Scaling-law relations characterize an immense number of natural processes, prominently in the form of
- scaling-law distributions,
- scale-free networks,
- cumulative relations of stochastic processes.
A scaling law, or power law, is a simple polynomial functional relationship, i.e.,
f(
x) depends on a power of
x. Two properties of such laws can easily be shown:
- a logarithmic mapping yields a linear relationship,
- scaling the function’s argument x preserves the shape of the functionf(x), called scale invariance.
See (Sornette, 2006).
Scaling-Law Distributions
Scaling-law distributions have been observed in an extraordinary wide range of natural phenomena: from physics, biology, earth and planetary sciences, economics and finance, computer science and demography to the social sciences; see (Newman, 2004). It is truly amazing, that such diverse topics as
- the size of earthquakes, moon craters, solar flares, computer files, sand particle, wars and price moves in financial markets,
- the number of scientific papers written, citations received by publications, hits on webpages and species in biological taxa,
- the sales of music, books and other commodities,
- the population of cities,
- the income of people,
- the frequency of words used in human languages and of occurrences of personal names,
- the areas burnt in forest fires,
are all described by scaling-law distributions. First used to describe the observed income distribution of households by the economist Pareto in 1897, the recent advancements in the study of complex systems have helped uncover some of the possible mechanisms behind this universal law. However, there is as of yet no real understanding of the physical processes driving these systems.
Processes following normal distributions have a characteristic scale given by the mean of the distribution. In contrast, scaling-law distributions lack such a preferred scale. Measurements of scaling-law processes yield values distributed across an enormous dynamic range (sometimes many orders of magnitude), and for any section one looks at, the proportion of small to large events is the same. Historically, the observation of scale-free or self-similar behavior in the changes of cotton prices was the starting point for Mandelbrot’s research leading to the discovery of fractal geometry; see (Mandelbrot, 1963).
It should be noted, that although scaling laws imply that small occurrences are extremely common, whereas large instances are quite rare, these large events occur nevertheless much more frequently compared to a normal (or Gaussian) probability distribution. For such distributions, events that deviate from the mean by, e.g., 10 standard deviations (called “10-sigma events”) are practically impossible to observe. For scaling law distributions, extreme events have a small but very real probability of occurring. This fact is summed up by saying that the distribution has a “fat tail” (in the terminology of probability theory and statistics, distributions with fat tails are said to be leptokurtic or to display positive kurtosis) which greatly impacts the risk assessment. So although most earthquakes, price moves in financial markets, intensities of solar flares, … will be very small, the possibility that a catastrophic event will happen cannot be neglected.
Scale-Free Networks
Another modern research field marked by the ubiquitous appearance of scaling-law relations is the study of complex networks. Many different phenomena in the physical (e.g., computer networks, transportation networks, power grids, spontaneous synchronization of systems of lasers), biological (e.g., neural networks, epidemiology, food webs, gene regulation), and social (e.g., trade networks, diffusion of innovation, trust networks, research collaborations, social affiliation) worlds can be understood as network based. In essence, the links and nodes are abstractions describing the system under study via the interactions of the elements comprising it.
In graph theory, the degree of a node (or vertex), k, describes the number of links (or edges) the node has to other nodes. The degree distribution gives the probability distribution of degrees in a network. For scale-free networks, one finds that the probability that a node in the network connects with k other nodes follows a scaling law. Again, this power law is characterized by the existence of highly connected hubs, whereas most nodes have small degrees.
Scale-free networks are
- characterized by high robustness against random failure of nodes, but susceptible to coordinated attacks on the hubs, and
- thought to arise from a dynamical growth process, called preferential attachment, in which new nodes favor linking to existing nodes with high degrees.
It should be noted, that another prominent feature of real-world networks, namely the so-called small-world property, is separate from a scale-free degree distribution, although scale-free networks are also small-world networks; (Strogatz and Watts, 1998). For small-world networks, although most nodes are not neighbors of one another, most nodes can be reached from every other by a surprisingly small number of hops or steps.
Most real-world complex networks - such as those listed at the beginning of this section - show both scale-free and small-world characteristics.
Some general references include (Barabasi, 2002), (Albert and Barabasi, 2001), and (Newman, 2003). Emergence of scale-free networks in the preferential attachment model (Albert and Barabasi, 1999). An alternative explanation to preferential attachment, introducing non-topological values (called fitness) to the vertices, is given in (Caldarelli et al., 2002).
Cumulative Scaling-Law Relations
Next to distributions of random variables, scaling laws also appear in collections of random variables, called stochastic processes. Prominent empirical examples are financial time-series, where one finds empirical scaling laws governing the relationship between various observed quantities. See (Guillaume et al., 1997) and (Dacorogna et al., 2001).
References
Albert R. and Barabasi A.-L., 1999, Emergence of Scaling in Random Networks, http://www.arXiv.org/abs/cond-mat/9910332.
Albert R. and Barabasi A.-L., 2001, Statistical Mechanics of Complex Networks, http://www.arXiv.org/abs/cond-mat/0106096.
Barabasi A.-L., 2002, Linked — The New Science of Networks, Perseus Publishing, Cambridge, Massachusetts.
Caldarelli G., Capoccio A., Rios P. D. L., and Munoz M. A., 2002, Scale- free Networks without Growth or Preferential Attachment: Good get Richer, http://www.arXiv.org/abs/cond-mat/0207366.
Dacorogna M. M., Gencay R., Müller U. A., Olsen R. B., and Pictet O. V., 2001, An Introduction to High-Frequency Finance, Academic Press, San Diego, CA.
Guillaume D. M., Dacorogna M. M., Dave R. D., Müller U. A., Olsen R. B., and Pictet O. V., 1997, From the Bird’s Eye to the Microscope: A Survey of New Stylized Facts of the Intra-Daily Foreign Exchange Markets, Finance and Stochastics, 1, 95–129.
Mandelbrot B. B., 1963, The variation of certain speculative prices, Journal of Business, 36, 394–419.
Newman M. E. J., 2003, The Structure and Function of Complex Networks, http://www.arXiv.org/abs/cond-mat/0303516.
Newman M. E. J., 2004, Power Laws, Pareto Distributions and Zipf ’s Law, http://www.arXiv.org/abs/cond-mat/0412004.
Sornette D., 2006, Critical Phenomena in Natural Sciences, Series in Synergetics. Springer, Berlin, 2nd edition.
Strogatz S. H. and Watts D. J., 1998, Collective Dynamics of ‘Small-World’ Networks,
Nature, 393, 440–442.
A single ant or bee isn’t smart, but their colonies are. The study of swarm intelligence is providing insights that can help humans manage complex systems.
In 1881 a result was published, based on the observation that the first pages of logarithm books, used at that time to perform calculations, were much more worn than the other pages. The conclusion was that computations of numbers that started with 1 were performed more often than others: if d denotes the first digit of a number the probability of its appearance is equal to log(d + 1).
The phenomenon was rediscovered in 1938 by the physicist F. Benford, who confirmed the “law” for a large number of random variables drawn from geographical, biological, physical, demographical, economical and sociological data sets. It even holds for randomly compiled numbers from newspaper articles. Specifically, Benford’s law, or the first-digit law, states, that the occurrence of a number with first digit 1 is with 30.1%, 2 with 17.6%, 3 with 12.5%, 4 with 9.7%, 5 with 7.9%, 6 with 6.7%, 7 with 5.8%, 8 with 5.1% and 9 with 4.6% probability. In general, the leading digit d ∈ [1, …, b−1] in base b ≥ 2 occurs with probability proportional to log_b(d + 1) − log_b(d) = log_b(1 + 1/d).
First explanations of this phenomena, which appears to suspend the notions of probability, focused on its logarithmic nature which implies a scale-invariant or power-law distribution. If the first digits have a particular distribution, it must be independent of the measuring system, i.e., conversions from one system to another don’t affect the distribution. (This requirement that physical quantities are independent of a chosen representation is one of the cornerstones of general relativity and called covariance.) So the common sense requirement that the dimensions of arbitrary measurement systems shouldn’t affect the measured physical quantities, is summarized in Benford’s law. In addition, the fact that many processes in nature show exponential growth is also captured by the law, which assumes that the logarithms of numbers are uniformly distributed.
So how come one observes random variables following normal and scaling-law distributions? In 1996 the phenomena was mathematically rigorously proven: if one repeatedly chooses different probability distribution and then randomly chooses a number according to that distribution, the resulting list of numbers will obey Benford’s law. Hence the law reflects the behavior of distributions of distributions.
Benford’s law has been used to detect fraud in insurance, accounting or expenses data, where people forging numbers tend to distribute their digits uniformly.
There is an interesting observation or conjecture to be made from the Mataphysics Map in the post
what can we know?, concerning the nature of infinity.
The Finite
Many observations reveal a finite nature of reality:
- Energy comes in finite parcels (quatum mechanics)
- The knowledge one can have about quanta is a fixed value (uncertainty)
- Energy is conserved in the universe
- The speed of light has the same constant value for all observers (special relativity)
- The age of the universe is finite
- Information is finite and hence can be coded into a binary language
Newer and more radical theories propose:
- Space comes in finite parcels
- Time comes in finite parcels
- The universe is spatially finite
- The maximum entropy in any given region of space is proportional to the regions surface area and not its volume (this leads to the holographic principle stating that our three dimensional universe is a projection of physical processes taking place on a two dimensional surface surrounding it)
So finiteness appears to be an intrinsic feature of the Outer Reality box of the diagram.
There is in fact a movement in physics ascribing to the finiteness of reality, called Digital Philosophy. Indeed, this finiteness postulate is a prerequisite for an even bolder statement, namely, that the universe is one gigantic computer (a Turing complete cellular automata), where reality (thought and existence) is equivalent to computation. As mentioned above, the selforganizing structure forming evolution of the universe can be seen to produce ever more complex modes of information processing (e.g., storing data in DNA, thoughts, computations, simulations and perhaps, in the near future, quantum computations).
There is also an approach to quantum mechanics focussing on information stating that an elementary quantum system carries (is?) one bit of information. This can be seen to lead to the notions of quantisation, uncertainty and entanglement.
The Infinite
It should be noted that zero is infinity in disguise. If one lets the denominator of a fraction go to infinity, the result is zero. Historically, zero was discovered in the 3rd century BC in India and was introduced to the Western world by Arabian scholars in the 10th century AC. As ordinary as zero appears to us today, the great Greek mathematicians didn’t come up with such a concept.
Indeed, infinity is something intimately related to formal thought systems (mathematics). Irrational numbers have an infinite number of digits. There are two measures for infinity: countability and uncountablility. The former refers to infinite series as 1, 2, 3, … Whereas for the latter measure, starting from 1.0 one can’t even reach 1.1 because there are an infinite amount of numbers in the interval between 1.0 and 1.1. In geometry, points and lines are idealizations of dimension zero and one, respectively.
So it appears as though infinity resides only in the Inner Reality box of the diagram.
The Interface
If it should be true that we live in a finite reality with infinity only residing within the mind as a concept, then there should be some problems if one tries to model this finite reality with an infinity-harboring formalism.
Perhaps this is indeed so. In chaos theory, the sensitivity to initial conditions (butterfly effect) can be viewed as the problem of measuring numbers: the measurement can only have a finite degree of accuracy, whereas the numbers have, in principle, an infinite amount of decimal places.
In quatum gravity (the, as yet, unsuccessful merger of quantum mechanics and gravity) many of the inherent problems of the formalism could by bypassed, when a theory was proposed (string theory) that replaced (zero space) point particles with one dimensionally extended objects. Later incarnations, called M-theory, allowed for multidimensional objects.
In the above mentioned information based view of quantum mechanics, the world appears quantised because the information retrieved by our minds about the world is inevitably quantised.
So the puzzle deepens. Why do we discover the notion of infinity in our minds while all our experiences and observations of nature indicate finiteness?
introduction
medical studies often contradict each other. results claiming to have “proven” some causal connection are confronted with results claiming to have “disproven” the link, or vice versa. this dilemma affects even reputable scientists publishing in leading medical journals.
the topics are divers:
- high-voltage power supply lines and leukemia [1],
- salt and high blood pressure [1],
- heart diseases and sport [1],
- stress and breast cancer [1],
- smoking and breast cancer [1],
- praying and higher chances of healing illnesses [1],
- the effectiveness of homeopathic remedies and natural medicine,
- vegetarian diets and health,
- low frequency electromagnetic fields and electromagnetic hypersensitivity [2],
- …
basically, this is understood to happen for three reasons:
- i.) the bias towards publishing positive results,
- ii.) incompetence in applying statistics,
- ii.) simple fraud.
i.)
publish or perish. in order the guarantee funding and secure the academic status quo, results are selected by their chance of being published.
an independent analysis of the original data used in 100 published studies exposed that roughly half of them showed large discrepancies in the original aims stated by the researchers and the reported findings, implying that the researchers simply skimmed the data for publishable material [3].
this proves fatal in combination with ii.) as every statistically significant result can occur (per definition) by chance in an arbitrary distribution of measured data. so if you only look long enough for arbitrary results in your data, you are bound to come up with something [1].
ii.)
often, due to budget reasons, the numbers of test persons for clinical trials are simply too small to allow for statistical relevance. ref. [4] showed next to other things, that the smaller the studies conducted in a scientific field, the less likely the research findings are to be true.
statistical significance - often evaluated by some statistics software package - is taken as proof without considering the plausibility of the result. many statistically significant results turn out to be meaningless coincidences after accounting for the plausibility of the finding [1].
one study showed that one third of frequently cited results fail a later verification [1].
another study documented that roughly 20% of the authors publishing in the magazine “nature” didn’t understand the statistical method they were employing [5].
iii.) a.)
two thirds of of the clinical biomedical research in the usa is supported by the industry - double as much as in 1980 [1].
it was shown that in 1000 studies done in 2003, the nature of the funding correlated with the results: 80% of industry financed studies had positive results, whereas only 50% of the independent research reported positive findings.
it could be argued that the industry has a natural propensity to identify effective and lucrative therapies. however, the authors show that many impressive results were only obtained because they were compared with weak alternative drugs or placebos. [6]
iii.) b.)
“Andrew Wakefield (born 1956 in the United Kingdom) is a Canadian trained surgeon, best known as the lead author of a controversial 1998 research study, published in the Lancet, which reported bowel symptoms in a selected sample of twelve children with autistic spectrum disorders and other disabilities, and alleged a possible connection with MMR vaccination. Citing safety concerns, in a press conference held in conjunction with the release of the report Dr. Wakefield recommended separating the components of the injections by at least a year. The recommendation, along with widespread media coverage of Wakefield’s claims was responsible for a decrease in immunisation rates in the UK. The section of the paper setting out its conclusions, known in the Lancet as the “interpretation” (see the text below), was subsequently retracted by ten of the paper’s thirteen authors.
[…]
In February of 2004, controversy resurfaced when Wakefield was accused of a conflict of interest. The London Sunday Times reported that some of the parents of the 12 children in the Lancet study were recruited via a UK attorney preparing a lawsuit against MMR manufacturers, and that the Royal Free Hospital had received £55,000 from the UK’s Legal Aid Board (now the Legal Services Commission) to pay for the research. Previously, in October 2003, the board had cut off public funding for the litigation against MMR manufacturers. Following an investigation of The Sunday Times allegations by the UK General Medical Council, Wakefield was charged with serious professional misconduct, including dishonesty, due to be heard by a disciplinary board in 2007.
In December of 2006, the Sunday Times further reported that in addition to the money given to the Royal Free Hospital, Wakefield had also been personally paid £400,000 which had not been previously disclosed by the attorneys responsible for the MMR lawsuit.”
wakefield had always only expressed his criticism of the combined triple vaccination, supporting single vaccinations spaced in time. the british tv station channel 4 exposed in 2004 that he had applied for patents for the single vaccines. wakefield dropped his subsequent slander action against the media company only in the beginning of 2007. as mentioned, he now awaits charges for professional misconduct. however, he has left britain and now works for a company in austin texas. it has been uncovered that other employees of this us company had received payments from the same attorney preparing the original law suit. [7]
epilog
should we be surprised by all of this? next to the innate tendency of human beings to be incompetent and unscrupulous, there is perhaps another level, that makes this whole endeavor special.
the inability of scientist to conclusively and reproducibly uncover findings concerning human beings is maybe better appreciated, if one considers the nature of the subject under study. life, after all, is an enigma and the connection linking the mind to matter is elusive at best (i.e., the physical basis of consciousness).
the bodies capability to heal itself, i.e., the placebo effect and the need for double-blind studies is indeed very bizarre. however, there are studies questioning, if the effect exists at all;-)
—
[1] This article in the magazine issued by the Neue Zürcher Zeitung by Robert Matthews
[2] C. Schierz; Projekt NEMESIS; ETH Zürich; 2000
[3] A. Chan (Center of Statistics in Medicine, Oxford) et. al.; Journal of the American Medical Association; 2004
[4] J. Ioannidis; “Why Most Published Research Findings Are False” ; University of Ioannina; 2005
[5] R. Matthews, E. García-Berthou and C. Alcaraz as reported in this “Nature” article; 2005
[6] C. Gross (Yale University School of Medicine) et. al.; “Scope and Impact of Financial Conflicts of Interest in Biomedical Research “; Journal of the American Medical Association; 2003
[7] H. Kaulen; “Wie ein Impfstoff zu Unrecht in Misskredit gebracht wurde”; Deutsches Ärzteblatt; Jg. 104; Heft 4; 26. Januar 2007
Introduction
Science, put simply, can be understood as working on three levels:
- i.) analyzing the nature of the object being considered/observed,
- ii.) developing the formal representation of the object’s features and its dynamics/interactions,
- iii.) devising methods for the empirical validation of the formal representations.
To be precise, level i.) lies more within the realm of philosophy (e.g., epistemology) and metaphysics (i.e., ontology), as notions of origin, existence and reality appear to transcend the objective and rational capabilities of thought. The main problem being:
“Why is there something rather than nothing? For nothingness is simpler and easier than anything.”; [1].
In the history of science the above mentioned formulation made the understanding of at least three different levels of reality possible:
- a.) the fundamental level of the natural world,
- b.) inherently random phenomena,
- c.) complex systems.
While level a.) deals mainly with the quantum realm and cosmological structures, levels b.) and c.) are comprised mostly of biological, social and economic systems.
Examples
a.) Fundamental
Many natural sciences focus on a.i.) fundamental, isolated objects and interactions, use a.ii.) mathematical models which are a.iii.) verified (falsified) in experiments that check the predictions of the model - with great success:
“The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious. There is no rational explanation for it.”; [2].
b.) Random
Often the nature of the object b.i.) being analyzed is in principle unknown. Only statistical evaluations of sets of outcomes of single observations/experiments can be used to estimate b.ii.) the underlying model, and b.iii.) test it against more empirical data. Often the approach taken in the fields of social sciences, medicine, and business.
c.) Complex
Moving to c.i.) complex, dynamical systems, and c.ii.) employing computer simulations as a template for the dynamical process, unlocks a new level of reality: mainly the complex and interacting world we experience at our macroscopic length scales in the universe. Here two new paradigms emerge:
- the shift from mathematical (analytical) models to algorithmic computations and simulations performed in computers,
- simple rules giving rise to complex behavior: “And I realized, that I had seen a sign of a quite remarkable and unexpected phenomenon: that even from very simple programs behavior of great complexity could emerge.”; [3].
However, things are not as clear anymore. What is the exact methodology, and how does it relate to underlying concepts of ontology and epistemology, and what is the nature of these computations per se? Or within the formulation given above, i.e., iii.c.), what is the “reality” of these models: what do the local rules determining the dynamics in the simulation have to say about the reality of the system c.i.) they are trying to emulate?
Outlook
There are many coincidences that enabled the structured reality we experience on this planet to have evolve: exact values of fundamental constants (initial conditions), emerging structure-forming and self-organizing processes, the possibility of (organic) matter to store information (after being synthesized in supernovae!), the right conditions of earth for harboring life, the emergent possibilities of neural networks to establish consciousness and sentience above a certain threshold, …
Interestingly, there are also many circumstances that allow the observable world to be understood by the human mind:
- the mystery allowing formal thought systems to map to patterns in the real world,
- the development of the technology allowing for the design and realization of microprocessors,
- the bottom-up approach to complexity identifying a micro level of simple interactions of system elements.
So it appears that the human mind is intimately interwoven with the fabric of reality that produced it.
But where is all this leading to? There exists a natural extension to science which fuses the notions from levels a.) to c.), namely
- information and information processing,
- formal mathematical models,
- statistics and randomness.
Notably, it comes from an engineering point-of-view and deals with quantum computers and comes full circle back to level i.), the question about the nature of reality:
“[It can be shown] that quantum computers can simulate any system that obeys the known laws of physics in a straightforward and efficient way. In fact, the universe is indistinguishable from a quantum computer.”; [4].
At first blush the idea of substituting reality with a computed simulation appears rather ad hoc, but in fact it does have potentially falsifiable notions:
- the discreteness of reality, i.e., the notion that continuity and infinity are not physical,
- the reality of the quantum realm should be contemplated from the point of view of information, i.e., the only relevant reality subatomic quanta manifest is that they register one bit of information: “Information is physical.”; [5].
—-
[1] von Leibniz, G. W., “Principes de la nature et de la grâce”, 1714
[2] Wigner, E. P., “Symmetries and Reflections”, MIT Press, Cambridge, 1967
[3] Wolfram, S., “A New Kind of Science”, Wolfram Media, pg. 19, 2002
[4] Lloyd, S., “Programming the Universe”, Random House, pgs. 53 - 54, 2006
[5] Landauer, R., Nature, 335, 779-784, 1988
Related posts:
Put bluntly, metaphysics asks simple albeit deep questions:
- Why do I exist?
- Why do I die?
- Why does the world exist?
- Where did everything come from?
- What is the nature of reality?
- What is the meaning of existence?
- Is there a creator or omnipotent being?
Although these questions may appear idle and futile, they seem to represent an innate longing for knowledge of the human mind. Indeed, children can and often do pose such questions, only to be faced with resignation or impatience of adults.
To make things simpler and tractable, one can focus on the question “What can we know?”.
When you wake up in the morning, you instantly become aware of your self, i.e., you experience an immaterial inner reality you can feel and probe with your thoughts. Upon opening your eyes, a structured material outer reality appears. These two unsurmountable facts are enough to sketch a small metaphysical diagram:
Outside
Focussing on the outer reality or physical universe, there exists an underlying structure forming and selforganizing process starting with an initial singularity or Big Bang (extremely low entropy state, i.e., high order, giving rise to the arrow or direction of time). Due to the exact values of physical constants in our universe, this organizing process yields structures eventually giving birth to stars, which, at the end of their lifecycle, explode (supernovae) allowing for nuclear reactions to fuse heavy elements.
One of these heavy elements brings with it novel bonding possibilities, resulting in a new pattern: organic matter. Within a couple of billion years, the structure forming process gave rise to a plethora of living organisms. Although each organism would die after a short lifespan, the process of life as a whole continued to live in a sustainable equilibrium state and survived a couple of extinction events (some of which eradicated nearly 90% of all species).
The second law of thermodynamics states, that the entropy of the universe is increasing, i.e., the universe is becoming an ever more unordered place. It would seem that the process of life creating stable and ordered structures violates this law. In fact, complex structures spontaneously appear where there is a steady flow of energy from a high temperature input source (the sun) to a low temperature external sink (the earth). So pumping a system with energy leads it to a state far from the thermodynamic equilibrium which is characterized by the emergence of ordered structures.
Viewed from an information processing perspective, the organizing process suddenly experienced a great leap forward. The brains of some organisms had reached a critical mass, allowing for another emergent behavior: consciousness.
Inside
The majority of people in industrialized nations take a rational and logicall outlook on life. Although one might think this is an inevitable mode of awareness, it actually is a cultural imprinting as there exist other civilization putting far less emphasis on rationality.
Perhaps the divide between Western and Eastern thinking illustrates this best. Whereas the former is locked in continuous interaction with the outer world, the latter focuses on the experience of an inner reality. A history of meditation techniques underlines this emphasis on the nonverbal experience of ones self. Thought is either totally avoided, or the mind is focused on repetitive activities, in effect deactivating it.
Recall from
fundamental that there are two surprising facts to be found. On the one hand, the physical laws dictating the fundamental behavior of the universe can be mirrored by formal thought systems devised by the mind. And on the other hand, real complex behavior can be emulated by computer simulations following simple laws (the computers themselves are an example of technological advances made possible by the successfull modelling of nature by formal thought systems).
Outlook
This conceptual map allows one to categorize a lot of stuff in a concise manner. Also, the interplay between the outer and inner realities becomes visible. However, the above mentioned questions remain unanswered. Indeed, more puzzles appear. So as usual, every advance in understanding just makes the question mark bigger…
Continued here:
infinity?
Arguably the most fruitful principle in physics has been the notion of symmetry.
Covariance and gauge invariance - two simply stated symmetry conditions - are at the heart of general relativity and the standard model (of particle physics).
This is not only aesthetically pleasing it also illustrates a basic fact: in coding reality into a formal system, we should only allow the most minimal reference to be made to this formal system. I.e. reality likes to be translated into a language that doesn’t explicitly depend on its own peculiarities (coordinates, number bases, units, …). This is a pretty obvious idea and allows for physical laws to be universal.
But what happens if we take this idea to the logical extreme? Will the ultimate theory of reality demand: I will only allow myself to be coded into a formal framework that makes no reference to itself whatsoever. Obviously a mind twister. But the question remains: what is the ultimate symmetry idea? Or: what is the ultimate invariant?
Does this imply “invariance” even with respect to our thinking? How do we construct a system that supports itself out of itself, without relying on anything external? Can such a magical feat be performed by our thinking?
Taken from this newsgroup message…
Outline
While physics has had an amazing success in describing most of the observable universe in the last 300 years, the formalism appears to be restricted to the fundamental workings of nature.
Only solid-state physics attempts to deal with collective systems. And only thanks to the magic of symmetry one is able to deduce fundamental analytical solutions.
In order to approach real life complex phenomena, one needs to adopt a more systems oriented focus. This also means that the interactions of entities becomes an integral part of the formalism.
Some ideas should illustrate the situation:
- Most calculations in physics are idealizations and neglect dissipative effects like friction
- Most calculations in physics deal with linear effect, as non-linearity is hard to tackle and is associated with chaos; however, most physical systems in nature are inherently non-linear
- The analytical solution of three gravitating bodies in classical mechanics, given their initial positions, masses, and velocities, cannot be found; it turns out to be a chaotic system which can only be simulated in a computer; however, there are an estimated hundred billion of galaxies in the universe
Systems Thinking
Systems theory is an interdisciplinary field which studies relationships of systems as a whole. The goal is to explain complex systems which consist of a large number of mutually interacting and interwoven parts in terms of those interactions.
A timeline:
- Cybernetics (50s): Study of communication and control, typically involving regulatory feedback, in living organisms and machines
- Catastrophe theory (70s): Phenomena characterized by sudden shifts in behavior arising from small changes in circumstances
- Chaos theory (80s): Describes the behavior of non-linear dynamical systems that under certain conditions exhibit a phenomenon known as chaos (sensitivity to initial conditions, regimes of chaotic and deterministic behavior, fractals, self-similarity)
- Complex adaptive systems (90s): The “new” science of complexity which describes emergence, adaptation and self-organization; employing tools such as agent-based computer simulations
In systems theory one can distinguish between three major hierarchies:
- Suborganic: Fundamental reality, space and time, matter, …
- Organic: Life, evolution, …
- Metaorganic: Consciousness, group dynamical behavior, financial markets, …
However, it is not understood how one can traverse the following chain: bosons and fermions -> atoms -> molecules -> DNA -> cells -> organisms -> brains. I.e., how to understand phenomena like consciousness and life within the context of inanimate matter and fundamental theories.
Illustrations
Category Theory
The mathematical theory called category theory is a result of the “unification of mathematics” in the 40s. A category is the most basic structure in mathematics and is a set of objects and a set of morphisms (maps). A functor is a structure-preserving map between categories.
This dynamical systems picture can be linked to the notion of formal systems mentioned above: physical observables are functors, independent of a chosen representation or reference frame, i.e., invariant and covariant.
Object-Oriented Programming
This paradigm of programming can be viewed in a systems framework, where the objects are implementations of classes (collections of properties and functions) interacting via functions (public methods). A programming problem is analyzed in terms of objects and the nature of communication between them. When a program is executed, objects interact with each other by sending messages. The whole system obeys certain rules (encapsulation, inheritance, polymorphism, …).
Some advantages of this integral approach to software development:
- Easier to tackle complex problems
- Allows natural evolution towards complexity and better modeling of the real world
- Reusability of concepts (design patterns) and easy modifications and maintenance of existing code
- Object-oriented design has more in common with natural languages than other (i.e., procedural) approaches
Algorithmic vs. Analytical
Perhaps the shift of focus in this new weltbild can be understood best when one considers the paradigm of complex system theory:
- The interaction of entities (agents) in a system according to simple rules gives rise to complex behavior: Emergence, structure-formation, self-organization, adaptive behavior (learning), …
This allows a departure from the equation-based description to models of dynamical processes simulated in computers. This is perhaps the second miracle involving the human mind and the understanding of nature. Not only does nature work on a fundamental level akin to formal systems devised by our brains, the hallmark of complexity appears to be coded in simplicity (”simple sets of rules give complexity”) allowing computational machines to emulate its behavior.
It is very interesting to note, that in this paradigm the focus is on the interaction, i.e., the complexity of the agent can be ignored. That is why the formalism works for chemicals in a reaction, ants in an anthill, humans in social or economical organizations, … In addition, one should also note, that simple rules - the epitome of deterministic behavior - can also give rise to chaotic behavior.
The emerging field of network theory (an extension of graph theory,yielding results such as scale-free topologies, small-worlds phenomena, etc. observed in a stunning veriety of complex networks) is also located at this end of the spectrum of the formal descriptions of the workings of nature.
Finally, to revisit the analytical approach to reality, note that in the loop quantum gravity approach, space-time is perceived as a causal network arising from graph updating rules (spin networks, which are graphs associated with group theoretic properties), where particles are envisaged as ‘topological defects’ and geometric properties of reality, such as dimensionality, are defined solely in terms of the network’s connectivity pattern.
March 13th, 2008 at 6:03 pm
July 1st, 2008 at 4:32 pm
fundamental
Ideas
In a nutshell:
Paradigm
Classical Mechanics
Mathematics of Symmetry
Physics of Non-Gravitational Forces
Physics of Gravity
Physics of Condensed Matter
Conclusion
The Success
The Future…
The efforts to quantize gravity and devise a unified theory, have taken a strange turn in the last 20 years. The problem is still unsolved, however, the mathematical formalisms engineered for this quest - namely string/M-theory and loop quantum gravity - have had a twofold impact:
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