of Big Data offers new opportunities for making science progress rapidly, particularly for investigating complex systems. Big Data Analytics can throw up unexpected correlations in the data, which may provide new leads for understanding a complex system. There is already a report that an artificial-intelligence system has been developed that is capable of formulating physical theories by recognizing patterns in complex data sets. So, we need an extended Scientific Method that makes science more inclusive in scope, that enables science to investigate complex systems by going beyond reductionism, and that incorporates the use of Big Data as an additional tool for doing science (data-driven science). In this book I explore the possibilities and make some suggestions. If insistence on empirical evidence limits the scope of science, then go for non-empirical and experiential evidence also. If the falsifiability requirement of the Scientific Method is too restrictive, then see what can be done to relax it in a carefully guarded and tentative manner. Big Data is offering wonderful new and highly unconventional ways of doing science. The training of scientists must now include awareness of this option, as also an exposure to the basics of how such research should be done. 

 Vinod Wadhawan New Delhi (January 2025)

Author’s response to Dr. Abhas Mitra’s review of the book THE 8-FOLD WAY OF THE SCIENTIFIC METHOD

Dr. Abhas Mitra, author of the bestseller book The Rise and Fall of the Blackhole Paradigm (2021), has published an Amazon-India review of my book, which I reproduce below. This is followed by my clarification of some of the points made by him. The points I respond to are marked with superscripts ⁽ⁱ⁾, ⁽ⁱⁱ⁾, etc.

Dr. Mitra’s review reads as follows:
==========
* * * * * One of the most valued books I have ever read, though the cheapest one
It may sound disappointing that many scientists practising science do so as a routine profession without worrying too much about “Scientific Methods”. Of course good scientists and good science students do possess some ideas about the scientific methods to be employed while doing science. However even the good scientists may not feel the requirement to delve into various latent dimensions of correct scientific methods and might occasionally be tempted to arrive at wrong scientific conclusions. For instance, many physicists now think that String Theory, one of the most hyped topics of theoretical physics ever, might have been degenerated into some sort of pseudo science even when most brilliant theoretical physicists have struggled to develop it over past four decades. And of course general public is liable to be enamored by even pseudo-sciences, for instance by astrology.

To the best of my knowledge, the present book might be the only one on the science and philosophy of scientific methods after the publication of The Logic of Scientific Discovery (1959), a book about the philosophy of science by the philosopher Karl Popper, in which the author argued that science should adopt a methodology based on falsifiability. This is so because no number of experiments can ever prove a theory, but a reproducible experiment or observation can refute one. In his book, the author first dwells on the eight tenets of the Scientific Method, namely, (1) Right questions, (2) Right (objective or empirical) observations, (3) Right hypothesis to explain the observations, (4) Right testing of predictions of the hypothesis, (5) Right theory, (6) Right language and logic, (7) Right (minimum number of) axioms and (8) Rightly worded (falsifiable) statements. And the author asserts that these eight tenets are someway analogous to the eight steps to Nirvana emphasized by Buddha: (1) Right beliefs, (2) Right intentions, (3) Right speech, (4) Right conduct, (5) Right livelihood, (6) Right effort, (7) Right mindfulness and finally (8) Right concentration.

As the author succinctly puts it: A crucial aspect of science is systematizing the knowledge acquired and, even more importantly, of making the knowledge available to everybody for scrutiny. Thus at least in principle, science is always self-correcting. For instance, in 2020, all top scientists believed that Covid-19 virus does not float in air for too long and hence the disease does not propagate through air. But by May 2021, it appears that Covid-19 viruses are likely to freely propagate through air. Such a logical virtue of science might however be dishonored by the intellectual arrogance of the proponents of hypotheses which are not verifiable by observations and experimentations, as is the case with several aspects of Theoretical Physics and in particular the String Theory.

But this book goes much beyond such mundane aspects as the author points out that all phenomena are natural and there is really nothing “supernatural”. If so, in some way religion and mysticism too come under the purview of science in a broader sense. Simultaneously, he delves deeper into some complex issues of scientific interest. Accordingly, the 2nd part of this book explores newer dimensions. For instance, there are small chapters entitled “How to Live Well Forever” and “Reversal of Chronic Diseases”. However, I have not yet gone through these chapters and feel that such topics do not gel well with the general character of this great book and were avoidable (they could be published separately).⁽ⁱ⁾

The author is bold and his book even contains a subsection “The end of theoretical physics as we know it?”. Unfortunately this subsection is based entirely on a recent popular physics book by the German theoretical physicist Sabine Hossenfelder. And though I personally like this content, I feel it has (not) been organically connected with his overall book.⁽ⁱⁱ⁾ The 2nd part also contains insightful long discussions on works of Stephan Wolfram on complexity. I am afraid, such sections though highly valuable in their own rights, look like add-ons and have been some sort of distraction from the central theme of the book.⁽ⁱⁱⁱ⁾

On the other hand, I feel that the section entitled “Which is the most scientific natural language?” is important for a comprehensive appreciation of scientific methods because computer science has started aiding science in a major way. The author points out that for artificial intelligence, and computational linguistics, there is a subfield called natural language processing (NLP), or computer linguistics, which is about using computational techniques to learn, understand and produce human language content. Here the author highlights that Sanskrit language whose literal meaning is ‘sculpted to perfection’ is the most scientific natural language. In this context, he explains why long ago, Charles Babbage (1791 – 1871), who is sometimes called as “father of computers’’ mentioned that “The structure of Pāṇinian Grammar is nothing but a computer program.”

The author is an atheist and rationalist in the true sense and not as an intellectual fashion statement so prevalent in modern India. Accordingly, the author has no inhibition in highlighting the ‘thought and philosophy’ behind Vedas the same way Ernest Schrodinger had no hesitation in comparing the weirdness of Quantum Mechanics to some of the oriental mysticism. In particular, the author points out that the ancient Indian school of philosophy, Nyāya, considers the five elements essential to correct reasoning, beginning with the statement that “The reason (evidence) must be present in the case under consideration.’

Overall, it’s a rich and exotic concoction of conventional studies on methodologies of science, history, philosophy both western and Indian. Though it is no easy read, this is one of most valuable books I have ever read. However, I have a complaint against the author. Such a unique and precocious book ought to be published through some reputable international publisher in order that it would garner real international traction that it deserves. Unfortunately, it has been self-published.^(iv) Yet I strongly recommend that all science lovers should enrich their collection by this gem whose e-version is available almost free (Rs. 149 or US $2.0).
==========

I am thankful to Dr. Mitra for his kind words about the book. Here is my response to some of the points raised by him (marked above with superscripts).

(i) Why are there as many as three chapters on human health and longevity in a book on the Scientific Method? There are at least two reasons for this:

(a) One underlying thought that influenced my entire planning and writing of this book was that science has to now find ways to go beyond its conventional reductionistic approach, so as to be able to investigate even complex systems effectively. The present century will be the century of complexity science. To bring home to the lay reader the fact that complex systems are all around us, I picked up the example of the human mindbody, which has consciousness as one of its ‘emergent’ properties. Our health and longevity issues are something that interest everybody. The chapter ‘How to Live Well Forever’ provides a powerful example of how modern science and technology (an outcome of the Scientific Method) enables us to realistically cherish the hope of living well forever, if we wish to. Similarly, the chapter ‘Reversal of Chronic Diseases’ serves to hold the hope that so many chronic diseases can indeed the reversed, thanks to the fruits of the application of the Scientific Method to the health sciences.

(b) By now it is well established that ancient Indian science and technology, as also philosophy and mathematics, got very bad treatment at the hands of Western historians of science, as also their misguided and/or politically motivated Indian cohorts (see, e.g., Kak (2021): ‘A Brief History of Indian Science’, (99+) (PDF) A Brief History of Indian Science | Subhash Kak - Academia.edu). In this book I have done my bit to restore the balance to some extent by including a chapter on the history of science. But the achievements of pre-modern India were so prolific that I had trouble keeping that chapter reasonably brief. One thing I have done is to move some part of the material to a chapter on the ‘The 8-Fold Yogic Way of Living’. So this is the third chapter on the health sciences. The three chapter together also served to enable me to compare the three approaches to human health and longevity. It goes without saying that the yogic way of living is the best: for the individual, for society, for world peace, and for living in complete harmony with Nature.

(ii) ‘The end of theoretical physics as we know it?’ is a section in the chapter ‘Going Beyond Reductionism in Science’. Dr. Mitra would agree with me readily that we have all been pining for long for the next golden age in theoretical physics. There has been a stalemate of sorts for the last several decades. No fundamental breakthroughs have come. People have been wondering why. Sabine Hossenfelder (2018) in her book has opined that perhaps it is because of our excessive obsession with the beauty and symmetry of the equations that embody our theories. That is one opinion. In fact I myself highlighted the crucial role played by conservation theorems in the discoveries of hidden (broken) symmetries (Wadhawan (2018): Latent, Manifest, and Broken Symmetry). So this approach has paid rich dividends. But perhaps we have already milked this cow too much. Other approaches are needed. The same chapter in the book gives a lot of space to the work of Stephen Wolfram, who has been advocating the use of cellular automata and local interactions for getting the hang of all sorts of research problems, including those in theoretical physics. I think theoretical physicists should take Wolfram’s claims more seriously. Who knows, they may end up getting the breakthrough ideas so sorely needed at the present juncture in the history of theoretical science.

(iii) Giving so much space to Wolfram’s work is also because of my agenda to highlight every possible approach that can help make progress in complexity science. So much so that I end up making some suggestions for relaxing the very strict Scientific Method a bit so as to bring the study of even highly complex systems (like the consciousness aspect of the human mindbody) into mainstream science.

(iv) This is perhaps the first and the only comprehensive book on the scientific method. And availability of such a book in high-school and college libraries can go a long way in promoting scientific temper in society. But there may well be scope for improvement of the book. For this I keep inviting suggestions and critical comments. Since I am also the publisher of the book, it is very easy for me to make corrections and improvements continuously. What is more, it would take me just a day or two to bring out a corrected / improved version, or even a new edition: All I have to do for this is to upload a revised pdf file. So please keep giving me your feedback.

Second Edition of Vinod Wadhawan’s Book on Symmetry (6 July 2018)

Title

Latent, Manifest, and Broken Symmetry: A Bottom-up Approach to Symmetry, with Implications for Complex Networks

Author

Vinod Wadhawan

Book details

Paperback: 210 pages 
Publisher: Createspace Independent Publishing Platform

Language: English
ISBN: 978-1463766718
Product dimension: 15.2 x 1.2 x 22.9 cm 

About the book

There is a subtle kind of symmetry called latent symmetry which manifests itself only when the conditions are right. It can occur in systems composed of equal or equivalent components. It lies dormant or latent, and becomes manifest when the components happen to have certain special mutual placements. Although the latent-symmetry idea has been around for more than a decade, not many natural manifestations have been observed to date. But a recognition of the possibility of latent symmetry enables us to formulate a comprehensive symmetry-composition principle enunciated in this book. The principle is applicable to any system composed of equal or equivalent sub-parts. And there are many such systems around. Crystals are an obvious example, the equal components being the unit cells. Several complex networks can also fall within the purview of this principle, if we take note of the approximate nature of their symmetry. This book presents such an all-inclusive view of symmetry in an accessible language.

We are surrounded by symmetry and broken symmetry. From the Big Bang onwards, as our universe cooled and expanded, a series of symmetry-breaking transitions occurred, resulting in a gradual evolution of the complexity of life we see today. By now it is well recognized that discovering new broken symmetries (particularly broken gauge symmetries) is the path science must take for going deeper into the mysteries of Nature. At a very fundamental level, laws of physics are all about symmetry. The present edifice of science in general, and physics in particular, would be unthinkable without symmetry. There is a lot of symmetry even in biological systems. This book celebrates symmetry in all its forms, including latent symmetry.

Foreword to first edition by Prof. A. M. Glazer

Wherever we look we see a variety of patterns and shapes that show different types of symmetry. Much of this is obvious, such as for instance when we look at the pyramids of Egypt, or crystals in a museum. However, what is not so obvious is just what exactly is symmetry and why is it so prevalent? In this unique and intriguing book, Professor Vinod Wadhawan has set about answering these sorts of questions. He takes us on a journey from very basic descriptions, such as the growth of a crystal, on to more esoteric and complex notions, demonstrating that, in fact, symmetry is even more pervasive than we thought before. Some symmetries are far from obvious, as illustrated by the idea of latent symmetry. This is said to manifest itself when one combines two or more ‘equal’ objects or systems, each with its own symmetry description, and the resulting composite system exhibits new symmetry elements that were not expected from the original systems. For instance, two identical right-angled isosceles triangles can be joined together to form a square, that has an unanticipated four-fold rotational symmetry. The notion of latent symmetry is relatively new and deserves further consideration.

Not only do we have the symmetry exhibited by living organisms and physical objects, but also by ideas themselves. As such this book has a strong philosophical content that will enable the reader to gain much more insight into the phenomenon than is normally got from a typical university education. Wadhawan shows us how even the concept of randomness is intricately bound up with notions of symmetry. Even the idea of predictability is an example of symmetry in action! And then, having explained what symmetry is, emphasis is placed on what happens when symmetry is broken. In a sense, pure symmetry could even be described as rather boring, since it implies a lack of change or progress. Nonetheless, we still need to understand it. It is when symmetry is broken that fun things start to happen and new ideas, progress and phenomena are created. This book explains how this comes about and why symmetry-breaking is so important. The book is written with an eye to explaining the fundamental
concepts of symmetry, rather than go into complex mathematical proofs and lemmas, which in any case can be found elsewhere for those who like those sorts of things. This means that Wadhawan is able instead to concentrate on the philosophical importance of understanding symmetry, and how it impacts on the world that we observe. Rather like the Second Law of Thermodynamics, symmetry is seen to play a vital role in what holds the universe together. You can see then that this book covers just about everything that we know about symmetry, and possibly that which we do not!

A.M. Glazer

Professor of Physics and Emeritus Fellow of Jesus College

University of Oxford

Author of Space Groups for Solid State Scientists

Preface to the First Edition 

The symmetry of any composite system made up of equal or equivalent components depends on at least two factors: The inherent symmetry of each component, and the symmetry imposed on the system by the manner in which the components are arranged with respect to one another (‘placement symmetry’). But if the composite system is found to have a higher symmetry than what can be accounted for by these two factors, then that extra, unexpected symmetry is what I call latent symmetry. It is as if this additional symmetry was lying latent or dormant in the equal or equivalent components, and became manifest only when the components came together to form the composite system. To accommodate such a possibility, I enunciate in this book a new symmetry composition principle. According to it: When the occurrence of a symmetry implies the coexistence of two or more equal or equivalent building blocks, the overall symmetry is either the product of the building-block-symmetry group and the placement-symmetry group, or there is an additional component which arises from the latent symmetry present in the building blocks.

The emergence of symmetry in thermodynamically open composite systems can be traced ultimately to the second law of thermodynamics, which is therefore the primary organizing principle. How this principle operates in various diverse systems is discussed in this book. It is argued that the same explanation holds, whether it is the symmetry of a crystal, or that of a complex social network.

Symmetry of complex networks is, in fact, another major theme of this book. That real-life networks should possess any symmetry at all may come as a surprise. But by now we should all be reconciled to the fact that there is something about symmetry which touches everything in our universe. The present edifice of science in general, and physics in particular, would be unthinkable without symmetry. There is a lot of symmetry even in biological systems.

We are surrounded not only by symmetry, but also broken symmetry. In fact, we see more of broken symmetry than intact symmetry. From the Big Bang onwards, as our universe cooled and expanded, a series of symmetry-breaking transitions occurred, leading eventually to the complexity of life we see today. This book is an attempt to explain, in an accessible language, the interplay between latent, manifest, and broken symmetry.

Vinod Wadhawan

Bengaluru

August 2011

Preface to the Second Edition

The book has been revised and updated substantially. In particular, gauge symmetry, which was discussed only briefly in the first edition, has been given the prominence it deserves. A new chapter has been added to deal with it in some detail.

Another new feature of this edition is the introduction of my notion of potential symmetry. It is similar to latent symmetry, but not identical to it. Latent symmetry is a kind of potential symmetry which becomes manifest symmetry when the conditions are just right. But potential symmetry is not always latent symmetry; in fact, it is only rarely so. Introduction of the notion of potential symmetry enables us to enunciate what I call the fundamental theorem of symmetry. It says that any spontaneously occurring symmetry of an object or system comprising of equal or equivalent subparts is nothing but a self-organized manifestation of the potential symmetry residing in its subparts.

Vinod Wadhawan

Bengaluru

July 2018

Contents

Foreword xi

 Preface xiii

1. Overview 1

2. Symmetry Fundamentals 9 

2.1 Definition of symmetry 9

2.2 Analogy and classification are symmetry 11

2.3 Reduction is symmetry 11

2.4 Reproducibility is symmetry 13

2.5 Predictability is symmetry 14

2.6 The symmetry principle 15

2.7 Thermodynamics and the symmetry principle 16

2.8 Ugly symmetry 17

3. Group-Theoretical Description of Symmetry 21

3.1 Discrete groups 21

3.2 Coset decomposition of a group 23

3.3 Lagrange theorem for subgroups 25

3.4 Symmetry group of a crystal 25

3.5 Continuous groups 27

3.6 Permutation groups 27

3.7 Special unitary groups 27

3.8 Topological space, open sets 28

3.9 Morphisms, categories 29

3.10 Semigroups, groupoids 30

3.11 Lie groups 32

4. Network Theory 39

4.1 Mathematical networks 39

4.2 Clustering coefficient 42

4.3 Permutation symmetry in graphs and networks 43

4.4 Real-life networks 45

4.5 Scale-free networks 46

5. Self-organization and Symmetry 47

5.1 Growth of a crystal as an ordering process 47

5.2 Similar linkage patterns and symmetry 49

5.3 Symmetry as a secondary organizing principle 50

5.4 Symmetry and biology 52

6. The Different Types of Exact and Approximate Symmetry 59

6.1 Crystallographic symmetry 59

6.2 Space symmetry and time symmetry 60

6.3 Permutational and more general symmetries of graphs 60

6.4 Approximate symmetry of graphs 61

6.5 Symmetry in real-life networks 62

    
    6.6 Structural vs. statistical equivalence and latent symmetry 69

7. Symmetry of Composite Systems 71

7.1 The Curie principle 71

7.2 The Curie-Shubnikov principle 73

7.3 Interplay between dissymmetrization and symmetrization 77

7.4 The Hermann theorem of crystal physics, and its applications 77

7.5 Hexply configurations for nanocomposites 79

8. Gauge Symmetry 81

8.1 Introduction 81

8.2 Gauge-symmetry groups 84

8.3 Noether’s theorems 86

9. Phase Transitions and Broken Symmetry 93

9.1 Liberal meanings of the term ‘phase transition’ 93

9.2 Spontaneous breaking of symmetry 94

9.3 The Landau theory of phase transitions 95

9.4 Ferroic phase transitions and domain structure 97

9.5 Prototype symmetry 98

9.6 The symmetry compensation law 98

9.7 Continuous broken symmetries 99

9.8 Discrete broken symmetries 105

9.9 Broken symmetry and biology 105

9.10 The principle of local activity 108

10. Particle Physics, Cosmology, and the Search for New Symmetries111

10.1 The Standard Model of Particle Physics 111

10.2 Beyond the Standard Model 121

10.3 Origin of our universe 124

11. Latent Symmetry, Potential Symmetry, and the Symmetry Composition Principle 129

11.1 Latent symmetry and potential symmetry 129

11.2 The distinction between potential symmetry and latent symmetry

11.3 The fundamental theorem of symmetry 133

11.4 The symmetry composition principle 133

11.5 Placement symmetry 136

11.6 Latent symmetry and algorithmic information 137

12. Group-Theoretical Determination of Latent Symmetry 139

12.1 Formal definition of latent symmetry 139

12.2 Litvin’s partition theorem for latent symmetry 140

12.3 Latent symmetry and domain-average engineered ferroic materials 144

12.4  An example of how ignorance about latent symmetry can

lead to errors 145

12.5 The role of placement symmetry in revealing latent symmetry 148

12.6 Concluding remarks 150

13. Symmetry of Complex Networks 151

13.1 Latent symmetry in complex networks 151

13.2 Measures of symmetry of networks 154

13.3 Origins of symmetry in complex networks 156

13.4 The similar-linkage-pattern model for symmetry 157

13.5 The free-energy landscape for biological networks 158

13.6 Social networks and the meaning of cohesive energy 160

14. Afterword 163

Bibliography 167

Index 179

Acknowledgements 187

About the Author 189