• part I: some history of science and logical empiricism,
• part II: problems of logical empiricism, critical rationalism and its problems.

After the unsuccessful attempts to found science on common sense notions as seen in the programs of logical empiricism and critical rationalism, people looked for new ideas and explanations.

The Kuhnian View

Thomas Kuhn’s enormously influential work on the history of science is called the Structure of Scientific Revolutions. He revised the idea that science is an incremental process accumulating more and more knowledge. Instead, he identified the following phases in the evolution of science:

• prehistory: many schools of thought coexist and controversies are abundant,
• history proper: one group of scientists establishes a new solution to an existing problem which opens the doors to further inquiry; a so called paradigm emerges,
• paradigm based science: unity in the scientific community on what the fundamental questions and central methods are; generally a problem solving process within the boundaries of unchallenged rules (analogy to solving a Sudoku),
• crisis: more and more anomalies and boundaries appear; questioning of established rules,
• revolution: a new theory and weltbild takes over solving the anomalies and a new paradigm is born.

Another central concept is incommensurability, meaning that proponents of different paradigms cannot understand the other’s point of view because they have diverging ideas and views of the world. In other words, every rule is part of a paradigm and there exist no trans-paradigmatic rules.

This implies that such revolutions are not rational processes governed by insights and reason. In the words of Max Planck (the founder of quantum mechanics; from his autobiography):

A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.

Kuhn gives additional blows to a commonsensical foundation of science with the help of Norwood Hanson and Willard Van Orman Quine:

• every human observation of reality contains an a priori theoretical framework,
• underdetermination of belief by evidence: any evidence collected for a specific claim is logically consistent with the falsity of the claim,
• every experiment is based on auxiliary hypotheses (initial conditions, proper functioning of apparatus, experimental setup,…).

People slowly started to realize that there are serious consequences in Kuhn’s ideas and the problems faced by the logical empiricists and critical rationalists in establishing a sound logical and empirical foundation of science:

• postmodernism,
• constructivism or the scoiology of science,
• relativism.

Postmodernism

Modernism describes the development of Western industrialized society since the beginning of the 19th Century. A central idea was that there exist objective true beliefs and that progression is always linear.

Postmodernism replaces these notions with the belief that many different opinions and forms can coexist and all find acceptance. Core ideas are diversity, differences and intermingling. In the 1970s it is seen to enter scientific and cultural thinking.

Postmodernism has taken a bad rap from scientists after the so called Sokal affair, where physicist Alan Sokal got a nonsensical paper published in the journal of postmodern cultural studies, by flattering the editors ideology with nonsense that sounds good.

Postmodernims has been associated with scepticism and solipsism, next to relativism and constructivism.

Notable scientists identifiable as postmodernists are Thomas Kuhn, David Bohm and many figures in the 20th century philosophy of mathematics. As well as Paul Feyerabend, an influential philosopher of science.

Constructivism

To quote the Nobel laureate Steven Weinberg on Kuhnian revolutions:

If the transition from one paradigm to another cannot be judged by any external standard, then perhaps it is culture rather than nature that dictates the content of scientific theories.

Constructivism excludes objectivism and rationality by postulating that beliefs are always subject to a person’s cultural and theological embedding and inherent idiosyncrasies. It also goes under the label of the sociology of science.

In the words of Paul Boghossian (in his book Fear of Knowledge: Against Relativism and Constructivism):

Constructivism about rational explanation: it is never possible to explain why we believe what we believe solely on the basis of our exposure to the relevant evidence; our contingent needs and interests must also be invoked.

The proponents of constructivism go further:

[…] all beliefs are on a par with one another with respect to the causes of their credibility. It is not that all beliefs are equally true or equally false, but that regardless of truth and falsity the fact of their credibility is to be seen as equally problematic.

From Barry Barnes’ and David Bloor’s Relativism, Rationalism and the Sociology of Knowledge.

In its radical version, constructivism fully abandons objectivism:

• Objectivity is the illusion that observations are made without an observer (from the physicist Heinz von Foerster; my translation)
• Modern physics has conquered domains that display an ontology that cannot be coherently captured or understood by human reasoning (from the philosopher Ernst von Glasersfeld); my translation

In addition, radical constructivism proposes that perception never yields an image of reality but is always a construction of sensory input and the memory capacity of an individual. An analogy would be the submarine captain who has to rely on instruments to indirectly gain knowledge from the outside world. Radical constructivists are motivated by modern insights gained by neurobiology.

Historically, Immanuel Kant can be understood as the founder of constructivism. On a side note, the bishop George Berkeley went even as far as to deny the existence of an external material reality altogether. Only ideas and thought are real.

Relativism

Another consequence of the foundations of science lacking commonsensical elements and the ideas of constructivism can be seen in the notion of relativism. If rationality is a function of our contingent and pragmatic reasons, then it can be rational for a group A to believe P, while at the same time it is rational for group B to believe in negation of P.

Although, as a philosophical idea, relativism goes back to the Greek Protagoras, its implications are unsettling for the Western mid: anything goes (as Paul Feyerabend characterizes his idea of scientific anarchy). If there is no objective truth, no absolute values, nothing universal, then a great many of humanity’s century old concepts and beliefs are in danger.

It should however also be mentioned, that relativism is prevalent in Eastern thought systems, and as an example found in many Indian religions. In a similar vein, pantheism and holism are notions which are much more compatible with Eastern thought systems than Western ones.

Furthermore, John Stuart Mill’s arguments for liberalism appear to also work well as arguments for relativism:

• fallibility of people’s opinions,
• opinions that are thought to be wrong can contain partial truths,
• accepted views, if not challenged, can lead to dogmas,
• the significance and meaning of accepted opinions can be lost in time.

From his book On Liberty.

Epilogue

But could relativism be possibly true? Consider the following hints:

• Epistemological
  • problems with perception: synaesthesia, altered states of consciousness (spontaneous, mystical experiences and drug induced),
  • psychopathology describes a frightening amount of defects in the perception of reality and ones self,
  • people suffering from psychosis or schizophrenia can experience a radically different reality,
  • free will and neuroscience,
  • synthetic happiness,
  • cognitive biases.
• Ontological
  • nonlocal foundation of quantum reality: entanglement, delayed choice experiment,
  • illogical foundation of reality: wave-particle duality, superpositions, uncertainty, intrinsic probabilistic nature, time dilation (special relativity), observer/measurment problem in quantum theory,
  • discreteness of reality: quanta of energy and matter, constant speed of light,
  • nature of time: not present in fundamental theories of quantum gravity, symmetrical,
  • arrow of time: why was the initial state of the universe very low in entropy?
  • emergence, selforganization and structureformation.

In essence, perception doesn’t necessarily say much about the world around us. Consciousness can fabricate reality. This makes it hard to be rational. Reality is a really bizarre place. Objectivity doesn’t seem to play a big role.

And what about the human mind? Is this at least a paradox free realm? Unfortunately not. Even what appears as a consistent and logical formal thought system, i.e., mathematics, can be plagued by fundamental problems. Kurt Gödel proved that in every consistent non-contradictory system of mathematical axioms (leading to elementary arithmetic of whole numbers), there exist statements which cannot be proven or disproved in the system. So logical axiomatic systems are incomplete.

As an example Bertrand Russell encountered the following paradox: let R be the set of all sets that do not contain themselves as members. Is R an element of itself or not?

If you really accede to the idea that reality and the perception of reality by the human mind are very problematic concepts, then the next puzzles are:

• why has science been so fantastically successful at describing reality?
• why is science producing amazing technology at breakneck speed?
• why is our macroscopic, classical level of reality so well behaved and appears so normal although it is based on quantum weirdness?
• are all beliefs justified given the believers biography and brain chemistry?

The Problems With Logical Empiricism

The programme proposed by the logical empiricists, namely that science is built of logical statements resting on an empirical foundation, faces central difficulties. To summarize:

• it turns out that it is not possible to construct pure formal concepts that solely reflect empirical facts without anticipating a theoretical framework,
• how does one link theoretical concepts (electrons, utility functions in economics, inflational cosmology, Higgs bosons,…) to experiential notions?
• how to distinguish science from pseudo-science?

Now this may appear a little technical and not very interesting or fundamental to people outside the field of the philosophy of science, but it gets worse:

• inductive reasoning is invalid from a formal logical point of view!
• causality defies standard logic!

This is big news. So, just because I have witnessed the sun going up everyday of my life (single observations), I cannot say it will go up tomorrow (general law). Observation alone does not suffice, you need a theory. But the whole idea here is that the theory should come from observation. This leads to the dead end of circular reasoning.

But surely causality is undisputable? Well, apart from the problems coming from logic itself, there are extreme examples to be found in modern physics which undermine the common sense notion of a causal reality: quantum nonlocality, delayed choice experiment.

But challenges often inspire people, so the story continues…

Critical Rationalism

OK, so the logical empiricists faced problems. Can’t these be fixed? The critical rationalists belied so. A crucial influence came from René Descartes’ and Gottfried Leibniz’ rationalism: knowledge can have aspects that do not stem from experience, i.e., there is an immanent reality to the mind.

The term critical refers to the fact, that insights gained by pure thought cannot be strictly justified but only critically tested with experience. Ultimate justifications lead to the so called Münchhausen trilemma, i.e., one of the following:

• an infinite regress of justifications,
• circular reasoning,
• dogmatic termination of reasoning.

The most influential proponent of critical rationalism was Karl Popper. His central claims were in essence

• use deductive reasoning instead of induction,
• theories can never be verified, only falsified.

Although there are similarities with logical empiricism (empirical basis, science is a set of theoretical constructs), the idea is that theories are simply invented by the mind and are temporarily accepted until they can be falsified. The progression of science is hence seen as evolutionary process rather than a linear accumulation of knowledge.

Sounds good, so what went wrong with this ansatz?

The Problems With Critical Rationalism

In a nutshell:

• basic formal concepts cannot be derived from experience without induction; how can they be shown to be true?
• deduction turns out to be just as tricky as induction,
• what parts of a theory need to be discarded once it is falsified?

To see where deduction breaks down, a nice story by Lewis Carroll (the mathematician who wrote the Alice in Wonderland stories): What the tortoise Said to Achilles.

If deduction goes down the drain as well, not much is left to ground science on notions of logic, rationality and objectivity. Which is rather unexpected of an enterprise that in itself works amazingly well employing just these concepts.

Explanations in Science

And it gets worse. Inquiries into the nature of scientific explanation reveal further problems. It is based on Carl Hempel’s and Paul Oppenheim’s formalisation of scientific inquiry in natural language. Two basic schemes are identified: deductive-nomological and inductive-statistical explanations. The idea is to show that what is being explained (the explanandum) is to be expected on the grounds of these two types of explanations.

The first tries to explain things deductively in terms of regularities and exact laws (nomological). The second uses statistical hypotheses and explains individual observations inductively. Albeit very formal, this inquiry into scientific inquiry is very straightforward and commonsensical.

Again, the programme fails:

• can’t explain singular causal events,
• asymmetric (a change in the air pressure explains the readings on a barometer, however, the barometer doesn’t explain why the air pressure changed),
• many explanations are irrelevant,
• as seen before, inductive and deductive logic is controversial,
• how to employ probability theory in the explanation?

So what next? What are the consequences of these unexpected and spectacular failings of the most simplest premises one would wish science to be grounded on (logic, empiricism, causality, common sense, rationality, …)?

The discussion is ongoing and isn’t expected to be resolved soon. See part III…

• common sense, i.e., rationalism,
• observation and experiments, i.e., empiricism.

Interestingly, both concepts turn out to be very problematic if applied to the question of what knowledge is and how it is acquired. In essence, they cannot be seen as a foundation for science.

But first a little history of science…

Classical Antiquity

The Greek philosopher Aristotle was one of the first thinkers to introduce logic as a means of reasoning. His empirical method was driven by gaining general insights from isolated observations. He had a huge influence on the thinking within the Islamic and Jewish traditions next to shaping Western philosophy and inspiring thinking in the physical sciences.

Modern Era

Nearly two thousand years later, not much changed. Francis Bacon (the philosopher, not the painter) made modifications to Aristotle’s ideas, introducing the so called scientific method where inductive reasoning plays an important role. He paves the way for a modern understanding of scientific inquiry.

Approximately at the same time, Robert Boyle was instrumental in establishing experiments as the cornerstone of physical sciences.

Logical Empiricism

So far so good. By the early 20th Century the notion that science is based on experience (empiricism) and logic, and where knowledge is intersubjectively testable, has had a long history.

The philosophical school of logical empiricism (or logical positivism) tries to formalise these ideas. Notable proponents were Ernst Mach, Ludwig Wittgenstein, Bertrand Russell, Rudolf Carnap, Hans Reichenbach, Otto Neurath. Some main influences were:

• David Hume’s and John Locke’s empiricism: all knowledge originates from observation, nothing can exist in the mind which wasn’t before in the senses,
• Auguste Comte’ and John Stuart Mills’ positivism: there exists no knowledge outside of science.

In this paradigm (see Thomas Kuhn a little later) science is viewed as a building comprised of logical terms based on an empirical foundation. A theory is understood as having the following structure: observation -> empirical concepts -> formal notions -> abstract law. Basically a sequence of ever higher abstraction.

This notion of unveiling laws of nature by starting with individual observations is called induction (the other way round, starting with abstract laws and ending with a tangible factual description is called deduction, see further along).

And here the problems start to emerge. See part II…

3d visualization of an ownership network. Using cuttlefish and processing.org.

How did the prevailing world view change from 1900 to 1905?

What connects the mathematicians Bachelier, Markov, Kolmogorov, Ito to the physicists Langevin, Fokker, Planck, Einstein and the economists Black, Scholes, Merton?

The Setting

• Science up to 1900 was in essence the study of solutions of differential equations (Newton’s heritage);
• Was very successful, e.g., Maxwell’s equations: four differential equations describing everything about (classical) electromagnetism;
• Prevailing world view:
  • Deterministic universe;
  • Initial conditions plus the solution of differential equation yield certain prediction of the future.

Three Pillars

By the end of the 20th Century, it became clear that there are (at least?) two additional aspects needed in a completer understanding of reality:

• Inherent randomness: statistical evaluations of sets of outcomes of single observations/experiments;
  • Quantum mechanics (Planck 1900; Einstein 1905) contains a fundamental element of randomness;
  • In chaos theory (e.g., Mandelbrot 1963) non-linear dynamics leads to a sensitivity to initial conditions which renders even simple differential equations essentially unpredictable;
• Complex systems (e.g., Wolfram 1983), i.e., self-organization and emergent behavior, best understood as outcomes of simple rules.

Stochastic Processes

• Systems which evolve probabilistically in time;
• Described by a time-dependent random variable;
• The probability density function describes the distribution of the measurements at time t;
• Prototype: The Markov process.

For a Markov process, only the present state of the system influences its future evolution: there is no long-term memory. Examples:

• Wiener process or Einstein-Wiener process or Brownian motion:
  • Introduced by Bachelier in 1900;
  • Continuous (in t and the sample path)
  • Increments are independent and drawn from a Gaussian normal distribution;
• Random walk:
  • Discrete steps (jumps), continuous in t;
  • Is a Wiener process in the limit of the step size going to zero.

To summarize, there are three possible characteristics:

1. Jumps (in sample path);
2. Drift (of the probability density function);
3. Diffusion (widening of the probability density function).

Probability distribution function showing drift and diffusion:

But how to deal with stochastic processes?

The Micro View

Einstein:

• Presented a theory of Brownian motion in 1905;
• New paradigm: stochastic modeling of natural phenomena; statistics as intrinsic part of the time evolution of system;
• Mean-square displacement of Brownian particle proportional to time;
• Equation for the Brownian particle similar to a diffusion (differential) equation.

Langevin:

• Presented a new derivation of Einstein’s results in 1908;
• First stochastic differential equation, i.e., a differential equation of a “rapidly and irregularly fluctuating random force” (today called a random variable)
• Solutions of differential equation are random functions.

However, no formal mathematical grounding until 1942, when Ito developed stochastic calculus:

• Langevin’s equations interpreted as Ito stochastic differential equations using Ito integrals;
• Ito integral defined to deal with non-differentiable sample paths of random functions;
• Ito lemma (generalized integration rule) used to solve stochastic differential equations.

Note:

• The Markov process is a solution to a simple stochastic differential equation;
• The celebrated Black-Scholes option pricing formula is a stochastic differential equation employing Brownian motion.

The Fokker-Planck Equation: Moving To The Macro View

• The Langevin equation describes the evolution of the position of a single “stochastic particle”;
• The Fokker-Planck equation describes the behavior of a large population of of “stochastic particles”;
  • Formally: The Fokker-Planck equation gives the time evolution of the probability density function of the system as a function of time;
• Results can be derived more directly using the Fokker-Planck equation than using the corresponding stochastic differential equation;
• The theory of Markov processes can be developed from this macro point of view.

The Historical Context

Bachelier

• Developed a theory of Brownian motion (Einstein-Wiener process) in 1900 (five years before Einstein, and long before Wiener);
• Was the first person to use a stochastic process to model financial systems;
• Essentially his contribution was forgotten until the late 1950s;
• Black, Scholes and Merton’s publication in 1973 finally gave Brownian motion the break-through in finance.

Planck

• Founder of quantum theory;
• 1900 theory of black-body radiation;
• Central assumption: electromagnetic energy is quantized, E = h v;
• In 1914 Fokker derives an equation on Brownian motion which Planck proves;
• Applies the Fokker-Planck equation as quantum mechanical equation, which turns out to be wrong;
• In 1931 Kolmogorov presented two fundamental equations on Markov processes;
• It was later realized, that one of them was actually equivalent to the Fokker-Planck equation.

Einstein

1905 “Annus Mirabilis” publications. Fundamental paradigm shifts in the understanding of reality:

• Photoelectric effect:
  • Explained by giving Planck’s (theoretical) notion of energy quanta a physical reality (photons),
  • Further establishing quantum theory,
  • Winning him the Nobel Prize;
• Brownian motion:
  • First stochastic modeling of natural phenomena,
  • The experimental verification of the theory established the existence of atoms, which had been heavily debate at the time,
  • Einstein’s most frequently cited paper, in the fields of biology, chemistry, earth and environmental sciences, life sciences, engineering;
• Special theory of relativity: the relative speeds of the observers’ reference frames determines the passage of time;
• Equivalence of energy and mass (follows from special relativity): E = m c^2.

Einstein was working at the Patent Office in Bern at the time and submitted his Ph.D. to the University of Zurich in July 1905.

Later Work:

• 1915: general theory of relativity, explaining gravity in terms of the geometry (curvature) of space-time;
  • Planck also made contributions to general relativity;
• Although having helped in founding quantum mechanics, he fundamentally opposed its probabilistic implications: “God does not throw dice”;
• Dreams of a unified field theory:
  • Spend his last 30 years or so trying to (unsuccessfully) extend the general theory of relativity to unite it with electromagnetism;
  • Kaluza and Klein elegantly managed to do this in 1921 by developing general relativity in five space-time dimensions;
  • Today there is still no empirically validated theory able to explain gravity and the (quantum) Standard Model of particle physics, despite intense theoretical research (string/M-theory, loop quantum gravity);
  • In fact, one of the main goals of the LHC at CERN (officially operational on the 21st of October 2008) is to find hints of such a unified theory (supersymmetric particles, higher dimensions of space).

Allow for predictions
• Dependent on only a small set of conditions (i.e., independent of very many conditions which could possibly have an effect)

 
…but why are there laws of nature and how can these laws be discovered and understood by the human mind?

No One Knows!

• G.W. von Leibniz in 1714 (Principes de la nature et de la grâce):
  • Why is there something rather than nothing? For nothingness is simpler and easier than anything

• E. Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences“, 1960:
  • […] the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and […] there is no rational explanation for it
    
  • […] it is not at all natural that “laws of nature” exist, much less that man is able to discover them
    
  • […] the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them
    
  • […] fundamentally, we do not know why our theories work so well
    

In a Nutshell

• We happen to live in a structured, self-organizing, and fine-tuned universe that allows the emergence of sentient beings (anthropic principle)
• The human mind is capable of devising formal thought systems (mathematics)
• Mathematical models are able to capture and represent the workings of the universe

See also this post: in a nutshell.

The Fundamental Level of Reality: Physics

• Classical mechanics: invariance of the equations under transformations (e.g., time => conservation of energy)
• Gravitation (general relativity): geometry and the independence of the coordinate system (covariance)
• The other three forces of nature (unified in quantum field theory): mathematics of symmetry and special kind of invariance

See also these posts: funadamental, invariant thinking.

Towards Complexity

• Physics was extremely successful in describing the inanimate world the in the last 300 years or so
• But what about complex systems comprised of many interacting entities, e.g., the life and social sciences?
• The rest is chemistry; C. D. Anderson in 1932; echoing the success of a reductionist approach to understanding the workings of nature after having discovered the positron
• At each stage [of complexity] entirely new laws, concepts, and generalizations are necessary […]. Psychology is not applied biology, nor is biology applied chemistry; P. W. Anderson in 1972; pointing out that the knowledge about the constituents of a system doesn’t reveal any insights into how the system will behave as a whole; so it is not at all clear how you get from quarks and leptons via DNA to a human brain…

Complex Systems: Simplicity

The Limits of Physics

• Closed-form solutions to analytical expressions are mostly only attainable if non-linear effects (e.g., friction) are ignored
• Not too many interacting entities can be considered (e.g., three body problem)

The Complexity of Simple Rules

• S. Wolfram’s cellular automaton rule 110: neither completely random nor completely repetitive
• [The] results [simple rules give rise to complex behavior] where were so surprising and dramatic that as I gradually came to understand them, they forced me to change my whole view of science […]; S. Wolfram reminiscing on his early work on cellular automaton in the 80s (”New Kind of Science”, pg. 19)

Complex Systems: The Paradigm Shift

• The interaction of entities (agents) in a system according to simple rules gives rise to complex behavior
• The shift from mathematical (analytical) models to algorithmic computations and simulations performed in computers (only this bottom-up approach to simulating complex systems has been fruitful, all top-down efforts have failed: try programming swarming behavior, ant foraging, pedestrian/traffic dynamics,… not using simple local interaction rules but with a centralized, hierarchical setup!)
• Understanding the complex system as a network of interactions (graph theory), where the complexity (or structure) of the individual nodes can be ignored
• Challenge: how does the macro behavior emerge from the interaction of the system elements on the micro level?

See also these posts: complex, swarm theory, complex networks.

Laws of Nature Revisited

• Yes: scaling (or power) laws
• Complex, collective phenomena give rise to power laws […] independent of the microscopic details of the phenomenon. These power laws emerge from collective action and transcend individual specificities. As such, they are unforgeable signatures of a collective mechanism; J.P. Bouchaud in “Power-laws in Economy and Finance: Some Ideas from Physics“, 2001

Scaling Laws

Scaling-law relations characterize an immense number of natural patterns (from physics, biology, earth and planetary sciences, economics and finance, computer science and demography to the social sciences) 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

f(x) = a x^k     <=>   Y = (X/C)^E

Scaling laws

• lack a preferred scale, reflecting their (self-similar) fractal nature
• are usually valid across an enormous dynamic range (sometimes many orders of magnitude)

See also these posts: scaling laws, benford’s law.

Scaling Laws In FX

• Event counts related to price thresholds
• Price moves related to time thresholds
• Price moves related to price thresholds
• Waiting times related to price thresholds

(Figs. by jbg under the Creative Commons Attribution-NonCommercial2.5 License)

Scaling Laws In Biology

So-called allometric laws describe the relationship between two attributes of living organisms as scaling laws:

• The metabolic rate B of a species is proportional to its mass M: B ~ M^(3/4)
• Heartbeat (or breathing) rate T of a species is proportional to its mass: T ~ M^(-1/4)
• Lifespan L of a species is proportional to its mass: L ~ M^(1/4)
• Invariants: all species have the same number of heart beats in their lifespan (roughly one billion)

(Fig. G. West)

G. West (et. al) proposes an explanation of the 1/4 scaling exponent, which follow from underlying principles embedded in the dynamical and geometrical structure of space-filling, fractal-like, hierarchical branching networks, presumed optimized by natural selection: organisms effectively function in four spatial dimensions even though they physically exist in three.

Conclusion

• The natural world possesses structure-forming and self-organizing mechanisms leading to consciousness capable of devising formal thought systems which mirror the workings of the natural world
• There are two regimes in the natural world: basic fundamental processes and complex systems comprised of interacting agents
• There are two paradigms: analytical vs. algorithmic (computational)
• There are ‘miracles’ at work:
  • the existence of a universe following laws leading to stable emergent features
  • the capability of the human mind to devise formal thought systems
  • the overlap of mathematics and the workings of nature
  • the fact that complexity emerges from simple rules
• There are basic laws of nature to be found in complex systems, e.g., scaling laws

We are not alone in our ability to invent or plan or to contemplate ourselves—or even to plot and lie.

Many scientists believed animals were incapable of any thought. They were simply machines, robots programmed to react to stimuli but lacking the ability to think or feel.

We’re glimpsing intelligence throughout the animal kingdom.


Copyright Vincent J. Musi, National Geographic


A dog with a vocabulary of 340 words. A parrot that answers “shape” if asked what is different, and “color” if asked what is the same, while being showed two items of different shape and same color. An octopus with “distinct personality” that amuses itself by shooting water at plastic-bottle targets (the first reported invertebrate play behavior). Lemurs with calculatory abilities. Sheep able to recognize faces (of other sheep and humans) long term and that can discern moods. Crows able to make and use tools (in tests, even out of materials never seen before). Human-dolphin communication via an invented sign language (with simple grammar). Dolphins ability to correctly interpret on the first occasion instructions given by a person displayed on a TV screen.

This may only be the tip of the iceberg…

Read the article Animal Minds in National Geographic`s March 2008 edition.

Ever think about vegetarianism?

• small-worlds property [1],
• and scale-free nature [2] (see also this older post: scaling laws).

Today, networks are ubiquitous: phenomena in the physical world (e.g., computer networks, transportation networks, power grids, spontaneous synchronization of systems of lasers), biological systems (e.g., neural networks, epidemiology, food webs, gene regulation), and social realms (e.g., trade networks, diffusion of innovation, trust networks, research collaborations, social affiliation) are best understood if characterized as networks.

The explosion of this field of research was and is coupled with the increasing availability of

• huge amounts of data, pouring in from neurobiology, genomics, ecology, finance and the Word-wide Web, …,
• computing power and storage facilities.

Paradigm

The new paradigm states that it is best to understand a complex system, if it is mapped to a network. I.e., the links represent the some kind of interaction and the nodes are stripped of any intrinsic quality. So, as an example, you can forget about the complexity of the individual bird, if you model the flocks swarming behavior. (See these older posts: complex, fundamental, swarm theory, in a nutshell.)

Weights

Only in the last years has the attention shifted from this topological level of analysis (either links are present or not) to incorporate weights of links, giving the strength relative to each other. Albeit being harder to tackle, these networks are closer to the real-world system it is modeling.

Nodes

However, there is still one step missing: also the vertices of the network can be assigned with a value, which acts as a proxy for some real-world property that is coded into the network structure.

The two plots above illustrate the difference if the same network is visualized [3] using weights and values assigned to the vertices (left) or simply plotted as a binary (topological) network (right)…

References

[1] Strogatz S. H. and Watts D. J., 1998, Collective Dynamics of ‘Small-World’ Networks,
Nature, 393, 440–442.

[2] Albert R. and Barabasi A.-L., 1999, Emergence of Scaling in Random Networks, www.arXiv.org/abs/cond-mat/9910332.

[3] Cuttlefish Adaptive NetWorkbench and Layout Algorithm: sourceforge.net/projects/cuttlefish/

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